Methods used in bbouretro for Estimating Survival, Recruitment, and Population Growth

There are multiple ways cited in literature to calculate survival, recruitment, and population growth. The traditional or retro methods used in bbouretro are outlined below.

Survival, $S$

The bbr_survival() function uses the staggered entry Kaplan-Meier method.

The staggered entry Kaplan-Meier, which uses discrete time steps, is defined algebraically below (Pollock et al. 1989). The probability of surviving in the $i$th time step (usually month) is given by the equation below

$$\hat{S}_{i} = 1 - \frac{d_{i}}{r_{i}}$$

where $d_{i}$ is the number of mortalities during the period and $r_{i}$ is the number of collared individuals at the start of period. The estimated survival for any arbitrary time period $t$ (usually year) is given by

$$\hat{S}(t) = \prod_{i=1}(\hat{S}_{i,t})$$

where $S$ is the survival during the $i$th time step of the $t$th period.

Standard error is estimated using the Greenwood’s formula (Cox and Oakes 1984).

$$SE(\hat{S}(t)) = \sqrt{[\hat{S}(t)]^2 \sum \frac{d_{i}}{r_{i}(r_{i} - d_{i})}}$$

Cox and Oakes (1984) propose a simple binomial variance formula “which is better in the tails of the distribution” (Pollock et al. 1989).

$$SE(\hat{S}(t)) = \sqrt{ \frac{[\hat{S}(t)]^2 - [1-\hat{S}(t)]}{r(t)}}$$

Logit-based confidence limits are estimated using the Wald method. Further details on the Kaplan-Meier approach as applied in boreal caribou studies are given in vignette("previous-methods").

Recruitment, $R$

The bbr_recruitment() function estimates recruitment using the following method.

Estimation of recruitment follows methods of DeCesare et al. (2012).

The age ratio, $X$, is commonly estimated as the number of calves, $n{_j}$, per adult female, $n_{af}$, observed at the end of a measured year, such that

$$X = \frac{n_j}{n_{af}}$$

where $X \cdot sex\_ratio$ estimates the number of female calves $(n_{jf})$ per adult female.

Recruitment is estimated using the equation below which accounts for recruitment of calves into the yearling/adult age class at the end of the caribou year.

$$R_{RM} = \frac{X \cdot sex\_ratio}{1 + X \cdot sex\_ratio}$$

Variance is estimated using a bootstrap approach or the binomial method.

The bootstrap approach randomly resamples groups for 1000 iterations to create 1000 estimates of calf cow ratio and recruitment. Percentile-based 95% confidence limits are then estimated from the 1000 estimates.

For the binomial method, variance is estimated as $(R_m \cdot (1 - R_{rm})/n)$ where $n$ is the number of adult females sampled during each yearly survey.

It is assumed that both survival and recruitment are independently distributed on the logit scale therefore bounding each value between 0 and 1. The bootstrap method is recommended as the most robust approach to obtain variance estimates (Manly 1997) . A full summary of methods is given in vignette("previous-methods").

Population Growth, $\lambda$

For bbr_growth() function uses the basic Hatter-Bergerud method (Hatter and Bergerud 1991).

The basic Hatter-Bergerud equation used to estimate $\lambda$ is given below.

$$\lambda = \frac{(S)}{(1 - R)}$$

The standard errors on estimates of survival and recruitment are then used to create simulated distributions of $S$ and $R$ and derive a distribution of $\lambda$ based on the simulated values. It is assumed both survival and recruitment are distributed on the logit scale therefore bounding each value between 0 and 1.

One thousand simulations are conducted. Percentile-based confidence limits are then derived. Also, the proportion of simulations where $\lambda$ is greater than 1 is tabulated to provide a p-value for the hypothesis test $(Ho: \lambda >= 1, Ha: \lambda < 1)$.

Output summaries also include mean values simulated for $S$, $R$, as well as mean and median $\lambda$ values from the simulation.

Users can generate plots of simulated lambda values using the bbr_plot_growth_distributions() or yearly estimates using bbr_plot_growth().

References

Cox, D. R., and David Oakes. 1984. Analysis of Survival Data. Monographs on Statistics and Applied Probability. London ; New York: Chapman; Hall.

DeCesare, Nicholas J., Mark Hebblewhite, Mark Bradley, Kirby G. Smith, David Hervieux, and Lalenia Neufeld. 2012. “Estimating Ungulate Recruitment and Growth Rates Using Age Ratios.” The Journal of Wildlife Management 76 (1): 144–53. https://doi.org/10.1002/jwmg.244.

Hatter, Ian, and Wendy Bergerud. 1991. “Moose Recuriment Adult Mortality and Rate of Change” 27: 65–73.

Manly, B. F. J. 1997. Randomization and Monte Carlo Methods in Biology. New York: Chapman; Hall.

Pollock, Kenneth H., Scott R. Winterstein, Christine M. Bunck, and Paul D. Curtis. 1989. “Survival Analysis in Telemetry Studies: The Staggered Entry Design.” The Journal of Wildlife Management 53 (1): 7. https://doi.org/10.2307/3801296.