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# 5. Four visualizations of sex differences/similarities. All plots are based on the same dataset with d = 1.0. (a) Relative density plot. This plot shows the relative male:female density at different quantiles of the female distribution (bottom axis); the corresponding values of the variable (X) are shown for references on the top axis. Dotted lines represent 95% pointwise confidence intervals. Assuming equal group sizes, a relative density of 1.0 (horizontal dashed line) indicates equal proportions of males and females. Under the same assumption, there are about five time as many males than females with values at the lower extreme of the female distribution (0.0 on the bottom axis; relative density » 5.0). At the median of the female distribution (0.5 on the bottom axis) there are about three times as many females than males (relative density » 0.3), approximately the same proportions found at the upper extreme (1.0 on the bottom axis). (b) Overlay density plot of the male and female distributions. This plot shows the shape of the distributions, their overlap, and the location of means (vertical dotted lines). (c) Normalized plot of the male and female distributions. This plot shows the standardized mean difference and the corresponding overlap assuming normality and equality of variances (in this case, OVL = .62 and OVL2 = .45). Horizontal bars represent 95% confidence intervals on d; the colors on the bottom bar can be reversed when the interval includes opposite-sign values. (d) Venn diagram of the overlap between the male and female distributions. This type of diagram can be used to intuitively communicate the overall size of effects in complex multivariate contexts.

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This chapter offers a concise, systematic introduction to quantification in sex differences research. The chapter reviews the main methods used to measure sex differences and similarities, including standardized distances (Cohen’s d and Mahalanobis’ D), indices of overlap, variance ratios, and tail ratios. Some less common approaches (e.g., relativ...

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... The same applies to many physical and physiological traits (Lehre et al., 2009). In the domain of personality, men's scores also tend to be somewhat more variable; the main exception is neuroticism/emotional stability, which shows signi cantly higher variability in women (see Del Giudice, 2015Giudice, , 2020Del Giudice et al., 2018b). Empirical con rmation has not made the hypothesis less incendiary, however. ...
... In all likelihood, the paper's visibility has contributed to entrench this mechanical practice even deeper in the literature (e.g., Zell et al., 2015); to illustrate, three of the meta-analyses I surveyed for this chapter interpreted their ndings based on the same thresholds (Kugler et al., 2018;Lauer et al., 2019;. Other limitations of Hyde's approach include averaging functionally distinct traits within the same category, neglecting measurement error, and failing to consider that differences can cumulate across traits yielding large multivariate distances between male and female pro les (see Del Giudice, 2020;Del Giudice et al., 2012). Be as it may, the conclusion that most sex differences are trivial to small struck a chord, and the paper has become a standard reference in the literature on gender stereotypes (e.g., Ellemers, 2018). ...
... The other major theme I have discussed is the deconstruction of gender and sex. Starting from the 1990s, the idea that masculinity and femininity are independent dimensions of variation has been challenged by research showing that, even if M-F is not a simple unitary construct, it is possible to derive robust and meaningful M-F dimensions from patterns of interest and personality (see Lippa, 2001Lippa, , 2010Del Giudice, 2020). The more radical project of disrupting the "sex binary" started in the 1970s and was still underway in the 1990s (e.g., Fausto-Sterling, 1993), but did not start to get serious traction until the mid-2010s, when it merged with fourth-wave feminism and transgender activism. ...
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This chapter discusses the influence of ideological bias in the psychological study of sex and gender, with a focus on academic psychology. After introducing the essence of the problem, the chapter examines the conceptual distinction between “sex” and “gender,” the binary nature of biological sex, and the challenges to the idea of a sex binary. The following sections provide a survey of recent introductory textbooks and generalist journals, as well as a historical overview of sex and gender in psychology from the late nineteenth century to the present day. The chapter ends with a consideration of the current state of the field, its future prospects, and suggestions to limit the growing influence of ideological bias in relation to sex and gender.
... Differences in central tendency, dispersion, and other aspects of the distributions (e.g., skewness) often combine to yield complex patterns of group differences, with different groups being over-or under-represented at different locations. For example, researchers are sometimes interested in directly comparing the representation of two groups at the distribution tails, which can be done with specialized indices such as tail ratios [1,2] or the newly proposed S-index [3]. As distributions grow in complexity (with features such as multiple peaks, thick tails, etc.), the resulting patterns of differences may become so intricate that they defy any attempt to describe them using summary indices. ...
... However, multivariate comparisons that consider several variables at once (as well as their patterns of covariation) can be quite informative in many scientific contexts. Drawing from my own interests, researchers who study psychological sex differences are increasingly tackling multivariate questions-for example about the overall distance between the sexes in multivariate space, or the proportion of overlap between the male and female distributions (e.g., [8][9][10]; see [1]). Even in these applications, researchers tend to focus mainly on differences between multivariate means or centroids (even though tail ratios and variance ratios can be easily generalized to the multivariate case; see [1]). ...
... Drawing from my own interests, researchers who study psychological sex differences are increasingly tackling multivariate questions-for example about the overall distance between the sexes in multivariate space, or the proportion of overlap between the male and female distributions (e.g., [8][9][10]; see [1]). Even in these applications, researchers tend to focus mainly on differences between multivariate means or centroids (even though tail ratios and variance ratios can be easily generalized to the multivariate case; see [1]). The logical next step is to move beyond means, and begin to investigate complex patterns of differences in the shape of multivariate distributions. ...
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This paper introduces relative density clouds, a simple but powerful method to visualize the relative density of two groups in multivariate space. Relative density clouds employ k-nearest neighbor density estimates to provide information about group differences throughout the entire distribution of the variables. The method can also be used to decompose overall group differences into the specific contributions of differences in location, scale, and covariation. Existing relative distribution methods offer a flexible toolkit for the analysis of univariate differences; relative density clouds bring some of the same advantages to fruition in the context of multivariate research. They can assist in the exploration of complex patterns of group differences, and help break them down into simpler, more interpretable effects. An easy-to-use R function is provided to make this visualization method widely accessible to researchers.
... Predictably, the effect is stronger when the domain is mapped with many narrow traits (e.g., the 30 facets of the Big Five) compared with a few broad traits (e.g., the Big Five). When personality is measured at the level of facets, the overall difference between the average male and female profiles in Englishspeaking countries is consistently larger than two standard deviations, corresponding to an overlap of less than 30% (Del Giudice, 2022;Del Giudice et al., 2012;Kaiser, 2019;Kaiser et al., 2020). For comparison, a detailed study of facial anatomy in males and females found an overall sex difference of approximately three standard deviations, corresponding to an overlap of about 10% between the distributions of male and female faces (Hennessy et al., 2005). ...
... If the assumptions of multivariate normality and equal covariance matrices hold, DM can be used to calculate the proportion of overlap between two distributions, in addition to several other indices of group difference (Del Giudice, 2022). Under the same assumptions, DM is the optimal criterion for classifying individuals into groups, and yields the same expected accuracy of LDA (see Ashby & Perrin, 1988;Del Giudice, 2022;Thomas, 1999Thomas, , 2003. In sum, it is plausible to expect that people will take correlations into account when dealing with group differences and classification; however, I do not know of any studies testing this prediction with respect to group differences in psychological traits. ...
... The average personality profiles of males and females in this sample are displayed in Figure 12 (raw score units). Consistent with the previous literature (see Del Giudice, 2015, 2022, females scored higher in Agreeableness and Neuroticism, with smaller differences in the other domains ( Figure 12A). Aspects and facets revealed a more nuanced picture-for example, within the Openness domain, males had higher scores in Intellect, while females scored higher in the more aesthetic-and imagination-oriented Openness and the corresponding facets ( Figures 12B and 12C; see Costa et al., 2001). ...
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The major domains of psychological variation are intrinsically multivariate, and can be mapped at various levels of resolution-from broad-band descriptions involving a small number of abstract traits to fine-grained representations based on many narrow traits. As the number of traits increases, the corresponding space becomes increasingly high-dimensional, and intuitions based on low-dimensional representations become inaccurate and misleading. The consequences for individual and group differences are profound, but have gone largely unrecognized in the psychological literature. Moreover, alternative distance metrics show distinctive behaviors with increasing dimensionality. In this paper, I offer a systematic yet accessible treatment of individual and group differences in multivariate domains, with a focus on high-dimensional phenomena and their theoretical implications. I begin by introducing four alternative metrics (the Euclidean, Mahalanobis, city-block, and shape distance) and reviewing their geometric properties. I also examine their potential psychological significance, because different metrics imply different cognitive models of how people process information about similarity and dissimilarity. I then discuss how these metrics behave as the number of traits increases. After considering the effects of measurement error and common methods of error correction, I conclude with an empirical example based on a large dataset of self-reported personality.
... In research on sex differences, in general, the subject of quantification is crucial in interpreting the results of the studies. Del Giudice's treatise [1] provides a concise and systematic review and comparison of the main methods used to measure sex differences and similarities, including standard metrics such as Cohen's d for comparing differences in means, and the variance ratio method discussed here, which is the standard for comparing differences in variation. ...
... There is no accepted rule for determining exactly how much larger than 1 the variance ratio must be to consider it significant, and of course, this depends on the goal of the study under question. As noted in [1], what counts as "small" or "large" depends entirely on the area of research, the variables under consideration, and the research question. Hyde and Mertz [8], for example, label variance ratios between 1.05 and 1.20 as not radically different from 1; similarly, Kane and Mertz's finding of VR = 1.08 for the mathematics performance led them to conclude that this is "pretty clear data debunking the greater male variability hypothesis" ([9] emphasis added). ...
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The past quarter century has seen a resurgence of research on the controversial topic of gender differences in variability, in part because of its potential implications for the issue of under- and over-representation of various subpopulations of our society, with respect to different traits. Unfortunately, several basic statistical, inferential, and logical errors are being propagated in studies on this highly publicized topic. These errors include conflicting interpretations of the numerical significance of actual variance ratio values; a mistaken claim about variance ratios in mixtures of distributions; incorrect inferences from variance ratio values regarding the relative roles of sociocultural and biological factors; and faulty experimental designs. Most importantly, without knowledge of the underlying distributions, the standard variance ratio test statistic is shown to have no implications for tail ratios. The main aim of this note is to correct the scientific record and to illuminate several of these key errors in order to reduce their further propagation. For concreteness, the arguments will focus on one highly influential paper.
... In the psychological literature, sex differences are typically measured by comparing male and female averages with univariate indices, such as Cohen's d (e.g., Hyde, 2014). 1 In recent years, researchers have started exploring alternative methods to reveal the full scope and complexity of sex-related patterns of cognition and behavior. These methods range from straightforward extensions of the standard approach (e.g., multivariate distances between means; see Del Giudice, 2009Giudice, , 2022Eagly & Revelle, 2022) to complex machine learning models for classification and prediction (e.g., Loesche, 2019). Meanwhile, the long-standing focus on averages has been broadened by a renewed interest in sex differences in variability, typically quantified as variance ratios (e.g., Borkenau et al., 2013;Gray et al., 2019;Johnson et al., 2008;see Del Giudice, 2022). ...
... At present, researchers have two main ways to quantify sex differences at the distribution extremes. Tail ratios measure the ratio of the two sexes in the region above or below a cutoff (see Del Giudice, 2022;Voracek et al., 2013). Relative distribution methods can be used to plot the relative density of males and females across the entire distribution, and separate the effect of mean differences from that of differences in dispersion and other aspects of distribution shape (Handcock & Morris, 1998, 1999see Del Giudice, 2022). ...
... Here I use "sex differences" as a descriptive label for differences between males and females, with no particular assumptions about their biological and/or cultural origins. For more discussion of this terminological issue see DelGiudice (2022Giudice ( , 2023. ...
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Sex differences researchers are becoming increasingly interested in how differences in averages and variability jointly influence male and female representation at the tails of the distribution. This paper introduces the S-index, a novel index that provides a simple and robust summary of the shape of sex differences at the distribution extremes. The use of S is illustrated with a selection of real-world datasets of personality and cognitive ability, and a R function is provided to calculate S and draw intuitive proportion plots of sex differences across the distribution. The S-index is not limited to the study of sex differences; it can be applied to other domains as long as the groups to be compared are about equally represented in the population and the variables of interest are approximately bell-shaped.
... Effect sizes can be meaningless without context (74), particularly with complex phenomena like sex differences (75). The effect of sex across the discovery and validation samples could be interpreted as a small or very small effect (76). ...
Article
... This suggests that aggregating self-protective reactions could more accurately describe the effect of greater self-protectiveness on females' than males' lives than simply comparing the sexes on only one self-protective response at a time. Thus, multivariate statistical techniques could provide a more qualitative distinction between the sexes (Del Giudice, 2022). ...
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The target article presented a plausible argument that females' susceptibility to threats might be self-protection for staying alive, but some evidence requires scrutiny. We need to consider (1) the biases of narrative reviews, (2) subjective life quality, and (3) the shadow side of extreme reactions to threats before concluding that females' threat-based response is a self-protection mechanism that promotes survival.
... This suggests that aggregating self-protective reactions could more accurately describe the effect of greater self-protectiveness on females' than males' lives than simply comparing the sexes on only one self-protective response at a time. Thus, multivariate statistical techniques could provide a more qualitative distinction between the sexes (Del Giudice, 2022). ...
Article
Extending Campbell's (1999) staying alive theory (SAT) beyond aggression, we reviewed evidence that females are more self-protective than males. Many commentators provided additional supporting data. Sex differences in life-history adaptations, in the optimal relation between survival and reproduction, and in the mechanisms underlying trade-offs involved with self-protection remain important topics with numerous opportunities for improved understanding.
... This suggests that aggregating self-protective reactions could more accurately describe the effect of greater self-protectiveness on females' than males' lives than simply comparing the sexes on only one self-protective response at a time. Thus, multivariate statistical techniques could provide a more qualitative distinction between the sexes (Del Giudice, 2022). ...
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We extend Benenson et al.'s hypothesis from the individual level to the societal level. Because women have highly limited reproductive rates, societies have generally prioritized female survival and regarded males as expendable. We describe various lines of evidence that are consistent with this hypothesis, and we offer additional predictions about differential attitudes toward male versus female endangerment.
... Therefore, in the present study, we adopted this approach to assess the possible multivariate sex differences in gray matter volume (GMVOL) without resorting to classification or its metrics. Because the sizes of univariate volumetric sex differences (Sanchis-Segura et al., 2019;van Eijk et al., 2021;Williams et al., 2021) and sex-classification accuracy (More et al., 2020;Sanchis-Segura et al., 2020) are strongly influenced by total intracranial volume (TIV), this assessment was conducted with raw estimates of GMVOL and after adjusting these estimates with the well-validated (Sanchis-Segura et al., 2020, 2019 power-corrected proportions method (Liu et al., 2014;PCP). More specifically, the raw and PCP-adjusted GMVOL estimates of the 116 brain areas defined by the AAL atlas (Tzourio-Mazoyer et al., 2002) were introduced as features of five different classification algorithms, which were trained and tested in two independent, sex-balanced, samples (n=288 and n=150 per group, respectively) in order to obtain the individuals' class probabilities (in this case, operationalized as the probability of being classified as male; PCAM). ...
... Previous studies have shown that the estimates of univariate and multivariate sex differences are largely dependent on TIV variation and that not all the currently used methods are equally effective and valid for removing TIV-variation (Sanchis-Segura et al., 2020, 2019. Therefore, in the present study, all analyses were conducted twice in the same subjects, without introducing any TIV adjustment ("raw" dataset) and after removing TIV variation with the well-validated power-corrected proportions (PCP) method (Liu et al., 2014). ...
... The PS is defined as the probability that a randomly sampled member of group A will have a higher score than the score attained by a randomly sampled member of group B. More specifically, the probability that males' PCAM scores would be higher (PS M ), equal to, or lower than those of females (PS F ), along with the Cliff's  statistic (Cliff, 1993) and its 95%CI, was obtained through the cidv2 function of the rogme package (Rousselet et al., 2017 Because no single score can properly summarize the differences between two distributions (Callaert, 1999;Cook et al., 2016;Del Giudice, 2019;Grissom and Kim, 2012;Handcock and Morris, 1999;Rousselet et al., 2017), male-female differences in the PCAM continuum were characterized by comparing their cumulative distribution functions (CDF; Callaert, 1999;Grissom and Kim, 2012). CDFs make it possible to directly estimate the proportion of cases in each group with PCAM values equal to or lower than any possible cutoff, but also the proportion of subjects in one group have PCAM values equal or lower than a given proportion of cases in another group (Callaert, 1999;Grissom and Kim, 2012). ...
Article
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Previous studies have shown that machine-learning (ML) algorithms can “predict” sex based on brain anatomical/ functional features. The high classification accuracy achieved by ML algorithms is often interpreted as revealing large differences between the brains of males and females and as confirming the existence of “male/female brains”. However, classification and estimation are different concepts, and using classification metrics as surrogate estimates of between-group differences may result in major statistical and interpretative distortions. The present study avoids these distortions and provides a novel and detailed assessment of multivariate sex differences in gray matter volume (GMVOL) that does not rely on classification metrics. Moreover, appropriate regression methods were used to identify the brain areas that contribute the most to these multivariate differences, and clustering techniques and analyses of similarities (ANOSIM) were employed to empirically assess whether they assemble into two sex-typical profiles. Results revealed that multivariate sex differences in GMVOL: 1) are “large” if not adjusted for total intracranial volume (TIV) variation, but “small” when controlling for this variable; 2) differ in size between individuals and also depends on the ML algorithm used for their calculation 3) do not stem from two sex-typical profiles, and so describing them in terms of “male/female brains” is misleading.