Insights from complete-incomplete brood sex-ratio disparity

Abstract

An assertion deeply rooted in the ornithological literature holds that sex-specific mortality causes a sex ratio disparity (SRD) between complete and incomplete broods. Complete broods are thought to reflect the primary sex ratio before any bias introduced by developmental mortality. Contrary to this view, however, complete and incomplete broods should exhibit identical sex ratio distributions even when the sexes experience differential mortality, as shown in the classic paper of Fiala (Am Nat 115: 442–444, 1980). Therefore, in partially unsexed samples, primary sex ratio biases cannot be distinguished from biases caused by differential mortality. In addition, complete broods do not represent primary sex ratio more accurately than incomplete ones and might even be misleading. Despite Fiala’s prediction, SRD does occur in some empirical studies. We show that this pattern could arise if (1) primary sex ratio affects chick mortality rates independently of sex (direct effect), (2) primary sex ratio covaries with a variable that also affects mortality rate, or (3) sex differential mortality covaries with overall mortality rate (indirect effects). Direct effects may cause stronger SRD than indirect ones with a smaller and opposite bias in the overall sex ratio and could also lead to highly inconsistent covariate effects on brood sex ratios. These features may help differentiate direct from indirect effects. Most interestingly, differences in covariate effects between complete and incomplete broods imply that influential variables are missing from the analysis.

Appendix

To simulate equivalent effects that can lead to SRD between complete and incomplete broods, i.e. that cause brood sex ratio and mortality rate to covary, we simulated 1,000 samples consisting of 50 broods each, where initial brood size was randomly assigned, ranging from 1 to 5 with probabilities 0.1, 0.2, 0.4, 0.2, and 0.1. Sex was taken at random from the binomial distribution, with p = 0.5 or according to the function given in scenario 1. Mortality risk for individuals in each brood was then assigned according to the following rules.

A.1 Scenario 1: Sex bias before mortality

A primary effect was simulated by increasing mortality risk, r, with initial brood size, n, by r = 1 − s = 0.075(n − 1) and by mimicking the sex ratio effect in scenario 2 by assigning primary sex according to p = s _m(t)/(s _m (t)+s _f (t))=(1−(n−1) · 0.15)/(2−(n−1) · 0.15). Hence, initial brood sex ratio decreased and mortality rate increased with brood size (Fig. 1a, upper panels). Due to this covariance, SRD results (Table 3). While in accordance with mortality being non-specific, brood sex ratios do not differ between complete and incomplete broods for each initial brood size (Fig. 1a, lower left) dependencies on final brood size differ for complete and incomplete broods (Fig. 1a, lower right; statistics not shown). This is because broods affected by mortality were originally larger than unaffected ones at each final brood size, carrying over their initially lower sex ratio.

A.2 Scenario 2: Sex-specific mortality rate

The sex-difference and overall rate of mortality were concomitantly increased with initial brood size by increasing mortality rate for males with r _m = 0.15(n − 1), but no female mortality, implying identical overall mortality per initial brood size as under scenario 1. By implication of our settings, average final brood size and sex ratio (Table 3), as well as brood sex ratio and mortality rate functions of initial brood size (Fig. 1b, upper panels) are identical to scenario 1. According to Fiala’s rationale outlined in the text (cf. Tables 1 and 2), this leads also to identical brood sex ratios at each initial brood size between complete and incomplete broods (Fig. 1b, lower left). As the sex ratio and mortality functions of initial brood size are identical to scenario 1, the covariance between brood sex ratio and mortality rate is also identical, causing identical SRD (Table 3) and a similar difference of effects with final brood size between complete and incomplete broods (Fig. 1a, lower right).

A.3 Scenario 3: Sex ratio-dependent mortality rate

Here, mortality rate increases with initial number of females per brood, $s = 1 - r = 1 - {{\left( {n - m} \right)} \mathord{\left/ {\vphantom {{\left( {n - m} \right)} {10}}} \right. \kern-\nulldelimiterspace} {10}}$. This did not only imply a correlation of brood sex ratio and mortality rate, which leads to SRD (Table 3) but implies also a correlation of overall mortality with initial brood size (Fig. 1c, upper right), resembling the relationships in scenarios 1 and 2 (Fig. 1a, b, upper right panels). Three important conclusions are substantiated by this simulation: (1) SRD can be much stronger than with indirect effects at comparable overall mortality and is unaffected by any covariate of mortality; (2) overall sex ratio bias is small and opposite in direction to the deviation of incomplete from complete broods; (3) covariations of variables with the sex ratio in complete and/or incomplete broods can vanish in the combined sample or could be spurious if they persisted.

• ad 1. The correlation between mortality and sex ratio is in the same direction but higher than in scenarios 1 and 2, leading to stronger SRD at similar overall mortality rate and stays significant when partialed against a correlate of mortality (i.e. brood size; Table 3).

• ad 2. The overall sex ratio bias is much smaller in amount and opposite in direction (i.e. deviation from 0.5) from scenarios 1 and 2. This follows from mortality per se selecting for sex ratio but not for sex. The slight deviation from unity (Table 3) is introduced because—while mortality rate is identical for males and females within broods—overall mortality rate differs between the sexes, as the average female occurs in broods with higher mortality rate than the average male. Overall brood sex ratio p _m  = s _m /(s _m  + s _f ) can be calculated with N the maximum of brood size n, s(t) a function of a brood parameter, p _n the proportion of broods of size n, f = n  −  m, p constant, and

  $$s_f = {{\sum\limits_{n = 1}^N {p_n \left( n \right)\left( {\sum\limits_{f = 1}^n {f\left( {\begin{array}{*{20}c} n \\ f \\ \end{array} } \right)} p^{n - f} \left( {1 - p} \right)^f s\left( t \right)} \right)} } \mathord{\left/ {\vphantom {{\sum\limits_{n = 1}^N {p_n \left( n \right)\left( {\sum\limits_{f = 1}^n {f\left( {\begin{array}{*{20}c} n \\ f \\ \end{array} } \right)} p^{n - f} \left( {1 - p} \right)^f s\left( t \right)} \right)} } n}} \right. \kern-\nulldelimiterspace} n} \cdot \left( {1 - p} \right){\text{ and }}s_m {\text{ accordingly}}.$$

In our case, s(t) = 1−f/10, p = 0.5, and there were 10, 20, 40, 20, and 10% of broods of sizes 1 to 5; hence p _m  = 0.5294, closely matched by the simulated overall sex ratio in scenario 3 (Table 3). Note that this effect leads to an extremely smooth increase in predicted sex ratio with brood size that will not reveal itself at realistic sample sizes (from size 1 to 5: 0.5263, 0.5278, 0.5294, 0.5313, 0.5333; cf. Fig. 1c, upper left).

• ad 3. If plotted for complete and incomplete broods, separately, initial brood size differentially affects brood sex ratios, by exhibiting different intercepts and slopes (Fig. 1c, lower panels)—although in the combined sample, sex ratios do not significantly depend on initial brood size (Fig. 1c, upper left; statistics not shown). The reason is that the proportion of broods containing females increases with brood size, which causes the proportion of broods affected by mortality to increase, but larger broods contain more males, on the average, for the same number of females. Hence, relatively more but less biased broods end up incomplete the larger the brood size. As a result, sex ratio functions diverge for complete and incomplete broods, while the weighted average sex ratios are nearly identical at each brood size, except for the very small effect noted above. The subsampled sex ratio effects are therefore spurious.

Final brood size is a variable correlating with mortality rate and initial brood size. Hence, sex ratio dependencies on final brood size in complete and incomplete broods are similar to those with initial brood size but not identical. Here, a sex ratio effect persists for the combined sample because the number of complete broods at each final brood size is identical to that of the respective initial brood size, but incomplete broods at each final brood size contain young from initially larger broods so that the proportion of affected broods is a function of brood size distribution and mortality rate function and does not necessarily exactly cancel out the sex ratio effect. For instance, at largest final brood size, all broods are complete, unless all broods from the originally largest size had been affected by mortality. With our simulation parameters, a significant increase of sex ratio with final brood size results for the combined sample (effect and analysis not shown), which is spurious in the sense, that this does not imply initial sex bias or sex-specific mortality to exist.