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New approaches to meta-analyze differences in skewness, kurtosis, and correlation [1]

['Pietro Pollo', 'Evolution', 'Ecology Research Centre', 'School Of Biological', 'Earth', 'Environmental Sciences', 'University Of New South Wales', 'Sydney', 'School Of Environmental', 'Life Sciences']

Date: 2026-02 

Abstract Biological differences between males and females are pervasive. Researchers often focus on sex differences in the mean or, occasionally, in variation, albeit other measures can be useful for biomedical and biological research. For instance, differences in skewness (asymmetry of a distribution), kurtosis (heaviness of a distribution’s tails), and correlation (relationship between two variables) might be crucial to improve medical diagnosis and to understand natural processes. Yet, there are currently no meta-analytic ways to measure differences in these metrics between two groups. We propose three effect size statistics to fill this gap: Δsk, Δku, and ΔZr, which measure differences in skewness, kurtosis, and correlation, respectively. Besides presenting the rationale for the calculation of these effect size statistics, we conducted a simulation to explore their properties and used a large dataset of mice traits to illustrate their potential. For example, in our case study, we found that females show, on average, a greater correlation between fat mass and heart weight than males. Although calculating Δsk, Δku, and ΔZr will require large sample sizes of individual data, technological advancements in data collection create increase opportunities to use these effect size statistics. Importantly, Δsk, Δku, and ΔZr can be used to compare any two groups, allowing a new generation of meta-analyses that explore such differences and potentially leading to new insights in multiple fields of study.

Citation: Pollo P, Drobniak SM, Haselimashhadi H, Lagisz M, Mizuno A, Wilson LAB, et al. (2026) New approaches to meta-analyze differences in skewness, kurtosis, and correlation. PLoS Biol 24(2): e3003653. https://doi.org/10.1371/journal.pbio.3003653 Academic Editor: Marcus R. Munafò, University of Bath, UNITED KINGDOM OF GREAT BRITAIN AND NORTHERN IRELAND Received: April 8, 2025; Accepted: January 29, 2026; Published: February 13, 2026 Copyright: © 2026 Pollo et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All data and code used in this study are available at https://github.com/pietropollo/new_effect_size_statistics and https://zenodo.org/records/18386956. Funding: SMD was supported by a National Science Centre (Poland; https://www.ncn.gov.pl/) grant (UMO-2020/39/B/NZ8/01274). ML was supported by an ARC (Australian Research Council; https://www.arc.gov.au/) Discovery Project grant (DP230101248). LABW was supported by an ARC Future Fellowship grant (FT200100822). DWAN was supported by an ARC Future Fellowship (FT220100276). SN was supported by an ARC Discovery Project grant (DP210100812) and the Canada Excellence Research Chair Program (CERC-2022-00074; https://www.cerc.gc.ca/). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist. Abbreviations: IMPC, International Mouse Phenotyping Consortium; ku, kurtosis; sk, sample skewness

Background Sex is a biological attribute that can strongly impact organisms’ traits, with differences between males and females being central to questions in the biological sciences [1,2]. In contrast, biomedical research has primarily focused on male subjects [3], posing a danger to female health [4,5]. Aware of these issues, the US National Institutes of Health and other health agencies have demanded using multiple sexes in animal studies when possible [6]. As a consequence, the number of biological and biomedical studies using both female and male animals as research subjects has increased in the last decade [7], leading to the accumulation of data that can be used to synthesize and quantify sex differences across biological domains. Realizing the accumulation of sex-specific data, many perspective pieces have encouraged researchers to investigate sex differences more carefully [8–10]. Yet, some of these pieces, and most of the biological literature, focus exclusively on mean differences between males and females. A fixation on mean differences has been present for a long time in science because researchers tend to focus on dimorphism in trait averages [11], lack sufficiently powerful data, or have limited statistical tools available (or difficulty to use them). Yet, measures such as variance, correlation, skewness, and kurtosis can be critical to understanding sex differences. For example, certain traits in mice may exhibit no disparity in average values between sexes, but substantial differences emerge in terms of variability [12,13]. These differences could be more easily assessed because of an effect size statistic that measures differences in variability between two groups (proposed by [14]), illustrating how novel statistical tools can expand possible research questions and provide new scientific insights, such as identifying sex differences in trait selection or canalization. Beyond variability, the relative shape of trait distributions to the normal distribution (measured by skewness and kurtosis, i.e., asymmetry of a distribution and heaviness of a distribution’s tails, respectively; Fig 1A and 1B) can also be crucial to understanding ecological and evolutionary processes and patterns [15–19], as well as improving medical diagnostics [20,21]. For instance, skewness can bias heritability estimates because evolutionary biologists assume that phenotypic components (genetic and environmental) are normally distributed [18]. Furthermore, kurtosis can be used to understand community assembly processes [16]. Besides the shape of trait distributions, evolutionary biologists and quantitative geneticists can quantify correlation matrices to understand trait plasticity and evolvability [22–24], which could then be used for group comparisons (as in [25]; Fig 1C). Although location-scale-shape models [26–28] may be used to explore between-group differences (e.g., males and females) in skewness, kurtosis, or within-group correlations, there are no effect size statistics that can easily measure such differences (but see also [29]). PPT PowerPoint slide

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TIFF original image Download: Fig 1. Simulated trait distributions for two groups with different shapes (A: distinct skewness, B: distinct kurtosis), and different correlations between two traits for two groups (C). The data and code needed to generate this Figure can be found in https://zenodo.org/records/18386956. https://doi.org/10.1371/journal.pbio.3003653.g001 Here, we propose three new effect size statistics to evaluate between-group differences in skewness (Δsk), kurtosis (Δku), and correlation (ΔZr), key moments of a distribution that are usually unexplored. These effect size statistics will be valuable to explore sex differences but can also be applied in other fields of study and used to compare differences between any two groups of interest. Meta-analyses using these new effect sizes will create multiple avenues for novel biological enquiries. The present moment is particularly conducive for analyses using these new effect sizes because the individual-level data (e.g., individual participant data [30,31]) required for their calculation are increasingly available from new technological advances that allow faster data collection and sharing (e.g., automated phenotyping).

Difference in skewness and kurtosis The mean and variance represent the first and second moments of a distribution, respectively. However, the third and fourth moments of a distribution (i.e., skewness and kurtosis, respectively) can also be valuable as they characterize the distribution’s shape. More specifically, skewness reflects the distribution’s asymmetry around its mean. While positive skewness indicates an elongated right tail with an excess of high values, negative skewness suggests an elongated left tail with an excess of low values. This asymmetry can influence the interpretation of means and variation, as the mean tends to be larger than the median in positively skewed distributions, while the mean tends to be smaller than the median in negatively skewed distributions. Note that a perfectly normal distribution is symmetric (i.e., skewness = 0), where the mean is equal to the median. Sample skewness (sk) [32] can be expressed as: (1) where x i is a raw data value, x̄ is the sample mean, and n is the sample size. Skewness sampling variance (s2 sk ) [32] can then be expressed as: (2) On the other hand, kurtosis measures tail heaviness: high kurtosis distributions have heavier tails (i.e., proportionally more extreme values than central values), whereas low kurtosis distributions have lighter tails. For comparison, a normal distribution is expected to have kurtosis = 3. Sample excess kurtosis (ku) [32] can be expressed as: (3) with sampling variance (s2 ku ) [32] as: (4) Evaluating skewness and kurtosis provides valuable insights into a variable distribution, which is crucial for interpreting means, assessing variability, and making informed decisions in statistical analyses. Although meta-analyses can use skewness (Eq 1) and kurtosis (Eq 3) to investigate single variables, effect size statistics that compare these metrics between two groups are lacking. Thus, we propose the difference between two groups in skewness (Δsk), expressed as: (5) and its sampling variance (s2 Δsk ) as: (6) where ρ sk represents the sampling correlation in skewness between the two groups (zero if assumed to be independent). Similarly, we propose the difference between two groups in kurtosis (Δku), expressed as: (7) and its sampling variance (s2 Δku ) as: (8) where ρ ku represents the sampling correlation in kurtosis between the two groups (zero if assumed to be independent). However, we note that Eqs 2 and 4 assume normality for sampling variances. When the underlying distributions are skewed or heavy-tailed, sampling error variances for skewness and kurtosis (Eqs 2 and 4) and, by extension, for their between-group contrasts (Eqs 5–8), can misestimate uncertainty. To assess robustness and to provide distribution-free alternatives, we complemented the analytic formulas with resampling-based estimators computed within each group and summed for the difference (i.e., jackknife [33]; see our simulation study below).

Difference in correlation Numerous meta-analyses estimate the correlation between two variables [34,35]. To do so, researchers use the effect size statistic Zr [36], which can be expressed as: (9) and its sampling variance (s2 Zr ) [36] as: (10) where r is Pearson’s correlation coefficient between two variables and n is the sample size. Although Zr alone remains extremely useful to test correlational hypotheses, researchers from all fields would benefit from being able to compare Zr values between two groups. Although Cohen [37] proposed the difference between two groups in Zr as q, he did not provide an equation to calculate its sampling variance. Consequently, this effect size statistic has not been used despite its potential. We therefore propose the difference between two groups in Zr with a new name (ΔZr), as: (11) and its sampling variance (s2 ΔZr ) as: (12) where ρ Zr represents the sampling correlation in Fisher’s Zr between the two groups (zero if assumed to be independent).

Simulation study We conducted Monte-Carlo simulations to evaluate bias and variance estimation for our new effect sizes Δsk, Δku, and ΔZr. For Δsk and Δku, we simulated independent samples for two groups from Pearson distributions with known moments using the rpearson function from the R package PearsonDS v. 1.3.2 [38]. We conducted two simulations: (1) by changing skewness between groups that involved moderate departures from normality in which group-specific skewness from sk ∈ {−1, −0.5, 0, 0.5, 1} and kurtosis was fixed at 3; (2) by holding skewness constant (sk = 0) while manipulating kurtosis from ku ∈ {2.5, 3, 4, 5, 6}. In all cases, we simulated scenarios where: (i) the variance between each group was the same ( = = 1) or different ( versus ); (ii) the mean between the two groups was the same (u 2 = u 1 = 0) or different (u 2 = 5, u 1 = 0). For simplicity, we assumed equal sample sizes between groups with sample size varying from n ∈ {10, 20, …, 100, 150, 500}. We created all unique combinations of the above scenarios resulting in 1,200 independent scenarios (when considering each of the 100 scenarios at each sample size). We estimated Δsk and Δku for each scenario using formulas for within-group sample skewness with small-sample correction (Eq 1) and excess kurtosis with small-sample correction (Eq 3) to estimate point estimates. To estimate associated sampling variance for Δsk and Δku we used the analytical variance estimators derived here (Eqs 2 and 4) and an associated re-sampling (jackknife) approach to compute group sampling variances separately followed by pooling. Importantly, our simulations assume no correlation between groups. For ΔZr simulations, we simulated two groups each containing two variables with known correlations within each group. For ΔZr, we drew bivariate normal data with target within-group correlations r ∈ {−0.8, −0.4, −0.2, 0, 0.2, 0.4, 0.6, 0.8} using the mvnorm function from the package MASS v. 7.3.61 [39]. Marginals were standard normal and group sizes varied from n ∈ {10, 20, …, 100, 150, 500}. We created all unique combinations of scenarios resulting in 768 unique scenarios. We estimated ΔZr using Fisher’s Z transformation Zr and calculating ΔZr as the difference of Zr across groups (Eqs 9–11). Sampling variance for ΔZr used Eq 10 and a jackknife approach. Again, we assumed no correlation between our groups. Note that our simulations did not explore differences in sample size between groups. However, many groups being compared in meta-analyses have the same or very similar sample size. Additionally, simulations often show relatively small impacts of unbalanced sample sizes [40,41], which is why we originally did not vary sample size between groups in our simulations. We resampled 2,500 times for each scenario across all simulations. Performance metrics were (a) bias of the point estimator, (b) relative bias of the sampling-variance estimator, (c) coverage (95%), and (d) Monte-Carlo standard errors (MCSEs). See Supporting information for full formulas. We also evaluated the performance of these effects for meta-analysis (see details in Sections 8.4 and 9.4 of the Supporting information).

Limitations Despite the enormous potential of the effect size statistics we proposed here, they are not free of limitations. For instance, skewness and kurtosis (and therefore the difference in these estimates between two groups; i.e., Δsk and Δku, respectively) are more likely to become extreme with small sample sizes and with variables with few unique values, either because the variable is discrete or because it is naturally constant (e.g., number of vertebrae in mice). We thus recommend that researchers only compute Δsk and Δku for continuous variables with a minimum sample size of 50 for each group (as shown in our simulations). Importantly, we found that Δku variance estimates can be biased in many situations, highlighting that exploring Δku should be a priority for future work. Because of this issue, meta-analyzing Δku requires sample size-based weights instead of the standard sampling variance (see supplementary material and [41]). Lastly, although Δsk, Δku, and ΔZr can be calculated, respectively, from reported skewness, kurtosis, or within-group correlations for different samples, empirical studies rarely report these estimates. Therefore, calculating these effect sizes will probably require raw data, which, fortunately, are now becoming more readily available.

Future opportunities The effect size statistics proposed in the present study can be useful across the life sciences, social sciences, and medicine. This is because skewness and kurtosis, and consequently differences between any two or more groups in these estimates (i.e., Δsk and Δku), may help researchers to understand epidemiological trends [53], genetic patterns relevant to medical diagnosis [20,21], disruptive selection on quantitative traits [54], body size patterns across individuals [55] and species [56], reproductive patterns [57], regime shifts in ecosystems [58], heritability [18], community assembly processes [16], and possibly many other topics. Meanwhile, comparisons regarding correlations have been used to explore memory processing during sleep [59], physiological patterns in patients with certain medical conditions [60], and selection patterns [22–24], to name a few. Because ΔZr can be used in virtually any comparison between two groups of correlational data, the opportunities for its use are endless. Most importantly, Δsk, Δku, and ΔZr are unitless measures, so they can be meta-analyzed to uncover patterns between two groups (e.g., males and females). Moreover, the growing availability of raw data and big data approaches, facilitated by technological advances, makes these effect size statistics particularly valuable for modern research.

Supporting information S1 File. An HTML file containing all steps to reproduce simulations and meta-analyses presented in our study. https://doi.org/10.1371/journal.pbio.3003653.s001 (HTML) S1 Fig. Bias in Δsk, Δku, and ΔZr effect estimates across simulations where samples ranged in group sample sizes between n ∈ {10, 20, …, 100, 150, 500}. A total of 100 simulated scenarios were assessed for Δsk and Δku whereas 64 simulated scenarios were assessed for ΔZr. We ran 2,500 simulations for each scenario. The data and code needed to generate this Figure can be found in https://zenodo.org/records/18386956. https://doi.org/10.1371/journal.pbio.3003653.s002 (TIF) S2 Fig. Bias of analytical point estimators in relation to the absolute difference in skewness and kurtosis between groups. (A) Skewness and (B) kurtosis. Color of points correspond to the sample size and each point is a single simulated scenario. The dotted line is the zero bias line. The data and code needed to generate this Figure can be found in https://zenodo.org/records/18386956. https://doi.org/10.1371/journal.pbio.3003653.s003 (TIF) S3 Fig. Bias for Δsk and Δku for simulated scenarios was not related to group means or variances being different. We ran 2,500 simulations for each scenario. The data and code needed to generate this Figure can be found in https://zenodo.org/records/18386956. https://doi.org/10.1371/journal.pbio.3003653.s004 (TIF) S4 Fig. Relative bias in Δsk, Δku, and ΔZr effect estimates across simulations where samples ranged in group sample sizes between n ∈ {10, 20, …, 100, 150, 500}. A total of 100 simulated scenarios were assessed for Δsk and Δku whereas 64 simulated scenarios were assessed for ΔZr. Note that for relative bias different combinations of point estimates and sampling variance estimates were used in their calculation as indicated in their titles which show the calculation. Notation is as follows ku and sk are the skewness and kurtosis calculated using original formulas. sk_sv and ku_sv are the sampling variance estimates using the original formulas. jack_skew_sv and jack_ku_sv are the sampling variance estimates for skewness and kurtosis using jackknife. jack_skew_bc and jack_ku_bc are the bias-corrected point estimates from the jackknife. We ran 2,500 simulations for each scenario. https://doi.org/10.1371/journal.pbio.3003653.s005 (TIF) S5 Fig. Coverage of 95% confidence intervals for Δsk, Δku, and ΔZr effect estimates across simulations where samples ranged in group sample sizes between n ∈ {10, 20, …, 100, 150, 500}. A total of 100 simulated scenarios were assessed for Δsk and Δku whereas 64 simulated scenarios were assessed for ΔZr. We ran 2,500 simulations for each scenario. The data and code needed to generate this Figure can be found in https://zenodo.org/records/18386956. https://doi.org/10.1371/journal.pbio.3003653.s006 (TIF) S6 Fig. Example sampling distributions of three different scenarios (Δku = 0, 1, or 2.5) for n = 10 and n = 500 samples for each group. We ran 2,500 simulations for each scenario. The data and code needed to generate this Figure can be found in https://zenodo.org/records/18386956. https://doi.org/10.1371/journal.pbio.3003653.s007 (TIF)

Acknowledgments We thank Yefeng Yang for his contribution in the early stage of this study. Declaration of AI use: The authors declare that they occasionally used GPT-4-turbo (OpenAI) to improve the clarity and readability of this work. After using these tools, the authors reviewed and edited the content as needed and took full responsibility for the content of the publication.

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