# Maximum Likelihood Estimation of Mark-recapture-recovery models in the Presence of Continuous Covariates

### Roland Langrock and Ruth King

### University of St. Andrews

## Summary

We consider mark-recapture-recovery (MRR) data of animals where the model parameters are a function of individual time-varying continuous covariates. For example, the survival probability of an individual may be a function of condition (with weight used as a proxy for this underlying condition). The relationship between the demographic parameters and covariates is often expressed in the form of a parametric regression. However, for time-varying individual covariates, if an individual is unobserved, the corresponding covariate value is also unobserved. For discrete-valued covariates, the corresponding likelihood can be expressed as a summation over all possible values of the unknown covariates, leading to the Arnason-Schwarz model and the maximum likelihood estimates of the parameters calculated numerically. For continuous-valued covariates, the corresponding likelihood can only be expressed in the form of an integral that is analytically intractable. We consider an approximate likelihood approach which essentially discretises the range of covariate values, reducing the integral to a summation, analogous to the Arnason-Schwarz model, but with a structured transition probability matrix. Assuming a first-order Markov structure for the covariate values, the (approximate) likelihood can be efficiently calculated using standard techniques for hidden Markov models. We initally assess the approach using simulated data before applying to real data relating to Soay sheep.

### Keywords:

Arnason-Schwarz model; Markov chain; maximum likelihood; missing values; numerical integration.