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Management of North American waterfowl relies heavily on survival rates (S) estimated from band recovery models (Brownie et al. 1985). Waterfowl are banded annually, primarily during late summer on the breeding grounds, and bands are recovered from hunters that harvest birds during the ensuing hunting season. Models used for estimating survival rates assume that there is no individual heterogeneity of the conditional band reporting rates, r. Seber (1970) defined band reporting rates (r) conditional on the bird dying, 1−S, so that the probability a band is reported is (1−S)r, which is equivalent to the band recovery rate (f) of Brownie et al. (1985).
Individual heterogeneity in r might occur for a myriad of reasons, including differential survival between time of banding and onset of hunting (Nichols et al. 1982; Zimmerman et al. 2010), differential vulnerability to harvest (Pollock and Raveling 1982; Pace and Afton 1999), or differential probability that hunters will report a recovered band (Henny and Burnham 1976; Reinecke et al. 1992). However, detection of individual heterogeneity in conditional band reporting rates is difficult, if not nearly impossible. The estimate of r is a purely binomial process, with each bird that dies having only a Bernoulli trial as to whether its band is reported. That is, a bird only dies once, and so there cannot be repeated trials of whether a band is reported that would allow detection of individual heterogeneity. In contrast, repeated trials of birds surviving annual intervals in a band reporting analysis provide the necessary information to detect individual heterogeneity in survival rates, and hence, estimation of σS in the models described below, and repeated recapture or resighting of living birds in Cormack–Jolly–Seber models allows for the detection of individual heterogeneity in detection probability (P).
One approach that might be used to detect individual heterogeneity of conditional band reporting rates would be to separate birds into independent analyses and then compare the estimates of r across these lots, in the sense of Burnham et al. (1987), Part 4. However, this approach is generally inefficient and would be unlikely to detect low amounts of heterogeneity. A second approach is the use of covariates or groups (Dorazio 1997). However, only covariates measured at the time of banding are available, because potential covariates at the time of recovery are only available for reported bands. Further, these covariates have to correlate with the conditional band reporting rate (r) which seems unrealistic except for a few easy to measure covariates such as age or body mass at time of banding (Pace and Afton 1999). Thus, the covariate approach is again not likely to provide much explanatory power for detecting individual heterogeneity of r.
Even if individual heterogeneity in r can be shown, the real issues are whether this effect causes bias in the estimates of survival and whether individual heterogeneity in r will result in the detection of nonexistent individual heterogeneity in S. Therefore, our objective was to evaluate the potential for incorrect inference about individual heterogeneity of S and the potential for biased estimates of S when considerable individual heterogeneity exists in r.
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Sixteen sets of simulations of Seber's (1970) single age class ring recovery model, each with 1000 replicates, were conducted as a factorial design. Four factors, each with 2 levels, were considered: (1) the focal effect of interest of identical or else individual variation in band reporting rate (r), (2) constant band reporting rate r(.) or time-specific band reporting rates r(t), (3) low 0.6 or high 0.8 levels of annual survival, and (4) annual survival assumed constant S(.) or time-specific S(t). Thus, half the simulations had constant reporting rate (r) of 0.196 for each individual, with the other half with a reporting rate drawn on the logit scale from a normal distribution with mean = −1.9459 with SD = 1.5, giving the random reporting rate with a back-transformed mean of 0.196 with SD = 0.196 and median = 0.125 (Fig. 1). Although time-specific effects were assumed for S and r in half of the analyses, actual values simulated were identical across time in all cases. Thus, only 4 different scenarios of data generation were performed, but with 4 estimation models applied to each scenario. For estimation model S(t),r(.) with time-specific and constant r, the mean of the 10 survival estimates was used as the overall estimate of , and for model S(t),r(t), the mean of the first nine survival estimates was used. All simulations were conducted with constant annual survival rates, that is, there was no temporal or individual heterogeneity in survival. Ten occasions were simulated, with 10,000 birds assumed banded and released on each occasion to provide high statistical power for selecting among models.
Figure 1. Distribution of the back-transformed value logit(r) ~ N(−1.9459, 1.5) used in our simulations, giving mean of 0.196 with SD = 0.196 and median = 0.125. Note that log[0.125/(1−0.125)] = −1.9459
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For each of the 16 sets of simulations, 2 models were estimated. The first modeled no individual variation in survival, that is, all individuals were assumed to have identical survival rates (σS = 0). The second more complex model assumed individual heterogeneity in survival (σS estimated) modeled as an individual random effect on the logit scale, following the approach of McClintock and White (2009), McClintock et al. (2009), and Gimenez and Choquet (2010). That is, for the ith individual, logit(Si) = μS + zi, where zi ~ N(0, σS). The parameter σS is estimated via maximum likelihood using numerical integration (Gaussian–Hermite quadrature, Givens and Hoeting 2005) to solve the integral numerically. For models assuming time-specific survival, a single parameter σS is estimated that applies to all of the survival estimates, that is, all the time-specific survival estimates for individual i have the same zi on the logit scale. For each replicate, these 2 models were compared using AICc to determine the selected model and the relative weight of each of the 2 models. There is 1 df of difference between the 2 models, depending on whether σS is estimated or assumed equal to zero; hence, models with σS > 0 can appear competitive even though σS = 0 (Burnham and Anderson 2002:131). All simulations and parameter estimates were produced using Program MARK (White and Burnham 1999).
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Overall, models with σS estimated were selected 8.9% of the time (Table 1). There was some evidence that rate of selection for model σS differed between data sets simulated with or without individual heterogeneity in conditional band reporting rates (i.e., σr = 1.5 or 0 on the logit scale), but the selection rate was higher for data sets with σr = 1.5 when survival was 0.6, but lower for data sets with σr = 1.5 when S was 0.8 (Table 2; Gadj = 10.68, df = 3, P = 0.01). In addition, σS was selected more frequently when it was included in model structures that included temporal variation in both S and r (Table 2; Gadj = 63.57, df = 3, P < 0.0001).
Table 1. Model selection results summarized for the model assuming individual heterogeneity in survival (σS estimated) or no heterogeneity (σS = 0) by whether there is individual heterogeneity in the conditional band reporting rate (σr = 1.5 on logit scale) or not (σr = 0). Values are the number of times a model was selected (percent of times selected) for each of the four combinations.
|True σr||Times selected||Total|
|σS estimated (%)||σS = 0 (%)|
|1.5||698 (8.7)||7302 (91.3)||8000|
|0||719 (9.0)||7281 (91.0)||8000|
|Total||1417 (8.9)||14,583 (91.1)||16,000|
Table 2. Percent of times out of 1000 simulations that models with σS estimated ranked higher than models with no heterogeneity (σS = 0) depending on true survival rate (0.6 or 0.8), whether there was individual heterogeneity in the conditional band reporting rate (σr = 1.5 or 0.0 on logit scale), and whether or not the estimating models included temporal variation in S(t) or r(t).
|True S||True σr||Percent times σS > 0 selected over σS = 0||Total|
|S(.), r(.)||S(.), r(t)||S(t), r(.)||S(t), r(t)|
Note that because there is only a single degree of freedom difference between models with and without σS, the expected difference between the −2log likelihood of the 2 models would be a χ2 random variable with 1 degree of freedom. Thus, the model with σS estimated would only be selected when the −2log likelihood of the σS = 0 model is 2 units larger (given that the effective sample size of these simulations is large, and AICc and AIC are nearly identical). The Pr(χ2(1) > 2) = 0.157, suggesting that the more complex model should have been selected 15.7% of the time, but our simulations show that in all cases, the more complex model is selected less often. However, the null hypothesis σS = 0 is on the boundary of the parameter space, and therefore, classical inference as described above no longer holds (Self and Liang 1987; Gimenez and Choquet 2010). The asymptotic null distribution of the LRT is a 50:50 mixture of χ2 distributions with 0 and 1 degrees of freedom (Stram and Lee 1994). Thus, our simulations are consistent with this theory in that models with σS estimated were selected just over half (8.9 vs. 15.7) of what is expected.
Average model weights were consistent across the levels of survival and whether or not individual heterogeneity is present in the conditional band reporting rate (Table 3). Given that the theoretical expected difference in the −2log likelihood value between the 2 models is χ2(1), the expected ΔAIC is 1. Given this expected ΔAIC, the expected weight of the more complex model would be 0.378. However, as described above, this theory does not hold, so we would expect our simulations to show better performance than expected by assuming the difference to be χ2(1).
Table 3. Model weight results summarized for the model assuming individual heterogeneity in survival (σS estimated) or no heterogeneity (σS = 0) by two levels of survival and whether there is individual heterogeneity in the conditional band reporting rate (σr = 1.5 on logit scale) or not (σr = 0). Values are the average model weight based on 4000 simulations for each scenario, with the SE of all entries ≤0.002.
|True S||True σr||Average model weights|
|σS estimated||σS = 0|
Table 4. Mean and SD of survival estimates and percent relative bias in under 16 data simulating models, each analyzed with two different estimating models (σS estimated, or σS = 0). Survival estimates were not affected by whether or not temporal variation in S or r was estimated, so results are collapsed. For estimation model S(t),r(.) with time-specific and constant r, the mean of the 10 survival estimates was used as the overall estimate of , and for model S(t),r(t), the mean of the first nine survival estimates was used. Values are the average estimated survival based on 4000 simulations for each scenario, all with SE <0.0005.
|True S||True σr||Estimation model|
|σS estimated||σS = 0|
|Mean||Bias (%)||SD||Mean||Bias (%)||SD|
Lastly, 1417 times out of 16,000 simulations (8.9%), the model that estimated σS would have provided an estimate. These estimates (Table 5) will always be >0 because σS is estimated on the log scale to maintain this constraint. As expected, for simulations with S = 0.8 and σS estimated, − S was positively correlated with (Fig. 2). The likelihood of selecting the minimum AICc model that estimates σS is not independent of the true S and the true σr: χ2(1) = 4.064, P = 0.044. The estimated σS is larger for true S = 0.8 than for S = 0.6, which makes sense because σS is on the logit scale. Had models where σS is estimated only been considered, the expected values again would be >0 because σS is estimated on the log scale to maintain this constraint (Table 6). However, in comparing the values in Table 5 with values in Table 6, we see that the expected value is nearer zero. We also see minor difference in Table 6 of the mean values of for simulations with σr > 0 versus σr = 0.
Table 5. Estimates of σS when the models producing this estimate were selected by minimum AICc.
|True S||True σr|| N ||Mean||Std. Error||Minimum||Maximum|
Table 6. Estimates of σS from all simulations for models producing this estimate.
|True S||True σr||Mean||Std. Error||Minimum||Maximum|
Figure 2. Relationship between estimated individual heterogeneity in survival and estimated mean survival rate for 1000 simulations where true S = 0.8, σS = 0, r = 0.196, and σr = 1.5. Estimation models were S(.),r(.) with σS estimated or fixed to zero. Blue data points indicate 923 data sets where the simpler model lacking σS was selected by AICc (wi of model [σS, S(.),r(.)] < 0.5), whereas red dots indicate 77 data sets where model σS was selected (wi > 0.5).
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As with any simulation study, and particularly one with the limited parameter space considered in this study, care must be taken in drawing inferences to other sets of parameter values. Our parameter values were focused on waterfowl because this is the primary assemblage of species where these models are used for management decisions. However, we are encouraged by how little effect individual heterogeneity in r had in causing the analyst to conclude that there was individual heterogeneity in S compared with the same error rate for data with no heterogeneity in r. But as shown in Table 5, these error rates differ depending on the magnitude of S, although the number of birds banded and mean r were constant. A wider range of parameter values, and probably more importantly, a wide range of number of birds banded and values of r, would be needed to evaluate what the error rate is in selecting the model estimating σS over the model with σS = 0, and how this error rate is affected. All simulations were conducted under the artificial scenario of no individual variation in S. Simulations encompassing a reasonable range of individual variation in S would be helpful, but were beyond the scope of this article.
Pollock and Raveling (1982) first claimed that heterogeneous band recovery rates would not bias estimates of survival rates, and our simulations further confirmed this result. The only scenarios where we found appreciable bias was when survival was high and models estimated individual heterogeneity in survival, even though data were simulated with σS = 0. However, there is undoubtedly real individual heterogeneity in S in most natural populations, for example Rexstad and Anderson (1992), and this individual heterogeneity is likely not always picked up by the use of covariates or group attributes as advanced by Dorazio (1997). Thus, assessing individual variation in S through the use of the individual heterogeneity model considered here should be routinely undertaken by managers. However, we advise caution, especially when survival rates are high, because overestimation of can lead to overestimation of . And we still encourage the collection of individual covariates at the time of banding, if only to compare these models against the models with σS estimated. Covariates attempt to explain the causes of individual heterogeneity, rather than just accommodate the heterogeneity as with models estimating σS.
Pollock and Raveling (1982) did find that heterogeneity in band recovery (f) or conditional reporting rates (r) could lead to bias if it was accompanied by individual heterogeneity in survival, especially if σS was correlated with σf (see also Nichols et al. 1982; Barker 1992). Such correlation might be expected if some individuals are more vulnerable to hunting (thus, higher f or r), and such vulnerability results in (or is correlated with) lower survival (Nichols et al. 1982; Sedinger and Herzog 2012). We did not consider individual heterogeneity in survival in our simulations, so analysts should beware of applying our results in cases where heterogeneity in conditional reporting rate might be correlated with heterogeneity in survival rates, and we recommend additional simulation studies to explore this condition.
There are likely two main sources of individual variation in conditional reporting rates of hunted waterfowl: (1) individual variation in vulnerability to hunting (Pace and Afton 1999; Arnold and Howerter 2012) and (2) individual variation in the probability that a recovered band will be reported to authorities (Henny and Burnham 1976, Royle and Garrettson 2005; Zimmerman et al. 2009). Our results are most relevant to the second scenario, as it is difficult to imagine how reporting rates would be correlated with survival. In North America, band inscriptions changed twice during 1993–1995, with the inclusion of zip codes in 1993 and toll-free phone numbers in 1995, and these changes have been considered problematic by analysts analyzing band recovery data during this time period (Royle and Garrettson 2005). However, our results suggest that mean survival rates and indeed mean band reporting rates are unlikely to be biased by heterogeneity in reporting rates caused by changing band inscriptions. Other studies have noted regional variation in band reporting rates (Zimmerman et al. 2009), and so long as the management goal is to estimate overall survival rate for a population, such heterogeneity is unlikely to lead to biases in survival estimation.