AppendixA1. Sample Size Required to Detect Sex Differences in Variability in a Single Study

Using Monte Carlo simulations, we calculated the power to detect differences in variability corresponding to lnCVR values of 0.02, 0.04, 0.06, and 0.09 in a single study, with α = 0.05 (two-tailed). For simplicity, we assumed equal means in males and females in the population (that is, lnCVR = lnVR); the female coefficient of variation was set to 0.08, which approximates the median value for body size across species (McKellar & Hendry, 2009). The total sample size was varied from 100 to 100,000; each sample included equal numbers of males and females. Figure 3 shows the simulation results. Note that these power calculations include all rejections of the null, including the probability of finding significant effects in the wrong direction (Type S error; Gelman & Carlin, 2014). This leads to somewhat higher estimates in the low-power region of the curve, but has no effect when power is moderate to high.

Fig. 3

Power to detect differences in variability in a single study (α = 0.05, two-tailed)

A2. Power of Harrison et al.’s Analysis in Mammals

Harrison et al.’s Table 1 reported confidence intervals for lnCVR. From the confidence interval reported for mammalian traits, we estimated the standard errors of estimate (where SE is roughly 1/4th of the width of a confidence interval). The SE was estimated to be 0.133. Using G*power 3.1 (Faul et al., 2009) and this estimated SE, we calculated power to detect true effect sizes of .02, .06, and .09.

A3. Attenuation of Variability Effect Sizes due to Measurement Error

Repeatability is defined as the between-individual variance (VI) divided by the total variance of the measured trait (VT), which in turn is the sum of VI and a residual variance term (VR) that includes within-individual variability, random noise, etc. The repeatabilities of males and females are:

$$_}=\frac_}}_} + _}},\ _}=\frac_}}_} + _}}.$$

(1)

If the residual variance (i.e., the measurement error) has the same magnitude in the two sexes (VRm = VRf = VR), greater male variability at the between-individual level (VIm > VIf) implies that males will show higher repeatability than females (Rm > Rf). Indeed, this is a common finding in the literature; in one meta-analysis, the effect seemed to be driven by mate choice behaviors (Bell et al., 2009), but other studies have found higher male repeatabilities in exploration, sociability, and other traits (e.g., Dingemanse et al., 2002; Strickland & Frère, 2018).

In what follows, we further assume equal means in males and females, so that lnCVR = lnVR. With an even sex ratio, the repeatability R calculated on the population as a whole is:

$$R=\frac_} +_}}}_} +_}} + _}}.$$

(2)

Rearranging Eq. (2) yields the residual variance VR:

$$_}=\frac_} + _}}\cdot \frac.$$

(3)

By definition, the true lnVR (in the population) is:

$$\mathrm=\mathrm\sqrt_}}_}}},$$

(4)

while the measured lnVR* is:

$$}^=\mathrm\sqrt_}}_}}}=\mathrm\sqrt_} + _}}_} + _}}}.$$

(5)

Define \(\uplambda\) as the true variance ratio VIm/VIf, so that VIm = λVIf. From Eq. (4),

Substituting Eq. (3) into Eq. (5) yields:

$$}^=\mathrm\sqrt_} + \frac_} + _}} \cdot \frac}}}_} + \frac_} + _}} \cdot \frac}}}}=\mathrm\sqrt \cdot \frac}}} \cdot \frac}}}},$$

(7)

which, together with Eq. (6), can be used to calculate the measured lnVR* from the true lnVR and the repeatability R.