%Format=BigLaTeX
% Run LaTeX twice.
%
\documentclass[a4paper,notitlepage]{report}

\begin{document}
%
% Setting some useful default definitions
\newcommand{\ma}[1]{\ensuremath{\mathbf{#1}}}
\newcommand{\masym}[1]{\ensuremath{\mathbf{#1}}}
\newcommand{\tr}{\mbox{tr }}
%
%produce title page

\title{PREFMAP-3 User's Guide\footnotetext{The PREFMAP-3 project was started when the two 
first authors were staying at Bell Labs in 1982. The authors would 
like to thank Joseph Kruskal for his comments and advice regarding 
the output of the computer program. Thanks are also due to Suzanne 
Winsberg and Sandra Pruzansky for their comments on an earlier version 
of the manuscript.}}
\author{Jacqueline Meulman$^*$ \and Willem J. Heiser\thanks{Department of 
Data Theory, University of Leiden, The Netherlands} \and J. Douglas 
Carroll\thanks{Bell Telephone Laboratories, Murray Hill, New Jersey 
07974, U.S.A.}}
\date{1986}

\maketitle
\pagenumbering{roman}
\tableofcontents
\pagenumbering{arabic}

\chapter*{Introduction}
%
Preference mapping (PREFMAP) is a basic statistical technique for the 
behavioral and social sciences, particularly psychology and 
marketing, and has wide potential for application in other areas as 
well (e.g., speech research, linguistics or psychoacoustics). Briefly 
speaking, its purpose is to relate preference information on a number 
of objects to a pre-existing spatial configuration of points, under 
the assumption of a simple class of models. Here ``preference'' is 
used as a generic name of any type of observations that indicate a 
conditional dominance relation among the objects; thus 
``properties'', ``attributes'', ``Q-sorts'', or -- in still another 
language -- ``variables'' defined on ``cases'' can also be modeled 
with preference mapping. What makes PREFMAP clearly distinct from a 
straightforward correlation approach is the possibility to go beyond 
monotonically increasing preference (or response-) functions, and to 
study various forms of {\it single-peakedness}. This concept has a long 
and complicated history in the behavioral sciences, some of which is 
touched upon in Coombs and Avrunin (1977), and Heiser (1981). The 
first regular description of the PREFMAP methodology as a hierarchy 
of models and techniques was given by Carroll (1972). Further 
useful references are Carroll (1980), Heiser and De Leeuw (1981), 
and Coxon (1982). 

This manual provides a comprehensive account of the PREFMAP models 
and techniques in connection with the PREFMAP-3 program. PREFMAP-3 is 
the successor to PREFMAP and PREFMAP-2 (Chang and Carroll, 1972), 
preserving most of their features, but completely redesigned and 
reprogrammed in order to obtain a greater flexibility, portability 
and capacity. An outline of the differences is included in Appendix 
\ref{chap:appendixA}, which also gives technical details and 
some instructions for getting the program running properly. Nowhere, 
is familiarity with the previous versions assumed.

The manual is organized into two parts. Part \ref{part_one} contains a 
general introduction to the PREFMAP hierarchy of models and the 
associated hierarchy of regression equations for fitting the models 
to data. It also contains a section on preliminary transformations of 
the spatial configuration to improve fit, and a short introduction to 
the way in which nonmetric analyses are performed. Part 
\ref{part_two} provides practical guidance in the use of the PREFMAP-3 
program. The so-called standard input stream is described, and the 
various mechanisms available to control the behavior of the program. 
There is also an explanation of the layout of the output, and of the 
trouble reports and program-specific warnings that may be issued 
during execution. Part \ref{part_two} concludes with some 
applications, merely giving a glimpse of the possibilities. 
Finally, Appendix \ref {chap:appendixB} is a detailed statement of the 
input records and their organization, attached there for ease of 
references.

We welcome feedback from the users of PREFMAP-3, such as applications 
of special interest, and novel but useful ways of using the program. 
We also welcome information about any errors which may remains in the 
program, especially if accompanied by sufficient information to 
permit tracking them down.


\part{PREFMAP models and techniques}
\label{part_one}


\chapter{Basic data and objectives}
\label{chap:chap1}
%
The basic data for a PREFMAP analysis consists of two parts. The first 
art is called the {\it external data matrix}, and can be depicted as 
follows.

\begin{figure}[ht]
\centerline{$
\begin{array}{cccccc}
           & 1 & \ldots & j & \ldots & m  \\ \cline{2-6}
    \multicolumn{1}{c|}{1}     & & &   &        &\multicolumn{1}{c|}{}\\
    \multicolumn{1}{c|}{\vdots}& & &   &        &\multicolumn{1}{c|}{}\\
    \multicolumn{1}{c|}{i}     & &&\delta_{ij}& &\multicolumn{1}{c|}{}\\
    \multicolumn{1}{c|}{\vdots}& & &   &        &\multicolumn{1}{c|}{}\\
    \multicolumn{1}{c|}{n}     & & &   &        &\multicolumn{1}{c|}{}\\ 
    \cline{2-6}
\end{array}$}
\caption{The external data matrix}
\label{fig:data}
\end{figure}

\noindent
The external data matrix contains entries $\delta_{ij}$ that are 
interpreted as measures of the {\it dissimilarity} between row-object 
$i$ and column-object $j$.  In a psychological context the row-objects 
are often called subjects, and the column objects stimuli.  The 
dissimilarities could be derived from preference judgments, in which 
case they could be interpreted as measures of relative distance from 
the subjects' ``ideal stimulus points'' in the space (a different 
``ideal point'' being assumed for each subject, which may, in the case 
of a model called the ``vector model'', be infinitely distant from the 
actual stimuli, so that only a direction -- indicated by a subject 
``vector'' -- is defined).  The analysis of this type of data, 
representing both the row- and the column-objects in a $p$-dimensional 
space such that the Euclidean distances $d_{ij}$ between row-point $i$ 
and column-point $j$ resemble as closely as possible the 
dissimilarities $\delta_{ij}$, has become known as {\it 
Multidimensional Unfolding}.  The type of analysis that is performed 
by the PREFMAP-3 program is sometimes called {\it External Unfolding}.  
Contrary to Internal Unfolding, where both the row-points and the 
column-points must be solved for, External Unfolding finds a 
representation for each row-object conditionally upon {\it given} 
points for the column- objects.

This situation defines the second part of the basic input as the 
target configuration (see Figure \ref{fig:target}).
\begin{figure}[ht]
\centerline{$
\begin{array}{lcccccc}
    &      & 1 & \ldots & s & \ldots & p  \\ \cline{3-7}
    &  \multicolumn{1}{c|}{1}     & & &      & &\multicolumn{1}{c|}{}\\
    &  \multicolumn{1}{c|}{\vdots}& & &      & &\multicolumn{1}{c|}{}\\
    \mbox{object-} 
    &  \multicolumn{1}{c|}{j}     & & &y_{js}& &\multicolumn{1}{c|}{}\\
    \mbox{points} 
    &  \multicolumn{1}{c|}{\vdots}& & &      & &\multicolumn{1}{c|}{}\\
    &  \multicolumn{1}{c|}{m}     & & &      & &\multicolumn{1}{c|}{}\\ 
    \cline{3-7}
    & & & \multicolumn{3}{c}{\mbox{dimensions}} & \\
\end{array}$}
\caption{The external data matrix}
\label{fig:target}
\end{figure}
The rows of the target configuration must correspond to the columns of 
the data matrix.  The entries $y_{js}$ are the coordinate values of 
the $m$ points in $p$-dimensional space.  These values may be obtained 
from any multidimensional data analysis technique, or be derived from 
some predefined structure.  For historical reasons, the technique that 
relates the external data to the given configuration is called 
PREFerence Mapping.  One should realize, however, that the entries of 
the data matrix can consist of any score, rating or ranking by the 
rows (subjects, scales, variables, properties) to the columns 
(stimuli, objects, cases).  From now on, we shall adhere to the 
convention to denote the row-entities as {\it individuals} and the 
column-entities as {\it objects}, and to assume that the scoring 
direction is: ``small value $\leftrightarrow$ high preference" and 
``large values $\leftrightarrow$ low preference" (hence that 
$\delta_{ij}$ are sometimes mnemonically called dispreferences).  If 
the original data are preferences (or other {\it similarity} measures) 
the values should be reversed in order to make them dispreferences (or 
{\it dissimilarities}) say by subtracting them all from some large 
number, or simply reversing their signs.  (At present, this step must 
be taken {\it before} input to PREFMAP-3.)

The individuals can be represented in the target configuration for the 
objects in two different ways: either as a {\it vector} (which can be 
thought of as indicating the direction of an infinitely distant ideal 
point, as discussed earlier), or as an {\it ideal point}.  In the 
former representation, preference increases monotonically along one 
single direction in space (indicated by the vector).  In the simplest 
case of the latter type of representation there is one single point of 
maximum preference (called the ideal point), and preference decreases 
along all directions in space as a function of the distance from the 
ideal point.  Thus in both cases the preference function is 
single-peaked, provided we imagine the vector representation as a 
"ridge of peaks" at infinity.  As we shall see shortly there are 
variations on this basic scheme that allow us to study single-dipped 
functions, when there exists a point of minimum preference, or 
mixtures of single-peakedness single-dippedness as well.  The whole 
range of models forms a nested sequence, or a hierarchy, and will be 
discussed in chapter \ref{chap:chap2} in order of complexity.

For each member of the hierarchy, a specific set of regression 
equations can be used to obtain least squares estimates of the model 
parameters.  A detailed derivation is given in chapter 
\ref{chap:chap3}.  For one member of the hierarchy - the simple 
Unfolding or ideal point model - it can be useful to transform the 
target configuration before the mapping process starts; these 
preliminary transformations are fully explained in chapter 
\ref{chap:chap4}.  Regardless of the chosen model, the analysis can be 
done metrically or nonmetrically (chapter \ref{chap:chap5}).  In the 
former case the relation between the values predicted by the 
preference function and the data is assumed to be linear, in the 
latter case merely the order of the data values is taken into account.  
This option again enlarges the class of possible preference functions 
a great deal (in anticipation we already freely used expressions like 
``monotonically increasing preference", it being understood that in a 
metric analysis monotonicity is constrained to be linear).

Due to the conditional nature of the preference mapping (i.e., a fixed 
target configuration), each individual can be modeled independently 
from the others.  There are definite advantages to considering groups 
of individuals simultaneously (therefore Figure \ref{fig:data} is a 
table, not a single row of dissimilarities).  But this circumstance 
does not necessarily urge us to choose the same model for each 
individual.


\chapter{The PREFMAP hierarchy of models}
\label{chap:chap2}
\section{The vector model}

The vector model is the simplest in the hierarchy.  The individual is 
represented in the $p$-dimensional configuration by a vector.  The 
direction of the vector is indicative of increasing preference.  More 
in particular, the dissimilarity values are approximated by the 
perpendicular projection of the object points onto the individual 
vector (where, in PREFMAP-3, the signs are adjusted to let a high 
projection value correspond with a low dissimilarity value, i.e., by 
convention, a high preference).

Although the representation of a row of the data matrix is very 
similar to the results for, say a variable in Principal Components 
analysis, we cannot conclude anything from a correlation point of view 
without restraint.  Only when the target configuration is orthonormal, 
i.e.  the coordinate values have sum of squares one and are 
uncorrelated, the angles between vectors and the axes of the 
configuration can be interpreted as correlations.  When this condition 
is not satisfied the vector must simply be interpreted as the best 
{\it direction} of overall increasing preference.


\section{The Unfolding model}
%
This model is also called the ideal point model, or the simple 
Euclidean model.  The individual is represented as a point, located at 
a position of an imaginary object point that would receive maximum 
preference value (called the ideal point).  The dissimilarities are 
approximated by the squared Euclidean distances between the ideal 
point and the object points.  Here it is quite natural that a small 
dissimilarity is represented by a small distance: the ideal point will 
be close to the objects the individual prefers most.  This explains 
the dissimilarity/dispreference scoring convention (the Unfolding 
model is, in many ways, most central in preference mapping).  The name 
Unfolding originates from the following metaphor.  If we imagine - in 
one or two dimensions - the object points as spots on a napkin, next 
pick it up at the ideal point position and {\it fold} it, then the spots 
will appear on the the folded napkin in order of preference.  The data 
analysis takes us in the reverse direction, hence the name Unfolding.


\section{The weighted Unfolding model}
%
The weighted Unfolding or weighted ideal point model also depicts the 
individual as a point, but now the model allows any individual to 
(re)weight the dimensions in his/her own way.  The weights can be 
thought of as importances of dimensions for a specific individual.  
Accordingly, the dissimilarities are approximated by weighted squared 
distances, with a distinct pattern of weights for every individual.

In terms of a preference function, more in particular a preference 
surface, the weighted Unfolding model still implies a single peak at 
the location of the ideal point, but now preference decreases more 
rapidly in the direction of a more heavily weighted axis (and less 
rapidly along a less heavily weighted one).  The contours of equal 
preference are ellipse (or ellipsoid) centered at the ideal point, 
with axes parallel to the coordinate axes, and lengths of axes 
inversely related to the weights.  The simple Unfolding model is of 
course the special case of equal weights, so that the ellipses 
(ellipsoids) become circles (spheres) centered at the ideal point.


\section{The general Unfolding model}
%
Distances between an ideal point and the object points are invariant 
under orthogonal rotation of the complete configuration.  They do 
become different, however, when we first allow an optimal rotation and 
then apply weights to this new set of reference axes (and, if desired, 
rotate back).  This idea constitutes the final ideal point model, 
sometimes called the general Euclidean model.  An individual is 
allowed to rotate the target configuration before weighting the new 
(rotated) axes.  Thus a group of individuals all modeled this way can 
show differences in three respects: ideal point location, axis 
orientation, and importance of the reoriented axes.

It will be clear that this generalization implies that the contours of 
equal preference still are ellipses (in more than two dimensions: 
ellipsoids) centered at the ideal point.  For the general Unfolding 
model is equivalent to the weighted Unfolding model {\it after} the 
individual reorientation has been effected.  Then, keeping the 
position of the target configurations constant, ellipses of different 
individuals will all have different orientations.


\section{Weights and their signs}
%
The parameters of the vector model are $p$ numbers fixing the 
direction of the individual vector with respect to the coordinate axes 
of the target configuration.  If normalized (sum of squares equal to 
1), these $p$ numbers are called direction cosines.  In another 
context, they are sometimes called ``weights'', but we have to be 
careful in PREFMAP not to confuse them with the weights of the 
weighted Unfolding model.  It is not true, for instance, that the 
vector model is a special case of the weighted Unfolding model in 
which the ideal point is located in the origin.  The unfolding model 
``weights" apply to coordinate-wise squared differences (between the 
ideal point and an object point); the vector model ``weights" weight 
contributions to a linear sum of coordinate values (for each object 
point).  If it is not in terms of ``weights", in what sense, then, is 
the vector model a special case of Unfolding model?  When we move the 
ideal point outwards along a fixed direction, say $v$, then the 
contours of equal preference become circles (or spheres in higher 
dimensions) with larger and larger radii.  If moved far enough, the 
circle (sphere) segments in the neighborhood of the object points are 
approximately straight lines (planes or hyperplanes) perpendicular to 
$v$, and this makes them indistinguishable from lines (hyperplanes) of 
equal preference for a vector model with vector $v$.

Actually, the three ideal point models all accommodate weights.  While 
in the weighted and the general Unfolding model a {\it weight pattern} 
is obtained, in the simple Unfolding model the {\it magnitude} of the 
weights is equal to {\it one} for each dimension.  Now, there is 
nothing in the method that fits the models to the external data that 
ensures the weights to be positive (see chapter \ref{chap:chap3}).  In 
the case of the simple ideal point model the weights for each 
dimension can become minus one.  We cannot interpret the individual 
point any longer as an ideal point, but have to appreciate it as an 
{\it anti-ideal point}: the (squared) distance between an individual 
point and an object point will be small when the dissimilarity is 
large.  Preferences are represented in a way that shows how much an 
individual {\it dislikes} an object.  For the more complicated ideal 
point models, the weight pattern can also turn out to be more 
complicated.  The same anti-ideal point interpretation holds when the 
weights for each dimension are negative.  When they are positive for 
some dimensions and negative for others, the individual point is 
called a {\it saddle point}.  Along the dimensions with positive 
weights distance decreases when preference increases, and for the 
dimensions with negative weights it is just the other way around.

Finally a situation must be anticipated that complicates the 
interpretation of the individual point for the simple ideal point.  
This might come out when a preliminary transformation of the target 
configuration is performed (see chapter \ref{chap:chap4}, for a 
description and a mathematical account).  When a preliminary 
transformation is allowed for, we obtain equal weights with a {\it 
sign pattern} for the ideal point model instead of a single weight 
with an equal sign for each dimension.  In that case we have to 
interpret the individual point in the same way as we would do for the 
weighted and the general model; i.e., we have to decide by inspecting 
the signs of the weights whether the individual point is a saddle 
point instead of simply an ideal or an anti-ideal point.  In all cases 
with mixed sign combinations the preference surface is no longer 
single-peaked or single-dipped; this may occasionally give problems in 
interpretation, but there seem to be no compelling general reasons for 
excluding these possibilities from the hierarchy altogether.  Another 
target configuration might be tried, which is often a good idea 
anyhow.


\chapter{The PREFMAP hierarchy of regression equations}
\label{chap:chap3}
%
Both the vector model and the Unfolding models can be expressed as a 
set of equations that are linear in the unknowns.  These equations can 
be approximately solved by ordinary least squares regression if we 
perform appropriate changes of variables and reparametrizations.  
Carroll (1972) is the major reference here, see also Carroll (1980) 
and Heiser and De Leeuw (1981).  It will be shown in the next 
paragraphs what operations have to be performed, starting again with 
the vector model.  In all models the data are assumed to be at the 
interval level, so that the obtained model values must remain 
invariant under linear transformations of the data (two parameters, 
$a_i$ and $b_i$, are introduced to take care of this).  The symbol 
$\simeq$ is used to indicate least squares approximation, and $R_i$ 
denotes the multiple correlation coefficient.  In all formulas the 
index $i$ is retained to refer to individual $i$, although strictly 
speaking it is superfluous (different individuals can be modeled 
differently and independently).  In particular, the introduction of 
some model parameter, e.g.  $w_{is}$, does not imply that it has to be 
there for all $i=1, \ldots, n$.  Whenever two parameters are 
confounded, the identification conditions selected in PREFMAP-3 are 
explicitly stated.


\section{Equations for the vector model}
%
In the vector model dissimilarities $\delta_{ij}$ are assumed to be 
approximated as
\begin{eqnarray}
  \delta_{ij} \simeq a_i \sum_{s=1}^p x_{is}y_{js} + b_i.
  \label{for:delta_approx}
\end{eqnarray}
Here $a_i$  is the slope parameter, $b_i$ is the intercept term, 
$x_{is}$ is the vector coordinate in dimension $s$, and $y_{js}$ is 
the target point coordinate; $\delta_{ij}$ and $y_{js}$ are known; 
$a_i, b_i$ and $x_{is}$ are unknown.  Now we re-express the system 
using the following convention: the index $q$  of the system, and $s$ or 
other indices of the old system are coupled in order of enumeration.
\begin{eqnarray*}
  {\tabcolsep2pt \begin{array}{r@{=}lr@{=}lr@{=}l}
    \multicolumn{2}{c}{\mbox{\underline{Change of variables}}} &
    \multicolumn{2}{c}{\mbox{\underline{Reparametrization}}} &
    \multicolumn{2}{c}{\mbox{\underline{Index ranges and definition}}} \\
    z_{j0}&1      & g_{i0}&b_i        & q&0  \\
    z_{jq}&y_{js} & g_{iq}&a_i x_{is} & q&s \ (s,q=1, \ldots,p) \\
  \end{array}}
\end{eqnarray*}
Substitution into equation (\ref{for:delta_approx}) gives us the 
transformed model
\begin{eqnarray}
  \delta_{ij} \simeq g_{i0}z_{j0} + \sum_{q=1}^p g_{iq}z_{jq} 
        = \sum_{q=0}^p g_{iq}z_{jq}.
  \label{for:delta_approx2}
\end{eqnarray}
Thus we end up with a set of $m$ nonhomogeneous linear equations in 
$p+1$ unknowns.  These can be approximately solved (in a least squares 
sense) by multiple regression techniques.  The predictor set contains 
a vector of ones plus $p$ vectors of target coordinates.  The 
dissimilarities $\delta_{ij}$ will function as the criterion values.  
Once the regression weights in equation (\ref{for:delta_approx2}) are 
determined, we find values for the parameters of the original model by 
applying
\begin{eqnarray*}
  a_{i} &=&\left({\sum_{q=1}^p g_{iq}^2/R_i^2 \max_j \sum_{s=1}^p 
  y_{js}^2}\right)^{1/2}, \\
  x_{is}&=& g_{iq}/a_i, \\
  b_i   &=& g_{i0}.
\end{eqnarray*}
In fact the normalization of the vector coordinates is free to be 
chosen.  In PREFPAM-3 it has been decided to normalize them in such a 
way that the length of the vector is proportional to the fit $R_i$.  The 
overall size is determined by the target point that has largest 
distance from the origin of the configuration.  In this way the length 
of the vectors will harmonize with the size of the target 
configuration, while still displaying the relative fit.


\section{Equations for the Unfolding model}
\label{sect:eqnunfolding}
%
Here the dissimilarities are assumed to be approximated as
\begin{eqnarray}
  \delta_{ij} \simeq a_id_{ij}^2 + b_i,
  \label{for:unfol}
\end{eqnarray}
with $d_{ij}^2$ the squared Euclidean distance between ideal point $i$ 
and target point $j$.  Rewriting (\ref{for:unfol}) with coordinates 
values $x_{is}$ and $y_{js}$ gives us
\begin{eqnarray}
  \delta_{ij} 
  \simeq a_i \left\{{\sum_{s=1}^p(x_{is}-y_{js})^2}\right\}+b_i
   =     a_i \left\{{\sum_{s=1}^px_{is}^2  
                   -2\sum_{s=1}^px_{is}y_{js}
                    +\sum_{s=1}^p y_{js}^2}\right\} + b_i.
  \label{for:unfol_exp}
\end{eqnarray}
\begin{eqnarray*}
  {\tabcolsep2pt \begin{array}{r@{=}lr@{=}lr@{=}l}
    \multicolumn{2}{c}{\mbox{\underline{Change of variables}}} &
    \multicolumn{2}{c}{\mbox{\underline{Reparametrization}}} &
    \multicolumn{2}{c}{\mbox{\underline{Index ranges and definition}}} \\
    z_{j0}&1      & g_{i0}&a_i \sum_{s=1}^px_{is}^2 + b_i & q&0  \\
    z_{jq}&y_{js} & g_{iq}&-2a_i x_{is} & q&s \ (s,q=1, \ldots,p) \\
    z_{j(p+1)}&\sum_{s=1}^p y_{js}^2 & g_{i(p+1)}&a_i & q&p+1 \\
  \end{array}}
\end{eqnarray*}

\noindent
{\it Transformed model}
\begin{eqnarray}
  \delta_{ij} 
  \simeq g_{i0}z_{j0} 
       + \sum_{q=1}^pg_{iq}z_{jq} + g_{i(p+1)}z_{j(p+1)} 
       = \sum_{q=0}^{p+1}g_{iq}z_{jq}.
\end{eqnarray}
Now the transformed model is a set of $m$ linear equations in $p+2$ 
unknowns.  The predictor set contains, in addition to the vector of 
ones and the target coordinates, the sum of squares of the target 
coordinates ($z_{j(p+1)}$).  The parameters of the original model are 
obtained as
\begin{eqnarray*}
  a_i &=&|g_{i(p+1)}|, \\
  x_{is}&=& -\frac{1}{2}g_{iq}/g_{i(p+1)}, \\
  b_i   &=& g_{i0}-a_i\sum_{s=1}^px_{is}^2.
\end{eqnarray*}
The slope is always identified as a nonnegative quantity; the sign of 
the regression weight $g_{i(p+1)}$ determines whether we deal with an 
ideal point or an anti-ideal point.  When this regression weight 
approaches zero, the ideal point will move to infinity.


\section{Equations for the weighted Unfolding model}
%
This model is equivalent to the Unfolding model in equation 
(\ref{for:unfol}), except that the distances are defined by
\begin{eqnarray}
  d_{ij}^2 = \sum_{s=1}^p w_{is}(x_{is}-y_{js})^2
\end{eqnarray}
and thus (\ref{for:unfol_exp}) can be written for the weighted model as
\begin{eqnarray}
  \delta_{ij} 
  \simeq a_i \left\{{\sum_{s=1}^p w_{is}x_{is}^2  
                   -2\sum_{s=1}^p w_{is}x_{is}y_{js}
                    +\sum_{s=1}^p w_{is} y_{js}^2}\right\} + b_i
\end{eqnarray}
giving for the new system
\begin{eqnarray*}
  {\tabcolsep2pt \begin{array}{r@{=}lr@{=}lr@{=}l}
    \multicolumn{2}{c}{\mbox{\underline{Change of variables}}} &
    \multicolumn{2}{c}{\mbox{\underline{Reparametrization}}} &
    \multicolumn{2}{c}{\mbox{\underline{Index ranges and definition}}} \\
    z_{j0}&1      & g_{i0}&a_i \sum_{s=1}^pw_{is}x_{is}^2 + b_i & q&0  \\
    z_{jq}&y_{js} & g_{iq}&-2a_i w_{is} x_{is} & q&s \ (s,q=1, \ldots,p) \\
    z_{jq}&y_{js}^2 & g_{iq}&a_i w_{is} & q&p+s \ (s=1,\ldots,p;  \\
    \multicolumn{5}{c}{}                   &q=p+1,\ldots,2p)  \\
  \end{array}}
\end{eqnarray*}
{\it Transformed model}
\begin{eqnarray}
  \delta_{ij} \simeq g_{i0}z_{j0} + \sum_{q=1}^{2p}g_{iq}z_{jq}  
       = \sum_{q=0}^{2p}g_{iq}z_{jq}.
\end{eqnarray}
Under the weighted Unfolding model we thus have to determine $2p+1$  
regression weights.  For that purpose the predictor matrix must 
contain, apart from the vector of ones and $p$  vectors with the 
target coordinates, another $p$  vectors with the squares of the 
target coordinates.  The solution for the original parameters is:
\begin{eqnarray*}
  \begin{array}{rcll}
    a_i   &=&\left({\frac{1}{p}\sum_{q=p+1}^{2p}g_{iq}^2}\right)^{1/2},& \\
    w_{is}&=& g_{iq}/a_i, & q=(p+1),\ldots,2p, \\
    x_{is}&=& -\frac{1}{2}g_{iq}/g_{i(q+p)}, & q=1,\ldots,p,\\
    b_i   &=& g_{i0}-a_i\sum_{s=1}^pw_{is}x_{is}^2.&
  \end{array}
\end{eqnarray*}
By choosing this identification for the $a_i$, the weights $w_{is}$ 
are normalized so that their sum of squares equals the number of 
dimensions $p$.  The sign pattern of the $w_{is}$'s indicates whether 
we deal with an ideal point (all signs positive), an anti-ideal point 
(all signs negative), or a saddle point (some signs positive, the 
others negative).


\section{Equations for the general Unfolding model}
%
The general Unfolding model is essentially the same as the weighted 
ideal point model, apart from the fact that both the ideal point and 
the target points are jointly reoriented by an orthogonal rotation 
matrix $\ma{T}_i$.  When we define $\ma{X}^*=\ma{XT}_i$ and 
$\ma{Y}^*=\ma{YT}_i$ the distances are given by
\begin{eqnarray}
  d_{ij}^2 = \sum_{s=1}^p w_{is}(x_{is}^*-y_{js}^*)^2.
\end{eqnarray}
Defining the transformation matrix 
$\ma{R}_i=\ma{T}_i\ma{W}_i\ma{T}_i'$, with $\ma{W}_i$ a diagonal 
matrix containing the individual dimension weights, the regression 
equations for the general Unfolding model are
\begin{eqnarray}
  \lefteqn{\delta_{ij} \simeq 
  a_i \left\{{\sum_{s=1}^p \sum_{u=1}^p x_{is}r_{su}^i x_{iu}} 
  \right.} \nonumber \\
  & &  \left.{-2\sum_{s=1}^p \sum_{u=1}^p x_{is}r_{su}^i y_{ju}
               +\sum_{s=1}^p \sum_{u=1}^p y_{js}r_{su}^i y_{ju}}\right\} + b_i.
\end{eqnarray}
(The notational convention used here is that the general entry in a 
matrix denoted by a bold capital letter with {\it sub}script $i$ will 
be indicated by the same letter, doubly subscripted and in small case 
with a {\it super}script $i$; e.g.  $r_{su}^i$ is the ($s,u$) element 
of $\ma{R}_i$.)
\begin{eqnarray*}
  {\tabcolsep2pt \begin{array}{r@{=}lr@{=}lr@{=}l}
    \multicolumn{2}{l}{\mbox{Change of}} & \multicolumn{2}{c}{} & 
    \multicolumn{2}{c}{} \\
    \multicolumn{2}{l}{\mbox{\underline{variables}}} &
    \multicolumn{2}{l}{\mbox{\underline{Reparametrization}}} &
    \multicolumn{2}{l}{\mbox{\underline{Index ranges and definition}}} \\
    z_{j0}&1      & 
    g_{i0}&{\displaystyle 
    a_i \sum_{s=1}^p\sum_{u=1}^p x_{is}r_{su}^i x_{iu} + b_i} & 
    q&0  \\
    z_{jq}&y_{js} & 
    g_{iq}&{\displaystyle -2a_i \sum_{u=1}^p x_{iu}r_{su}^i} & 
    q&s \ (s,q=1, \ldots,p) \\
    z_{jq}&y_{js}^2 & 
    g_{iq}&a_i r_{ss}^i & q&p+s \ (s=1,\ldots,p;  \\
    \multicolumn{5}{c}{}&q=p+1,\ldots,2p)  \\
    z_{jq}&y_{js}y_{ju} & 
    g_{iq}&2a_ir_{su}^i(=2a_ir_{us}^i) & 
    q&2p+s(s-1)/2+u-1 \\ 
    \multicolumn{5}{c}{}&(s<u=1,\ldots,p;  \\
    \multicolumn{5}{c}{}& q=2p+1,\ldots,p(p+3)/2)  \\
  \end{array}}
\end{eqnarray*}
{\it Transformed model}
\begin{eqnarray}
  \delta_{ij} 
  &\simeq& g_{i0}z_{j0} 
       + \sum_{q=1}^{p}           g_{iq}z_{jq} 
       + \sum_{q=p+1}^{2p}        g_{iq}z_{jq}
       + \sum_{q=2p+1}^{p(p+3)/2} g_{iq}z_{jq} \nonumber \\
 &\simeq& \sum_{q=0}^{p(p+3)/2} g_{iq}z_{jq}.
\end{eqnarray}
For the general model we can estimate the values of parameters by 
determining $p(p+3)/2$ regression weights.  In addition to the 
predictor matrix for the weighted Unfolding model, we have $p(p - 
1)/2$ vectors of cross product terms of the target coordinates.  The 
solution in terms of the original parameters is
\begin{eqnarray*}
  \begin{array}{r@{=}lr@{=}ll}
  a_i&\left({\frac{1}{p}\sum_{q=p+1}^{2p}g_{iq}^2}\right)^{1/2} \\
  r_{ss}^i& g_{iq}/a_i, &q&p+1,\ldots,2p &(s=q)\\
  r_{su}^i(& r_{us}^i)=\frac{1}{2} g_{iq}/a_i, &q&2p+1,\ldots,p(p+3)/2 &
  (s=\mbox{integer}\\
  \multicolumn{4}{c}{}&s'\ni s'(s'-1)/2<q' \mbox{ and}\\
  \multicolumn{4}{c}{}&s'(s'+1)/2\geq q' \mbox{ while}\\
  \multicolumn{4}{c}{}&u= q'-s, \mbox{ where}\\
  \multicolumn{4}{c}{}&q'= q-2p)\\
  b_i& \multicolumn{3}{l}{g_{i0}-a_i\sum_{s=1}^p\sum_{u=1}^p x_{is}r_{su}^ix_{iu}}.
  \end{array}
\end{eqnarray*}

The transformation matrix $\ma{R}_i$ can be decomposed to obtain the 
weighted matrix $\ma{W}_i$ and the rotation matrix $\ma{T}_i$, by 
solving for the eigenvalues and eigenvectors of 
$\ma{R}_i=\ma{T}_i\ma{W}_i\ma{T}_i'$.  Next the ideal point coordinates 
can be obtained as
\begin{eqnarray*}
  x_{is}= -\frac{1}{2a_i}\sum_{q=1}^p g_{iq}
      \sum_{u=1}^p t_{qu}^it_{su}^i/w_{uu}^i, \ s=1,\ldots,p.
\end{eqnarray*}
This settles the calculations for the general Unfolding model.


\section{Model testing: F-statistics}
\label{sect:ModelTesting}
%
Summarizing the results of the previous sections we obtain the 
following table of variables that are involved in the parameter 
estimation of the various models (Table \ref{tab:pred_vars}).

\begin{table}[h]
  \caption{The set of predictor variables.}
  \label{tab:pred_vars}
  \begin{center}{\footnotesize
  \begin{tabular}{lcccccc} \hline
              & vector of & Coordinates & SSQ's & SQ's & Cross- & 
              Total number \\ 
              & ones      &             &       &      & products &
              of parameters \\ \hline
    Model     & 1         & $y_{js}$    &$\sum_{s=1}^p y_{js}^2$&
                $y_{js}^2$& $y_{js}y_{ju}$&k \\ \hline
    Vector    & x         & x           &       &      &  & $p+1$ \\
    Unfolding & x         & x           & x     &      &  & $p+2$ \\
    Weighted  & x         & x           &       & x    &  & $2p+1$ \\
    General   & x         & x           &       & x    & x& $p(p+3)/2$ \\
    \hline
  \end{tabular} }
  \end{center}
\end{table}

\noindent
Note that for any model the total number of parameters, $k$, must be 
smaller than or equal to the number, $m$, of target points, $y_j\ 
(j=1,\ldots,m)$.  If this is not the case, and the model is 
nevertheless asked for, the PREFMAP-3 program will print a warning 
message and will not apply the model.

Since for the models we use regression equations that are linear in 
the parameters, we could use the squared multiple correlation 
coefficient (the squared fit) to compute F-ratios in order to test 
null hypotheses.  Some precautions are warranted, however, because the 
observations might not be independent (e.g., preference judgments 
obtained from one individual).  Lack of independence will generally 
inflate the significance levels; therefore one should be very 
conservative about them.  Moreover, significance tests are only one 
criterion among many others to decide which model fits the best data.

The F-statistics are computed by PREFMAP-3 are based on the squared 
{\it metric} fit and the number of fitted parameters involved ($k$, as 
defined in Table \ref{tab:pred_vars}).  The total number of data 
values is $m$.  For testing the null hypothesis $\rho_k = 0$ against 
the alternative hypothesis $\rho_k > 0$, under the assumption of 
normally distributed errors, the following F-ratio and accompanying 
degrees of freedom apply:
\begin{eqnarray*}
   \frac{R_k^2/(k-1)}{(1-R_k^2)/(m-k)}, 
   \mbox{ with }k-1\mbox{ and } m-k\ df.
\end{eqnarray*}
Here $R_k$ is the empirical estimate of $\rho_k$, the multiple 
correlation with $k$ predictors.  When $m = k$ we will obtain perfect 
fit and the ratio is not defined.  For testing the hypothesis 
$\rho_a=\rho_b$, where model $a$ is the more complex model, the 
following ratio applies:
\begin{eqnarray*}
   \frac{(R_a^2-R_b^2/(k_a-k_b)}{(1-R_a^2)/(m-k_a)}, 
   \mbox{ with }k_a-k_b\mbox{ and } m-k_a\ df,
\end{eqnarray*}
which can also be compared with the tabulated values for the 
F-distribution.


\chapter{Preliminary transformations of the target configuration}
\label{chap:chap4}
\section{Room for improvement under the simple Unfolding model}

In some cases it is suitable to allow the target configuration to be 
transformed before a group of individuals is fitted into it.  There 
are two possible linear transformations, which in spirit resemble the 
weighted and the general Unfolding model: the original axes of the 
target configuration can be differentially ``stretched", or a new set 
of reference axes can be obtained by weighting after orthogonal 
rotation.  Either transformation will apply to all individuals, and is 
optimal in the sense that the {\it proportion of total variance 
accounted for} by the simple Unfolding model will be maximal.  The 
proportion of total variance accounted for equals the average squared 
fit across individuals, for the {\it metric} case.

There are a number of situations where this type of preliminary 
transformation seems suitable.  When, e.g., the target configuration 
is the group stimulus space from an INDSCAL individual differences 
scaling analysis (cf.  Carroll and Chang, 1970), the object point 
coordinates are normalized for each axis.  By performing a preliminary 
weighting of the axes the configuration might become more meaningful.  
Note that the full transformation, i.e.  rotation and weighting, would 
not be wise in this case since the orientation of axes obtained by an 
INDSCAL analysis is uniquely related to the individual weights from 
that analysis.  Another application might be the case where we want to 
use a target configuration that has been constructed from a set of 
hypothetical variables.  Since we would not be sure of their proper 
orientation and their relative importance, it might be enlightening to 
allow a preliminary rotation and stretching (reweighting) of the axes.

It is important to bear in mind that both options for the 
transformation are especially designed for the simple ideal point 
model.  The weighted and the general ideal point model already provide 
optimal weights by themselves, while the vector model will absorb 
weights in the vector coordinates.  Therefore we will obtain the same 
fit when applying, e.g., the weighted Unfolding model directly 
compared to performing optimal weighting beforehand and next fitting 
the weighted model.  The full transformation will affect both the 
weighted and the simple ideal point model, but it is optimal only for 
the latter.  The vector model and the general model are not affected 
by it, at least as far as the fit is concerned.

When a preliminary transformation is called for, the optimal 
configuration is always determined across all rows, no matter what 
models are fitted in the external analysis.  This also implies that a 
special feature of external analyses is lost when applying a 
preliminary transformation.  When we do not ask for a transformation, 
the results across individuals are invariant under different 
selections of subgroups of individuals in the analysis (since 
individual results are obtained by separate multiple regressions).  
When we do ask for a transformation, this is no longer true, because 
the optimal configuration is solved for {\it given the selection of 
individual data} in the analysis.  Thus although the actual preference 
mapping consists of separate regressions, the preliminary 
transformation is determined ``jointly" across individuals.

In the next two sections we show how the transformed target 
configuration is obtained.  The solution for the full transformation 
can also be found in Carroll (1980), the explicit solution for the 
weighted case is new.


\section{The preliminary full transformation}
%
The problem that has to be solved can be written as 
\begin{eqnarray}
   \delta_{ij} \simeq a_i d_{ij}^2+b_i^*,
   \label{for:fullproblem}
\end{eqnarray}
with
\begin{eqnarray}
   d_{ij}^2 = (\ma{x}_i - \ma{Ty}_j)'(\ma{x}_i - \ma{Ty}_j),
   \label{for:distgenlin}
\end{eqnarray}
where \ma{T} is a general linear transformation matrix defined as 
$\ma{T} = \ma{WR}$, with \ma{R} an orthogonal rotation matrix and 
\ma{W} a diagonal weights matrix.  We use the notational convention 
here to denote the $i$'th row of \ma{X} and the $j$'th row of \ma{Y} 
by column vectors $\ma{x}_i$ and $\ma{y}_j$, resp.  Substituting 
(\ref{for:distgenlin}) into (\ref{for:fullproblem}) gives us:
\begin{eqnarray}
   \delta_{ij} \simeq a_i \ma{x}_i'\ma{x}_i - 2a_i\ma{x}_i'\ma{Ty}_j
   +a_i\ma{y}_j'\ma{T}'\ma{Ty}_j+b_i^*.
   \label{for:fullproblem2}
\end{eqnarray}
When we define $\ma{C}=\ma{T}'\ma{T}$, and in addition the matrices 
\ma{V}, \ma{B}, \ma{U}, and the vector \ma{w} (using the same coupling 
of indices $q$ and $s$ as before):
\begin{eqnarray*}
  \begin{array}{r@{=}lr@{=}ll}
    v_{qj}&y_{js}      & 
    b_{iq}&-2a_i \ma{x}_i'\ma{T} & 
    q=s \ (s=1,\ldots,p) \\
    v_{(p+1)j}&1      & 
    b_{i(p+1)}&a_i \ma{x}_i'\ma{x}_i +b_i^* & 
    (q=p+1) \\  
    u_{qj}&y_{js}^2      & 
    w_{q}&c_{ss} & 
    q=s \ (s=1,\ldots,p) \\
    u_{qj}&2y_{js}y_{jt}      & 
    w_{q}&c_{st} & 
    \left\{{\begin{array}{l} 
       q=p-1+s(s-1)/2+t \\
       s<t=1,\ldots,p;  \\
       q=p+1,\ldots,p(p+1)/2
    \end{array}}\right. \\
  \end{array}
\end{eqnarray*}
then equation (\ref{for:fullproblem2}) can be written as:
\begin{eqnarray}
   \delta_{ij} \simeq \sum_{q=1}^{p+1}b_{iq}v_{qj}
        +a_i \sum_{q=1}^{p(p+1)/2}w_{q}u_{qj}
   \label{for:fullproblemrep}
\end{eqnarray}
or, in matrix notation
\begin{eqnarray}
   \ma{\Delta} \simeq \ma{BV} + \ma{aw}'\ma{U}.
   \label{for:fullproblemmat}
\end{eqnarray}
Note that \ma{\Delta}, \ma{U}, and \ma{V} are given, while \ma{a}, 
\ma{w}, and \ma{B} are to be solved for so as to yield a least squares 
fit to \ma{\Delta}.  Note, also, that \ma{B} can be {\it any} matrix 
whatever (so long as \ma{T} is nonsingular); i.e., given any \ma{B} we 
can find a set of $x_i$'s, $a_i$'s and $b_i^*$'s that would produce it.  
Therefore the complete squares problem
\begin{eqnarray}
   \min_{\ma{a,w,B}}SSQ(\ma{\Delta} - \ma{BV} - \ma{aw}'\ma{U})
   \label{for:loss_matrix}
\end{eqnarray}
[where $SSQ$ is a ``sum of squares" operator, i.e.  $SSQ(\ma{A}) = \tr 
\ma{AA}'$] can be simplified by splitting the sum of squares in 
(\ref{for:loss_matrix}) into two additive parts.  Defining the 
projector matrices \ma{H}  and \ma{G}, $(\ma{H}  + \ma{G}  = \ma{I})$ as:
\begin{eqnarray}
   \ma{H} =\ma{I}- \ma{V}'(\ma{VV}')^{-1}\ma{V} = \ma{I}- \ma{G}
   \label{for:projector}
\end{eqnarray}
where $\ma{G}= \ma{V}'(\ma{VV}')^{-1}\ma{V}$, the additive decomposition is:
\begin{eqnarray}
   \lefteqn{SSQ(\ma{\Delta} - \ma{BV} - \ma{aw}'\ma{U})} \nonumber \\
   &=&SSQ(\ma{\Delta} - \ma{BV} - \ma{aw}'\ma{U})(\ma{H}+\ma{G}) \nonumber \\
   &=&SSQ(\ma{\Delta H} - \ma{aw}'\ma{UH}) + \nonumber \\
   & &SSQ(\ma{\Delta V}'(\ma{VV}')^{-1}\ma{V} - \ma{BV} 
    -\ma{aw}'\ma{UV}'(\ma{VV}')^{-1}\ma{V}).
\end{eqnarray}
Now, the second sum of squares on the right-hand side of 
(\ref{for:projector}) can be made zero for any choice of \ma{a} and 
\ma{w} by defining \ma{B} appropriately.  Therefore the unknowns of 
primary concern, \ma{w}, can be found by minimizing the first part of 
the decomposition.  The minimum is attained by choosing \ma{w} as the 
vector maximizing the following ratio of quadratic forms:
\begin{eqnarray}
   \frac{\ma{w}'(\ma{UHH}'\ma{\Delta}'\ma{\Delta HH}'\ma{U}')\ma{w}}
        {\ma{w}'(\ma{UHH}'\ma{U}')\ma{w}}.
\end{eqnarray}
Since \ma{H} is a projector matrix it is symmetric and idempotent; 
therefore the problem can be simplified into
\begin{eqnarray}
   \max_{\ma{w}}\frac{\ma{w}'(\ma{UH}\ma{\Delta}'\ma{\Delta HU}')\ma{w}}
        {\ma{w}'(\ma{UHU}')\ma{w}}
\end{eqnarray}
The maximum is found by solving a generalized eigenvalue problem.  
Once \ma{w} is chosen as the eigenvector associated with the largest 
eigenvalue, we can apply the inverse substitution
\begin{eqnarray*}
  \begin{array}{lll}
    c_{ss}=w_q          & \mbox{for }q=1, \ldots,p \ (s=q) \\
    c_{st}(=c_{ts})=w_q & \mbox{for }q=p+1, \ldots,p(p+1)/2 \\
                        & (s=\mbox{integer }s' \ni s'(s'-1)/2<q' \\
                        & \mbox{ while }s'(s'+1)/2\geq q' \\
                        & \mbox{ while }t=q'-s \mbox{ where }s<t=1, \ldots,p \\
                        & \mbox{ and }q'=q-p).
  \end{array}
\end{eqnarray*}

Next \ma{C} can be decomposed into $\ma{K\Lambda K}'$, and we solve 
for \ma{R} and \ma{W} by taking $\ma{R} = \ma{K}$ and $\ma{W} = 
\ma{\Lambda}^{1/2}$.  Finally \ma{W} is normalized so that its sum squares 
equals $p$.  When \ma{C} is not positive semidefinite, some diagonal 
values of \ma{\Lambda} are negative.  In that case we take $w_q = 
|\lambda_q|^{1/2}$ and apply the negative sign when computing the 
solution for the row points $\ma{x}_i$, as described in section 
\ref{sect:eqnunfolding}.  For example: we have found the sign pattern 
``+ + --" from \ma{\Lambda}, and from the regression we obtain a 
negative regression weight for the quadratic term.  The final weight 
pattern shall be ``-- -- +", indicating a saddle point.  It will be 
obvious from (\ref{for:fullproblemrep}) that we at least should have 
$p(p + 1)/2$ points $\ma{y}_j$ in order to be able to find a 
preliminary full transformation.


\section{The preliminary weighting of axes}
%
Under the assumption of no rotation the problem of the previous 
section simplifies somewhat.  Equation (\ref{for:fullproblem}) 
combined with (\ref{for:distgenlin}) can now be written as
\begin{eqnarray}
   \delta_{ij} \simeq a_i 
   (\ma{x}_i - \ma{Wy}_j)'(\ma{x}_i - \ma{Wy}_j)+b_i^*,
\end{eqnarray}
with \ma{W} a diagonal weight matrix, with diagonal entries positive.  
In this case we use the following definitions:
\begin{eqnarray*}
  \begin{array}{r@{=}lr@{=}lr@{=}l}
    v_{qj}&y_{js}   & b_{iq}&-2a_i\ma{x}_i'\ma{W} & q&s \ (s=1,\ldots,p) \\
    v_{(p+1)j}&1    & b_{i(p+1)}&a_i\ma{x}_i'\ma{x}_i+b^*_i & (q&p+1) \\
    g_{qj}&y_{js}^2 & f_q&w_{ss}^2 & q&s \ (s=1,\ldots,p) \\
  \end{array}
\end{eqnarray*}
which gives us
\begin{eqnarray}
   \delta_{ij} \simeq \sum_{q=1}^{p+1}b_{iq}v_{qj} 
                 + a_i\sum_{q=1}^p f_q g_{qj}
\end{eqnarray}
or, in matrix notation:
\begin{eqnarray}
   \ma{\Delta} \simeq \ma{BV} + \ma{af}'\ma{G}.
\end{eqnarray}
We end up with a problem that is completely analogous to 
(\ref{for:fullproblemmat}) in the previous section.  We now have to 
solve a (smaller) generalized eigenvalue problem to obtain the 
solution of
\begin{eqnarray}
   \max_{\ma{f}}\frac{\ma{f}'(\ma{GH \Delta}'\ma{\Delta HG}')\ma{f}}
        {\ma{f}'(\ma{GHG}')\ma{f}}.
\end{eqnarray}
From the solution for \ma{f} we obtain the diagonal entries of \ma{W} 
by defining $w_{ss} = |f_q|^{1/2}$.  Taking absolute values is again 
necessary in the case that some elements of \ma{f} are negative, and, 
as before, the sign pattern of \ma{f} is transmitted to the point in 
PREFMAP-3 where the coordinate values of $\ma{x}_i$ are computed.  The 
final solution for $w_{ss}$ as given by the program again has sum of 
squares equal to $p$.  To keep the projector matrix \ma{H} 
well-defined, there have to be at least $p + 1$ points $\ma{y}_j$ when 
preliminary weighting of axes is desired.


\chapter{Metric and nonmetric analyses}
\label{chap:chap5}
%
This terminology corresponds to the notions that are known from 
Multidimensional Scaling (Shepard, 1962; Kruskal, 1964; Kruskal and 
Shepard, 1974), and from generalizations of the Multivariate Analysis 
to what is called Nonlinear Multivariate Analysis (Gifi, 1981) or 
Optimal Scaling (cf.  Young, 1981).  Most closely related to the 
PREFMAP situation is the MONANOVA approach of Kruskal (1965).

The differences between metric and nonmetric is reflected in the way 
the dissimilarities are assumed to be related to the squared distances 
in the Unfolding models, or to the perpendicular projections in the 
vector model.  In a metric analysis we assume this relationship to be 
linear, in a nonmetric analysis we allow a {\it monotone 
transformation} of the original dissimilarities.  Since this 
transformation should be {\it optimal} for the squared distances (or 
the projections) that are fitted, we do not only have to solve for the 
ideal point (or vector coordinates), but for a monotone function that 
yields a best least squares fit as well.  This chapter provides only a 
very brief introduction to the major ideas involved.

To keep the notation simple, we now drop nearly all reference to
individual $i$, it being understood that everything is repeated for 
each $i = 1,\ldots,n$.  Suppose $\{\mu_1,\ldots,\mu_m\}$ denotes the 
model values predicted by any member of the hierarchy of PREFMAP 
models, and let $\Omega_k$ denote the set of all possible model values 
under model $k$.  In addition, let $\{\gamma_1,\ldots,\gamma_m\}$ 
denote a set of {\it substitute dissimilarity values}, and let 
$\Gamma_i$ be the set of all monotonic transformations of row $i$  of 
the external data matrix.  A feasible substitute is a 
$\{\gamma_1,\ldots,\gamma_m\}\in\Gamma_i$, which is 
simply another notation for the requirement
\begin{eqnarray}
   \gamma_j \leq \gamma_l \mbox{ if } \delta_{ij} < \delta_{il}.
   \label{for:orderrestr}
\end{eqnarray}
Thus we are satisfied with any substitute that is monotonically 
increasing with the data.  Now the complete, {\it nonmetric} PREFMAP 
problem is to find
\begin{eqnarray}
   \min_{\{\mu_1,\ldots,\mu_m\}\in \Omega_k} 
   \min_{\{\gamma_1,\ldots,\gamma_m\}\in\Gamma_i}
   \sum_{j=1}^m (\gamma_j - \mu_j)^2
   \label{for:nonmetric}
\end{eqnarray}
under the condition that the $\gamma_j$'s are normalized (to avoid 
trivial solutions with $\gamma_j = \mu_j = 0$ for all $j$).  This is 
done by using an algorithm that solves iteratively for one of the two 
sets of parameters at a time.  Such an algorithm has become known as 
an ALS algorithm, the acronym for {\it Alternating Least Squares}. 
PREFMAP fits, for fixed $\gamma$ (initialized as $\delta$), the vector 
and distance models by solving for the parameters in a regression 
equation (see chapter \ref{chap:chap3}, the outer minimization in 
(\ref{for:nonmetric})).  These multiple regression calculations are 
then alternated with a so-called {\it monotone regression} of the 
predicted values (the inner minimization in (\ref{for:nonmetric})).  
The monotone regression part is performed by an algorithm as described 
in Kruskal (1964).

Apart from the fact that we no longer need to assume the external data 
to be at the interval level (the nonmetric analysis will remain 
invariant under all (row-wise) monotonic transformations of the 
dissimilarities), there can also be a consideration of simplicity.  
The model might be kept more simple when the data are transformed, 
because part of the nonlinearity (or ``interaction") can be absorbed 
in the transformation.  In terms of preference surfaces, e.g., the 
difference between linearly increasing and quadratically increasing 
slopes disappears under nonmetricity, so that the vector and the ideal 
point model will sooner appear to be equivalent.  (That is, an ideal 
point sufficiently distant, but not infinitely distant, from the 
stimuli, will yield an order of predicted dispreferences identical to 
the order induced by projections on a vector in that direction.)

If the external data contain ties, i.e.\ equal values, then two 
different approaches are available to be used.  The first is called 
the {\it primary approach}, and it completes (\ref{for:orderrestr}) 
with:
\begin{eqnarray}
   \mbox{if }\delta_{ij}=\delta_{il} \mbox{ then }
   \left\{{\begin{array}{rl}
     \mbox{either} & \gamma_j \leq \gamma_l \\
     \mbox{or }    & \gamma_j > \gamma_l
     \end{array}}\right.. 
\end{eqnarray}
Thus ties in the data may become untied by the monotone regression, in 
either direction; the primary approach puts no additional constraints 
on the modeling process.  In contrast, the {\it secondary approach} 
does constrain equal values to remain equal:
\begin{eqnarray}
   \mbox{if }\delta_{ij}=\delta_{il} \mbox{ then }\gamma_j = \gamma_l . 
\end{eqnarray}
It primarily depends on the precision and reliability of the data 
which option is to be preferred; the secondary approach assumes more 
precise and reliable data, and consequently will always give a worse 
(or, at best, an equally bad) fit for a given model choice from the 
PREFMAP hierarchy.


\part{Use of the PREFMAP-3 program}
\label{part_two}
\chapter{Description of the input}
%
As explained in chapter \ref{chap:chap1}, there are two basic pieces 
of data for PREFMAP-3 to work with: an external data matrix, and a 
target configuration.  In addition, of course, the program has to be 
told what exactly to do in one single run.  In section 
\ref{sect:generalsetup} a general description of the input 
organization is given, while some parts of it are further explained in 
sections \ref{sect:options}--\ref{sect:unitnumbers}.  Finally, section 
\ref{sect:sample} provides a small working example, which will also 
serve in the discussion of the PREFMAP-3 output (chapter 
\ref{chap:chap7}).

Since there any many ways to communicate with a computer, and many 
control languages in current use, it is useful to state explicitly 
what is meant by some of the words and phrases used throughout this 
part of the PREFMAP-3 User's Guide.  The program has been written in 
ANSI-FORTRAN, and it obeys all rules and conventions from this 
standard language.

First, the input must be coded on a special type of record, called 
{\it card}, which is a record of fixed length and 80 positions long.  
Of course, it does not have to be an actual ``card", but in some file 
systems care must be taken to ensure that the input has this fixed 
form.  Secondly, cards are read by the program from a {\it unit}, 
which is an input device as defined by certain control phrases in the 
operating system.  In what will be called the {\it standard input 
stream} it is assumed that all cards come from the same unit 
(PREFMAP-3 is able to read parts of the input from distinct units, cf.  
\ref{sect:generalsetup}, \ref{sect:unitnumbers}).  Always make sure that a 
card contains {\it blanks} on whatever positions where nothing else is 
intentionally specified (although often PREFMAP-3 will not react 
strangely to unexpected symbols).  If the program expects a 
specification on a certain position, but encounters a blank, it will 
mostly perform a prechosen action called the {\it default}.  If in the 
sequel it is not indicated what the default action is, this can either 
mean that PREFMAP-3 will not be able to respond, lacking essential 
information, or that it will perform the action called `0' (zero).

Similarly, on output PREFMAP-3 generates a special type of records, 
called {\it lines}, which have 132 positions if routed to a line 
printer, and sometimes 133 positions if displayed on a screen (the 
first extra character controls the line printer behavior).  It is 
possible for PREFMAP-3 to write different pieces of output to 
different units, in the form of cards (again, not necessarily actual 
cards, depending on the type of output device).  However, this is not 
assumed to be the case in the {\it standard output stream} as 
described in chapter \ref{chap:chap7}.  Cards and lines are discussed 
in groups called {\it blocks}, independently from the ``blocks" that 
might be present in a file system; a number of cards or blocks 
together form a {\it deck}.  The organization and interpretation of 
the symbols on a card is ruled by a {\it format}, positions are also 
called {\it columns}, and a group of columns a {\it field}.  Constants 
given to the program are called {\it parameters}, whereas the 
statistical parameter estimates calculated by the program will always 
be called by their specific name, as defined in Part \ref{part_one} of 
this User's Guide.


\section{General set-up}
\label{sect:generalsetup}
%
The standard input stream consists of five blocks, which have a fixed, 
predetermined order (the order of cards within blocks is fixed as 
well).  Schematically, we must have:

\begin{center}
\begin{tabular}{ll} \\ \\
   \cline{1-1}
   \multicolumn{1}{|l|}{Title card}  & TITLE \\
   \cline{1-1} & \\ \\
   \cline{1-1}
   \multicolumn{1}{|l|}{Data specification card}  & CONTROL BLOCK \\
   \multicolumn{1}{|l|}{Analysis specification card}  & (
   \ref{sect:DataSpecifications},
   \ref{sect:AnalysisSpecifications},
   \ref{sect:PrintPlot},
   \ref{sect:unitnumbers}) \\
   \multicolumn{1}{|l|}{Print/plot options card}  &  \\
   \multicolumn{1}{|l|}{Unit number card}  &  \\
   \cline{1-1} & \\ \\   
   \cline{1-1}
   \multicolumn{1}{|l|}{Format card}  & DATA BLOCK 1 \\
   \multicolumn{1}{|l|}{\{Target configuration\} cards}  & \\
   \cline{1-1} & \\ \\
   \cline{1-1}
   \multicolumn{1}{|l|}{Format card}  & DATA BLOCK 2 \\
   \multicolumn{1}{|l|}{\{External data matrix\} cards}  & \\
   \cline{1-1} & \\ \\
   \cline{1-1}
   \multicolumn{1}{|l|}{\{Option table\} cards}  & MODEL OPTION BLOCK 
   (\ref{sect:options}) \\
   \cline{1-1} & \\ \\
\end{tabular}
\end{center}

\noindent
The {\it Title card} forms the first block.  It simply identifies the 
job, and may contain any alphanumeric symbols (up to 80 characters).  
The PREFMAP-3 output will be labeled with this information.

Next, there must be a {\it Control block} formed by four cards.  In the 
Control block all details of the job are specified, including 
information on the size and type of the three remaining blocks: Data 
block 1, Data block 2, and the Model option block.  The {\it Model 
option block} contains specific information on the models to be fitted 
for each individual, or group of individuals.  An understanding of its 
organization is needed for a proper specification of parameters in the 
Control block, and therefore the Model option block will be fully 
discussed first (section \ref{sect:options}); after that, each card of 
the Control block is explained in a separate section 
(\ref{sect:DataSpecifications}--\ref{sect:unitnumbers}).

{\it Data block 1} must contain the {\it target configuration}.  The first 
card is a format card, in particular a FORTRAN F-format card.  Any of 
the standard FORTRAN specifications may be used freely, but in many 
cases the card will look like, e.g., 
\begin{eqnarray*}
  \mbox{(3F10.4)}
\end{eqnarray*}
indicating that each following target configuration card will contain 
three numbers of the {\it F}loating type, occupying 10 positions each, 
with a decimal point in the sixth position, so that there remain four 
positions for the fractional portion.  After the format card there 
must follow $m$ (number of objects) cards, each with $p$ (number of 
dimensions) coordinate values.  Any configuration can be given as 
input to the program.  If the coordinates are not in column deviation 
form (column means equal to zero), PREFMAP-3 will perform a centering 
operation.

{\it Data block 2} must contain the {\it external data matrix}.  Like 
in Data block 1, the first card is a format card, and the earlier 
remarks about its construction apply here too.  Next there must be $n$ 
(number of individuals) cards, each with $m$ (number of objects) 
dissimilarity values.  Make sure that the scoring direction is: 
``small values $\leftrightarrow$ high preferences" and ``large value 
$\leftrightarrow$ low preference", regardless of the model to be 
fitted.  If the data happen to be coded in the other direction, a 
recoding facility {\it outside} PREFMAP-3 must be used.  Most common 
is to subtract all values from the largest possible one, or to change 
all signs in the matrix.

Through specifications on the Unit number card of the Control block it 
is possible to change the standard input stream in the sense that 
PREFMAP-3 will read parts of the input from different sources (units).  
Either the target configuration, and/or the external data matrix, 
and/or the option table may be dropped from the standard input stream.  
Notice that the Title card, the Control block, and the two format 
cards {\it always} remain on the same, predetermined unit (with number 
5, although it will mostly not be necessary for the user to specify 
this anywhere).

A concise description of the complete deck set-up is given in {\it 
Appendix \ref{chap:appendixB}}, which provides sufficient guidance for 
the advanced user.  The next sections implicitly refer to Appendix 
\ref{chap:appendixB}; it is important to note that the order of 
description in consecutive sections {\it does not correspond} to the 
order of the blocks in the standard input stream (the Model option 
block is explained first; the Title and Data blocks have already been 
fully explained here).


\section{The option table}
\label{sect:options}
%
To the large extent, the option table embodies the flexibility of 
PREFMAP-3; it monitors the variety of analyses to be done in a single 
run.  The rows of the table are called {\it option sets}, its columns 
{\it analyses}, and each entry contains an {\it option}.  An option, 
in this context, is a particular combination of a model from the 
PREFMAP hierarchy and a regression type.  Since there are four 
possible models and three possible regression types, there are 12 
options to choose from.  They are characterized by two-letter acronyms, 
as shown in Table \ref{tab:ModRegr}.  Both an option set, and
\begin{table}
  \caption{Acronyms used for model-regression combinations.}
  \protect\label{tab:ModRegr}
  \begin{center}
  {\footnotesize
    \begin{tabular}{l|ccc}
    \hline
     & \multicolumn{3}{c}{Type of regression}  \\
    \cline{2-4}
    Model                    &        & Nonmetric     & Nonmetric  \\
                         &{\it M}etric& {\it P}rimary & {\it S}econdary \\
                             &        & approach      & approach  \\
    \hline
    {\it V}ector             & VM     & VP            & VS  \\
    {\it U}nfolding          & UM     & UP            & US  \\
    {\it W}eighted Unfolding & WM     & WP            & WS  \\
    {\it G}eneral Unfolding  & GM     & GP            & GS  \\
    \hline
  \end{tabular} }
  \end{center}
\end{table}
an analysis is a series of chosen options; the former refers to the 
things to be done for any individual, the latter refers to the 
presentation of results for a group of individuals.  Let's first 
consider the situation in which there is only one individual.

In the case of a single individual the option table can have only one 
row (one option set), and at most four columns (four analyses).  There 
is no restriction on the order of the options.  Examples are given in 
Table \ref{tab:4M}, \ref{tab:one} and \ref{tab:3U}.  Table 
\ref{tab:4M} will inform PREFMAP-3 to successively apply the vector, 
the Unfolding, the weighted Unfolding, and the general Unfolding model 
all metrically.

\begin{table}
  \caption{Option table for four metric analyses.}
  \protect\label{tab:4M}
  \begin{center}{\footnotesize
    \begin{tabular}{|c|cccc|}
      \hline
       & \multicolumn{4}{c|}{analysis}  \\
      \cline{2-5}
      option set & 1  & 2  & 3  & 4  \\
      \hline
      1          & VM & UM & WM & GM  \\
      \hline
    \end{tabular}}
  \end{center}
\end{table}
\begin{table}
  \caption{Option table for one single analysis.}
  \protect\label{tab:one}
  \begin{center}{\footnotesize
    \begin{tabular}{|c|cccc|}
      \hline
       & \multicolumn{4}{c|}{analysis}  \\
      \cline{2-5}
      option set & 1  & 2  & 3  & 4  \\
      \hline
      1          & UP &    &    &    \\
      \hline
    \end{tabular}}
  \end{center}
\end{table}
\begin{table}
  \caption{Option table for three different Unfolding analyses.}
  \protect\label{tab:3U}
  \begin{center}{\footnotesize
    \begin{tabular}{|c|cccc|}
      \hline
       & \multicolumn{4}{c|}{analysis}  \\
      \cline{2-5}
      option set & 1  & 2  & 3  & 4  \\
      \hline
      1          & UM & UP & US &    \\
      \hline
    \end{tabular}}
  \end{center}
\end{table}

Table \ref{tab:one} indicates that only one analysis has to be done, 
with the Unfolding model, nonmetrically with primary approach to ties 
(i.e., ties may become untied).  Table \ref{tab:3U} will cause 
PREFMAP-3 to perform three different unfolding analyses, first 
metrically, next nonmetrically with primary approach to ties, and 
finally nonmetrically with secondary approach to ties (i.e., ties must 
remain tied in the latter analysis).  If more than four analyses on a 
single individual are desired, the data could be repeated and treated 
as a case of multiple individuals.

If there are $n$ individuals in the external data matrix (as is most 
commonly the case), the following three possibilities can be 
distinguished:
\begin{itemize}
\item[(a)] every individual gets a different option set;
\item[(b)] there are groups of individuals that share the same option set;
\item[(c)] every individual gets same option set.
\end{itemize}
The most general situation is (a); each individual becomes associated 
with one row of the option table.  An example is given in Table 
\ref{tab:GenOpt}, in which for every individual an entirely different 
series of options is specified.  PREFMAP-3 executes the options in 
rowwise order: first all options of option set 1, next all options of 
option set 2, etc.
\begin{table}
  \caption{Option table, general type.}
  \protect\label{tab:GenOpt}
  \begin{center}{\footnotesize
    \begin{tabular}{|c|cccc|}
      \hline
       & \multicolumn{4}{c|}{analysis}  \\
      \cline{2-5}
      option set & 1  & 2  & 3  & 4  \\
      \hline
      1          & VM & UM & GM & WM \\
      2          & VM & UM &    &    \\
      3          & UM & UP &    &    \\
      4          & UP &    &    &    \\
      5          & UP &    & GP &    \\
      6          & UP & US &    & WS \\
      7          & VP & US &    &    \\
      \hline
    \end{tabular} }
  \end{center}
\end{table}
After all rows have been processed (and interim results have been 
printed), the program considers all options in a column of the option 
table as a separate group, as one analysis.  Clearly, the number of 
individuals in each analysis may be different.  According to Table 
\ref{tab:GenOpt}, all individuals are in analysis 1; PREFMAP-3 will 
give, upon request, a joint plot of the target configuration with 
vectors for individuals 1, 2, and 7, and ideal points for the others.  
Analysis 2 gives the unfolding results of individuals 1, 2, 3, 6, and 
7; analysis 3 gives the general Unfolding results of 1 and 5; finally, 
analysis 4 gives the weighted unfolding results of 1 and 6.  Notice 
that it is no problem to leave an intermediate entry in a row 
unspecified.  For instance, when PREFMAP-3 reaches option set 5, it 
simply executes UP and GP consecutively; in analysis 2, it will give 
individual 5 coordinates zero (marked with ``N.A.", not applied); the 
GP results of individual 5 are presented in analysis 3.

It is also not a problem to have {\it redundancies} in the option 
table.  An example is Table \ref{tab:Redun}: there is a horizontal 
redundancy in the last three option sets (VP under analysis 3 and 4), 
and a vertical one in rows (1, 2, 3), (4, 5, 6, 7) and (8, 9, 10).  
Apparently, there are three groups of data (for instance: preferences 
from male subjects, preferences from female subjects, and a number of 
properties characterizing the objects).  Analysis 1, 2, and 3 each 
focus on a separate group while analysis 4 gives a joint representation 
of all groups under the same model.  The horizontal redundancy will 
cause PREFMAP-3 to compute VP twice in the last group (somewhat 
superfluously, indeed, but no harm is done).
\begin{table}
  \caption{Option table showing redundancies.}
  \protect\label{tab:Redun}
  \begin{center} {\footnotesize
    \begin{tabular}{|c|cccc|}
      \hline
       & \multicolumn{4}{c|}{analysis}  \\
      \cline{2-5}
      \multicolumn{1}{|c|}{option set}  & 1  & 2  & 3  & 4  \\
      \hline
      1          & UP &    &    & VP \\
      2          & UP &    &    & VP \\
      3          & UP &    &    & VP \\
      4          &    & UP &    & VP \\
      5          &    & UP &    & VP \\
      6          &    & UP &    & VP \\
      7          &    & UP &    & VP \\
      8          &    &    & VP & VP \\
      9          &    &    & VP & VP \\
     10          &    &    & VP & VP \\
      \hline
    \end{tabular} }
  \end{center}
\end{table}
\begin{table}
  \caption{Reduced version of Table \protect\ref{tab:Redun}.}
  \protect\label{tab:Red}
  \begin{center}{\footnotesize
    \begin{tabular}{|c|cccc|}
      \hline
       & \multicolumn{4}{c|}{analysis}  \\
      \cline{2-5}
      option set & 1  & 2  & 3  & 4  \\
      \hline
      1          & UP &    &    & VP \\
      2          &    & UP &    & VP \\
      3          &    &    & VP & VP \\
      \hline
    \end{tabular} }
  \end{center}
\end{table}
The vertical redundancy is in fact possibility (b) mentioned above, 
and can be communicated to PREFMAP-3 in a more economical way: through 
a proper specification on the first card of the Control block (see 
section \ref{sect:DataSpecifications}), Table \ref{tab:Red} will be 
sufficient information for PREFMAP-3 to be able to execute each option 
set repeatedly for all individuals within a group.  The reduced form 
of the table explains the introduction of the term option set: a row 
may correspond with one individual, or with a group.  Obviously, 
possibility (c) --- every individual gets the same option set --- is a 
special case of (b), and single-row tables like Table \ref{tab:4M}, 
\ref{tab:one} and \ref{tab:3U} suffice.  A very small number of 
distinct option sets (compared to the number of individuals) is most 
common in applications of PREFMAP-3.  The possibility to have a short 
cut specification has been limited to the range of one up to four 
groups.  If more than four groups of individuals share the same 
options, the user will still have to specify a complete option table.

The option table must be coded in the Model option block with one card 
for each option set.  Each two-letter acronym must occupy the first 
two positions of four-column fields.  For example:
  \begin{center}{\tt \footnotesize\setlength{\tabcolsep}{0pt}
    \begin{tabular}{*{17}{l}l}
      \cline{1-16}
       & & & & & & & & &1&1&1&1&1&1&1& ~~~column number \\
      1&2&3&4&5&6&7&8&9&0&1&2&3&4&5&6&  \\
      \cline{1-16}
      U&M& & &W&M& & &G&M& & &V&M& & & ~~~option codes \\
      \cline{1-16}
    \end{tabular}}
  \end{center}
Empty cells of the option table must be coded as blanks; there are no 
default options.  The fact that the option acronyms have to link up to 
the left might be confusing for some users, as it is in contrast to 
the usual FORTRAN I-format convention.  The options may be coded in 
either upper to lower case characters.


\section{Data specifications}
\label{sect:DataSpecifications}
%
The data specification card is the first card of the Control block, 
and may contain from 3 up to 7 parameters, coded in five-column fields 
(linked up to the right).  The parameters will each be described in 
turn.

The first parameter indicates the {\it number of rows} in the external 
data matrix, i.e.  the number of individuals $n$.  It has no default 
value.

The second parameter indicates the {\it number of columns} in the 
external data matrix, i.e.  the number of objects $m$.  It corresponds 
to the number of rows in the target configuration, and has no default 
value.

The third parameter indicates the {\it option set selection}: it 
describes the way in which option sets in the option table are to be 
linked to the individuals.  If it is given value 0 (zero), PREFMAP-3 
will apply the options in the first (and possibly only) row of the 
option table to each and every individual.  The value 1 designates the 
situation that all individuals are to be analyzed with a different 
option set; thus the program will expect an option table with $n$
rows.  The value 2 tells PREFMAP-3 that some individuals should have 
the same option set applied to them, in a way to be specified through 
the next parameters.

The last four parameters are only needed when the third one (option 
set selection) has value 2 (subgroups with the same option set).  
Otherwise, they are ignored.  Their values designate the {\it starting 
points} in the external data matrix for which a new option set 
applies.  Thus each parameter must equal the row number of the first 
individual of each subgroup sharing the same option set.  As an 
example, consider Table \ref{tab:Redun} as the complete specification 
for what has to be done with 10 individuals.  Under option set 
selection 2 we can work with the reduced option Table \ref{tab:Red}, 
and have to give the 4th parameter value 1, the 5th parameter value 4, 
and the 6th parameter value 8.  The maximum number of subgroups is 
four (hence at most four parameters).  PREFMAP-3 expects an increasing 
sequence of values.  Whenever the specified sequence is not increasing 
(e.g., the numbers 1, 5, 3, and 2 are given, in that order), the 
conflicting starting points (3 and 2) will be ignored, and the program 
acts as if a smaller number of subgroups has been specified (two 
subgroups, starting with 1 and 5).  If the same rule by which 
PREFMAP-3 is able to determine the number of subgroups in general; 
e.g., `` 1 4 8 0 " can only mean that there are three subgroups 
(therefore, no separate parameter for number of subgroups is needed).

The rows of the external data matrix must be in the right order 
according to the subgroups intended.  If they are not, say we would 
like to apply the vector model and the Unfolding model alternatingly, 
we should specify that all rows have different option sets, and 
alternate a ``vector card" with an ``Unfolding card" in the Model 
option block.


\section{Analysis specifications}
\label{sect:AnalysisSpecifications}
%
The analysis specification card is the second card of the Control 
block, and contains 10 parameters.  It controls the general 
characteristics of the analyses, and enables PREFMAP-3 to set up the 
right amount of working area.  The first nine parameters must be coded 
as integer numbers in five-column fields, the last one is a floating 
point number occupying 10 positions.

The first parameter indicates the {\it number of dimensions} of the 
object space ($p$).  It can be any number between 1 and $m- 1$ (the 
number of objects minus one).  Note that PREFMAP-3 will refuse to 
attempt specific analyses for which the number of free parameters (a 
function of $p$, see Table \ref{tab:pred_vars} in section 
\ref{sect:ModelTesting}) exceeds $m$.  Also note that the program 
expects to be able to read $p$ coordinate values from the target 
configuration cards in Data block 1 (which may contain more, but never 
less than that number of values).

The second parameter indicates the {\it maximum number of analyses in 
any option set}, or the number of columns in the option table.  This 
need not be the same as the maximum number of options in any option 
set.  For example, when using Tables \ref{tab:one}, \ref{tab:3U}, 
\ref{tab:GenOpt} or \ref{tab:Redun}, the value of this parameter must 
be 1, 3, 4, and 4, resp.

The third parameter controls the application of a {\it preliminary 
transformation of the configuration}.  The default value (zero) will 
leave the target configuration unchanged.  A value of 1 designates 
preliminary weighting of axes, a value of 2 designates preliminary 
rotation and rotation and weighting (the full transformation).

The fourth parameter controls the {\it standardization of the external 
data}.  Since standardization does not affect the fit, and the 
individual data are more comparable when they have zero mean and unit 
variance, the data are standardized row-wise by default.  In addition, 
it is possible to apply only centering (zero mean) by specifying 1, or 
only normalizing (sum of squares equal to $m$) by specifying 2.  The 
value 3 designates no standardization at all.

The next four parameters indicate simply whether or not a model is 
going to be applied in any option set.  This is necessary for 
efficient array allocation.  When a model, say the Unfolding model, is 
not referred to on this card (by a `1'), but later on the program 
encounters an Unfolding model in some option set, the Unfolding model 
will not be applied, and a warning message will be printed.  These 
parameters are ordered according to the complexity of the models:
\begin{itemize}
\item[--] the fifth parameter: {\it application of the vector model};
\item[--] the sixth parameter: {\it application of the Unfolding model};
\item[--] the seventh parameter: {\it application of the weighted Unfolding model};
\item[--] the eighth parameter: {\it application of the general Unfolding model}.
\end{itemize}
In all cases, a value of 1 designates the affirmative, and a 0 (zero) 
the negative specification.

The ninth parameter sets a {\it limit to the number of nonmetric 
iterations}.  Obviously, it is needed only if there are nonmetric 
options specified.  Its value is the maximum number of ALS cycles 
allowed in each nonmetric analysis (cf.  chapter \ref{chap:chap5}); 
the default value is 50.  Usually PREFMAP-3 will be ready long before 
this number of iterations when the standard convergence criterion 
applies.  If not, the maximum number of iterations generally has to be 
adjusted, because it is not a proper stopping rule but merely a 
safeguard.

The tenth parameter is the {\it convergence criterion}.  for nonmetric 
iterations, needed to decide when to stop the ALS process.  When the 
rate of change in the function (\ref{for:nonmetric}) drops below the 
value of this parameter, the program concludes that the process has 
converged.  The default value is .00001; with a more stringent 
criterion the number of iterations might increase beyond the limit of 
50, the default value of the previous parameter.  The user is advised 
against making the criterion more lenient, unless it is absolutely 
imperative to economize on computation time.  The convergence 
criterion must be coded with format F10.8.


\section{Print/plot options}
\label{sect:PrintPlot}
%
The print/plot options card is the third card of the Control block, 
and contains 9 parameters of the five-column integer type.  It 
controls the output of PREFMAP-3.  A detailed description of the 
output will be given in chapter \ref{chap:chap7}; described here is 
merely the way of getting various parts of it.

The first parameter controls how much of the {\it input data} is 
printed.  By default the first 10 rows of the external data matrix (at 
most) are printed, to enable checking up on correct transfer.  When a 
value of 2 is specified, the complete external data matrix is printed.  
The target configuration can be obtained by specifying 1 (to get it in 
addition to the first 10 rows of the external data), or 3 (to get it 
in addition to the complete external data).

The second parameter indicates what is to be printed {\it for each 
option}. Complete results (value 1) 
include the following:
\begin{itemize}
\item[--] the metric fit and the variance accounted for;
\item[--] the nonmetric fit (when applicable);
\item[--] the coordinates of the vector or the ideal point;
\item[--] the normalized weights (when applicable);
\item[--] the orthogonal rotation matrix (when applicable);
\item[--] the criterion and predicted values;
\item[--] the slope and the intercept.
\end{itemize}
The terms criterion values and predicted values refer to the fact that 
PREFMAP-3 solves a regression problem.  The criterion values either 
are the (possibly standardized) external data, or the external data 
transformed by a monotone function (for the nonmetric case).  The 
predicted values are the best approximation of the criterion values 
under the chosen model (the right hand sides of the regression 
equations, including the slope and the intercept terms).  The 
(nonmetric) fit is the correlation between the criterion values and 
the predicted values.  The slope and intercept terms make it possible 
to construct the predicted values from the squared Euclidean distances 
between an ideal point and the object points.  Part of the results can 
be suppressed by specifying a 2 (only the fit will be printed), or a 3 
(fit as well as criterion and predicted values will be printed).

The third parameter controls the {\it selected results for each 
analysis}.  An analysis is a series of models for different 
individuals brought together (as specified in the option table coded 
in the Model option block).  A value of 1 will cause PREFMAP-3 to 
print a table of coordinate values, next a table of weights (if 
applicable), and finally a series of rotation matrices (if 
applicable).  This is done for each analysis (column of the option 
table) in turn.  Individuals are always identified with their original 
row number and the model that has been applied.

The fourth parameter indicates that a {\it scatter and transformation 
plot} must be made for each fitted model of the first $N$ individuals; 
the parameter value $N$ can be any integer in the range from 0 up to 
$n$, the number of individuals.  In a single plot the criterion values 
are plotted against the data (the transformation), along with the 
predicted values (showing the vertical scatter around the 
transformation).  Note that the individuals for whom the plots are to 
be obtained should come first in the data matrix.  In metric analyses, 
the transformation will always be a straight line; in nonmetric 
analyses, it will be the optimal monotonically increasing step 
function.  The scatter not only visualizes the fit, it provides in 
fact a pictorial breakdown of fit into its $m$ components, each 
associated with one of the objects.

The fifth parameter calls for a {\it plot of ideal points} and/or {\it 
vectors} in the target configuration, again for each analysis in turn.  
The ideal points and vectors are labeled by integers 
$(1,2,...,8,9,0,1,...)$, and the object points are labeled by 
characters (A,B,...,H,I,J,A,...).  When the weighted Unfolding model 
or the general Unfolding model has been applied for more than one 
individual, the ideal point plot is followed by a plot of the weights.  
Further details are given in section \ref{sect:OverviewOutput}.  The 
value of this parameter, $K$, indicates that all pairs of the first 
$K$ dimensions will be plotted.  Thus if $K = 3$, then three plots are 
obtained: dimension 1 versus 2, 1 versus 3, and 2 versus 3.  When $K = 
2$, simply one plot is obtained; when $K = 1$, a two- dimensional plot 
of the first dimension is made (a straight line in a square box).  Of 
course, $K$ should not be larger than $p$, the dimensionality of the 
analysis.

The sixth parameter indicates printing of a {\it history of 
computation in the nonmetric regression} (1 = yes, 0 = no).  When 
requested, a history is given for each nonmetric option.  It will 
provide an impression of the course of the iterative process, which 
can be worthwhile looking at, especially when convergence is suspected 
to be slow or irregular.

The seventh parameter indicates computation and printing of the {\it 
F-statistic} (1 = yes, 0 = no).  The F-ratios for significance testing 
will be printed for each model in an option set, while each model will 
also be compared to the nearest simpler model from the hierarchy 
available.  This is done for each individual in turn.  For example: 
when the weighted Unfolding model and the vector model have been 
applied, the F-statistics for testing the null hypotheses $\rho_W = 0, 
\rho_V = 0$, and $\rho_U = \rho_V$ are given (cf.  section 
\ref{sect:ModelTesting}).  In case the simple Unfolding model has been 
applied too, the program gives -- in addition to the single model 
F-statistics -- the F-ratios for the hypotheses $\rho_W = \rho_U$ and 
$\rho_U = \rho_V$.  These F- ratios are always computed on the basis of 
the metric fit, and when nonmetric regressions were performed.  In the 
latter case the F-statistics approach is no longer meaningful.  When a 
model fits the data perfectly ($R_2 = 1$), the F-ratio is undefined 
and thus will not be given.

The last two parameters arrange {\it storing of the individual 
results} on an output device to be specified on the unit number card 
(section \ref{sect:unitnumbers}).  The eighth parameter indicates 
storage of coordinates (code 1), of coordinates and weights (code 2), 
or of coordinates, weights and rotation matrices (code 3).  When in 
operation, the results are always preceded by the target coordinates.  
The ninth parameter indicates storage of the external data together 
with the predicted values (code 1), or the external data together with 
the predicted values and the criterion values (code 2).

The prime usage of the storage facility is to be able to produce 
graphics on a plotting device outside PREFMAP-3.  However, there can 
be another reason for using it.  This is because the third and fifth 
parameter of the print/plot options card affect the efficiency of 
array allocation by the program itself.  When a very large number of 
individuals must be dealt with, and the core memory area available to 
the program is limited, the storage facilities can be utilized as a 
substitute for printing.  In general, then, it can be advised:
\begin{itemize}
  \item[(a)] When the number of individuals is small, to ask for
  \begin{itemize}
    \item[--] complete results for each option (2nd parameter, 
       especially if one is interested in the predicted values and 
       the monotone transformation of the data);
    \item[--] selected results for each analysis (3rd parameter);
    \item[--] plotting of ideal points and/or vectors in the target 
       configuration (5th parameter).
  \end{itemize}
\item[(b)] When the number of individuals is large, to ask for
  \begin{itemize}
    \item[--] the fit for each option only (2nd parameter);
    \item[--] selected results for each analysis (3rd parameter);
    \item[--] routing of individual results to other output units 
      (8th and 9th parameter);
    \item[--] plotting of ideal points and/or vectors in the target 
      configuration (5th parameter).
  \end{itemize}
\item[(c)] When a number of individuals is very large, to ask for
  \begin{itemize}
    \item[--] routing of individual results to other output 
      units (8th and 9th parameter).
  \end{itemize}
\end{itemize}
If the default values for storage are used on the unit number card, 
PREFMAP-3 will actually print, rather than store, the individual 
results.  Nevertheless, the program does not need additional array 
area for doing that, and the above-mentioned recommendations remain 
valid.


\section{unit numbers for I/O blocks}
\label{sect:unitnumbers}

The unit number card is the fourth card of the Control block, and 
contains 9 parameters of the five-column integer type.  The first {\it 
three} parameters (with default value 5) all refer to {\it input 
units}, and can be used to alter the standard input stream (cf.  
section \ref{sect:generalsetup}).  The next {\it two} parameters 
(having no default value) refer to {\it scratch files}, which are 
needed during execution of the program.  The last {\it four} 
parameters (with default value 6) refer to {\it output units} for 
receiving part of the results outside the standard output stream.  See 
Appendix B for details.  Whenever a unit number specification deviates 
from the default value, the user has to define a file in the operating 
system associated with the same number.

Note that the codes on the input files should satisfy the format 
specification given on the format cards, which always reside in the 
same unit as the parameter cards (section \ref{sect:generalsetup}).  
If the target configuration, the external data and the option table 
share the same unit number, they should always be there in that order.  
It is also possible to have the target configuration and the option 
table on one unit, say 22, and the external data on another, say 21.

The required space for the two scratch files depends upon the size of 
the problem and the organizational features of the operating system.

Whether or not the output files are really used is controlled by the 
last two parameters of the print/plot options cards (section 
\ref{sect:PrintPlot}).  Various partitions of the results are possible 
by specifying different output unit numbers.  Since the results are 
written in the order of computation, we would otherwise obtain 
coordinates, weights, and rotation matrices alternatingly on the same 
output file, which would make them rather inaccessible for plotting 
afterwards.


\section{Sample input stream}
\label{sect:sample}

A sample input is given here to illustrate the complete input to 
PREFMAP-3 for a single run; the output of this small example will be 
discussed in section \ref{sect:SampleOutput}.  The column numbers are 
given on top, for easy counting out.

\noindent
{\footnotesize 
\begin{verbatim}
    5   10   15   20   25   30   35   40   45   50   55
-------------------------------------------------------
*** TEST PREFMAP 3 ***
    5    5    2    1    3
    3    3   02    0    1    1    0    0   50    0
    3    1    1    2    2    1    1    0    0
    5    5    5    9    8    0    0    0    0
(8X,3F12.7)
           0.2863523   0.1391261  -0.4
           0.2459524  -0.0714838  -0.2
           0.0495586   0.1090570   0.0
          -0.1291228  -0.1841551   0.2
          -0.4527405   0.0074558   0.4
(5F7.3)
  1.500  3.500  1.500  1.500  3.000
  6.500  6.000  4.895  5.273  1.000
 -9.000 -7.677 -8.115 -7.625 -5.182
  3.000  3.000  3.000  3.000  3.000
  9.000  8.667  8.077  8.375  8.000
VM  VP
VS  UP  WM  
-------------------------------------------------------
\end{verbatim}}


\chapter{Description of output}
\label{chap:chap7}

\section{General output structure}
%
The standard output stream consists of three major blocks, with 
contents highly dependent upon the specification of the print/plot 
options card.  Schematically, we get:

\noindent\centerline{\footnotesize
\begin{tabular}{ll} \\ \\
   \cline{1-1}
   \multicolumn{1}{|l|}{Heading}  &  \\
   \multicolumn{1}{|l|}{Overview of chosen parameters}  & JOB INFORMATION BLOCK \\
   \multicolumn{1}{|l|}{Report of data \& options}  &  \\
   \cline{1-1} \\ \\
   \cline{1-1}
   \multicolumn{1}{|l|}{Results for individual 1}  & \\
   \multicolumn{1}{|l|}{\hspace{1.5cm}$\vdots$}      & \\
   \multicolumn{1}{|l|}{Results for individual $i$}& INDIVIDUAL MODELS BLOCK \\
   \multicolumn{1}{|l|}{\hspace{1.5cm}$\vdots$}      & \\
   \multicolumn{1}{|l|}{Results for individual $n$}  &  \\
   \multicolumn{1}{|l|}{Summary of results}  &  \\
   \cline{1-1} \\ \\   
   \cline{1-1}
   \multicolumn{1}{|l|}{Summary tables for consecutive analyses} & ANALYSIS BLOCK \\
   \multicolumn{1}{|l|}{Joint plots for consecutive analyses}  & \\
   \cline{1-1} \\ \\
\end{tabular}}

\noindent
The output routed to other units has a form comparable to the 
Individual models block, and will actually be inserted there if the 
default unit numbers are in operation.


\section{Overview of the output blocks}
\label{sect:OverviewOutput}

The {\it Job information block} is the least elastic part of the 
output; it is only the report of the data that can be compressed 
(first parameter of the print/plot options card).  The overview of 
chosen parameters is given in two forms; as an {\it echo} of the 
parameters cards, enabling the user to verify the literal input 
instructions, and as a {\it list of interpreted instructions} 
providing the user with feedback on the actions actually selected.  
The subheadings of the list are numbered with the corresponding card 
numbers printed in form of the echo.  When the target configuration 
turns out not to be centered at the mean, this is remedied by the 
program.  The mean for each original dimension will be printed.  When 
a preliminary transformation has been requested (third parameter of 
the analysis specification card) there will always be a print of the 
transformation target configuration, as well as the orthogonal 
rotation matrix and/or the normalized weights.

After the job information block PREFMAP-3 starts giving results for 
each individual in turn, across options, except when {\it all} 
parameters on the print/plot options card have default values.  In the 
latter case only the last part of the {\it Individual models block} is 
given: a summary of results consisting of the {\it average fit} across 
individuals sharing the same option (for each of the options present, 
not necessarily in the same column of the option table).  For the 
metric options the total variance, the total variance accounted for, 
and the proportion of total variance accounted for are printed as 
well.  The user is referred to section \ref{sect:PrintPlot} for an 
explanation of the selections of individual results possible; here 
merely a number of additional details are mentioned.

The vector coordinates are normalized such that their length is 
proportional to the individual fit.  Their overall size is determined 
by the size of the target configuration.  For each Unfolding model it 
is indicated whether the fitted point is an ideal or an anti-ideal 
point.  Saddle points are recognized by inspection of the sign pattern 
of the fitted weights.  When a coordinate value ``9999.0'' is printed 
for an ideal point this indicates that the weight for the squared 
predictor term in the regression equation (cf.  chapter 
\ref{chap:chap3}) has become almost zero.  In such a case the ideal 
point is located at infinity (or is behaving like a vector, one could 
say).  Thus it might be wise to reanalyse the individual with the 
vector model.  For the weighted models, the weights are normalized 
such that their sum squares equals the number of dimensions.

All parts of the {\it Analysis block} are optional.  The summary 
tables for consecutive analyses are controlled by the third parameter 
of the print/plot options card, the joint plots by the fifth 
parameters on that card.  Firstly all coordinates for the first 
analysis (e.g., the general Unfolding model for each individual) are 
given, followed by individual normalized weights and rotation 
matrices.  These results are printed next to the row number and the 
model that has been applied (especially useful for mixed models within 
an analysis).  If an individual did not participate in an analysis, 
the coordinates will be given as ``0.0 0.0" with the label ``N.A." 
(not applied) as the option description.  When plotted, such an 
individual is to be found in the origin, the natural point of 
inconspicuousness.  After the results for analysis 1 the program 
continues with the results for subsequent analysis, if any.

In the joint plots for consecutive analyses the ideal points and 
vectors are labeled by integers (1,2,...,8,9,0,1,...) and the object 
points are labeled by characters (A,B ...,H,I,J,A,...).  The 
convention to plot non-participating individuals in the origin is also 
maintained for fitted anti-ideal points or saddle points.  These will 
be displayed in a separate plot to prevent interpretational confusion.  
In all cases the original row number is retained to label the points.  
When two or more points coincide, the location is labeled with ``M" 
for ``more points".  As was mentioned before, an ideal point can be 
far outside the configuration of objects points (maximally ``9999.0").  
Including such a point in the joint plot would make it impossible to 
adequately display the other points.  To prevent such a ``degenerate" 
plot, all points that are far outside the range of the object space 
are omitted from the plot, and a warning is issued.

For the weighted and the general unfolding model the individual points 
are plotted in the so-called {\it common space}, i.e.  the original 
target configuration.  Individually reshaped plots could be made 
outside PREFMAP-3 by applying the appropriate (rotation and) weights 
to both the target configuration and the coordinates of the individual 
point.  Finally, the weights are displayed in a separate plot.  Under 
the general model we have to bear in mind that the weights pertain to 
differently oriented reference axes.  We can still compare, however, 
the relative importance of preferred directions across individuals 
(e.g., individual 1 has a ratio of 2:1, whereas individual 2 has a 
ratio of 3:1 for the first axis against the second in an idiosyncratic 
orientation).



\section{Sample output stream}
\label{sect:SampleOutput}

This section will show the output of the program in response to the 
sample input stream from section \ref{sect:sample}. The example has 
not been designed to demonstrate a serious analysis. On the contrary, 
the example was created to show a number of possible peculiarities of 
which the user should be aware when using the program. The complete 
printout is presented at the end of this section.

In the first place the full preliminary transformation of the target 
configuration is asked for.  This transformation cannot be performed: 
the number of target points should at least be 7 to fit the $p(p+1)/2 
= 6$ parameters of the preliminary transformation in $p = 3$ 
dimensions.  Next the input data is read, and the program detects that 
one row of data matrix has zero variance; hence this row will be 
omitted from the analysis.  The other rows will maintain their 
original row number for easy identification.

When the program reads the option sets, where the first two rows of 
the data matrix share the same set, it finds the option WM in the 
second option set.  However, the weighted Unfolding model cannot be 
applied: in the first place, there are too many parameters to be 
estimated $(2p + 1 = 7)$, and, moreover, the weighted model has not 
been referred to on the analysis specification card.

The results for individual 1 show the following peculiarities.  The 
first option, the vector model applied metrically, gives a very poor 
fit (.306).  This result is depicted in the accompanying scatter plot, 
where the points for the criterion values (labeled by a star) and the 
predicted values (labeled by a ``D") are far apart.  When the vector 
model is next applied nonmetrically, the fit proves considerably 
(.985); but the criterion values, which are the external data 
transformed by a monotone function, are hardly informative.  There 
remain only two distinct values; -2.0 (the star in the lower left 
corner of the plot) and 0.5 (the ``M" in the upper left corner, in this 
case indicating two stars on the right).  The fact that the ``D" points 
are relatively close to the stars is reflected in the relatively high 
correlation.  One should always mistrust such a dramatic difference in 
fit between the metric and the nonmetric application of a model.  The 
analysis for individual 2 shows a rather high fit for the metric 
vector model (.961).  When next the nonmetric model is applied, the 
fit is perfect, at the cost of obtaining a three-step transformation 
shown in the corresponding scatter/transformation plot.  The ``M" 
labels indicate that the criterion values and the predicted values 
coincide completely.

For individual 3 the metric Unfolding model fits perfectly.  Hence 
there is no improvement possible for the nonmetric model, and the 
F-statistics are not defined.  The same applies to the analysis for 
individual 5, and these results could already have been anticipated 
since the number of target points equals exactly the number of 
parameters that has to be estimated $(p+2 = 5)$.  When we inspect 
the selected results across analyses, we see that for individual 4 
zeros have been assigned to the coordinates and weights, while for 
analysis 3 all individuals obtain zeros (since 1 and 2 were not 
included in this analysis, and the option ``WM" for 3 and 5 could not 
be applied).

The first joint plot of the target points and the vectors from 
analysis 1 shows the coordinates for individual 4 plotted in the 
origin (``M").  In the joint plot for analysis 2, which shows vectors 
for individuals 1 and 2, and an ideal point for individual 3, we do 
not find the ideal point for individual 5.  Because individual 5 
obtained negative weights under the Unfolding model, the point should 
be depicted in a separate plot as an anti-ideal point.  However, we do 
not actually obtain this plot, because the coordinates of the 
anti-ideal point are too much outside the range of the target 
coordinates.

\newpage
{\scriptsize \def\baselinestretch{.75}
\begin{verbatim}
P R E F M A P - 3
                                  *** EXTERNAL UNFOLDING ***               JACQUELINE MEULMAN
 VERSION - 1.0                    ***         &          ***               WILLEM HEISER
 FEBRUARY 1985                    *** PROPERTY FITTING   ***               J. DOUGLAS CARROLL
                                                                           BELL LABORATORIES
                                                                           MURRAY HILL, NJ



1 JOB TITLE: *** TEST PREFMAP 3 ***                                                          


ECHO OF PARAMETER CARDS:


2     5    5    2    1    3    0    0
3     3    3    2    0    1    1    0    0   500.00000000
4     3    1    1    2    2    1    1    0    0
5     5    5    5    9    8    0    0    0    0



2 DATA SPECIFICATIONS:

    NUMBER OF ROW POINTS (THESE ARE FITTED) IS      5
    NUMBER OF COLUMN POINTS (THESE ARE GIVEN) IS    5
    OPTION SET SELECTION:                           2
             0 = ALL ROWS SAME OPTION SET       
             1 = ALL ROWS DIFFERENT OPTION SETS 
             2 = SPECIFIED ROWS SAME OPTION SET 
    OPTION SETS START WITH ROWS                     1    3



3 ANALYSIS SPECIFICATIONS:

    THE NUMBER OF DIMENSIONS IS                     3
    MAXIMUM NUMBER OF ANALYSES IN ANY OPTION SET    3
    PRELIMINARY TRANSFORMATION OF CONFIGURATION     2
             0 = REMAINS UNCHANGED              
             1 = WEIGHTED                       
             2 = ROTATED AND WEIGHTED           
    STANDARDIZE EXTERNAL DATA                       0
             0 = YES (=1+2)                     
             1 = CENTER ONLY                    
             2 = NORMALIZE ONLY                 
             3 = NONE OF THE ABOVE              
    MODELS:  0 = NOT APPLIED, 1 = APPLIED       
                 VECTOR MODEL                       1
                 UNFOLDING MODEL                    1
                 WEIGHTED UNFOLDING MODEL           0
                 GENERAL UNFOLDING MODEL            0
    THE NUMBER OF NON-METRIC ITERATIONS IS         50
    THE CONVERGENCE CRITERION IS             0.10E-04



4 PRINT/PLOT OPTIONS: 0 = NONE              

    PRINT INPUT:                                    3
             1 = TARGET CONFIGURATION           
             2 = EXTERNAL DATA                  
             3 = 1 & 2                          
    PRINT RESULTS ACROSS OPTION SETS                1
             1 = COMPLETE RESULTS               
             2 = FIT                            
             3 = 2 & CRITERION-PREDICTED VALUES 
    SELECTED RESULTS ACROSS ANALYSES                1
    SCATTER&TRANSFORMATION PLOT FOR N ROWS,  N =    2
    PAIRWISE PLOTS IDEAL POINTS/VECTORS,   DIM =    2
    HISTORY OF NON-METRIC REGRESSION                1
    STATISTICS FOR METRIC ANALYSES                  1
    STORE OUTPUT:                                   0
             1 = COORDINATES                    
             2 = LIKE 1 + WEIGHTS               
             3 = LIKE 2 + ROTATION MATRICES     
    STORE OUTPUT:                                   0
             1 = PREDICTED VALUES               
             2 = LIKE 1 + CRITERION             
\end{verbatim}
\newpage

\begin{verbatim}
5 UNIT NUMBERS FOR INPUT/OUTPUT

    THE CONFIGURATION WILL BE READ FROM UNIT        5
    THE EXTERNAL DATA WILL BE READ FROM UNIT        5
    THE OPTIONS WILL BE READ FROM UNIT              5
    UNIT NUMBER FOR SCRATCH FILE 1                  9
    UNIT NUMBER FOR SCRATCH FILE 2                  8
    OUTPUT UNIT FOR THE COORDINATES                 0
    OUTPUT UNIT FOR THE WEIGHTS                     0
    OUTPUT UNIT FOR THE ROTATION MATRICES           0
    OUTPUT UNIT FOR PREDICTED&CRITERION VALUES      0




*** WARNING *** THE PRELIMINARY TRANSFORMATION WILL NOT BE PERFORMED, BECAUSE THE     
                NUMBER OF TARGET POINTS IS NOT SUFFICIENT.


THE CONFIGURATION WILL BE READ WITH FORMAT (8X,3F12.7)                                                                     

THE TARGET CONFIGURATION WILL BE CENTERED: MEAN ORIGINAL DIMENSION    1      0.000
THE TARGET CONFIGURATION WILL BE CENTERED: MEAN ORIGINAL DIMENSION    2      0.000
THE TARGET CONFIGURATION WILL BE CENTERED: MEAN ORIGINAL DIMENSION    3      0.000


TARGET CONFIGURATION (CENTERED)
-------------------------------
             1        2        3

     1      0.286    0.139   -0.400
     2      0.246   -0.071   -0.200
     3      0.050    0.109    0.000
     4     -0.129   -0.184    0.200
     5     -0.453    0.007    0.400


THE EXTERNAL DATA WILL BE READ WITH FORMAT (5F7.3)                                                                         



    5 ROWS OF THE EXTERNAL DATA
-------------------------------
             1        2        3        4        5
     1      1.500    3.500    1.500    1.500    3.000
     2      6.500    6.000    4.895    5.273    1.000
     3     -9.000   -7.677   -8.115   -7.625   -5.182
     4      3.000    3.000    3.000    3.000    3.000


ROW    4 WILL BE OMITTED FROM THE ANALYSIS SINCE IT HAS ZERO VARIANCE


     5      9.000    8.667    8.077    8.375    8.000

MODEL:      V  = VECTOR            
            U  = UNFOLDING         
            W  = WEIGHTED UNFOLDING
            G  = GENERAL UNFOLDING 
REGRESSION:  M = METRIC            
             P = NON-METRIC, PRIMARY APPROACH TO TIES
             S = NON-METRIC, SECONDARY APPROACH TO TIES


              ANALYSIS
OPTION SET:   1   2   3
          1  VM  VP      
          2  VS  UP  WM  

*** WARNING *** OPTION 3 WILL NOT BE APPLIED. EITHER THE MODEL HAS NOT BEEN REFERRED TO
                ON CARD 3 OR THE NUMBER OF TARGET POINTS IS NOT SUFFICIENT TO FIT THIS MODEL.
\end{verbatim}
\newpage

\begin{verbatim}
ROW=    1 ANALYSIS= 1 VECTOR MODEL              METRIC REGRESSION
=================================================================


           METRIC FIT          0.306  VARIANCE      1.000    V.A.F.      0.094


           COORDINATES
           -----------

             1        2        3

     1      0.108    0.112    0.100


           CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
           ----------------

             1        2        3        4        5

     1     -0.803    1.491   -0.803   -0.803    0.918


           PREDICTED VALUES
           ----------------

             1        2        3        4        5

     1     -0.176    0.040   -0.480    0.397    0.219


           SLOPE   27.33366   INTERCEPT    0.00000
\end{verbatim}
\newpage

\begin{verbatim}
SCATTER PLOT. DATA (X-AXIS) VS CRITERION (*) AND PREDICTED VALUES (D) (M=*+D), ROW  1 VECTOR MODEL            
            .--+------+------+------+------+------+------+------+------+------+------+---.
      1.491 I                                                                        *   I
      1.450 I                                                                            I
      1.408 I                                                                            I
      1.367 I                                                                            I
      1.325 I                                                                            I
      1.283 I                                                                            I
      1.242 I                                                                            I
      1.200 I                                                                            I
      1.159 I                                                                            I
      1.117 I                                                                            I
      1.076 I                                                                            I
      1.034 I                                                                            I
      0.993 I                                                                            I
      0.951 I                                                                            I
      0.909 I                                                      *                     I
      0.868 I                                                                            I
      0.826 I                                                                            I
      0.785 I                                                                            I
      0.743 I                                                                            I
      0.702 I                                                                            I
      0.660 I                                                                            I
      0.619 I                                                                            I
      0.577 I                                                                            I
      0.535 I                                                                            I
      0.494 I                                                                            I
      0.452 I                                                                            I
      0.411 I  D                                                                         I
      0.369 I                                                                            I
      0.328 I                                                                            I
      0.286 I                                                                            I
      0.245 I                                                                            I
      0.203 I                                                      D                     I
      0.162 I                                                                            I
      0.120 I                                                                            I
      0.078 I                                                                            I
      0.037 I                                                                        D   I
     -0.005 I                                                                            I
     -0.046 I                                                                            I
     -0.088 I                                                                            I
     -0.129 I                                                                            I
     -0.171 I  D                                                                         I
     -0.212 I                                                                            I
     -0.254 I                                                                            I
     -0.296 I                                                                            I
     -0.337 I                                                                            I
     -0.379 I                                                                            I
     -0.420 I                                                                            I
     -0.462 I  D                                                                         I
     -0.503 I                                                                            I
     -0.545 I                                                                            I
     -0.586 I                                                                            I
     -0.628 I                                                                            I
     -0.670 I                                                                            I
     -0.711 I                                                                            I
     -0.753 I                                                                            I
     -0.794 I  M                                                                         I
            .--+------+------+------+------+------+------+------+------+------+------+---.
            -0.803 -0.574 -0.344 -0.115  0.115  0.344  0.574  0.803  1.032  1.262  1.491
\end{verbatim}
\newpage

\begin{verbatim}
ROW=  1 ANALYSIS= 2 VECTOR MODEL        NON-METRIC REGRESSION    PRIMARY APPROACH TO TIES
=========================================================================================


           METRIC FIT          0.306  VARIANCE      1.000    V.A.F.      0.094


               HISTORY OF NON-METRIC REGRESSION
               --------------------------------
                                              DIFFERENCE WITH
               ITERATION       FIT            PRECEDING ITERATION

                        1      0.92946        0.62341

                        2      0.94096        0.01150

                        3      0.94956        0.00860

                        4      0.95847        0.00891

                        5      0.96705        0.00858

                        6      0.97473        0.00769

                        7      0.98120        0.00646

                        8      0.98526        0.00406

                        9      0.98549        0.00023

                       10      0.98549        0.00000


           NON-METRIC FIT      0.985


           COORDINATES
           -----------

             1        2        3

     1      0.369    0.258    0.390


           CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
           ----------------

             1        2        3        4        5

     1      0.500    0.500   -2.000    0.500    0.500


           PREDICTED VALUES
           ----------------

             1        2        3        4        5

     1      0.612    0.241   -1.942    0.714    0.375


           SLOPE   41.88269   INTERCEPT    0.00000
\end{verbatim}
\newpage

\begin{verbatim}
SCATTER PLOT. DATA (X-AXIS) VS CRITERION (*) AND PREDICTED VALUES (D) (M=*+D), ROW  1 VECTOR MODEL            
            .--+------+------+------+------+------+------+------+------+------+------+---.
      0.714 I       D                                                                    I
      0.665 I                                                                            I
      0.616 I       D                                                                    I
      0.566 I                                                                            I
      0.517 I       M                                           *              *         I
      0.468 I                                                                            I
      0.419 I                                                                            I
      0.370 I                                                   D                        I
      0.321 I                                                                            I
      0.272 I                                                                            I
      0.222 I                                                                  D         I
      0.173 I                                                                            I
      0.124 I                                                                            I
      0.075 I                                                                            I
      0.026 I                                                                            I
     -0.023 I                                                                            I
     -0.073 I                                                                            I
     -0.122 I                                                                            I
     -0.171 I                                                                            I
     -0.220 I                                                                            I
     -0.269 I                                                                            I
     -0.318 I                                                                            I
     -0.367 I                                                                            I
     -0.417 I                                                                            I
     -0.466 I                                                                            I
     -0.515 I                                                                            I
     -0.564 I                                                                            I
     -0.613 I                                                                            I
     -0.662 I                                                                            I
     -0.712 I                                                                            I
     -0.761 I                                                                            I
     -0.810 I                                                                            I
     -0.859 I                                                                            I
     -0.908 I                                                                            I
     -0.957 I                                                                            I
     -1.006 I                                                                            I
     -1.056 I                                                                            I
     -1.105 I                                                                            I
     -1.154 I                                                                            I
     -1.203 I                                                                            I
     -1.252 I                                                                            I
     -1.301 I                                                                            I
     -1.351 I                                                                            I
     -1.400 I                                                                            I
     -1.449 I                                                                            I
     -1.498 I                                                                            I
     -1.547 I                                                                            I
     -1.596 I                                                                            I
     -1.646 I                                                                            I
     -1.695 I                                                                            I
     -1.744 I                                                                            I
     -1.793 I                                                                            I
     -1.842 I                                                                            I
     -1.891 I                                                                            I
     -1.940 I       D                                                                    I
     -1.990 I       *                                                                    I
            .--+------+------+------+------+------+------+------+------+------+------+---.
            -1.013 -0.741 -0.470 -0.199  0.073  0.344  0.616  0.887  1.158  1.430  1.701



         STATISTICS ACROSS OPTIONS FOR ROW    1
         --------------------------------------
         THE STATISTICS GIVEN BELOW DEPEND ONLY ON THE METRIC FIT, NOT ON THE NON-METRIC FIT.

         VECTOR MODEL,       P.VAF. = 0.094, F =   0.034 WITH   3 AND   1 DEGREES OF FREEDOM
\end{verbatim}
\newpage

\begin{verbatim}
ROW=    2 ANALYSIS= 1 VECTOR MODEL              METRIC REGRESSION
=================================================================


           METRIC FIT          0.961  VARIANCE      1.000    V.A.F.      0.923


           COORDINATES
           -----------

             1        2        3

     2     -0.520    0.250   -0.060


           CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
           ----------------

             1        2        3        4        5

     2      0.907    0.650    0.083    0.277   -1.916


           PREDICTED VALUES
           ----------------

             1        2        3        4        5

     2      0.724    1.074   -0.012   -0.073   -1.712


           SLOPE    8.02532   INTERCEPT    0.00000
\end{verbatim}
\newpage

\begin{verbatim}
SCATTER PLOT. DATA (X-AXIS) VS CRITERION (*) AND PREDICTED VALUES (D) (M=*+D), ROW  2 VECTOR MODEL            
            .--+------+------+------+------+------+------+------+------+------+------+---.
      1.074 I                                                                D           I
      1.020 I                                                                            I
      0.965 I                                                                            I
      0.911 I                                                                      *     I
      0.857 I                                                                            I
      0.803 I                                                                            I
      0.749 I                                                                      D     I
      0.695 I                                                                            I
      0.640 I                                                                *           I
      0.586 I                                                                            I
      0.532 I                                                                            I
      0.478 I                                                                            I
      0.424 I                                                                            I
      0.370 I                                                                            I
      0.316 I                                                                            I
      0.261 I                                                       *                    I
      0.207 I                                                                            I
      0.153 I                                                                            I
      0.099 I                                                  *                         I
      0.045 I                                                                            I
     -0.009 I                                                  D                         I
     -0.064 I                                                       D                    I
     -0.118 I                                                                            I
     -0.172 I                                                                            I
     -0.226 I                                                                            I
     -0.280 I                                                                            I
     -0.334 I                                                                            I
     -0.388 I                                                                            I
     -0.443 I                                                                            I
     -0.497 I                                                                            I
     -0.551 I                                                                            I
     -0.605 I                                                                            I
     -0.659 I                                                                            I
     -0.713 I                                                                            I
     -0.768 I                                                                            I
     -0.822 I                                                                            I
     -0.876 I                                                                            I
     -0.930 I                                                                            I
     -0.984 I                                                                            I
     -1.038 I                                                                            I
     -1.093 I                                                                            I
     -1.147 I                                                                            I
     -1.201 I                                                                            I
     -1.255 I                                                                            I
     -1.309 I                                                                            I
     -1.363 I                                                                            I
     -1.417 I                                                                            I
     -1.472 I                                                                            I
     -1.526 I                                                                            I
     -1.580 I                                                                            I
     -1.634 I                                                                            I
     -1.688 I   D                                                                        I
     -1.742 I                                                                            I
     -1.797 I                                                                            I
     -1.851 I                                                                            I
     -1.905 I   *                                                                        I
            .--+------+------+------+------+------+------+------+------+------+------+---.
            -2.000 -1.701 -1.402 -1.103 -0.804 -0.505 -0.206  0.093  0.392  0.691  0.990
\end{verbatim}
\newpage

\begin{verbatim}
ROW=  2 ANALYSIS= 2 VECTOR MODEL        NON-METRIC REGRESSION    PRIMARY APPROACH TO TIES
=========================================================================================


           METRIC FIT          0.961  VARIANCE      1.000    V.A.F.      0.923


               HISTORY OF NON-METRIC REGRESSION
               --------------------------------
                                              DIFFERENCE WITH
               ITERATION       FIT            PRECEDING ITERATION

                        1      0.99313        0.03253

                        2      0.99790        0.00478

                        3      0.99934        0.00143

                        4      0.99979        0.00045

                        5      0.99993        0.00014

                        6      0.99998        0.00005

                        7      0.99999        0.00001

                        8      1.00000        0.00000


           NON-METRIC FIT      1.000


           COORDINATES
           -----------

             1        2        3

     2     -0.247    0.405    0.374


           CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
           ----------------

             1        2        3        4        5

     2      1.034    1.034   -0.200   -0.200   -1.667


           PREDICTED VALUES
           ----------------

             1        2        3        4        5

     2      1.033    1.037   -0.201   -0.202   -1.666


           SLOPE    6.29833   INTERCEPT    0.00000
\end{verbatim}
\newpage

\begin{verbatim}
SCATTER PLOT. DATA (X-AXIS) VS CRITERION (*) AND PREDICTED VALUES (D) (M=*+D), ROW  2 VECTOR MODEL            
            .--+------+------+------+------+------+------+------+------+------+------+---.
      1.096 I                                                                            I
      1.045 I                                                                 M      M   I
      0.994 I                                                                            I
      0.943 I                                                                            I
      0.891 I                                                                            I
      0.840 I                                                                            I
      0.789 I                                                                            I
      0.738 I                                                                            I
      0.687 I                                                                            I
      0.636 I                                                                            I
      0.585 I                                                                            I
      0.534 I                                                                            I
      0.482 I                                                                            I
      0.431 I                                                                            I
      0.380 I                                                                            I
      0.329 I                                                                            I
      0.278 I                                                                            I
      0.227 I                                                                            I
      0.176 I                                                                            I
      0.125 I                                                                            I
      0.073 I                                                                            I
      0.022 I                                                                            I
     -0.029 I                                                                            I
     -0.080 I                                                                            I
     -0.131 I                                                                            I
     -0.182 I                                                   M    M                   I
     -0.233 I                                                                            I
     -0.285 I                                                                            I
     -0.336 I                                                                            I
     -0.387 I                                                                            I
     -0.438 I                                                                            I
     -0.489 I                                                                            I
     -0.540 I                                                                            I
     -0.591 I                                                                            I
     -0.642 I                                                                            I
     -0.694 I                                                                            I
     -0.745 I                                                                            I
     -0.796 I                                                                            I
     -0.847 I                                                                            I
     -0.898 I                                                                            I
     -0.949 I                                                                            I
     -1.000 I                                                                            I
     -1.051 I                                                                            I
     -1.103 I                                                                            I
     -1.154 I                                                                            I
     -1.205 I                                                                            I
     -1.256 I                                                                            I
     -1.307 I                                                                            I
     -1.358 I                                                                            I
     -1.409 I                                                                            I
     -1.460 I                                                                            I
     -1.512 I                                                                            I
     -1.563 I                                                                            I
     -1.614 I                                                                            I
     -1.665 I  M                                                                         I
     -1.716 I                                                                            I
            .--+------+------+------+------+------+------+------+------+------+------+---.
            -1.916 -1.634 -1.352 -1.069 -0.787 -0.505 -0.223  0.060  0.342  0.624  0.907



         STATISTICS ACROSS OPTIONS FOR ROW    2
         --------------------------------------
         THE STATISTICS GIVEN BELOW DEPEND ONLY ON THE METRIC FIT, NOT ON THE NON-METRIC FIT.

         VECTOR MODEL,       P.VAF. = 0.923, F =   3.981 WITH   3 AND   1 DEGREES OF FREEDOM
\end{verbatim}
\newpage

\begin{verbatim}
ROW=  3 ANALYSIS= 1 VECTOR MODEL        NON-METRIC REGRESSION  SECONDARY APPROACH TO TIES
=========================================================================================


           METRIC FIT          0.898  VARIANCE      1.000    V.A.F.      0.807


               HISTORY OF NON-METRIC REGRESSION
               --------------------------------
                                              DIFFERENCE WITH
               ITERATION       FIT            PRECEDING ITERATION

                        1      0.98603        0.08762

                        2      0.99826        0.01223

                        3      0.99979        0.00153

                        4      0.99997        0.00019

                        5      1.00000        0.00002

                        6      1.00000        0.00000


           NON-METRIC FIT      1.000


           COORDINATES
           -----------

             1        2        3

     3      0.488    0.224    0.277


           CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
           ----------------

             1        2        3        4        5

     3     -0.898   -0.724   -0.724    0.728    1.617


           PREDICTED VALUES
           ----------------

             1        2        3        4        5

     3     -0.897   -0.725   -0.724    0.729    1.617


           SLOPE   14.88134   INTERCEPT    0.00000
\end{verbatim}
\newpage

\begin{verbatim}
ROW=  3 ANALYSIS= 2 UNFOLDING MODEL     NON-METRIC REGRESSION    PRIMARY APPROACH TO TIES
=========================================================================================


           METRIC FIT          1.000  VARIANCE      1.000    V.A.F.      1.000


               HISTORY OF NON-METRIC REGRESSION
               --------------------------------
                                              DIFFERENCE WITH
               ITERATION       FIT            PRECEDING ITERATION

                        1      1.00000        0.00000


           NON-METRIC FIT      1.000


           COORDINATES IDEAL POINT
           -----------------------

             1        2        3

     3     -0.891   -0.162   -0.903


           CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
           ----------------

             1        2        3        4        5

     3     -1.167   -0.124   -0.469   -0.083    1.843


           PREDICTED VALUES
           ----------------

             1        2        3        4        5

     3     -1.167   -0.124   -0.469   -0.083    1.843


           SLOPE   15.93727   INTERCEPT  -28.74530



        STATISTICS ACROSS OPTIONS FOR ROW    3
        --------------------------------------
        THE STATISTICS GIVEN BELOW DEPEND ONLY ON THE METRIC FIT, NOT ON THE NON-METRIC FIT.


        PERFECT FIT FOR OPTION 2. NO STATISTICS.

        VECTOR MODEL,       P.VAF. = 0.807, F =   1.395 WITH   3 AND   1 DEGREES OF FREEDOM
\end{verbatim}
\newpage

\begin{verbatim}
ROW=  5 ANALYSIS= 1 VECTOR MODEL        NON-METRIC REGRESSION  SECONDARY APPROACH TO TIES
=========================================================================================


           METRIC FIT          0.997  VARIANCE      1.000    V.A.F.      0.994


               HISTORY OF NON-METRIC REGRESSION
               --------------------------------
                                              DIFFERENCE WITH
               ITERATION       FIT            PRECEDING ITERATION

                        1      1.00000        0.00285

                        2      1.00000        0.00000


           NON-METRIC FIT      1.000


           COORDINATES
           -----------

             1        2        3

     5      0.311    0.232    0.463


           CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
           ----------------

             1        2        3        4        5

     5      1.502    0.770   -0.960   -0.227   -1.086


           PREDICTED VALUES
           ----------------

             1        2        3        4        5

     5      1.502    0.770   -0.960   -0.227   -1.086


           SLOPE   23.54395   INTERCEPT    0.00000



ROW=  5 ANALYSIS= 2 UNFOLDING MODEL     NON-METRIC REGRESSION    PRIMARY APPROACH TO TIES
=========================================================================================


           METRIC FIT          1.000  VARIANCE      1.000    V.A.F.      1.000


               HISTORY OF NON-METRIC REGRESSION
               --------------------------------
                                              DIFFERENCE WITH
               ITERATION       FIT            PRECEDING ITERATION

                        1      1.00000        0.00000


           NON-METRIC FIT      1.000


           COORDINATES ANTI-IDEAL POINT
           ----------------------------

             1        2        3

     5     -2.369   -1.185   -2.932


           CRITERION VALUES (EXTERNAL DATA, POSSIBLY TRANSFORMED BY MONOTONE FUNCTION)
           ----------------

             1        2        3        4        5

     5      1.547    0.653   -0.931   -0.131   -1.138
\end{verbatim}
\newpage

\begin{verbatim}
           PREDICTED VALUES
           ----------------

             1        2        3        4        5

     5      1.547    0.653   -0.931   -0.131   -1.138


           SLOPE    2.73786   INTERCEPT   43.19877



        STATISTICS ACROSS OPTIONS FOR ROW    5
        --------------------------------------
        THE STATISTICS GIVEN BELOW DEPEND ONLY ON THE METRIC FIT, NOT ON THE NON-METRIC FIT.


        PERFECT FIT FOR OPTION 2. NO STATISTICS.

        VECTOR MODEL,       P.VAF. = 0.994, F =  58.233 WITH   3 AND   1 DEGREES OF FREEDOM







SUMMARY OF RESULTS: (PROPORTION OF TOTAL VARIANCE ACCOUNTED FOR ONLY GIVEN FOR METRIC OPTIONS)


MODEL '  VM  ' N =    2  AVERAGE FIT = 0.633
               ROOT MEAN SQUARED FIT = 0.713
TOTAL VARIANCE =    2.000   TOTAL VARIANCE ACCOUNTED FOR =    1.016
PROPORTION OF TOTAL VARIANCE ACCOUNTED FOR = 0.508

MODEL '  UP  ' N =    2  AVERAGE FIT = 1.000
               ROOT MEAN SQUARED FIT = 1.000

MODEL '  VP  ' N =    2  AVERAGE FIT = 0.993
               ROOT MEAN SQUARED FIT = 0.993

MODEL '  VS  ' N =    2  AVERAGE FIT = 1.000
               ROOT MEAN SQUARED FIT = 1.000




       ANALYSIS 1 COORDINATES
       ----------------------
                          1     VM      0.108    0.112    0.100
                          2     VM     -0.520    0.250   -0.060
                          3     VS      0.488    0.224    0.277
                          4     VS      0.000    0.000    0.000
                          5     VS      0.311    0.232    0.463




       ANALYSIS 2 COORDINATES
       ----------------------
                          1     VP      0.369    0.258    0.390
                          2     VP     -0.247    0.405    0.374
                          3     UP     -0.891   -0.162   -0.903
                          4     UP      0.000    0.000    0.000
                          5     UP     -2.369   -1.185   -2.932



       ANALYSIS 2 WEIGHTS
       ------------------
                          3     UP      1.000    1.000    1.000
                          4     UP      0.000    0.000    0.000
                          5     UP     -1.000   -1.000   -1.000




       ANALYSIS 3 COORDINATES
       ----------------------
                          1     N.A.    0.000    0.000    0.000
                          2     N.A.    0.000    0.000    0.000
                          3     N.A.    0.000    0.000    0.000
                          4     N.A.    0.000    0.000    0.000
                          5     N.A.    0.000    0.000    0.000
\end{verbatim}
\newpage

\begin{verbatim}
IDEAL POINTS AND/OR VECTORS (INTEGERS) IN TARGET CONFIGURATION. ANALYSIS 1 DIM 1 (X-AXIS) VS DIM 2 (Y-AXIS)
            .--+------+------+------+------+------+------+------+------+------+------+---.
      0.537 I                                                                            I
      0.519 I                                                                            I
      0.500 I                                                                            I
      0.482 I                                                                            I
      0.464 I                                                                            I
      0.446 I                                                                            I
      0.427 I                                                                            I
      0.409 I                                                                            I
      0.391 I                                                                            I
      0.373 I                                                                            I
      0.354 I                                                                            I
      0.336 I                                                                            I
      0.318 I                                                                            I
      0.300 I                                                                            I
      0.281 I                                                                            I
      0.263 I                                                                            I
      0.245 I  2                                                                         I
      0.226 I                                                           5            3   I
      0.208 I                                                                            I
      0.190 I                                                                            I
      0.172 I                                                                            I
      0.153 I                                                                            I
      0.135 I                                                         A                  I
      0.117 I                                         C   1                              I
      0.099 I                                                                            I
      0.080 I                                                                            I
      0.062 I                                                                            I
      0.044 I                                                                            I
      0.026 I                                                                            I
      0.007 I      E                               M                                     I
     -0.011 I                                                                            I
     -0.029 I                                                                            I
     -0.048 I                                                                            I
     -0.066 I                                                       B                    I
     -0.084 I                                                                            I
     -0.102 I                                                                            I
     -0.121 I                                                                            I
     -0.139 I                                                                            I
     -0.157 I                                                                            I
     -0.175 I                             D                                              I
     -0.194 I                                                                            I
     -0.212 I                                                                            I
     -0.230 I                                                                            I
     -0.248 I                                                                            I
     -0.267 I                                                                            I
     -0.285 I                                                                            I
     -0.303 I                                                                            I
     -0.322 I                                                                            I
     -0.340 I                                                                            I
     -0.358 I                                                                            I
     -0.376 I                                                                            I
     -0.395 I                                                                            I
     -0.413 I                                                                            I
     -0.431 I                                                                            I
     -0.449 I                                                                            I
     -0.468 I                                                                            I
            .--+------+------+------+------+------+------+------+------+------+------+---.
            -0.520 -0.420 -0.319 -0.218 -0.117 -0.016  0.085  0.186  0.286  0.387  0.488
\end{verbatim}
\newpage

\begin{verbatim}
IDEAL POINTS AND/OR VECTORS (INTEGERS) IN TARGET CONFIGURATION. ANALYSIS 2 DIM 1 (X-AXIS) VS DIM 2 (Y-AXIS)
            .--+------+------+------+------+------+------+------+------+------+------+---.
      0.740 I                                                                            I
      0.718 I                                                                            I
      0.695 I                                                                            I
      0.672 I                                                                            I
      0.649 I                                                                            I
      0.626 I                                                                            I
      0.603 I                                                                            I
      0.581 I                                                                            I
      0.558 I                                                                            I
      0.535 I                                                                            I
      0.512 I                                                                            I
      0.489 I                                                                            I
      0.467 I                                                                            I
      0.444 I                                                                            I
      0.421 I                                                                            I
      0.398 I                                     2                                      I
      0.375 I                                                                            I
      0.352 I                                                                            I
      0.330 I                                                                            I
      0.307 I                                                                            I
      0.284 I                                                                            I
      0.261 I                                                                        1   I
      0.238 I                                                                            I
      0.216 I                                                                            I
      0.193 I                                                                            I
      0.170 I                                                                            I
      0.147 I                                                                   A        I
      0.124 I                                                                            I
      0.101 I                                                      C                     I
      0.079 I                                                                            I
      0.056 I                                                                            I
      0.033 I                                                                            I
      0.010 I                          E                        M                        I
     -0.013 I                                                                            I
     -0.035 I                                                                            I
     -0.058 I                                                                            I
     -0.081 I                                                                 B          I
     -0.104 I                                                                            I
     -0.127 I                                                                            I
     -0.150 I                                                                            I
     -0.172 I  3                                                                         I
     -0.195 I                                            D                               I
     -0.218 I                                                                            I
     -0.241 I                                                                            I
     -0.264 I                                                                            I
     -0.287 I                                                                            I
     -0.309 I                                                                            I
     -0.332 I                                                                            I
     -0.355 I                                                                            I
     -0.378 I                                                                            I
     -0.401 I                                                                            I
     -0.423 I                                                                            I
     -0.446 I                                                                            I
     -0.469 I                                                                            I
     -0.492 I                                                                            I
     -0.515 I                                                                            I
            .--+------+------+------+------+------+------+------+------+------+------+---.
            -0.891 -0.765 -0.639 -0.513 -0.387 -0.261 -0.135 -0.009  0.117  0.243  0.369



  ANALYSIS 2 *** NOT PLOTTED ***   5   UP   -2.369   -1.185 *** SEVERELY OUT OF RANGE ***
\end{verbatim}
\def\baselinestretch{1} }

\chapter{Some applications of PREFMAP-3}

In this chapter three applications of PREFMAP-3 will be presented.  
because it is not possible to cover the whole range of applications 
here, the user might find it useful to consult one of the following 
applications: Coxon (1974), Cermak and Cornillon (1976), Davison and 
Jones (1976), Falbo (1977), Nygren and Jones (1977), Seligson (1977), 
Coxon and Jones (1978), Davison {\it et al.} (1980), and Kuyper (1980).

Heiser and De Leeuw (1981) have made an attempt to classify social 
science applications of preference mapping into prototypical groups.  
They distinguish:

\begin{itemize}
\item[--] {\it general Thurstonian attitude scaling}: scale values for 
  the attitude items are determined by a separate experimental procedure 
  (cf.  Torgerson (1958), chapter 4), and the problem of finding 
  attitude scores for the individuals, given their list of 
  ``endorsements" of preference ratings, can be solved by PREFMAP (also 
  if we switch from one-dimensional to multidimensional representations 
  of the attitude items);

\item[--] {\it trade-off studies}: suppose we have a collection of objects 
  known to differ on two negatively correlated desirable traits, e.g.  a 
  set of insurance policies varying in prize and in cover.  We now may 
  want to characterize customers in terms of their {\it safety bias} on 
  the basis of reported preferences;


\item[--] {\it multidimensional psychophysics}: suppose we have a collection 
  of objects selected as to differ on two physical attributes, e.g.  a 
  set of taste mixtures (say, alanine and glutamic acid combined in 
  various concentrations), which are to be judges as to their {\it 
  sweet-sourness}; or a set of odour mixtures (say, jasmine bergamot in 
  various concentrations), to be judges on their {\it hedonic tone};

\item[--] {\it multidimensional psychophysics}: here the objects are 
  varied systematically on psychological attributes.  Typically, one 
  confronts subjects with hypothetical ``stimulus persons", differing on, 
  e.g., intelligence and dominance, and asks for a judgment of overall 
  {\it likeableness}.  A large amount of research has been dedicated to 
  the discovery of the rule by which a subject combines different pieces 
  of information into one final impression.

\end{itemize}
It will be clear that in all cases the prime advantage of the PREFMAP 
methodology is to be able to go beyond the assumption of monotonicity 
of the dependent variable with respect to the independent (i.e., 
varied or selected by the experimenter) variables.  For a discussion 
of applying PREFMAP as a kind of quality control for Multidimensional 
Scaling and Internal Unfolding (study of {\it discriminant-, 
convergent-, or cross-validity}) the user is referred to Heiser and de 
Leeuw's original (1981) paper, and to Bechtel (1976, 1981).


\section{Facial expressions}

Our first application is a reanalysis of a classical example 
concerning facial expressions.  The target configuration has been 
obtained by performing a nonmetric Multidimensional Scaling analysis 
of the dissimilarities collected by Abelson and Sermat (1962).  The 
objects are selected photographs from the Lightfoot-series, in which 
an actress expresses 13 different emotions.  They are listed in the 
stub of Table \ref{tab:facial}.  The external data are the median 
ratings by 96 male undergraduates on three nine-point scales, obtained 
for the same 13 emotions by Engen {\it et al.} (1958), and reflecting 
judgments on the attributes ``pleasant-unpleasant" (P-U), 
``attention-rejection" (A-R), and ``tension-sleep" (T-S).

\begin{table}
  \caption{Median rating scale values of 13 facial expressions.}
  \protect\label{tab:facial}
  \begin{center}{\footnotesize
  \begin{tabular}{rl|rrr} \hline
      & \multicolumn{1}{c|}{Expression}                  & P-U & A-R & T-S \\ \hline
   1. & Grief at death of mother                         & 3.8 & 4.2 & 4.1 \\
   2. & Savoring a Coke                                  & 5.9 & 5.4 & 4.8 \\
   3. & Very pleasant surprise                           & 8.8 & 7.8 & 7.1 \\
   4. & Maternal love -- baby in arms                    & 7.0 & 5.9 & 4.0 \\
   5. & Physical exhaustion                              & 3.3 & 2.5 & 3.1 \\
   6. & Anxiety -- something is wrong with her plane     & 3.5 & 6.1 & 6.8 \\
   7. & Anger at seeing dog beaten                       & 2.1 & 8.0 & 8.2 \\
   8. & Physical strain -- pulling hard on seat of chair & 6.7 & 4.2 & 6.6 \\
   9. & Unexpectedly meets old boy friend                & 7.4 & 6.8 & 5.9 \\
  10. & Revulsion                                        & 2.9 & 3.0 & 5.1 \\
  11. & Extreme pain                                     & 2.2 & 2.2 & 6.4 \\
  12. & Knows her plane will crash (disaster)            & 1.1 & 8.6 & 8.9 \\
  13. & Light sleep                                      & 4.1 & 1.3 & 1.0 \\ \hline
  \end{tabular} }
  \end{center}
\end{table}

The data in Table \ref{tab:facial} are to be interpreted 
dissimilarities with respect to the right pole of the bipolar 
attribute scales; e.g., emotion 12, disaster, is judged to be very 
unpleasant, and emotion 3, very pleasant surprise, is indeed judged to 
be very pleasant, i.e.  very dissimilar to unpleasant.  On the other 
hand, remember the convention in PREFMAP-3 to let a small 
dissimilarity correspond to a {\it large} projection score under the 
vector model (and to a {\it small} distance under the Unfolding 
model).  For these data we consider the vector model as the most 
appropriate, because it most readily incorporates the bipolarity of 
the scales (viz., as two opposing directions along the vector).  Only 
when the fit would be very poor we might consider to apply a more 
complex model, in which one of the poles would have to be favoured as 
the central one (to be mapped as the peak/dip of the preference 
function).

The data were analyzed both metrically and nonmetrically, and the 
resulting fit measures are collected in Table \ref{tab:facial_fit}.  Notice that the 
metric fit is already quite good for all attributes, and that it is 
only slightly improved by allowing an optimal monotone transformation.  
We may conclude that the attribute judgments are fairly coherent with 
the dissimilarity judgments, a conclusion also reached by Heiser and 
Meulman (1983), who analyzed the two sets of data simultaneously.  The 
results for the present analysis are presented in Figure \ref{fig:facial}, which 
displays the points for the facial expressions and the vectors for the 
scales in a joint plot.

\begin{table}
  \caption{Fit measures for facial expressions, vector model.}
  \protect\label{tab:facial_fit}
  \begin{center}{\footnotesize
  \begin{tabular}{c|ccc} \hline
    Regression type & P-U  & A-R  & T-S  \\ \hline
    Metric          & .961 & .861 & .947 \\
    Nonmetric       & .988 & .922 & .991 \\ \hline
  \end{tabular} }
  \end{center}
\end{table}

\begin{figure}
  \centerline{\footnotesize
  \setlength{\unitlength}{0.7cm}
  \begin{picture}(14.8,14.8)(0,-1)
    \linethickness{0.3pt}
    \put(0,-1){\framebox(14.8,14.8){}}
    \put( 2.3 ,11.7 ){\makebox(0,0){disaster}}
    \put(13.5 ,10.45){\makebox(0,0){surprise}}
    \put( 9.8 , 9.9 ){\makebox(0,0){strain}}
    \put(  .7 , 9.1 ){\makebox(0,0){anger}}
    \put( 4.05, 8.0 ){\makebox(0,0){anxiety}}
    \put(12.1 , 8.1 ){\makebox(0,0){meeting}}
    \put(10.1 , 7.5 ){\makebox(0,0){savor}}
    \put(12.25, 5.75){\makebox(0,0){maternal}}
    \put( 4.2 , 5.1 ){\makebox(0,0){pain}}
    \put( 5.35, 4.55){\makebox(0,0){grief}}
    \put( 4.6 , 3.3 ){\makebox(0,0){revulsion}}
    \put( 7.7 , 3.2 ){\makebox(0,0){exhaust}}
    \put( 9.05, 1.5 ){\makebox(0,0){sleep}}
    \put( 4.7 ,13.3 ){\makebox(0,0){T}}
    \put(10.3 ,  .2 ){\makebox(0,0){S}}
    \put( 6.5 ,13.3 ){\makebox(0,0){A}}
    \put( 8.45,  .2 ){\makebox(0,0){R}}
    \put(13.8 , 9.55){\makebox(0,0){P}}
    \put( 1.0 , 4.0 ){\makebox(0,0){U}}
    \bezier{204}( 4.7 ,13.3 )(7.5  ,6.75 )(10.3 ,  .2 )
    \bezier{  6}( 9.85,  .5 )(10.075,.35 )(10.3 ,  .2 )
    \bezier{  6}(10.4 ,  .7 )(10.35, .45 )(10.3 ,  .2 )
    \bezier{200}( 6.5 ,13.3 )(7.475,6.75 )( 8.45,  .2 )
    \bezier{  6}( 8.1 ,  .4 )(8.275, .3  )( 8.45,  .2 )
    \bezier{  6}( 8.75,  .5 )(8.6  , .35 )( 8.45,  .2 )
    \bezier{190}(13.8 , 9.55)(7.4  ,6.775)( 1.0 , 4.0 )
    \bezier{  6}( 1.3 , 4.4 )(1.15 ,4.2  )( 1.0 , 4.0 )
    \bezier{  6}( 1.5 , 3.9 )(1.25 ,3.95 )( 1.0 , 4.0 )
  \end{picture} } 
  \caption{PREFMAP-3 solution for three attributes of facial 
  expression}
  \protect\label{fig:facial}
\end{figure}

It is evident from Figure \ref{fig:facial} that ``attention-rejection" 
and ``tension-sleep" partition the facial expressions in much the same 
way, and that both are psychologically independent from the 
``pleasant-unpleasant" contrast.


\section{Parliament 1972}
  
Parliament 1972\protect\footnote{The data of the Parliament Survey 
  were collected by a team of political scientists of the Department of 
  Political Science at Leiden University.  The project was supported in 
  part by the Netherlands Organization for the Advancement of Pure 
  Research (ZWO), under grants 43-03 and 43-09.} 
is a questionnaire study among the members of the 
Second Chamber of the Dutch Parliament (comparative with the House of 
Representatives in the U.S.).  The members (MPs) expressed their 
preferences for the political parties residing in Parliament, and also 
their position with respect to 7 political issues on a nine-point 
scale.  The political parties are given in Table \ref{tab:party_abbr}, 
and the issues in Table \ref{tab:pol_issues}.  The objects in the 
PREFMAP-3 analysis are the Members, identified only by their party 
allegiance; the issues serve as the PREFMAP-3 individuals, thus the 
issue self-ratings are the external data of this application.  To 
obtain a target the preference data have been analyzed with a metric 
{\it internal} Unfolding program (SMACOF3, Heiser and de Leeuw, 1979).  
From the SMACOF3 results the {\it coordinates for the MPs} served as 
the target configuration for the PREFMAP-3 analysis.

\begin{table}
  \caption{Party allegiance of MPs in the Parliament 1972 study.}
  \protect\label{tab:party_abbr}
  \begin{center}{\footnotesize
  \begin{tabular}{llcc} \hline
    Party & Description            & No. of MPs  
                                        & MP-label  \\ \hline \\
    PSP   & pacifist-socialist     &  2 & P \\
    PPR   & radical                &  1 & R \\
    PvdA  & labor                  & 36 & L \\
    D'66  & pragmatic liberal      &  8 & 6 \\
    ARP   & protestant             & 11 & A \\
    KVP   & catholic               & 32 & K \\
    CHU   & protestant             & 10 & U \\
    VVD   & conservative-liberal   & 15 & V \\
    GPV   & calvinist              &  2 & G \\
    SGP   & calvinist              &  1 & S \\
    DS'70 & conservative-socialist &  6 & 7 \\  \\ \hline
  \end{tabular} }
  \end{center}
\end{table}

\begin{table}
  \caption{Political issues in the Parliament 1972 study.}
  \protect\label{tab:pol_issues}
  \begin{center}{\footnotesize
  \begin{tabular}{lp{4cm}p{4cm}} \hline
    Issue & 
    \multicolumn{2}{c}{lower end (1) \protect\dotfill \ (9) upper end}\\ \hline
    \\
    1. DEVELOPMENT {\it AID}   & 
    Government should spend {\it less} money on aid to developing 
    countries &
    Government should spend {\it more} money on aid to developing 
    countries \\ \\
    2. {\it ABORTION} &
    A {\it woman has the right} to decide for herself about abortion &
    Government should {\it prohibit} abortion completely \\ \\
    3. LAW \& {\it ORDER} &
    Government should take {\it stronger} action against public 
    disturbances &
    Government takes {\it too strong} action against public 
    disturbances \\ \\
    4. {\it INCOME} DIFFERENCES &
    Income differences should become {\it much less} &
    Income differences should {\it remain} as they are \\ \\
    5. {\it PARTICIP}ATION &
    {\it Workers too} must have participation in decisions important 
    in industry &
    {\it Only management} should decide important matters in industry 
    \\ \\
    6. {\it TAXES} &
    Taxes should be {\it decreased} so that people can decide for 
    themselves &
    Taxes should be {\it increased} for general welfare \\ \\
    7. {\it ARMIES} &
    Government should insist on {\it maintaining} strong Western 
    armies &
    Government should insist on {\it shrinking} the Western 
    armies \\ \\ \hline
  \end{tabular} }
  \end{center}
\end{table}


Since it can be argued that at least for some issues the vector model 
adage ``the more the better" is not too suitable (more money to 
developing countries, but not the whole budget; taxes could be 
increased, but not to 100\%) both the vector model and the Unfolding 
model have been tried.  The fit measures are collected in Table 
\ref{tab:parl_fit}.

\begin{table}
  \caption{Fit measures for Parliament 1972.}
  \protect\label{tab:parl_fit}
  \begin{center}{\footnotesize
  \begin{tabular}{l|cccc} \hline
    Issue     &  VM  &  VP  &  UM  &  UP \\ \hline
    AID       & .692 & .845 & .714 & .870 \\
    ABORTION  & .609 & .836 & .665 & .839 \\ 
    ORDER     & .793 & .925 & .794 & .930 \\
    INCOME    & .703 & .867 & .710 & .876 \\
    PARTICIP  & .654 & .875 & .727 & .877 \\
    TAXES     & .791 & .902 & .813 & .908 \\
    ARMIES    & .772 & .872 & .772 & .880 \\ \hline
  \end{tabular} }
  \end{center}
\end{table}


On the basis of the fit measures we would decide that the vector 
vector is yet the more appropriate model; the Unfolding model, either 
metrically or nonmetrically, does not fit much better.  When 
inspecting the issue points in the Unfolding analysis it became clear 
that all of them are located at the outskirts of, or outside the 
target configuration.  This implies that the regression weight for the 
adequate term is small.  Besides, ABORTION, ORDER, PARTICIP, and 
ARMIES turned out to be mapped as ideal-points.  These are easy to 
interpret in this context, since we only have to reverse the 
interpretation of a small value on the issue scales.

Nevertheless, the combination of the just slightly better fit and the 
small (negative) regression weights for the quadratic term leads us to 
prefer the vector model over the Unfolding model.  We see the joint 
plot in Figure \ref{fig:parl_joint}.  Apart from the target points 
(MP's, labeled with small capitals) the column points from the 
SMACOF3 internal Unfolding analysis are plotted as well (the political 
parties, labeled with large capitals).  The latter have not been used 
in the PREFMAP-3 analysis, but are given for completeness; they are in 
fact located at the centroid of the MPs belonging to that party (a 
restriction used in the internal Unfolding analysis).  The vectors 
should be interpreted according to the lower end descriptions of the 
issue scales.  The horizontal axis clearly reflects the left-right 
distinction in Dutch politics (plotted in reversed order), coherent 
with issues like income differences (``much less" versus ``remain"), 
armies (``shrinking" versus ``maintaining"), participation (``workers 
too" versus ``management only"), and law \& order (``too strong" versus 
``stronger").  But the second dimension should not be disregarded.  
Here the abortion issue brings together the left-wing parties (PSP, 
PVA, D66) and the (economically conservative) liberal parties D70 and 
VVD.  They are opposed to the denominational parties (KVP, ARP, CHU, 
SGP, GPV) and, although to a lesser extent, to the left wing party 
PPR.  The presence of this latter party on the ``prohibit" side of the 
abortion vector can be easily explained by the fact that this MP used 
to belong to the catholic party KVP.

\begin{figure}
  \centerline{\footnotesize
  \setlength{\unitlength}{0.7cm}
  \begin{picture}(15.3,14.8)(0,0)
    \linethickness{0.3pt}
    \put( -.5 , 0   ){\framebox(15.3,14.8){}}
    \put( 9.3 ,12.8 ){\makebox(0,0){ABORTION}}
    \put( 1.6 ,10.1 ){\makebox(0,0){TAXES}}
    \put( 2.1 ,10.1 ){\makebox(0,0){AID}}
    \put( 1.1 , 7.4 ){\makebox(0,0){ARMIES}}
    \put(  .7 , 6.85){\makebox(0,0){ORDER}}
    \put(13.4 , 6.0 ){\makebox(0,0){PARTICIP}}
    \put(13.25, 5.4 ){\makebox(0,0){INCOME}}
    \bezier{200}(7.35,7.35)( 8.325,10.075)( 9.3 ,12.8 )
    \bezier{221}(7.35,7.35)( 4.475, 8.725)( 1.6 ,10.1 )
    \bezier{207}(7.35,7.35)( 4.725, 8.725)( 2.1 ,10.1 )
    \bezier{217}(7.35,7.35)( 4.225, 7.375)( 1.1 , 7.4 )
    \bezier{228}(7.35,7.35)( 4.025, 7.1  )(  .7 , 6.85)
    \bezier{214}(7.35,7.35)(10.375, 6.675)(13.4 , 6.0 )
    \bezier{214}(7.35,7.35)(10.3  , 6.375)(13.25, 5.4 )    
    \put( 4.4 , 9.45){\makebox(0,0){D70}}
    \put( 4.15, 9.3 ){\makebox(0,0){VVD}}
    \put( 9.55, 9.0 ){\makebox(0,0){D66}}
    \put(10.3 , 7.2 ){\makebox(0,0){PVA}}
    \put( 6.9 , 6.8 ){\makebox(0,0){KVP}}
    \put(13.75, 6.8 ){\makebox(0,0){PSP}}
    \put( 4.6 , 6.5 ){\makebox(0,0){CHU}}
    \put( 6.75, 5.8 ){\makebox(0,0){ARP}}
    \put(12.2 , 5.3 ){\makebox(0,0){PPR}}
    \put( 2.5 , 2.05){\makebox(0,0){SGP}}
    \put( 3.15, 2.15){\makebox(0,0){GPV}}
    \put(13.6 , 7.0 ){\makebox(0,0){\tiny P}}
    \put(13.85, 6.6 ){\makebox(0,0){\tiny P}}
    \put(12.2 , 5.3 ){\makebox(0,0){\tiny R}}
    \put( 8.2 , 8.5 ){\makebox(0,0){\tiny L}}
    \put(10.65, 8.5 ){\makebox(0,0){\tiny L}}
    \put(10.15, 8.2 ){\makebox(0,0){\tiny L}}
    \put(10.95, 8.25){\makebox(0,0){\tiny L}}
    \put(10.95, 8.15){\makebox(0,0){\tiny L}}
    \put( 9.3 , 7.85){\makebox(0,0){\tiny L}}
    \put( 9.45, 7.65){\makebox(0,0){\tiny L}}
    \put( 9.5 , 7.5 ){\makebox(0,0){\tiny L}}
    \put(10.15, 7.75){\makebox(0,0){\tiny L}}
    \put(10.15, 7.65){\makebox(0,0){\tiny L}}
    \put(10.4 , 7.75){\makebox(0,0){\tiny L}}
    \put(10.4 , 7.7 ){\makebox(0,0){\tiny L}}
    \put(10.4 , 7.6 ){\makebox(0,0){\tiny L}}
    \put( 9.75, 7.4 ){\makebox(0,0){\tiny L}}
    \put(10.9 , 7.4 ){\makebox(0,0){\tiny L}}
    \put(11.0 , 7.4 ){\makebox(0,0){\tiny L}}
    \put( 9.45, 7.25){\makebox(0,0){\tiny L}}
    \put( 9.45, 7.25){\makebox(0,0){\tiny L}}
    \put(10.7 , 7.3 ){\makebox(0,0){\tiny L}}
    \put( 9.5 , 7.1 ){\makebox(0,0){\tiny L}}
    \put( 9.55, 6.9 ){\makebox(0,0){\tiny L}}
    \put(10.75, 7.2 ){\makebox(0,0){\tiny L}}
    \put(10.65, 7.1 ){\makebox(0,0){\tiny L}}
    \put(10.8 , 7.0 ){\makebox(0,0){\tiny L}}
    \put(10.55, 6.75){\makebox(0,0){\tiny L}}
    \put(10.75, 6.8 ){\makebox(0,0){\tiny L}}
    \put(11.2 , 7.0 ){\makebox(0,0){\tiny L}}
    \put(11.5 , 6.85){\makebox(0,0){\tiny L}}
    \put(10.45, 6.6 ){\makebox(0,0){\tiny L}}
    \put(10.45, 6.5 ){\makebox(0,0){\tiny L}}
    \put(10.8 , 6.55){\makebox(0,0){\tiny L}}
    \put( 9.4 , 6.6 ){\makebox(0,0){\tiny L}}
    \put(10.2 , 6.4 ){\makebox(0,0){\tiny L}}
    \put(10.35, 6.05){\makebox(0,0){\tiny L}}
    \put(10.45, 5.7 ){\makebox(0,0){\tiny L}}
    \put(11.15, 5.8 ){\makebox(0,0){\tiny L}}
    \put( 7.4 ,10.1 ){\makebox(0,0){\tiny 6}}
    \put( 8.55, 9.3 ){\makebox(0,0){\tiny 6}}
    \put( 9.2 , 9.25){\makebox(0,0){\tiny 6}}
    \put( 9.9 , 9.2 ){\makebox(0,0){\tiny 6}}
    \put( 9.9 , 8.7 ){\makebox(0,0){\tiny 6}}
    \put(10.1 , 9.1 ){\makebox(0,0){\tiny 6}}
    \put(10.4 , 8.1 ){\makebox(0,0){\tiny 6}}
    \put(10.85, 8.65){\makebox(0,0){\tiny 6}}
    \put( 4.35, 4.4 ){\makebox(0,0){\tiny A}}
    \put( 4.95, 6.1 ){\makebox(0,0){\tiny A}}
    \put( 5.45, 6.4 ){\makebox(0,0){\tiny A}}
    \put( 5.55, 6.3 ){\makebox(0,0){\tiny A}}
    \put( 6.1 , 6.1 ){\makebox(0,0){\tiny A}}
    \put( 7.8 , 5.95){\makebox(0,0){\tiny A}}
    \put( 8.55, 6.2 ){\makebox(0,0){\tiny A}}
    \put( 8.55, 5.8 ){\makebox(0,0){\tiny A}}
    \put( 7.1 , 5.6 ){\makebox(0,0){\tiny A}}
    \put( 7.65, 5.55){\makebox(0,0){\tiny A}}
    \put( 7.95, 5.0 ){\makebox(0,0){\tiny A}}
    \put( 9.05, 3.6 ){\makebox(0,0){\tiny K}}
    \put( 9.2 , 4.55){\makebox(0,0){\tiny K}}
    \put( 5.15, 5.1 ){\makebox(0,0){\tiny K}}
    \put( 7.75, 9.6 ){\makebox(0,0){\tiny K}}
    \put( 7.65, 8.7 ){\makebox(0,0){\tiny K}}
    \put( 6.9 , 7.95){\makebox(0,0){\tiny K}}
    \put( 8   , 8   ){\makebox(0,0){\tiny K}}
    \put( 9.6 , 5.45){\makebox(0,0){\tiny K}}
    \put( 8.8 , 6.05){\makebox(0,0){\tiny K}}
    \put( 9.3 , 6.45){\makebox(0,0){\tiny K}}
    \put( 7.6 , 5.75){\makebox(0,0){\tiny K}}
    \put( 7.55, 6.2 ){\makebox(0,0){\tiny K}}
    \put( 7.65, 6.4 ){\makebox(0,0){\tiny K}}
    \put( 8.05, 6.9 ){\makebox(0,0){\tiny K}}
    \put( 5.25, 6   ){\makebox(0,0){\tiny K}}
    \put( 6.45, 6.45){\makebox(0,0){\tiny K}}
    \put( 6.05, 6.65){\makebox(0,0){\tiny K}}
    \put( 3.9 , 6.95){\makebox(0,0){\tiny K}}
    \put( 5.55, 6.9 ){\makebox(0,0){\tiny K}}
    \put( 5.85, 6.85){\makebox(0,0){\tiny K}}
    \put( 6.2 , 6.8 ){\makebox(0,0){\tiny K}}
    \put( 6.5 , 7.1 ){\makebox(0,0){\tiny K}}
    \put( 6.7 , 7.05){\makebox(0,0){\tiny K}}
    \put( 5.35, 7.1 ){\makebox(0,0){\tiny K}}
    \put( 5.65, 7.25){\makebox(0,0){\tiny K}}
    \put( 5.95, 7.25){\makebox(0,0){\tiny K}}
    \put( 6.1 , 7.4 ){\makebox(0,0){\tiny K}}
    \put( 7   , 7.35){\makebox(0,0){\tiny K}}
    \put( 5.35, 7.85){\makebox(0,0){\tiny K}}
    \put( 5.85, 7.65){\makebox(0,0){\tiny K}}
    \put( 6.5 , 7.8 ){\makebox(0,0){\tiny K}}
    \put( 7   , 7.5 ){\makebox(0,0){\tiny K}}
    \put( 4.4 , 4.35){\makebox(0,0){\tiny U}}
    \put( 3.75, 6.05){\makebox(0,0){\tiny U}}
    \put( 3.4 , 6.3 ){\makebox(0,0){\tiny U}}
    \put( 3.95, 6.35){\makebox(0,0){\tiny U}}
    \put( 4.1 , 6.45){\makebox(0,0){\tiny U}}
    \put( 4.35, 6.7 ){\makebox(0,0){\tiny U}}
    \put( 4.95, 7.1 ){\makebox(0,0){\tiny U}}
    \put( 5.7 , 7.0 ){\makebox(0,0){\tiny U}}
    \put( 6.45, 7.4 ){\makebox(0,0){\tiny U}}
    \put( 4.9 , 6.85){\makebox(0,0){\tiny U}}
    \put( 2.55, 7.85){\makebox(0,0){\tiny V}}
    \put( 3.35, 8.3 ){\makebox(0,0){\tiny V}}
    \put( 2.85, 8.55){\makebox(0,0){\tiny V}}
    \put( 3.25, 8.6 ){\makebox(0,0){\tiny V}}
    \put( 3.0 , 8.7 ){\makebox(0,0){\tiny V}}
    \put( 2.25, 8.8 ){\makebox(0,0){\tiny V}}
    \put( 3.6 , 9.4 ){\makebox(0,0){\tiny V}}
    \put( 3.7 , 9.4 ){\makebox(0,0){\tiny V}}
    \put( 4.6 ,10.2 ){\makebox(0,0){\tiny V}}
    \put( 5.15, 9.8 ){\makebox(0,0){\tiny V}}
    \put( 5.2 , 9.85){\makebox(0,0){\tiny V}}
    \put( 5.35,10   ){\makebox(0,0){\tiny V}}
    \put( 5.5 ,10   ){\makebox(0,0){\tiny V}}
    \put( 5.55,10   ){\makebox(0,0){\tiny V}}
    \put( 6.3 ,10.65){\makebox(0,0){\tiny V}}
    \put( 2.85, 2.35){\makebox(0,0){\tiny G}}
    \put( 3.5 , 1.9 ){\makebox(0,0){\tiny G}}
    \put( 2.5 , 2.05){\makebox(0,0){\tiny S}}
    \put( 2.8 , 8.9 ){\makebox(0,0){\tiny 7}}
    \put( 3.3 , 9.05){\makebox(0,0){\tiny 7}}
    \put( 6.1 , 8.85){\makebox(0,0){\tiny 7}}
    \put( 4.85,10.1 ){\makebox(0,0){\tiny 7}}
    \put( 5.4 ,10.6 ){\makebox(0,0){\tiny 7}}
    \put( 3.75, 9.3 ){\makebox(0,0){\tiny 7}}
  \end{picture} } 
  \caption{Joint plot of the MPs, parties and issues for the 
  Parliament 1972 study.}
  \protect\label{fig:parl_joint}
\end{figure}

On the whole, the representation of MPs, parties and issues is quite 
convincing, because it accumulates evidence against the idea of a 
one-dimensional polity on the one hand, and, on the other hand, it 
demonstrates the apparent possibility of predicting the stands on 
seven major political issues from a two-dimensional preference space.  
It summarizes, with little loss of information, 2232 observations in 
284 parameters.  The close resemblance to Heiser's (1981) analysis is 
also worth mentioning.


\section{Preference for family composition}

Our final application is a partial reanalysis of preference data 
collected by Delbeke (1978)\footnote{We are indebted to Luc Delbeke for 
kindly making these data available to us.}.  The objects in this study 
are 16 family types, the individuals 82 undergraduates at Leuven 
University in Belgium, who ranked the family types in order of 
preference.  Family types are defined as all combinations of {\it 
number of sons} and {\it number of daughters}, each ranging from 0 to 
3.  Thus (2,1) indicates two sons and one daughter, and (0,3) means no 
sons and three daughters.  The number of sons and the number of 
daughters have been used to create two variables, defining the target 
configuration (see Figure \ref{fig:family_target}).  When we would 
connect the points horizontally, we find family types with an equal 
number of daughters.  Connecting the points vertically gives families 
with an equal number of boys.

\begin{figure}
  \centerline{\footnotesize
  \setlength{\unitlength}{1.5cm}
  \begin{picture}(4,4)(-.5,-.5)
    \linethickness{0.3pt}
    \put(-.5,-.5){\framebox(4,4){}}
    \put(0,0){\makebox(0,0){(0,0)}}
    \put(0,1){\makebox(0,0){(0,1)}}
    \put(0,2){\makebox(0,0){(0,2)}}
    \put(0,3){\makebox(0,0){(0,3)}}
    \put(1,0){\makebox(0,0){(1,0)}}
    \put(1,1){\makebox(0,0){(1,1)}}
    \put(1,2){\makebox(0,0){(1,2)}}
    \put(1,3){\makebox(0,0){(1,3)}}
    \put(2,0){\makebox(0,0){(2,0)}}
    \put(2,1){\makebox(0,0){(2,1)}}
    \put(2,2){\makebox(0,0){(2,2)}}
    \put(2,3){\makebox(0,0){(2,3)}}
    \put(3,0){\makebox(0,0){(3,0)}}
    \put(3,1){\makebox(0,0){(3,1)}}
    \put(3,2){\makebox(0,0){(3,2)}}
    \put(3,3){\makebox(0,0){(3,3)}}
  \end{picture} } 
  \caption{Target configuration for the analysis of family composition 
  preference.}
  \protect\label{fig:family_target}
\end{figure}

In Coombs' ({\it et al.}, 1973) theory regarding family composition 
preferences two new variables are defined in terms of the old ones.  
These are the {\it size} and the {\it sex} of the families.  The main 
idea of the present analysis is to apply PREFMAP-3 with a {\it 
preliminary full transformation} of the target configuration.  This 
might give us a clue on the question which of the two possibilities is 
more appropriate for describing Delbeke's family composition 
preferences.

\begin{figure}
  \centerline{\footnotesize
  \setlength{\unitlength}{0.7cm}
  \begin{picture}(14.8,14.8)(0,0)
    \linethickness{0.3pt}
    \put(0,0){\framebox(14.8,14.8){}}
    \put( 0.95, 7.6 ){\makebox(0,0){(0,0)}}
    \put( 2.5 , 9.05){\makebox(0,0){(0,1)}}
    \put( 4.05,10.55){\makebox(0,0){(0,2)}}
    \put( 5.6 ,12   ){\makebox(0,0){(0,3)}}
    \put( 2.6 , 6.2 ){\makebox(0,0){(1,0)}}
    \put( 4.15, 7.65){\makebox(0,0){(1,1)}}
    \put( 5.75, 9.15){\makebox(0,0){(1,2)}}
    \put( 7.3 ,10.65){\makebox(0,0){(1,3)}}
    \put( 4.3 , 4.8 ){\makebox(0,0){(2,0)}}
    \put( 5.85, 6.3 ){\makebox(0,0){(2,1)}}
    \put( 7.45, 7.8 ){\makebox(0,0){(2,2)}}
    \put( 9.1 , 9.25){\makebox(0,0){(2,3)}}
    \put( 6   , 3.35){\makebox(0,0){(3,0)}}
    \put( 7.6 , 4.9 ){\makebox(0,0){(3,1)}}
    \put( 9.15, 6.4 ){\makebox(0,0){(3,2)}}
    \put(10.7 , 7.85){\makebox(0,0){(3,3)}}
    \put( 9.85, 2.80){\circle*{.07}}
    \put(11.05, 3.65){\circle*{.07}}
    \put(12.80, 4.60){\circle*{.07}}
    \put(12.65, 4.75){\circle*{.07}}
    \put(13.00, 6.90){\circle*{.07}}
    \put(13.65, 8.80){\circle*{.07}}
    \put(12.35, 8.70){\circle*{.07}}
    \put(11.50, 8.85){\circle*{.07}}
    \put(10.80, 8.65){\circle*{.07}}
    \put(10.65, 8.65){\circle*{.07}}
    \put( 7.60, 5.90){\circle*{.07}}
    \put( 7.00, 5.70){\circle*{.07}}
    \put( 6.85, 5.75){\circle*{.07}}
    \put( 6.60, 5.90){\circle*{.07}}
    \put( 5.80, 5.70){\circle*{.07}}
    \put( 7.90, 4.80){\circle*{.07}}
    \put( 7.55, 5.00){\circle*{.07}}
    \put( 6.95, 5.95){\circle*{.07}}
    \put( 6.90, 6.30){\circle*{.07}}
    \put( 6.00, 6.05){\circle*{.07}}
    \put(11.25, 7.00){\circle*{.07}}
    \put(10.40, 6.95){\circle*{.07}}
    \put( 9.75, 6.90){\circle*{.07}}
    \put( 9.70, 6.95){\circle*{.07}}
    \put( 9.25, 6.60){\circle*{.07}}
    \put( 6.95, 9.80){\circle*{.07}}
    \put( 3.80, 6.85){\circle*{.07}}
    \put( 2.25, 6.20){\circle*{.07}}
    \put( 1.55, 7.25){\circle*{.07}}
    \put(11.30, 8.15){\circle*{.07}}
    \put(10.95, 8.15){\circle*{.07}}
    \put(10.35, 7.65){\circle*{.07}}
    \put( 9.65, 7.10){\circle*{.07}}
    \put( 9.20, 7.40){\circle*{.07}}
    \put( 9.00, 7.95){\circle*{.07}}
    \put( 5.30, 6.80){\circle*{.07}}
    \put( 5.15, 7.10){\circle*{.07}}
    \put( 5.10, 7.15){\circle*{.07}}
    \put( 4.95, 7.35){\circle*{.07}}
    \put( 5.05, 7.50){\circle*{.07}}
    \put( 4.80, 7.60){\circle*{.07}}
    \put( 4.90, 7.90){\circle*{.07}}
    \put( 5.05, 8.20){\circle*{.07}}
    \put( 8.20, 6.90){\circle*{.07}}
    \put( 8.00, 6.90){\circle*{.07}}
    \put( 6.85, 6.95){\circle*{.07}}
    \put( 6.55, 7.10){\circle*{.07}}
    \put( 5.55, 7.20){\circle*{.07}}
    \put( 5.55, 7.35){\circle*{.07}}
    \put( 7.05, 7.30){\circle*{.07}}
    \put( 7.00, 7,25){\circle*{.07}}
    \put( 6.50, 7.15){\circle*{.07}}
    \put( 6.15, 7.25){\circle*{.07}}
    \put( 6.30, 7.85){\circle*{.07}}
    \put( 8.45, 7.70){\circle*{.07}}
    \put( 8.30, 7.35){\circle*{.07}}
    \put( 8.10, 7.25){\circle*{.07}}
    \put( 7.45, 7.40){\circle*{.07}}
    \put( 6.80, 7.60){\circle*{.07}}
    \put( 6.75, 7.55){\circle*{.07}}
    \put( 6.55, 7.55){\circle*{.07}}
    \put( 9.05, 9.10){\circle*{.07}}
    \put( 8.55, 8.50){\circle*{.07}}
    \put( 7.55, 8.10){\circle*{.07}}
    \put( 7.15, 8.05){\circle*{.07}}
    \put( 6.85, 7.95){\circle*{.07}}
    \put( 6.30, 8.55){\circle*{.07}}
    \put( 7.70, 7.75){\circle*{.07}}
    \put( 7.25, 7.85){\circle*{.07}}
    \put( 8.90, 7.05){\circle*{.07}}
    \put( 8.85, 7.05){\circle*{.07}}
  \end{picture} } 
  \caption{Transformed target configuration and ideal points from the analysis 
  of preferences for family composition.}
  \protect\label{fig:family_trans}
\end{figure}

The average fit for the 82 subjects, analyzing their preferences with 
the Unfolding model, metrically, with preliminary full transformation, 
is .906; the proportion of the total variance accounted for is .823.  
Without a preliminary full transformation, these figures are .899 and 
.814, resp.  The increase in fit is obviously not dramatic.  For the 
preliminary transformation weights of 1.060 and .936 were obtained, 
for the first new axis and the second, resp.  Thus the axes are only 
slightly differentially weighted.  But the new axes' orientation is 
interesting (see Figure \ref{fig:family_trans}).  Here the subjects 
points are labeled with a ``x" (nine individuals are not included in 
the plot because their coordinates were severely out of range).  It is 
clear that the transformation recovered the ``size/sex" set of 
variables.  Looking at the family points we now have a large bias 
towards daughters at the top, moving to a large bias towards sons at 
the bottom.  The horizontal axis coincides with the total number of 
children.  Evidently, this sample of Belgium students has high 
preference for large families, while sons are preferred over 
daughters.

For further analyses on this kind of data, we refer to Coxon (1974), 
Bechtel (1976), and Rodgers and Young (1981).

\vspace{3cm}
Jacqueline Meulman

Willem J. Heiser

J. Douglas Carroll




\chapter*{References}

\begin{description}
\item
Abelson, R.P. \& Sermat, V.  (1962).  Multidimensional scaling of 
facial expressions.  {\it J. Exp. Psych.}, {\it 63}, 546-554.

\item
Bechtel, G.G.  (1976).  {\it Multidimensional Preference Scaling}.  The 
Hague: Mouton.

\item
Bechtel, G.G.  (1981).  Metric information for group representations.  
In I.  Borg (Ed.), {\it Multidimensional Data Representations: When \& Why}.  
Ann Arbor, Mich.: Mathesis Press, 441-478.

\item
Carroll, J.D.  (1972).  Individual differences and multidimensional 
scaling.  In R.N.  Shepard {\it et al.}  (Eds.), {\it Multidimensional Scaling}, 
{\it Vol.  I: Theory}.  New York: Seminar Press, 105-155.

\item
Carroll, J.D.  (1980).  Models and methods for multidimensional 
analysis of preferential choice (or other dominance) data.  In E.D.  
Lantermann \& H.  Feger (Eds.), {\it Similarity and Choice}.  Bern: Hans 
Huber Publ., 234-289.

\item
Carroll, J.D.  \& Chang, J.J.  (1970).  Analysis of individual 
differences in multidimensional scaling via an N-way generalization of 
``Eckart-Young" decomposition.  {\it Psychometrika}, {\it 35}, 283-319.

\item
Cermak, G.W.  \& Cornillon, P.C.  (1976).  Multidimensional analyses of 
judgments about traffic noise.  {\it J. Acoust. Soc. Am.}, {\it 59}, 1412-1420.

\item
Chang, J.J.  \& Carroll, J.D.  (1972).  {\it How to use PREFMAP and 
PREFMAP-2} -- Programs which relate preference data to multidimensional 
scaling solutions.  Unpublished manuscript, Bell Telephone Labs, 
Murray Hill, NJ.

\item
Coombs, C.H.  \& Avrunin, G.S.  (1977).  Single-peaked functions and 
the theory of preference.  {\it Psych. Rev.}, {\it 84}, 216-230.

\item
Coombs, C.H.,  McClelland, G.H.  \& Coombs, L.  (1973).  {\it The measurement 
and analysis of family composition preferences}.  Internal Report, 
Mich.  Math.  Psych.  Progr.  no.  73-5, University of Michigan.

\item
Coxon, A.P.M.  (1974).  The mapping of family composition preferences: 
a scaling analysis.  {\it Social Science Research}, {\it 3}, 191-210.

\item
Coxon, A.P.M.  (1982).  {\it The User's Guide to Multidimensional 
Scaling}.  London: Heinemann Educ.  Books.

\item
Coxon, A.P.M.  \& Jones, C.L.  (1978).  {\it The images of Occupational 
Prestige}.  New York: MacMillan.

\item
Davison, M.L.  \& Jones, L.E.  (1976).  A similarity-attraction model 
for predicting sociometric choice from perceived group structure.  
{\it J. Pers. Soc. Psych.}, {\it 33}, 601-612.

\item
Davison, M.L., King, P.M., Kitchener, K.S.  \& Parker, C.A.  (1980).  
The stage sequence concept in cognitive and social development.  {\it 
Developm. Psych.}, {\it 16}, 121-131.

\item
Delbeke, L.  (1979). {\it  Enkele analyses op voorkeuroordelen voor 
gezinssamenstellingen}.  Internal Report, Centrum voor Math.  Psych.  
en Psych.  Meth., University of Leuven, Leuven, Belgium.

\item
Engen, T., Levy, N.  \& Schlosberg, H. (1958). The dimensional analysis of a 
new series of facial expressions.  {\it J. Exp. Psych.}, {\it 55}, 454-458.

\item
Falbo, T.  (1977).  Multidimensional scaling of power strategies.  
{\it J. Pers. Soc. Psych.}, {\it 35}, 537-547.

\item
Gifi, A.  (1981).  {\it Nonlinear Multivariate Analysis}.  Leiden, The 
Netherlands: Department of Data Theory, University of Leiden.

\item
Heiser, W.J.  (1981).  {\it Unfolding Analysis of Proximity Data}.  Doctoral 
Dissertation, University of Leiden, Leiden, The Netherlands.

\item
Heiser, W.J.  \& De Leeuw, J.  (1979).  {\it How to use SMACOF-3}.  Internal 
Report, Department of Data Theory, University of Leiden, Leiden, The 
Netherlands.

\item
Heiser, W.J.  \& De Leeuw, J.  (1981).  Multidimensional mapping of 
preference data.  {\it Math. Sci. Hum.}, {\it 19}, 39-96.

\item
Heiser, W.J.  \& Meulman, J.  (1983).  Constrained multidimensional 
scaling, including confirmation.  {\it Appl. Psych. Meas.}, {\it 7}, 381-404.

\item
Kruskal, J.B.  (1964).  Multidimensional scaling by optimizing 
goodness of fit to a nonmetric hypothesis. {\it  Psychometrika}, {\it 
29}, 1-28.

\item
Kruskal, J.B.  (1965).  Analysis of factorial experiments by 
estimating monotone transformation of the data.  {\it J.  Roy.  Stat.  
Soc.}, {\it 27B}, 251-263.

\item
Kruskal, J.B.  \& Shepard, R.N.  (1974).  A nonmetric variety of 
linear factor analysis.  {\it Psychometrika}, {\it 39}, 123-157.

\item
Kuyper, H.  (1980).  {\it About the Saliency of Social Comparison 
Dimensions}.  Doctoral Dissertation, University of Groningen, 
Groningen, The Netherlands.

\item
Nygren, T.E.  \& Jones, L.E.  (1977).  Individual differences in 
perceptions and preferences for political candidates.  {\it J. Exp. 
Soc. Psych.}, {\it 13}, 182-197.

\item
Rodgers, J.L.  \& Young, F.W.  (1981).  Successive unfolding of family 
preferences.  {\it Appl. Psych. Meas.}, {\it 5}, 51-62.

\item
Seligson, M.A.  (1977).  Prestige among peasants: a multidimensional 
analysis of preference data.  {\it Am. J. Sociol.}, {\it 83}, 632-652.

\item
Shepard, R.N.  (1962).  The analysis of proximities, I \& II.  
{\it Psychometrika}, {\it 27}, 125-140 and 219-246.

\item
Torgerson, W.S. (1958). {\it Theory and Methods of Scaling}. New York: Wiley.

\item
Young, F.W. (1981). Quantitative analysis of qualitative data. {\it 
Psychometrika}, {\it 46}, 357-387.
\end{description}

\appendix
\chapter{Comparison with PREFMAP-2 and implementation}
\label{chap:appendixA}
%Comparison with PREFMAP-2 and  miscellaneous technical information for 
%implementation of PREFMAP-3.

\begin{sloppypar}
Only the most important differences between the PREFMAP-2 and the 
PREFMAP-3 program will be described here; there are many disparities 
in details, most notably in the input and the output, but these will 
be evident to anyone switching from one program to the other.
\end{sloppypar}

From a practical point of view the PREFMAP-3 program has been designed 
to be as {\it flexible} as possible.  The array allocation is done 
dynamically, so that -- given a large enough array to start with -- 
there are no severe limitations to the number of individuals, objects, 
or dimensions to be analyzed (see Table \ref{tab:declar} for 
examples).  It is even possible to analyze an ``{\it infinite}" number 
of individuals without any increase in space beyond what is required 
to analyze one single individual.  In that case the fact that every 
row of the external data matrix is analyzed separately is maximally 
exploited.  Although we don't yet have experience with it, we expect 
this feature will make it feasible to run PREFMAP-3 smoothly on a 
Personal Computer (or other machines with modest core memory).  Note, 
however, that it will still require space to obtain print output 
dealing with all individuals simultaneously (e.g., plotting the ideal 
points in the target configuration); thus one should refrain from 
those specifications when the data matrix is very large (and array 
area is limited).  However, the individual results will never be 
completely lost, since they can be routed to other output units to 
provide for the possibility of constructing composite plots 
afterwards.  Specific suggestions for dealing with various kinds of 
situations are given in section \ref{sect:PrintPlot}.

A second major difference with the PREFMAP-2 program consists in the 
choice of the {\it organization of the model fitting process}.  In the 
PREFMAP-2 program the same model must necessarily be fitted, at each 
phase of the analysis, to each row of the data matrix.  One has to 
start at one stage in the hierarchy and to stop at another, solving 
for all intermediate models.  Moreover, the choice of the starting 
point affects the results for the simpler models in the PREFMAP-2 
program.  Those organizational features have been changed quite 
radically in the present program.  Any ordered combination of models 
can be chosen, for example the most complex (the general ideal point 
model) and the simplest (the vector model), and they will be applied 
to the same (transformed) target configuration.  In addition, the 
separate rows need not be fitted by the same models, although that is 
still possible.  The possibility exists to give each row its own set 
of models.

The final major difference between PREFMAP-2 and PREFMAP-3 that we 
want to mention consists in the fact that the present program performs 
only {\it purely external analyses}.  In the PREFMAP-2 program one out 
of a number of possible object configurations could be generated.  
This object configuration was obtained from the very same data matrix, 
whereas the rows would be fitted in afterwards in the second stage of 
the analysis.  This two-stage process in fact renders an internal 
Unfolding solution.  Since there are many general MDS and PCA programs 
available that perform internal analyses under various criteria, this 
possibility does not exist anymore in PREFMAP-3.  If the user desires 
to employ, in PREFMAP-3, an object configuration derived from the 
external data, this configuration must be calculated with some 
auxiliary procedure, and must then be inserted in the input stream to 
the PREFMAP-3 program.


\subsection*{Implementation}

PREFMAP-3 is a portable ANSI FORTRAN-IV program.  It has been 
successfully tested on IBM, Honeywell, and VAX computers.  The program 
contains a FORTRAN array allocation routine, called DECLAR.  This 
routine allocates a superarray of a certain specified length.

\begin{table}
  \caption{Examples of analyses that can be performed within a 
  specified superarray size of 25100.}
  \protect\label{tab:declar}
  \begin{center}{\footnotesize\setlength{\tabcolsep}{3pt}
  \begin{tabular}{rr*{9}{c}|c} \hline
    NROW& NCOL& NDIM& NANA& IPRE& VP& UP& WP& GP& IPRI& IPLO& NWORDS \\ \hline
     350&  20 &  3  &  4  &  2  & 1 & 1 & 1 & 1 &  1  &   3 &  24033 \\
     500&  65 &  2  &  4  &  2  & 1 & 1 & 1 & 1 &  1  &   2 &  24934 \\
    1000&  75 &  2  &  3  &  2  & 1 & 1 & 1 & 0 &  1  &   2 &  24307 \\
    1000& 115 &  2  &  1  &  2  & 0 & 1 & 0 & 0 &  1  &   2 &  24831 \\
    2000&  85 &  2  &  1  &  2  & 0 & 1 & 0 & 0 &  1  &   2 &  24643 \\
    3000&  40 &  2  &  1  &  2  & 0 & 1 & 0 & 0 &  1  &   2 &  24263 \\
    3500&  10 &  2  &  1  &  2  & 0 & 1 & 0 & 0 &  1  &   2 &  25093 \\
   99999& 130 &  3  &  1  &  2  & 0 & 1 & 0 & 0 &  0  &   0 &  24266 \\
   99999& 730 &  4  &  1  &  0  & 0 & 1 & 0 & 0 &  0  &   0 &  24934 \\
   99999& 890 &  2  &  1  &  0  & 0 & 1 & 0 & 0 &  0  &   0 &  24794 \\
 \hline
  \end{tabular}  
  \begin{tabular}{llll} 
    \\ \\
    {\it Legend:} \\ \\
    NROW   & = number of rows                  & 
    UP     & = Unfolding model, primary app.   \\
    NCOL   & = number of columns               & 
    WP     & = weighted Unfolding model, ditto \\
    NDIM   & = number of dimensions            & 
    GP     & = general Unfolding model, ditto  \\
    NANA   & = number of analyses              & 
    IPRI   & = print selected results          \\
    IPRE   & = preliminary transformation      & 
    IPLO   & = plot pairwise dimensions        \\
    VP     & = vector model, primary approach  & 
    NWORDS & = total array area                \\
  \end{tabular}}
  \end{center}
\end{table}

The required size of the superarray depends on a number of parameters, 
some of which vary with the size of the problem (number of objects, 
dimensions), while others depend on the user-selected analysis and 
print specifications.  If the superarray is not large enough, the 
program will return from DECLAR with an error message and with the 
correct size of the superarray.  Table \ref{tab:declar} gives an overview of 
different analyses that can be performed within the internally fixed 
upper bound of the superarray (25100 words).  This upper bound can, of 
course, be enlarged when implementing the program.  If a genuine 
dynamic storage allocation facility is derived, subroutine DECLAR 
should be replaced by a machine assembler routine.

On some installations the plots produced by the program might come out 
rectangular instead of square.  Several statements in the subroutine 
PRPLOT should be adapted according to the comments in this subroutine.  
The logical unit numbers to read the standard input stream and to 
write the standard output stream are defined in the main routine by 
the statements INPARA = 5 and IWRITE = 6, respectively.


\subsection*{Program structure}

In terms of its subroutines, the program is structured as indicated in 
Figure \ref{fig:flow}.
% (unfortunately, the name PLOT appears in the 
%Figure instead of the correct subroutine name PRPLOT).  
Here, successive subroutine calls are given from top to bottom, 
whereas further calls into a deeper level are depicted horizontally.  
The subroutine MAIN3 controls the flow of the program.  PREREC is used 
to perform printing of the input data and configuration, PREPO 
performs the preliminary transformation, PRED1 up to PRED4 construct 
predictor matrices suitable for the selected models, and PSINV 
computes their pseudo inverses.

\begin{figure}
  \centerline{\footnotesize
  \setlength{\unitlength}{0.5cm}
  \begin{picture}(27,28)(-11,-1)
    \linethickness{0.3pt}
    \put(-11,-1){\framebox(27,28){}}
    \put( 0  , 0  ){\line(0,1){26}}
    \put( 0  , 0  ){\line(1,0){1}}
    \put( 1  , 0  ){\makebox(0,0)[l]{~OUTP3}}
    \put( 4  , 0  ){\line(1,0){2}}
    \put( 6  , 0  ){\makebox(0,0)[l]{~PRPLOT}}
    %
    \put( 0  , 2  ){\line(1,0){1}}
    \put( 1  , 2  ){\makebox(0,0)[l]{~STATIS}}
    \put( 4  , 2  ){\line(1,0){2}}
    \put( 6  , 2  ){\makebox(0,0)[l]{~SHEL9}}
    %
    \put( 0  , 4  ){\line(1,0){1}}
    \put( 1  , 4  ){\makebox(0,0)[l]{~RESUL2}}
    \put( 4  , 4  ){\line(1,0){1}}
    \put( 5  , 3.5){\line(0,1){1}}
    \put( 5  , 3.5){\line(1,0){1}}
    \put( 6  , 3.5){\makebox(0,0)[l]{~PRPLOT}}
    \put( 5  , 4.5){\line(1,0){1}}
    \put( 6  , 4.5){\makebox(0,0)[l]{~PREREC}}
    %
    \put( 0  , 9  ){\line(1,0){1}}
    \put( 1  , 9  ){\makebox(0,0)[l]{~UNRAV}}
    \put( 4  , 9  ){\line(1,0){2}}
    \put( 5  , 7  ){\line(0,1){4}}
    \put( 6  , 9  ){\makebox(0,0)[l]{~UNRA3}}
    \put( 5  ,11  ){\line(1,0){1}}
    \put( 6  ,11  ){\makebox(0,0)[l]{~UNRA1}}
    \put( 5  ,10  ){\line(1,0){1}}
    \put( 6  ,10  ){\makebox(0,0)[l]{~UNRA2}}
    \put( 5  , 8  ){\line(1,0){1}}
    \put( 6  , 8  ){\makebox(0,0)[l]{~UNRA4}}
    \put( 5  , 7  ){\line(1,0){1}}
    \put( 6  , 7  ){\makebox(0,0)[l]{~RESUL1}}
    \put( 9  , 7  ){\line(1,0){1}}
    \put(10  , 7  ){\makebox(0,0)[l]{~PREREC}}
    %
    \put( 0  ,15  ){\line(1,0){1}}
    \put( 1  ,15  ){\makebox(0,0)[l]{~MONO}}
    \put( 4  ,15  ){\line(1,0){2}}
    \put( 5  ,13  ){\line(0,1){4}}
    \put( 6  ,15  ){\makebox(0,0)[l]{~PROJEC}}
    \put( 5  ,17  ){\line(1,0){1}}
    \put( 6  ,17  ){\makebox(0,0)[l]{~RANK1}}
    \put( 9  ,17  ){\line(1,0){.5}}
    \put( 9.5,16.5){\line(0,1){1}}
    \put( 9.5,16.5){\line(1,0){.5}}
    \put(10  ,16.5){\makebox(0,0)[l]{~TIEBL}}
    \put( 9.5,17.5){\line(1,0){.5}}
    \put(10  ,17.5){\makebox(0,0)[l]{~SHEL9}}    
    \put( 5  ,13  ){\line(1,0){1}}
    \put( 6  ,13  ){\makebox(0,0)[l]{~MONOR}}
    \put( 5.5,14  ){\vector(0,-1){1}}
    \put( 5.5,14  ){\vector(0, 1){1}}
    \put( 9  ,13  ){\line(1,0){1}}
    \put( 9.5,12  ){\line(0,1){2}}
    \put(10  ,13  ){\makebox(0,0)[l]{~SCAR}}
    \put( 9.5,12  ){\line(1,0){.5}}
    \put(10  ,12  ){\makebox(0,0)[l]{~MRMNH}}
    \put( 9.5,14  ){\line(1,0){.5}}
    \put(10  ,14  ){\makebox(0,0)[l]{~PRAR}}
    \put(12  ,14  ){\line(1,0){1}}
    \put(13  ,14  ){\makebox(0,0)[l]{~SHEL9}}
    %
    \put( 0  ,17  ){\line(1,0){1}}
    \put( 1  ,17  ){\makebox(0,0)[l]{~ESTIMA}}
    %
    \put( 0  ,18  ){\line(1,0){1}}
    \put( 1  ,18  ){\makebox(0,0)[l]{~ZEROI}}
    %
    \put( 0  ,19  ){\line(1,0){1}}
    \put( 1  ,19  ){\makebox(0,0)[l]{~PSINV}}
    \put( 4  ,19  ){\line(1,0){2}}
    \put( 6  ,19  ){\makebox(0,0)[l]{~SIVAD}}
    %
    \put( 0  ,20  ){\line(1,0){1}}
    \put( 1  ,20  ){\makebox(0,0)[l]{~PRED4}}
    %
    \put( 0  ,21  ){\line(1,0){1}}
    \put( 1  ,21  ){\makebox(0,0)[l]{~PRED3}}
    %
    \put( 0  ,22  ){\line(1,0){1}}
    \put( 1  ,22  ){\makebox(0,0)[l]{~PRED2}}
    %
    \put( 0  ,23  ){\line(1,0){1}}
    \put( 1  ,23  ){\makebox(0,0)[l]{~PRED1}}
    %
    \put( 0  ,24  ){\line(1,0){1}}
    \put( 1  ,24  ){\makebox(0,0)[l]{~PREPRO}}
    \put( 4  ,24  ){\line(1,0){2}}
    \put( 5  ,22  ){\line(0,1){4}}
    \put( 6  ,24  ){\makebox(0,0)[l]{~TRED2}}
    \put( 5  ,26  ){\line(1,0){1}}
    \put( 6  ,26  ){\makebox(0,0)[l]{~PRED4}}
    \put( 5  ,25  ){\line(1,0){1}}
    \put( 6  ,25  ){\makebox(0,0)[l]{~PSINV}}
    \put( 5  ,23  ){\line(1,0){1}}
    \put( 6  ,23  ){\makebox(0,0)[l]{~IMTQL2}}
    \put( 5  ,22  ){\line(1,0){1}}
    \put( 6  ,22  ){\makebox(0,0)[l]{~PREREC}}
    %    
    \put( 0  ,25  ){\line(1,0){1}}
    \put( 1  ,25  ){\makebox(0,0)[l]{~ZERO}}
    %
    \put( 0  ,26  ){\line(1,0){1}}
    \put( 1  ,26  ){\makebox(0,0)[l]{~PREREC}}
    %
    \put( 0  ,20.5){\line(-1,0){1}}
    \put(-3  ,20.5){\makebox(2,0){MAIN3}}
    \put(-3  ,20.5){\line(-1,0){1}}
    \put(-6  ,20.5){\makebox(2,0){MAIN2}}
    \put(-6  ,20.5){\line(-1,0){1}}
    \put(-10 ,20.5){\makebox(0,0)[l]{~DECLAR}}
    %
    \put(-8.5,21  ){\line(0,1){1}}
    \put(-8.5,22.5){\oval(3,1)}
    \put(-10 ,22  ){\makebox(3,1){START}}
    %
    \put(-8.5,20  ){\line(0,-1){1}}
    \put(-8.5,18.5){\oval(3,1)}
    \put(-10 ,18  ){\makebox(3,1){STOP}}
    %
    \put(-1   ,17.5){\vector(1,0){1}}
    \put(-1   , 2.5){\vector(1,0){1}}
    \put(-1   , 2.5){\line(0,1){15}}
    \put(-0.7 , 4  ){\shortstack{O\\P\\T\\I\\O\\N\\S}}    
    %
    \put(-2   ,18.5){\vector(1,0){2}}
    \put(-2   , 1  ){\vector(1,0){2}}
    \put(-2   , 1  ){\line(0,1){17.5}}
    \put(-2.7 ,10  ){\shortstack{L\\O\\O\\P\\ \\A\\C\\R\\O\\S\\S}}    
    \put(-1.7 , 8  ){\shortstack{R\\O\\W\\S}}    
  \end{picture} } 
  \caption{Flow of the PREFMAP-3 program.}
  \protect\label{fig:flow}
\end{figure}

The program contains two major loops.  The inner one is across 
options, the outer one is across rows of the data matrix.  The inner 
loop computes for each option successively:
\begin{itemize}
  \item[(1)] the predicted values for the metric case (ESTIMA);

  \item[(2)] the optimal monotone function for the nonmetric case (by cycling 
  between PROJEC and MONOR, the Alternating Least Squares iterations);

  \item[(3)] the predicted values for the nonmetric case (ESTIMA);

  \item[(4)] the parameter estimates of the original model (UNRA1 up to UNRA4);

  \item[(5)] RESUL1 and RESUL2 take care of the various output options.
\end{itemize}
When no composite plotting or printing is asked for, the individual 
results will be lost from here on.  After the loop across options is 
finished, statistics across options are given (STATIS).  Then the 
program continues by finishing the outer loop across individuals.  
Composite results are provided for by OUTP3.  Of course, the program 
skips subroutines when the activities performed by them are not called 
for by the user.

\chapter{Deck set-up PREFMAP-3}
\label{chap:appendixB}

{\footnotesize
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
  \multicolumn{4}{c}{{\it CARD 1: TITLE CARD} (section \ref{sect:generalsetup})}  \\
  \multicolumn{4}{c}{} \\
  \multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} & 
  \multicolumn{1}{l}{{\underline{Information}}} \\
  1&80 & 20A4 & Any alphanumeric information to title the printout  \\
\end{tabular}

\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
  \multicolumn{4}{c}{{\it CARD 2: DATA SPECIFICATION} (section 
  \ref{sect:DataSpecifications})}  \\
  \multicolumn{4}{c}{} \\
  \multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} & 
  \multicolumn{1}{l}{{\underline{Information}}} \\
   1 &  5 & I5 & number of row points  \\
   6 & 10 & I5 & number of column points  \\
  11 & 15 & I5 & option set selection  \\
  \multicolumn{3}{c}{}& 0 = all rows same option set \\
  \multicolumn{3}{c}{}& 1 = all rows different option set \\
  \multicolumn{3}{c}{}& 2 = specified rows same option set \\
  16 & 20 & I5 & row number to start first option set (only needed if 
  col. 11--15 equals 2; ditto for the next three parameters) \\
  21 & 25 & I5 & row number to start second option set \\
  26 & 30 & I5 & row number to start third option set \\
  31 & 35 & I5 & row number to start fourth option set \\
\end{tabular}

\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
  \multicolumn{4}{c}{{\it CARD 3: ANALYSIS SPECIFICATION} (section 
  \ref{sect:AnalysisSpecifications})}  \\
  \multicolumn{4}{c}{} \\
  \multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} & 
  \multicolumn{1}{l}{{\underline{Information}}} \\
   1 &  5 & I5 & number of dimensions  \\
   6 & 10 & I5 & maximum number of analyses in any option set  \\
  11 & 15 & I5 & preliminary transformation of configuration  \\
  \multicolumn{3}{c}{}& 0 = remains unchanged \\
  \multicolumn{3}{c}{}& 1 = weighted \\
  \multicolumn{3}{c}{}& 2 = rotated and weighted \\
  16 & 20 & I5 & standardize external data \\
  \multicolumn{3}{c}{}& 0 = yes (default, amounts to 1 + 2) \\
  \multicolumn{3}{c}{}& 1 = center only \\
  \multicolumn{3}{c}{}& 2 = normalize only \\
  \multicolumn{3}{c}{}& 3 = none of the above \\
  21 & 25 & I5 & application vector model (in any option set) \\
  \multicolumn{3}{c}{}& 0 = no \\
  \multicolumn{3}{c}{}& 1 = yes \\
  26 & 30 & I5 & application Unfolding model (in any option set)  \\
  \multicolumn{3}{c}{}& 0 = no \\
  \multicolumn{3}{c}{}& 1 = yes \\
  31 & 35 & I5 & application weighted Unfolding model (in any option set)  \\
  \multicolumn{3}{c}{}& 0 = no \\
  \multicolumn{3}{c}{}& 1 = yes \\
  36 & 40 & I5 & application general Unfolding model (in any option set)  \\
  \multicolumn{3}{c}{}& 0 = no \\
  \multicolumn{3}{c}{}& 1 = yes \\
  41 & 45 & I5 & number of nonmetric iterations (default = 50) \\
  46 & 55 & F10.8 & convergence criterion for nonmetric regression (default = 
  .10E-04) \\
\end{tabular}

\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
  \multicolumn{4}{c}{{\it CARD 4: PRINT/PLOT OPTIONS} (section 
  \ref{sect:PrintPlot})}  \\
  \multicolumn{4}{c}{} \\
  \multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} & 
  \multicolumn{1}{l}{{\underline{Information}}} \\
   1 &  5 & I5 & print input  \\
  \multicolumn{3}{c}{}& 0 = 10 rows of the external data at most \\
  \multicolumn{3}{c}{}& 1 = target configuration \\
  \multicolumn{3}{c}{}& 2 = external data \\
  \multicolumn{3}{c}{}& 3 = target configuration as well as external data \\
   6 & 10 & I5 & print results for each option  \\
  \multicolumn{3}{c}{}& 0 = none \\
  \multicolumn{3}{c}{}& 1 = complete results \\
  \multicolumn{3}{c}{}& 2 = fit only \\
  \multicolumn{3}{c}{}& 3 = fit + criterion and predicted values \\
  11 & 15 & I5 & print selected results for each analysis  \\
  \multicolumn{3}{c}{}& 0 = none \\
  \multicolumn{3}{c}{}& 1 = coordinates, weights, rotation matrices \\
  16 & 20 & I5 & scatter and transformation plots \\
  \multicolumn{3}{c}{}& 0 = no plots \\
  \multicolumn{3}{c}{}& N = plot for first N rows \\
  21 & 25 & I5 & plot ideal points in target configuration \\
  \multicolumn{3}{c}{}& 0 = no plots \\
  \multicolumn{3}{c}{}& 1 = plot first dimension only \\
  \multicolumn{3}{c}{}& 2 = plot first two dimensions \\
  \multicolumn{3}{c}{}& K = plot K dimensions (pairwise) \\
  26 & 30 & I5 & print history of nonmetric regression  \\
  \multicolumn{3}{c}{}& 0 = no print \\
  \multicolumn{3}{c}{}& 1 = print for each row \\
  31 & 35 & I5 & print F-statistic for metric analyses \\
  \multicolumn{3}{c}{}& 0 = no statistics \\
  \multicolumn{3}{c}{}& 1 = F-statistic for each row \\
  36 & 40 & I5 & store output 1  \\
  \multicolumn{3}{c}{}& 0 = no storage \\
  \multicolumn{3}{c}{}& 1 = store coordinates \\
  \multicolumn{3}{c}{}& 2 = store coordinates and weights \\
  \multicolumn{3}{c}{}& 3 = store coordinates, weights and rotation 
  matrices \\
  41 & 45 & I5 & store output 2  \\
  \multicolumn{3}{c}{}& 0 = no storage \\
  \multicolumn{3}{c}{}& 1 = store predicted values \\
  \multicolumn{3}{c}{}& 2 = store predicted values and criterion values \\
\end{tabular}

\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
  \multicolumn{4}{c}{{\it CARD 5: UNIT NUMBERS FOR INPUT/OUTPUT} (section 
  \ref{sect:unitnumbers})}  \\
  \multicolumn{4}{c}{} \\
  \multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} & 
  \multicolumn{1}{l}{{\underline{Information}}} \\
   1 &  5 & I5 & unit number to read the target configuration from 
   (default = 5) \\
   6 & 10 & I5 & unit number to read the external data from (default = 5) \\
  11 & 15 & I5 & unit number to read the option table (default = 5) \\
  16 & 20 & I5 & unit number for scratch file 1 \\
  21 & 25 & I5 & unit number for scratch file 2 \\
  26 & 30 & I5 & unit number to store the coordinates (default = 6) \\
  31 & 35 & I5 & unit number to store the weights (default = 6) \\
  36 & 40 & I5 & unit number to store the rotation matrices (default = 6) \\
  41 & 45 & I5 & unit number to store the predicted and the criterion 
  values (default = 6) \\
\end{tabular}

\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
  \multicolumn{4}{c}{{\it CARD 6: FORMAT CARD FOR THE CONFIGURATION} (section 
  \ref{sect:generalsetup})}  \\
  \multicolumn{4}{c}{} \\
  \multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} & 
  \multicolumn{1}{l}{{\underline{Information}}} \\
   1 & 80 & 20A4 & FORTRAN F-format to read the configuration \\ 
  \multicolumn{4}{c}{}\\
  \multicolumn{3}{c}{}& {\it TARGET CONFIGURATION CARDS} (section 
  \ref{sect:generalsetup}) \\
  \multicolumn{3}{c}{}& The target configuration must appear here if 
  the input medium is unit 5. The configuration must have the form as 
  given in Figure \ref{fig:target}, i.e. as many (groups of) cards as 
  there are objects, and as many fields as there are dimensions \\ 
\end{tabular}

\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
  \multicolumn{4}{c}{{\it FORMAT CARD FOR THE EXTERNAL DATA} (section 
  \ref{sect:generalsetup})}  \\
  \multicolumn{4}{c}{} \\
  \multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} & 
  \multicolumn{1}{l}{{\underline{Information}}} \\
   1 & 80 & 20A4 & FORTRAN F-format to read the external data \\ 
  \multicolumn{4}{c}{}\\
  \multicolumn{3}{c}{}& {\it EXTERNAL DATA MATRIX CARDS} (section 
  \ref{sect:generalsetup}) \\
  \multicolumn{3}{c}{}& The external data must appear here if 
  the input medium is unit 5. The data matrix must have the form as 
  given in Figure \ref{fig:data}, i.e. as many (groups of) cards as 
  there are individuals, and as many fields as there are objects \\ 
\end{tabular}

\vspace{1cm}
\noindent
\begin{tabular}{r@{--}lcp{8cm}}
  \multicolumn{4}{c}{{\it OPTION TABLE CARDS} (section 
  \ref{sect:options})}  \\
  \multicolumn{4}{c}{} \\
  \multicolumn{2}{c}{\underline{Col.}} & {\underline{Format}} & 
  \multicolumn{1}{l}{{\underline{Information}}} \\
   1 & 16 & 4A4 & The option table must appear here if 
  the input medium is unit 5. Each card contains one option set, 
  consisting of 1 up to 4 options. Each option is coded with two 
  characters; the first option code starts in column 1, the second 
  starts in column 5, etc.\\ 
  \multicolumn{4}{c}{}\\
  \multicolumn{3}{c}{}& First characters may be: \\   
  \multicolumn{4}{c}{}\\
  \multicolumn{3}{c}{}& 
  {\begin{tabular}{r@{=}l}
    V & Vector model \\ 
    U & Unfolding model \\ 
    W & Weighted Unfolding model \\ 
    W & General Unfolding model \\   
  \end{tabular}} \\
  \multicolumn{3}{c}{}&\\
  \multicolumn{3}{c}{}& Second characters may be: \\    
  \multicolumn{4}{c}{}\\
  \multicolumn{3}{c}{}&{   
  \begin{tabular}{r@{=}l}
    M & Metric regression \\ 
    P & Nonmetric regression, primary approach to ties \\ 
    S & Nonmetric regression, secondary approach to ties \\ 
  \end{tabular} }\\
\end{tabular}
}

\end{document}

.
