compositionBiasCorrection: Calculate bias correction term for calf:cow composition...

In LandSciTech/caribouMetrics: Models and Metrics of Boreal Caribou Demography and Habitat Selection

Calculate bias correction term for calf:cow composition survey.

Description

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.

Usage

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: caribouBayesianIPM(), 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)

compositionBiasCorrection(w = 6,
                          q = runif(nr, 0, 0.6),
                          u = runif(nr, 0, 0.2),
                          z = runif(nr, 0, 0.2),
                          approx = TRUE)

LandSciTech/caribouMetrics documentation built on Feb. 3, 2024, 9:41 p.m.