Hidden Markov Model for Dependent Mark Loss and Survival Estimation

Abstract

Mark-recapture estimators assume no loss of marks to provide unbiased estimates of population parameters. We describe a hidden Markov model (HMM) framework that integrates a mark loss model with a Cormack–Jolly–Seber model for survival estimation. Mark loss can be estimated with single-marked animals as long as a sub-sample of animals has a permanent mark. Double-marking provides an estimate of mark loss assuming independence but dependence can be modeled with a permanently marked sub-sample. We use a log-linear approach to include covariates for mark loss and dependence which is more flexible than existing published methods for integrated models. The HMM approach is demonstrated with a dataset of black bears (Ursus americanus) with two ear tags and a subset of which were permanently marked with tattoos. The data were analyzed with and without the tattoo. Dropping the tattoos resulted in estimates of survival that were reduced by 0.005–0.035 due to tag loss dependence that could not be modeled. We also analyzed the data with and without the tattoo using a single tag. By not using.

Supplementary materials accompanying this paper appear on-line.

Appendix

The probability structure for mark loss is equivalent to capture-recapture (mark-recapture) for two occasions with a closed population, which has been used with two observers to measure detection probability in visual surveys. When detection probability is measured solely with the mark-recapture data, it is necessary to assume independence between the detections by the two observers because those missed by both observers ($n_{00})$ are obviously not included in the sample (Borchers 1996). Recently, the independence assumption was weakened (Laake 1999; Laake and Borchers 2004; Borchers et al. 2006) in the combined mark-recapture and distance sampling by including a dependence measure $\delta (x)$ which was estimated as the discrepancy between the detection probability at distance $x$ measured by the mark-recapture (double observer) data (based on independence) and the distance sampling data. If $\delta (x)$ = 1 then independence at all distances is achieved. Because detection probability at $x$ = 0 cannot be measured from the distance sampling data, the independence assumption for the mark-recapture data was required for $x$ = 0 ($\delta$(0) = 1) but not for the other distances.

The dependence structure we have defined for mark loss can be expressed in terms of the $\delta$ dependence of Borchers et al. (2006). Under the independence model, the probability that an animal would retain at least one mark is:

$$\begin{aligned} 1-\tau _{00}^{*}=\frac{1+e^{\beta _{1}}+e^{\beta _{2}}}{1+e^{\beta _{1}} +e^{\beta _{2}}+e^{\beta _{1}+\beta _{2}}} \end{aligned}$$

Likewise for the dependence model:

$$\begin{aligned} 1-\tau _{00}=\frac{1+e^{\beta _{1}}+e^{\beta _{2}}}{1+e^{\beta _{1}} +e^{\beta _{2}}+e^{\beta _{1}+\beta _{2}+\beta _{3}}} \end{aligned}$$

The dependence measure of Borchers et al. (2006) is a ratio that measures the distortion between the joint probabilities from the independence model ($\beta _{3}=0$) and the dependence model ($\beta _{3}\ne 0$) which can be expressed as:

$$\begin{aligned} \delta =\frac{1-\tau _{00}^{*}}{1-\tau _{00}}=\frac{K}{K^{*}}=1 +\frac{e^{\beta _{1}+\beta _{2}}(e^{\beta _{3}}-1)}{1+e^{\beta _{1} +\beta _{2}}+e^{\beta _{1}}+e^{\beta _{2}}}=1+(e^{\beta _{3}}-1) \tau _{00}^{*} \end{aligned}$$

The same relationship can be obtained using conditional and marginal probabilities. Defining $\tau _{i}^{(s_{3-i})}$ to be the conditional probability that the $i$th mark is lost given the other mark status is $s_{3-i}$:

$$\begin{aligned} \tau _{i}^{(s_{3-i})}=Pr\left( S_{i}=0\vert S_{3-i}=s_{3-i}\right) =\frac{e^{\beta _{i}+\beta _{3}(1-s_{3-i})}}{1+e^{\beta _{i} +\beta _{3}(1-s_{3-i})}}. \end{aligned}$$

If $q_{i}^{(s_{3-i})}=1-\tau _{i}^{(s_{3-i})}$and $q_{i}=1-\tau _{i}$ is the marginal mark retention rate, then $\delta =q_{i}^{(1)}/q_{i}=\frac{K}{K^{*}}$. Likewise the same ratio for any of the joint probabilities based on independence and dependence other than for the (0,0) event which is not used in the independence model. The dependence measure can also be expressed in terms of covariance (Borchers 1996):

$$\begin{aligned} \delta =1+\frac{\mathrm {cov}(S_{1},S_{2})}{q_{1}q_{2}}=1 +\frac{\frac{1}{K}-q_{1}q_{2}}{q_{1}q_{2}}= \frac{\frac{1}{K}}{\frac{K^{*}}{K^{2}}} =1+(e^{\beta _{3}}-1)\tau _{00}^{*} \end{aligned}$$

In general there will likely be positive dependence in mark loss which means $\beta _{3}> 0$ and $\delta > 1$ but negative dependence $(\beta _{3} < 0)$ is possible with a lower bound of $\delta > 1-\tau _{00}^{*}$.

The joint probabilities can be rewritten in terms of $\delta$ as: $\tau _{11}=\delta q_{1}q_{2}, \, \tau _{10}=q_{1}(1-\delta q_{2})$ and $\tau _{01}=q_{2}\left( 1-\delta q_{1}\right)$ or as $\tau _{11}=q_{1}^{(1)}q_{2}^{(1)}/\delta ,\, \tau _{10}=q_{1}^{(1)}\tau _{2}^{(1)}/\delta$ and $\tau _{01}=q_{2}^{(1)}\tau _{1}^{(1)}/\delta$. The latter form makes it obvious that once you exclude $n_{00}$ and condition on the observed data, the $\delta$ will cancel from the rescaled joint probabilities which will only be functions of the $q_{i}^{1}$. This is also obvious by noting that the joint probabilities for the observed set of data ($n_{11}n_{10}n_{01})$ would only be functions of $\beta _{i}$ after conditioning on the exclusion of $n_{00}$. The same result was shown by Borchers et al. (2006) for mark-recapture distance sampling but in that case $\delta$ could be estimated from the observed distances.