Appendix E. Results of a simulation study exploring the effects (i.e., absolute and relative bias) of an uneven sex ratio at metamorphosis on juvenile survival, adult survival, recapture probability and probability to reproduce at an age of 1 year.
Since spadefoot toads can only be sexed when captured as breeders (they cannot be sexed at metamorphosis), we assigned a sex to the metamorphs that were not recaptured as adults. We assigned a sex in such a way that the sex ratio was even at metamorphosis. This approach works well if the assumption of the even sex ratio is fulfilled (Nichols et al. 2004). In order to check the consequences of the violation of the assumption of a 1:1 sex ratio at metamorphosis, we performed a simulation study. We were interested in computing the bias of age-specific survival, probability to start to reproduce and recapture probability when the sex ratio is uneven. A possibility to study bias is the analytical-numerical approach (Burnham et al. 1987). The principle is to create a data set using predefined parameter estimates and to analyse this data set. If the number of released individuals is very large, resulting maximum likelihood estimates and standard errors represent approximate expected values of the estimators and their standard errors, respectively (Burnham et al. 1987). This approach was used here. We only report mean and relative bias but not standard errors or coverage probabilities.
Simulating model
We considered a structurally identical model as the one presented in Appendix B. We assumed that most males and half of the females started to reproduce at an age of 1 year (i.e., α_{1,M} = 0.9; α_{1,F} = 0.5). In both sexes we assumed that so far inexperienced breeders start to reproduce at an age of 2 years (i.e., a_{2,M} = α_{2,F} = 1). Further, we assumed juvenile and adult survival as well as recapture probability to be sex-specific (values see table D1). We further assumed that at each of the 7 occasions a number of metamorphic males (N_{m}) and females (N_{f}) are released. By varying the ratio of N_{m} / N_{f} we vary the true underlying sex ratio. At each occasion 48000 individuals were released, and we considered 7 scenarios regarding the true underlying sex ratio. In the first scenario we considered an even sex ratio. This scenario was included to evaluate whether all parameters were unbiased. Following Nichols et al. (2004), we expected a priori that survival and recapture should be unbiased, but this was not clear for the age-specific probability to start to reproduce. The next six scenarios considered uneven sex ratios (extreme: 2:1; strong: 7:5; moderate: 13:11) each either in favour of males or of females. Capture histories with these properties were constructed in R.
The sex of the individuals that have never been recaptured is known in the constructed capture histories. Yet, in reality we do not know the sex of unsexed metamorphs that are never recaptured and therefore we have to recalculate the number of released males and females that were never recaptured. Let n_{i} be the total number of individuals that are released at occasion i, and rf_{i} (rm_{i}) denote the number of these recaptured at least once and determined to be females (males), then - under the assumption of an even sex ratio - the number of females (males) released and never recaptured is [0.5 × n_{i} - rf_{i}; 0.5 × n_{i} - rm_{i}]. For each constructed capture history file we recalculated these numbers and changed the input file accordingly. Finally, we analysed the data with program MARK (White & Burnham 1999).
Results
We computed absolute and relative bias of all the estimated parameters (Table D1). For the interpretation of the results it is important to note that some bias is always expected with the method that we used. This is because the computed values refer to one particular simulated data set. The random variation (i.e. “small sample size bias”) is smoothed largely by the consideration of a large number of released individuals, but it is still persistent to some degree. We consider relative bias smaller than 3% as due to random variation.
If the sex ratio was even, then bias in the estimates of all parameters was negligible (relative bias < 3% for all parameters; Table E1). If the true underlying sex ratio is uneven and the data are analysed as if the sex ratio was even, then adult survival, recapture and the probability to start to reproduce still had negligible bias (relative bias < 3% for all these parameters and all scenarios). However, juvenile survival was biased. If the true sex ratio is in favour of males, juvenile survival of males is underestimated and that of females overestimated. If the true sex ratio is in favour of females, juvenile survival of males is overestimated and that of females is underestimated. The relative bias was more or less symmetrical for the two sexes, i.e. males survival was overestimated to the same degree as was females survival underestimated. The bias declined the closer the true sex ratio was to an even one. The relative bias was relatively small (< 8%) when the sex ratio was 13:11 (54% males vs 46% females).
The simulation study deals with the case where a sex is assigned with a probability of 50% to a metamorph. In our mark-recapture analysis, we used information on size at metamorphosis (male and female metamorphs differ in mean size at metamorphosis) to assign sex with a probability that depended on the size at metamorphosis. That is, our assignment of sex is not random but based on information in the data. Therefore, we assume that bias in our case study is smaller than in these simulations.
TABLE E1. Absolute (B) and relative bias (rB, in %) of male juvenile survival (φ_{juv, M}), female juvenile survival (φ_{juv, F}), male adult survival (φ_{ad, M}), female adult survival (φ_{ad, F}), male recapture (p_{M}), female recapture (p_{F}), probability of males to start to reproduce at an age of 1 year (α_{1,M}), and probability of males to start to reproduce at an age of 1 year (α_{1,F}) under variable true sex ratios, when the data are analysed as if the sex ratio were even. For each scenario a data set was computed where at each of 7 capture occasions 48000 individuals were released.
Parameter | True value | Even sex ratio (1:1) | Uneven sex ratio (1:2) | Uneven sex ratio (2:1) | Uneven sex ratio (5:7) | Uneven sex ratio (7:5) | Uneven sex ratio (11:13) | Uneven sex ratio (13:11) | |||||||
B | rB | B | rB | B | rB | B | rB | B | rB | B | rB | B | rB | ||
φ_{juv, M} | 0.2 | -0.0001 | -0.05 | 0.0670 | 33.49 | -0.0653 | -32.65 | -0.0319 | -15.95 | 0.0341 | 17.04 | -0.0144 | -7.20 | 0.0171 | 8.53 |
φ_{juv, F} | 0.15 | 0.0044 | 2.96 | -0.0484 | -32.26 | 0.0501 | 33.41 | 0.0224 | 14.94 | -0.0256 | -17.09 | 0.0133 | 8.86 | -0.0100 | -6.64 |
φ_{ad, M} | 0.4 | -0.0047 | -1.18 | 0.0020 | 0.50 | -0.0012 | -0.30 | 0.0053 | 1.33 | -0.0036 | -0.91 | -0.0031 | -0.78 | 0.0025 | 0.63 |
φ_{ad, F} | 0.45 | -0.0055 | -1.23 | -0.0034 | -0.75 | 0.0024 | 0.54 | 0.0056 | 1.24 | -0.0034 | -0.75 | -0.0099 | -2.20 | -0.0001 | -0.03 |
p_{M} | 0.6 | 0.0121 | 1.72 | 0.0050 | 0.72 | 0.0045 | 0.64 | -0.0108 | -1.54 | 0.0005 | 0.07 | 0.0107 | 1.53 | 0.0072 | 1.03 |
p_{F} | 0.7 | 0.0048 | 0.80 | 0.0043 | 0.72 | -0.0009 | -0.15 | 0.0034 | 0.56 | 0.0016 | 0.26 | 0.0037 | 0.61 | -0.0133 | -2.21 |
α_{1,M} | 0.9 | -0.0171 | -1.91 | -0.0067 | -0.75 | -0.0088 | -0.98 | 0.0187 | 2.08 | -0.0116 | -1.29 | -0.0216 | -2.40 | -0.0046 | -0.51 |
α_{1,F} | 0.5 | -0.0125 | -2.49 | -0.0049 | -0.98 | -0.0066 | 1.33 | 0.0127 | 2.53 | -0.0025 | -0.49 | -0.0066 | -1.31 | 0.0033 | 0.65 |
Proportion males | 0.5 | 0.33 | 0.67 | 0.42 | 0.58 | 0.46 | 0.54 |
LITERATURE CITED
Burnham, K. P., D. R. Anderson, G. C. White, C. Brownie, and K. H. Pollock. 1987. Design and analysis of fish survival experiments based on release-recapture data. American Fisheries Society, Monograph 5. Bethesda, Maryland. 437 pp.
Nichols, J. D, W. L. Kendall, J. E. Hines, and J. A. Spendelow. 2004. Estimation of sex-specific survival from capture-recapture data when sex is not always known. Ecology 85: 3192–3201.
White, G. C., and K. P. Burnham. 1999. Program MARK: survival estimation from populations of marked animals. Bird Study 46 (suppl.):120–139.