DO BIRDS SHOW UNIQUE MACROEVOLUTIONARY PATTERNS OF SEXUAL SIZE DIMORPHISM COMPARED TO OTHER AMNIOTES?

Body size is undoubtedly one of the most useful measures of sexual dimorphism and, by proxy, sexual selection. Here, I examine large, published datasets of average sexual size dimorphism (SSD) in four clades of amniotes: birds, mammals, squamates, and turtles. Most sexual variation is of subtle magnitude; attempts to discretely categorize species as monomorphic may overlook genuine and common sexual variations of small magnitude (e.g., <10–20% difference). Mammals, squamates, and turtles have unimodal SSD distributions centered close to zero that vary in skew. Mammals skew towards a preponderance of taxa with larger males than females, and mammals with the most extreme SSD have larger males than females. Turtles, however, skew strongly towards a preponderance of taxa with larger females than males, and turtles with the most extreme SSD have larger females than males. Squamates are intermediate to these two clades. Birds are unique in that they 1) are noticeably deficient in taxa near monomorphism, 2) have a bimodal distribution with peaks closely and roughly equidistantly straddling either side of monomorphism, and 3) have a high preponderance of taxa with larger males than females. This suggests stronger disruptive selection or constraints against monomorphism in birds compared to other amniotes. Bird data from Dunning (2007) yields bimodality, while other datasets do not, possibly due to data artefacts/errors. Although Rensch’s rule (RR) is difficult to apply to broad clades, scaling patterns were nevertheless examined here. While turtles and squamates show full adherence to RR, mammals show weaker adherence. Mammal scaling is comparatively less male-biased with increased size than scaling in squamates and turtles, and sex-role reversed mammals instead approach isometry between male and female size. Although bird taxa with larger males than females follow RR, sex-role reversed birds show the converse RR pattern. In birds, increasing size leads to increased dimorphism magnitude regardless of the direction of dimorphism, even though regression of the entire clade deceptively suggests they scale isometrically. This paradoxical scaling explains their unusual bimodal SSD distribution, as shown here through simulation. Equidistant bimodality from monomorphism might suggest disruptive selection where both mating systems have mirrored sexual selection dynamics of comparable effect. Scaling patterns between dimorphism magnitude and overall taxon size in non-reversed and reversed systems might not be readily apparent when examining the whole clade. Large mammals have disproportionately male-biased and more extreme SSD magnitudes. In comparison, large birds have relatively numerous sex-role reversed taxa as well as more extreme SSD magnitudes. These results deserve further testing with tighter phylogenetic controls and comparison of data sources. Additional ecological, physiological, and behavioral variables should also be examined in relation to SSD (e.g., altriciality vs. precociality, oviparity vs. viviparity, clutch size, neonate mass).


INTRODUCTION
Sexual selection theory has been one of the most influential and active areas of evolutionary biology research since Charles Darwin's Descent of Man in 1871 (Darwin 1871).Since then, researchers have grown an appreciation for the nuances and complexities of mating systems (Hunt et al. 2009) beyond the initial framework provided by pioneers such as Darwin.This is especially true when it comes to the role females play in mate choice (Clutton-Brock & McAuliffe 2009) and intrasexual competition (Young & Bennett 2013) as well as 'sex-role reversed' mating systems (Fritzsche et al. 2021).
When studying sexual selection, a commonly used proxy is sexual dimorphism.Although numerous characteristics can exhibit sexual dimorphism, body size is one of the most practical given its universality and its board impacts upon the entire biology of an organism, such as its ecology, behavior, and physiology (Peters 1986;LaBarbera 1989).Ideally, body mass is measured since it is independent of the shape of the organism.However, for some clades, most of the available data is of length.For example, in non-bird sauropsids, the snout-vent length (SVL) is commonly reported (Cox et al. 2007).Many studies of sexual size dimorphism (SSD), regardless of the precise metric, have been undertaken on a variety of research questions.Perhaps one of the most intriguing is Rensch's rule (RR), which states that in closely related taxa, as the overall size of a species increases, SSD becomes more male-biased, and as overall size of a species decreases, SSD becomes more female-biased (Rensch 1950;Dale et al. 2007).In other words, female size scales allometrically to male size.
Here, SSD is examined in four clades of amniotes for which appreciably large datasets have been published: birds, mammals, squamates (i.e., lizards and snakes), and turtles.Do these clades show differences in their SSD distributions?How closely do they adhere to RR at these broad phylogenetic scales?

MATERIALS & METHODS
Data & Considerations -For each of the four clades, no taxa were dropped from their published secondary sources (see secondary sources of Cox et al. [2007], Dunning [2008], and Tombak et al. [2024] for details about primary sources for the data).As such, I attempt to account for uncertainty in mean male and female sizes by maximizing sample size in terms of the number of taxa in each clade's dataset.In other words, no form of error bars (e.g., confidence intervals, standard deviations) are added to individual taxa as a way of data quality control prior to analyzing SSD distributions.Additionally, no further phylogenetic controls are added within the four clades.Size is measured as body mass in birds and mammals, while in squamates and turtles, size is measured as SVL.
Data in grams (g) for mean male and female masses in birds was downloaded from CD-ROMs associated with Dunning (2007), as modified by Saitta et al. (2020).Every taxon for which both male and female masses were reported are included in the dataset, without any changes to the taxonomy as reported in Dunning (2007).This bird dataset contains 2,576 taxa, the largest of the four clades examined here.Although many of these taxa might be considered sub-species or geographic populations, the dataset has the benefit of broadly covering much of crown bird diversity.
Data in grams (g) for mean male and female masses in mammals comes from Tombak et al. (2024).Unlike Tombak et al. (2024), no taxa were dropped.In the mammal dataset, 691 taxa are included.No taxonomic changes were made.
Data in centimeters (cm) for mean male and female SVL in squamates and turtles was provided by Cox (pers. comm. 2024) and correspond to Cox et al. (2007).Specifically, the data used to examine RR in Cox et al. (2007) were used here, leading to datasets with 1,027 taxa for squamates and 197 taxa for turtles.Although the dataset for turtles is appreciably smaller than the others, given that that are only about 357 total turtle species (Rhodin et al. 2021), this dataset is a sizeable proportion of turtle diversity.No taxonomic changes were made to the squamate or turtle data.
Future analyses (e.g., phylogenetic generalized least squares regressions) can address the above limitations surrounding the degree of phylogenetic control.Similarly, the uncertainty of each reported value for male and female mean size can also be further explored in future work.
Analyses -All statistical analysis was performed using R (version 4.3.3).See supplemental material for both data files and R code.
SSD was first calculated as mean male size (i.e., either mass or SVL) divided by mean female size, and this ratio was then log10-transformed.As such, taxa with larger males have positive values, while taxa with larger females have negative values.Summary statistics were gathered, and histograms were produced.When calculating the percentage of taxa in which males are larger than females on average, taxa that were reported in the datasets as having exactly equal male and female mean sizes were excluded from the total taxa count (i.e., denominator).
Next, kernel density functions were produced on this SSD metric for the four datasets using the default settings in R for the command density().
Then, rather than examining the log10-transformed ratio of male to female size as above, mean male and female sizes were log10-transformed and then plotted directly against each other.Note that it is challenging, and at times inappropriate, to consider RR across broad clades due to various confounding selective forces (Székely pers. comm. 2024).With that in mind, I nevertheless thought it was worthwhile to examine broad scaling patterns within these four datasets.Linear regressions were applied, such that negative allometry (i.e., slope <1) corresponded to increasing male-biased SSD with larger taxon size (i.e., RR) and positive allometry (i.e., slope >1) corresponded to increasing female-biased SSD with larger taxon size (i.e., converse RR).During this regression analysis, each dataset was also reanalyzed after dividing into sex-role reversed (SSD less than or equal to 0) and non-reversed (SSD greater than or equal to 0) subsets.While this might seem unconventional, I think that it might be useful to examine each mating system in isolation in order to judge how the absolute value of SSD changes according to overall taxon size.Non-reversed and reversed mating systems might in some ways be diametrically distinct in their selection dynamics.For example, polyandry, male-only parental care, and male mate choice are disproportionately common in sex-role reversed taxa (Vincent et al. 1992;Kvarnemo & Ahnesjo 1996;Ah-King & Ahnesjö 2013).Also, sex-role reversals are not evenly distributed across phylogenies within clades (e.g., see Tombak et al. 2024 for percentages of reversed mammals by order).Therefore, sex-role reversal may represent independent evolution to a degree.In other words, my approach here assumes that the two mating systems can be roughly approximated as distinct evolutionary lineages or trajectories.
The log10-transformed mean male and female sizes were next analyzed with principal component analysis (PCA) in R using the prcomp() function with scale set to "TRUE".With only two variables analyzed with PCA, this is similar to producing a residual plot -deviation from the major axis of size variation can be visualized.See supplemental R code for full summaries of the PCAs.
Finally, hypothetical data was simulated for male and female mass, with parameters inspired largely by birds, to examine the connection between scaling relationships and SSD distribution.See the supplemental material for full parameter details within the R code used to simulate the data.
Uncertainty and sensitivity of the results was judged by comparing standard deviations and sample sizes between the mammal and bird datasets, while also controlling for mean size, and by altering the bandwidths of kernel density functions.Likewise, I attempted to judge the precision of the four datasets by estimating the number of digits to which each dataset rounded their mean size values, using the R command roundedSigDigits().I also compared Dunning (2007) bird data to previously published bird SSD datasets (Székely et al. 2022;Myhrvold et al. 2015) in order to determine if any systematic/random errors or data artefacts might exist within them; I modelled their consequences in simulated data to determine if they might alter the Dunning (2007) SSD distribution to appear more like the other datasets.Again, see the supplemental material for full parameter details in the R code used to simulate the data.
On the terms monomorphic and dimorphic -Rather than assign an arbitrary threshold to categorize taxa as either monomorphic or dimorphic, I instead treat these terms more like descriptions of the magnitude of sexual variation.I think that doing so helps to reduce subjective interpretation of the underlying data in the SSD distributions caused by an a priori arbitrary threshold.A taxon with greater sexual variation than another therefore shows 'greater dimorphism' relative to the other.At times, I might refer to a calculation from a dataset that yields SSD = 0, or to the position on an SSD axis at SSD = 0, as 'precise' or 'exact monomorphism'.Summary statistics & histograms -Summary statistics are provided in Table 1.Birds and mammals show similar ranges of SSD to each other.The most extreme SSD (i.e., in absolute value) are positive for birds and mammals, while the most extreme SSD are negative for squamates and turtles.Turtles especially show some very extreme values of negative SSD, even though they are measured in SVL.Birds and mammals show mean and median SSD values that are positive, squamates have near zero mean and median SSD, and turtles have negative mean and median SSD.

Clade
Turtle SSD has noticeably high standard deviation, even compared to squamates, which are also measured in SVL.Mammals show right-skewness, while squamates and especially turtles show leftskewness.All four clades have Pearson's kurtosis values greater than three, indicating thicker tails than would be expected from a normal distribution and a greater prevalence of extreme values.However, squamates and turtles have noticeably lower kurtosis than do birds or mammals, bearing in mind that this might be driven by SVL being measured as opposed to mass.
All clades, expect turtles, have a preponderance of taxa where males are larger than females.In particular, birds are far more likely to have larger males than females than the other clades (e.g., ~72% in birds versus ~59% in mammals).Furthermore, the dataset of birds contains hardly any taxa for which the mean male mass is reported to be precisely equal to mean female mass.
When the data is plotted as histograms, birds (Fig. 1A) appear clearly bimodal in their SSD distribution, with a readily apparent paucity of taxa near monomorphism.The two modes straddle either side of SSD = 0, and they appear to be roughly equidistant from it.The positive mode of birds, however, is about 3 times higher than the negative mode.
The histograms of mammals (Fig. 1A), squamates, and turtles (Fig. 1B) are all unimodal and centered near zero.Although it is hard to compare mass SSD to SVL SSD, one can see that mammals have a slight right-skewness, turtles have a clear left-skewness, and squamates are intermediate.
Kernel density functions -When kernel density functions are fit to the data, the patterns seen in the histograms above are supported.Birds (Fig. 2A) show bimodality, with a positive mode at least 3 times greater than the negative mode, and both modes are equidistant from monomorphism.They do, however, lie close to monomorphism, at SSD = +/-0.04.These modes translate to males being ~110% and ~91% the size of females on average (i.e., ~10% difference).Mammals, in comparison (Fig. 2A), are modelled as unimodal with right-skewness and centered near monomorphism.When the mammal kernel density function is subtracted by the bird kernel density function (Fig 2B), mammals appear to show a greater proportion of taxa near monomorphism than birds do.Additionally, mammals appear to show a greater proportion of taxa with prominent (i.e., SSD > 0.1) positive SSD values than do birds.However, birds appear to show a greater proportion of taxa with prominent (i.e., SSD < -0.1) negative SSD values than to mammals.
Kernel density functions of SVL in squamates and turtles (Fig. 2C) show that both clades have unimodal SSD centered near monomorphism.Turtles show a kernel density function with a prominent left-skewness.2) of log10-transformed mean male size versus mean female size show that broader scaling trends across a whole clade might become complicated when the datasets are divided according to mating system (i.e., sex-role reversed versus non-reversed).Remember though, examining scaling patterns across broad and disparate clades is challenging, so these results should be considered carefully.At the level of the entire clade (Fig. 3A-B) birds appear to be approximately isometric, while mammals, squamates, and turtles all appear to follow RR (designated here by a slope < 1).Although, mammals (slope ~ 0.969) are noticeably closer to isometry than are squamates (slope ~ 0.897) or turtles (slope ~ 0.811).If, however, taxa with larger females than males are dropped to look only at non-reversed mating systems (Fig. 3C-D), then birds become consistent with RR; mammals, squamates, and turtles remain consistent with RR.If taxa with larger males than females are dropped to look only at sex-role reversed mating systems (Fig. 3E-F), then mammals become nearly isometric and birds become converse RR; squamates and turtles remain RR.Table 2. Regression slope summary.When data is divided according to mating system (i.e., sex-role reversed or not), overall patterns become complicated in birds and mammals.All slopes rounded to three decimal places.

Regressions -Regressions (Fig 3; Table
If the log10-transformed mean male and female sizes are analyzed with PCA (Fig. 4), one can better visualize the relationship between SSD (~PC2) and overall taxon size (~PC1).See supplemental material for loadings of the male and female size variables.PC1 explains about 99.7%, 99.9%, 98.9%, and 92.8% of the variation in log10-transformed male and female mean masses in birds, mammals, squamates and turtles, respectively.This indicates that variation in size between taxa is far greater than SSD between sexes, especially in the masses of birds and mammals.
Birds show clear divergence/bifurcation away from monomorphism as size increases, and this is true in both male-biased and female-biased directions of dimorphism.Bird taxa with male-biased SSD reach larger overall size and greater SSD than do sex-role reversed bird taxa.Mammals, squamates, and turtles lack this obvious bifurcation.Like in birds, mammals show a tendency for increasing spread in SSD as overall taxon size increases, but this is mostly biased towards larger males than females, rather than allometrically bifurcating away from monomorphism in both reversed and non-reversed mating systems.Combined with the above regression slopes, mammals are perhaps the closest to approaching the scaling condition seen in birds compared to squamates and turtles.

DISCUSSION
SSD distributions: monomorphism & sex biases -Previous studies have attempted to dichotomously designate taxa into discrete 'monomorphic' or 'dimorphic' categories.Some studies have used thresholds of <10% (Lindenfors et al. 2007) or even <20% (Ruckstuhl & Neuhaus 2002) difference between mean male and female size in order to define a 'monomorphic' category.In all four of the clades studied here, most taxa show subtle sexual variation with modes either near monomorphism or, in the case of birds, at ~ +/-10% difference.The fact that birds show clear underrepresentation near monomorphism (i.e., SSD = 0) and bimodality symmetric around monomorphism suggests that taxa with low SSD values represent genuine and common sexual variation.I think that arbitrary categorization of such taxa as 'monomorphic' might obscure much, or even most, of the true sexual variation present in a clade.
Despite their sex-role reversed mode, birds show the greatest preponderance of taxa in which males are larger than females.Mammals then show the next most male bias and have right-skewness in their SSD distribution, followed my squamates, which show the least skewness among the four clades.In contrast, turtles have a clear preponderance of female-biased taxa and left-skewness.
Size metrics & kurtosis -The general patterns described here for squamate and turtle SSD are expected to largely hold, even if mass were used instead of SVL.Since mass scales allometrically to length (i.e., linear measures compared to volumetric measures), the expectation is that using mass SSD for squamates and turtles might increase the spread of the data and exaggerate the tails of the distributionsincreasing standard deviation, skewness, and kurtosis.
As for kurtosis, all four clades show Pearson's kurtosis values greater than three (i.e., leptokurtic), meaning that all have thicker tails than would be expected in a normal distribution (i.e., random evolution under Brownian motion) -possibly an indicator of runaway sexual selection (Fisher 1930).
Rensch's rule?-Although it is difficult to apply RR to broad clades, I nevertheless thought it was worthwhile to examine scaling relationships in these four datasets.Furthermore, while dividing each clade according to mating system might seem unconventional to some, I think the bird results ultimately show that this approach is useful.Simply running regressions on the clade as a whole might fail to give a full picture of evolutionary patterns within it, a point similar to that of phylogenetic comparative methodology.While birds as a whole appear to fall close to isometry between male and female sizes, the reality is that as the overall size of the taxon increases, SSD in birds can increase in both positive and negative magnitude.Larger birds tend to have greater absolute values of SSD, no matter if they are sex-role reversed or not.The same is not true for squamates and turtles, which show full adherence to RR no matter how the data is subdivided, even if only sex-role reversed taxa are considered.
Mammals appear to follow RR as a whole, but they are the closest of the clades examined here to approaching the scaling relationships seen in birds, as seen by their higher slope values than those of squamates and turtles.When sex-role reversed mammal taxa are regressed alone, they approach isometry.As mammals get larger in overall taxon size, only non-reversed systems show an increase in SSD.Therefore, large mammal species will be disproportionately biased towards taxa with larger males than females, and these taxa are the likeliest to show the most extreme SSD magnitudes.
Larger species might be more commonly studied than smaller species.As Tombak et al. (2024) suggest, this might explain why mammals are sometimes hastily characterized as being very male-biased in SSD, even though their SSD distribution is centered near monomorphism and they do not show as high of a percentage of taxa with male-biased SSD as do birds.Likewise, birds such as the jacana are often offered as prime examples of sex-role reversed taxa (Emlen & Wrege 2004), even though birds have a very high percentage of taxa with male-biased SSD.This thinking is possibly due to the fact that as birds get larger, both sex-role reversed and non-reversed systems increase their magnitude of SSD in absolute value.Therefore, despite their high preponderance of taxa with male-biased SSD, birds have many examples of relatively large sex-role reversed species amenable to study.

Do scaling relationships explain SSD bimodality in birds?
-To me, the two most salient observations of this study are that bird SSD is, perhaps uniquely, 1) bimodal and 2) scales positively to overall taxon size within both sex-role reversed and non-reserved systems.It is therefore worthwhile to consider how these two dynamics are related.Might birds converge upon some sort of optimal SSD under both mating systems, as seen by the fact that their positive and negative SSD modes are roughly equidistant from monomorphism?This shift away from monomorphism would lead to a pattern of disruptive selection in their SSD distribution at the macroevolutionary level (Fig 5).Other clades, which more fully adhere to either RR or converse RR, might be able to occupy monomorphism under such a scaling dynamic in a way that the diverging pattern of birds cannot.One can attempt to model this dynamic (Fig. 6).I use parameters inspired by birds, such as the proportion of sex-role reversed taxa, and scaling slopes of reasonable effect based on the observed slopes in the clades above.I first model female mass as normally distributed in both sex-role reversed (Fig. 6B) and non-reversed lineages (Fig. 6A), centered around a mass of 10 g for both mating systems.Then, using bifurcating scaling relationships based on birds, where sex-role reversed taxa scale according to the converse of RR and non-reversed taxa scale according to RR, I can generate male masses (Fig. 6C-D).I also add random noise to induce variability beyond just a perfectly linear relationship between female and male mass.By plotting males versus females, one can see that both RR non-reversed systems (Fig. 6E) and converse RR sex-role reversed systems (Fig. 6F) were successfully simulated.
Ultimately, this modeling shows that by simply taking normally distributed female masses and applying two separate evolutionary scaling relationships corresponding to reversed and non-reversed lineages, one can produce a bimodal SSD distribution similar to that seen in birds (Fig. 6G).It also suggests that the effect of scaling (i.e., slope) can influence the position of the SSD modes in the resulting histogram.The fact that the two modes in birds are roughly equidistant from monomorphism might suggest that similarly sized effects of sexual selection exist in both mating systems, driving both systems away from monomorphism.
Figure 6 (Above).Simulated male and female masses that follow the scaling relationships like those in birds.Sexrole reversed taxa, B,D,F, show converse RR, while non-reversed taxa, A,C,E, show RR.A-B, first, female masses are randomly simulated based on normal distributions.C-D, second, the slopes of the scaling relationship are used to generate male masses, with random noise added to prevent an unrealistic perfectly linear relationship, E-F, third, male versus female mass can be plotted to compare against isometry (red line, slope = 1).G, finally, bimodality results when SSD is calculated from the combined males and females of both mating systems (red vertical line, SSD = 0).The scaling relationships dictate the position of the two modes.The ratio of the number of taxa in each of the two mating systems dictates the ratio of the heights of the resulting modes.Also shown as insets are the histograms of the male and female data combined from both mating systems.
Uncertainty -Here, I attempted to account for uncertainty in male and female mean sizes by maximizing the number of taxa per dataset.However, one can examine this more closely, especially in the bird dataset, where I propose a possibly unique bimodal SSD distribution.
One obvious line of investigation might be to compare the magnitude of SSD to standard deviations of the mean masses of the sexes (or some other measure, such as confidence intervals, that contains information about uncertainty in the mean mass value [e.g., Tombak et al. 2024]), particularly in taxa with low SSD.However, I worry about such a comparison because many taxa have SSD of subtle magnitude, which I hypothesize is genuine, where male and female distributions will naturally show extensive overlap.For example, human height is sexually dimorphic, but human height distribution is not bimodal.As Schilling et al. (2002, p. 233) describe, "a mixture of equally weighted normal distributions with common standard deviation σ is bimodal if and only if the difference between the means of the distributions is greater than 2σ".
Instead, if 1) bird SSD data is most comparable to mammal SSD data because both are measured using mass and 2) bird SSD distribution is suspected to be bimodal and mammal SSD distribution unimodal, then one can compare factors related to uncertainty between the two datasets.If the bird data show similar signs of uncertainty to the mammal data, then one can be more confident in the conclusions here.
First, one can examine standard deviation and mean mass of the sex.Note that the mammal dataset contains several taxa for which a standard deviation of the mean mass for a given sex is reported as zero (3 male masses and 2 female masses from 5 different taxa).In the bird dataset, many taxa have a single mean mass reported from either "unknown" or "both" sexes, and I ignore those bird entries.Further, some mean values of male or female mass do not have a corresponding standard deviation reported.To compensate for this reduction in datapoints, I include all taxa in Dunning ( 2007) for which male or female standard deviations are reported, even if only one sex is reported (i.e., they do not contribute to the bird SSD data above).
Bird standard deviations, when reported, more often represent a lower proportion of their mean mass for a given sex compared to mammals (Fig. 7A, as seen by the modes).In other words, while standard deviations correlate positively with mean mass of a sex in both clades, birds tend to have lower standard deviations for a given mass than do mammals (Fig. 7B).These results suggest that the bird data might show uncertainty in mean masses at least comparable to that of the mammal data, if not lower uncertainty.
Second, one can examine sample size of the sex and mean mass of the sex.The sample sizes of males and females used to calculate mean masses in both birds and mammals are strongly right-skewed, with most entries having very small sample sizes (Fig. 7C-F).Although birds have many entries where male or female masses are based upon a sample size of one, where standard deviations are undefined, the mode of mammals is just two.Both mammals and birds have similarly shaped distributions of sample sizes.Mammals have a higher median sample size than do birds, but birds have a larger range of sample sizes.Little correlation exists between mean mass and its sample size (Fig. 7G).In summary, although the mammal dataset tends to have better sample sizes on average, both birds and mammals have similarly skewed sample size distributions.There is also little correlation between mean size and sample size that might bias the scaling relationships between SSD and taxon size reported above.
Finally, one can reexamine the kernel density functions to see how sensitive modality is to changes in bandwidth for the bird and mammal data.The difference in bi-versus unimodality between the two clades is robust against sizeable changes to bandwidth in the kernel density functions (Fig 7H ), when bandwidth for bird SSD is increased and bandwidth for mammal SSD is decreased from before (Fig. 2A-B).Bird SSD still appears bimodal, while mammal SSD still appears unimodal.

Figure 7 (Above).
Comparisons of factors related to uncertainty between bird and mammal mass SSD datasets.A, the ratio of the standard deviation to the mean mass for each sex entry.The number of male and female entries for each clade is shown.B, regression of log10-transformed mean mass versus standard deviation for each sex entry.Slopes are reported.Note that some mammal datapoints cannot be log10-transformed if their standard deviation is reported as zero.Histograms of bird sample sizes for each sex entry from zero to, C, 30 or, D, 100.Histograms of mammal sample sizes for each sex entry from zero to, E, 30 or, F, 100.The red asterisks indicate a sample size of one for many bird sex entries, meaning that standard deviation cannot be defined.Shown are the number of male and female entries for each clade, as well as the summary statistics for the distributions of sample sizes in male and female masses (minimum, 1 st quartile, median, mean, 3 rd quartile, maximum).G, regression of log10-transformed mean mass versus sample size for each sex entry.Slopes are reported.H, sensitivity analysis of kernel density functions for SSD where bandwidth (BW) is increased in birds and decreased in mammals compared to the default settings of the density() function in R as shown in Fig. 2A.

CONCLUSIONS
Key takeaways -Most sexual variation is of subtle magnitude.Attempts to create dichotomous categories of strictly 'dimorphic' or 'monomorphic' taxa might be overlooking genuine patterns related to sexual selection.Mammals, squamates, and turtles show unimodal SSD distributions centered near monomorphism with squamates as intermediate between right-skewed mammals and left-skewed turtles.Birds are not just heavily weighted toward male-biased SSD, they also have a bimodal SSD distribution and clear underrepresentation of taxa near monomorphism.When looking at very broad scaling relationships, turtles and squamates fully adhere to RR, sex-role reversed mammals approach isometry, while sex-role reversed birds show converse RR.
Modelling shows that diametrically distinct scaling relationships between sex-role reversed and non-reversed birds can drive their bimodality away from monomorphism.Possible disruptive selection against monomorphism might be linked to sexual selection dynamics in sex-role reversed birds conversely mirroring those of non-reversed birds.As birds get bigger, they become more dimorphic, regardless of mating system.
Previous work -Despite using an even greater number of bird taxa (up to 4,497 taxa with reported SSD) as I do here, Székely et al. (2007) and an updated dataset (Székely et al. 2022) did not recover bimodality in bird mass SSD and instead modelled the distribution as unimodal.If my dataset here is incorrect and the distribution of bird mass SSD is indeed unimodal, the cause might be my use of a smaller dataset (only 2,576 taxa), the specific taxa included, outdated taxonomy, inclusion of subspecies/geographic populations, and/or errors in transcribing the data from CD-ROM to my analyzed data file.However, given the stark equidistant positive and negative bimodality observed in the downloaded Dunning (2007) data and the associated scaling patterns, this would imply rather specific systematic errors.For example, it could imply that Dunning (2007) contains data biased against birds with SSD near monomorphism but biased in favor of birds with SSD around +/-10%; I do not know why that would be the case though.

Figure 8 (Below).
A, Bird SSD recorded in Székely et al. (2022) versus SSD that I recalculated as log10(male mass/female mass) using their own data of mean male and female mass.B, histograms of the original and recalculated SSD.C, kernel density functions of the original and recalculated SSD.Red vertical line is SSD = 0.
Instead, if my dataset here is correct and the distribution of bird mass SSD is indeed bimodal, the cause might be differences or errors in data sourcing, transcribing, calculating, and/or rounding/precision.For example, Székely et al. (2022) directly use 143 different sources for male and female bird masses in addition to Dunning (2007), with the two sexes sometimes reported from different sources for the same taxon.Rounding to fewer significant digits (i.e., lower precision) of mean male and female bird masses in some sources could, for example, lead to more instances of exact monomorphism reported (i.e., SSD = 0), given that most sexual variation is subtle.Also, the types of models fit to the data might influence their interpretation.I used kernel density functions in R to model the data, which might have provided greater model fitting that allowed for bimodality to be detected.Székely et al. (2007) used Minitab and SPSS software and fit unimodal distributions to the data (Székely pers. comm. 2024).
I recalculated SSD as log10(male mass/female mass) using the Székely et al. (2022) data and plotted those results against the originally reported SSD in the dataset (Fig. 8).While many taxa are the same in SSD value (i.e., slope = 1 in Fig. 8A), Székely et al. (2022) might also systematically underestimate the absolute value of dimorphism magnitude in many of the taxa, which could further obscure the disruptive selection at monomorphism observed here.The discrepancy might derive from the use of linear measurements to calculate SSD when mass is not available, producing a scaling effect between linear and volumetric dimensionalities (i.e., slope < 1 in Fig. 8A).Plotting the original and recalculated SSD as histograms and kernel density plots (Fig. 8B-C; kernel density using default values in R density() function) shows that recalculated SSD does indeed show fewer taxa with SSD near monomorphism and greater spread towards larger absolute values of SSD magnitude.The histogram of recalculated mass SSD (Fig. 8B) also shows an unnatural-looking spike right near zero SSD, possibly an artefact of data sourcing and precision/rounding.This spike is about twice as high as what would otherwise be the natural-looking mode at a positive SSD (similar in position to the positive mode I detected in the Dunning (2007) data).This spike is due to fact that, even when SSD is recalculated, 72 taxa are reported in the Székely et al. (2022) dataset as having SSD of exactly zero.Other datasets, such Myhrvold et al. (2015), show a similar unimodal distribution with a positive mode (Fig. 9A) and a possibly unnatural spike near SSD of zero (Fig. 9B) - Myhrvold et al. (2015) have 109 taxa reported with SSD of exactly zero.Myhrvold et al. (2015) also used diverse data sources that might be prone to error and low precision, like Székely et al. (2022).2007) is reflective of the true population distribution of all birds, then I can simulate reversed and non-reversed SSD using parameters inspired by birds to create a bimodal distribution with positive and negative peaks equidistant from SSD = 0. D, I can simulate rounding/precision error by simulating a smaller number of taxa, and then inappropriately rounding them to only one significant figure, leading to an unnatural spike at SSD = 0. E, I can also simulate the process of obtaining data from disparate sources by accurately simulating as many new taxa as before and then introducing appreciable random error to each male and female mass.All the data is then combined and viewed as a histogram, E, or fit with a kernel density function, F (bandwidth is the same as Fig. 7H, which would normally still show bimodality in the real Dunning [2007] dataset).Vertical red line indicates SSD of zero.
Again, one can attempt to model (Fig. 9C-F) this sampling dynamic using parameters inspired by the Dunning (2007) bird mass SSD distribution.See supplemental material for full code and parameters.The imprecision caused by rounding some of the taxa to only one significant figure leads to the calculation of SSD = 0 and produces an artificial spike at monomorphism (Fig. 9D).Random error introduced from the mixing of disparate datasets can also be simulated (Fig. 9E).Even if one assumes that the true population distribution of all birds is indeed bimodal (Fig. 9C), adding these two types of mistakes onto a correct dataset (i.e., accurate data plus imprecisely rounded data plus differently sourced data) can lead to a distribution (Fig. F-G) very similar to those of previous datasets (e.g., Székely et al. 2007Székely et al. , 2022;;Myhrvold et al. 2015).The random error from mixing different sources gives the SSD distribution greater spread and a more unimodal appearance, especially when combined with an artificial spike of taxa with reported SSD of exactly zero.By collating data from many different sources, these other datasets have an advantage over my use of Dunning (2007) alone in that they sample considerably more taxa.However, does this increase in taxa also come with a tradeoff, whereby studies that are not perfectly comparable (for any number of reasons) introduce random error, which increases data spread and obscures smaller modes?This is like the phenomenon seen in the unimodality of human height SSD.As the spread of the two peaks increases, the greater overlap between them can eliminate bimodality (Schilling et al. 2002).In fact, unequal weighting of the two peaks (here, the numbers of reversed and non-reversed bird taxa) and unequal standard deviations of the two peaks (possibly higher in non-reversed birds, given that the largest SSD taxa are non-reversed) would require even further separation of their means in order to yield a bimodal SSD distribution when combined.This would mean that less random error is required to induce unimodality than would otherwise (e.g., if there were equal numbers of reversed and non-reversed taxa or if the standard deviations of reversed and non-reversed taxa were equal).To be clear, I am not faulting any prior researchers (e.g., Székely et al. 2007Székely et al. , 2022;;Myhrvold et al. 2015) for their commendable and very useful efforts in collating these sorts of large datasets.If my ideas here are correct, then I am simply suggesting that the impact of combining disparate data sources might need to be considered in the analyses that use such datasets.
If my bird dataset is indeed more reliable, the obvious question that follows is whether the lack of bimodality in mammals, squamates, and turtles described here is also possibly reflective of factors such as precision/rounding of the sourced data.Indeed, these three clades have a greater percentage of taxa reported with SSD of precisely zero than do birds -could this partly be an artefactual spike?However, 1) all four datasets used here often report mean sizes of small taxa to one decimal place and 2) the mammal dataset (i.e., the most appropriate comparison to the bird mass data) often reports mean masses at higher precisions than does the bird dataset.Using the command roundedSigDigits() with default settings in R, I estimated the range of the number of digits that the datasets rounded male and female mean sizes to across their spans of taxon size (Table 3).As for the potential issue of combining linear and mass variables for SSD estimates, the squamate and turtle data use only linear SVL, while mammal and bird data are limited to mass.Still, the impact of obtaining data from multiple sources, such that error is introduced into SSD, might still be present in the mammal, squamate, and turtle datasets studied here.Table 3. Summary statistics for the distributions of the estimated number of digits to which mean male and female sizes were rounded in the four datasets using the command roundedSigDigits() in R.
Future work -Still, future work could more thoroughly compare data sourcing, precision/rounding, and quality between clades and between studies.Researchers should also add better phylogenetic controls through phylogenetic generalized least squares regression (PGLS) and evolutionary rate modeling.Do the scaling relationships at the broader macroevolutionary scale, especially those in birds, still appear at smaller phylogenetic scales within monophyletic lineages?For example, sexual selection likely drives RR in shorebirds (Charadriides), and this result is supported by phylogenetic comparative methods (Székely et al. 2004).Finally, additional ecological, behavioral, and physiological variables should be compared to SSD in light of these results.Of particular interest might be precociality versus altriciality in birds, oviparity versus viviparity in squamates, clutch/litter sizes, neonate/hatchling sizes, etc.

Figure 2 (
Figure 2 (Below).Kernel density functions of SSD for, A, bird (black) and mammal (orange) body mass, B, mammal minus bird, and, C, turtle (purple) and squamate (blue) SVL.Vertical red line indicates an SSD value of 0 (i.e., exact monomorphism).The x and y coordinates for the modes are shown in corresponding colors to the clades.Median values of the density function, along with their bandwidth (BW), are also presented.All reported values rounded to two decimal places, except BW.

Figure 3 .
Figure 3. Mean female versus male size regressions on log10-transformed data for, A, C, E, bird (black) and mammal (orange) mass and for, B, D, F, squamate (blue) and turtle (purple) SVL.A-B, all taxa.C-D, only non-reversed systems are plotted (i.e., SSD greater than or equal to 0), where female-biased taxa are dropped.E-F, only sex-role reversed systems are plotted (i.e., SSD less than or equal to 0), where male-biased taxa are dropped.The red line indicates isometry (i.e., slope = 1).All slopes are rounded to three decimal places.Slopes are categorized as either consistent with RR, converse RR, or approximately isometric.

Figure 4 .
Figure 4. PCA of log10-transformed male and female mean sizes of birds (black), mammals (orange), squamates (blue), and turtles (purple), allowing for visualization of SSD (PC2) and overall taxon size (PC1).Arrows indicate increasing taxon size along PC1, and male and female symbols indicate the direction of SSD along PC2.

Figure 5 .
Figure 5. Histogram of bird mass SSD and the PCA depiction of the scaling relationship between overall taxon size and SSD in birds.Is there intense selection against monomorphism in birds compared to other amniotes?Is this SSD distribution related to the scaling relationships?Modified from Saitta et al. (2020).

Figure 9 (
Figure 9 (Below).Myhrvold et al. (2015) bird mass SSD presented as, A, a histogram with potential artificial spike, and B, a kernel density function (bandwidth shown) with positive unimodality.C-F, Data simulation to examine the impact of errors on the SSD distribution.C, if I assume the SSD distribution in Dunning (2007) is reflective of the true population distribution of all birds, then I can simulate reversed and non-reversed SSD using parameters inspired by birds to create a bimodal distribution with positive and negative peaks equidistant from SSD = 0. D, I can simulate rounding/precision error by simulating a smaller number of taxa, and then inappropriately rounding them to only one significant figure, leading to an unnatural spike at SSD = 0. E, I can also simulate the process of obtaining data from disparate sources by accurately simulating as many new taxa as before and then introducing appreciable random error to each male and female mass.All the data is then combined and viewed as a histogram, E, or fit with a kernel density function, F (bandwidth is the same as Fig.7H, which would normally still show bimodality in the realDunning [2007]  dataset).Vertical red line indicates SSD of zero.