--- title: "Methods used in `bbouretro` for Estimating Survival, Recruitment, and Population Growth" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Methods used in `bbouretro` for Estimating Survival, Recruitment, and Population Growth} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} bibliography: bibliography.bib --- ```{r, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) ``` There are multiple ways cited in literature to calculate survival, recruitment, and population growth. The traditional or **retro** methods used in bbou**retro** 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_survival_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_analysis_1984]. $$SE(\hat{S}(t)) = \sqrt{[\hat{S}(t)]^2 \sum \frac{d_{i}}{r_{i}(r_{i} - d_{i})}}$$ Cox and Oakes [-@cox_analysis_1984] propose a simple binomial variance formula “which is better in the tails of the distribution” [@pollock_survival_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. [-@decesare_estimating_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_1984] . 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_moose_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