Calculate bias correction term for calf:cow composition survey.

When composition surveys are conducted there is a possibility of bias in calf cow ratios due to misidentifying young bulls as adult females and vice versa or missing calves.

compositionBiasCorrection(w, q, u, z, approx = F)

Arguments

w

number. The apparent number of adult females per collared animal in composition survey.

q

number in 0, 1. Ratio of bulls to cows in composition survey groups.

u

number in 0, 1. Probability of misidentifying young bulls as adult females and vice versa in composition survey.

z

number in 0, <1. Probability of missing calves in composition survey.

approx

logical. If TRUE approximate the uncertainty about the value of the composition bias correction value (c) with the log-normal distribution of c given all the supplied values of q, u, and z. If FALSE the composition bias correction value (c) is returned for each value of q, u, and z

Value

number or tibble. If approx = FALSE a vector of composition bias correction values (c) of the same length as q, u, and z. If approx = TRUE a tibble with on row per unique value of w and columns w, m, v, sig2, mu representing w, mean c, variance of c, and parameters for a log-normal approximation of the distribution of c.

Model of bias in recruitment estimates from calf:cow surveys

We assume each group of animals in a calf:cow composition survey contains one or more collared adult females (\(T\)), and may also include: uncollared adult females misidentified as young bulls or unknown sex (\(U\)); correctly identified uncollared adult females (\(V\)); young bulls correctly identified as male or unknown sex (\(O\)); young bulls misidentified as uncollared adult females (\(P\)); observed calves (\(J\)); and unobserved calves (\(K\)). The apparent number of adult females in the group is \(T+V+P=Tw\), where \(w\) is a multiplier that defines the apparent number of adult females as a function of the number of collared animals. The ratio of young bulls to uncollared adult females in the group is: $$q = \frac{P+O}{U+V}$$. Assuming an equal probability \(u\) of misidentifying young bulls as adult females and vice versa, we get \(V=(U+V)(1-u)\) and \(P=(O+P)u\). Given a probability \(z\) of missing calves, we get \(J=(J+K)(1-z)\).

Our objective is to model the sex and bias-corrected recruitment rate \(X=\frac{J+K}{2(T+U+V)}\) as a function of the observed calf:cow ratio \(R=J/(T+V+P)\), the cow multiplier \(w\), the ratio of young bulls to adult females \(q\), and the misidentification probabilities \(u\) and \(z\). We start by solving for \(T+U+V\) as a function of \(q,w,u\) and \(T\). Recognize that \(P=Tw-T-V\), \(U+V=V/(1-u)\), and \(P+O=P/u\) to write \(q\) as $$q=\frac{Tw-T-V}{uV/(1-u)}.$$ Rearrange to get $$V=\frac{T(w-1)(1-u)}{qu+1-u}.$$ Recognize that \(U=Vu/(1-u)\) to write \(T+U+V\) as a function of \(q,w,u\) and \(T\): $$T+U+V=T\frac{qu+w-u}{qu+1-u}.$$ Recognize that the number of observed calves \(J\) is the product of the apparent recruitment rate and the apparent number of adult females \(J=RTw\), and that therefore \(J+K=RTw/(1-z)\) to rewrite the bias corrected recruitment rate \(X=\frac{J+K}{2(T+U+V)}\) as a function of \(w,u,z\) and \(R\): $$X=R\frac{w(1+qu-u)}{2(w+qu-u)(1-z)}.$$ For simplicity, we write \(X\) as a function of a bias correction term \(c\): $$c=\frac{w(1+qu-u)}{(w+qu-u)(1-z)}; X=cR/2.$$ If we also adjust for delayed age at first reproduction (DeCesare et al. 2012; Eacker et al. 2019), the adjusted recruitment rate becomes $$X=\frac{cR/2}{1+cR/2}.$$

Uncertainty about the value of the bias correction term \(c\) can be approximated with a Log-normal distribution. Given the apparent number of adult females per collared animal \(w\) the mean and standard deviation of \(\log{c}\) can be calculated for samples from the expected range of values of \(q\), \(u\) and \(z\).

See also

Caribou demography functions: caribouBayesianPM(), caribouPopGrowth(), demographicCoefficients(), demographicProjectionApp(), demographicRates(), getOutputTables(), getPriors(), getScenarioDefaults(), getSimsNational(), plotRes(), popGrowthTableJohnsonECCC, runScnSet(), simulateObservations()

Examples

# number or reps
nr <- 10

compositionBiasCorrection(w = 6,
                          q = runif(nr, 0, 0.6),
                          u = runif(nr, 0, 0.2),
                          z = runif(nr, 0, 0.2),
                          approx = FALSE)
#>  [1] 0.9001006 1.0139583 1.0039392 0.9471162 0.9248250 1.0703274 1.2075070
#>  [8] 1.1245819 1.0172426 1.0894222

compositionBiasCorrection(w = 6,
                          q = runif(nr, 0, 0.6),
                          u = runif(nr, 0, 0.2),
                          z = runif(nr, 0, 0.2),
                          approx = TRUE)
#> # A tibble: 1 × 5
#>       w     m       v    sig2     mu
#>   <dbl> <dbl>   <dbl>   <dbl>  <dbl>
#> 1     6  1.04 0.00518 0.00480 0.0348