## Introduction

Time-varying, individual covariates like body mass or fitness present a significant problem in modelling mark–recapture and mark–recapture–recovery data. Such quantities can only be observed when an individual is captured and a large proportion of the values may be unknown, particularly when capture probabilities are low. Moreover, the unknown values are not missing at random–the probability that a value of the covariate is observed may depend on the value itself – and cannot simply be ignored. This makes it necessary to model the distribution of the missing covariate values to construct the full-likelihood function. Evaluating the likelihood then requires computing high-dimensional integrals that can only be estimated numerically which makes classical, maximum likelihood (ML) estimation based on the full-likelihood impractical.

Catchpole, Morgan & Tavecchia (2008) presents a solution to this problem by constructing a reduced, conditional likelihood based only on the events that depend on observed covariate information. Instead of modelling the full capture history for each individual, the likelihood considers only the events that directly follow the releases of each marked individual with three possible outcomes–the individual is captured alive on the next occasion, recovered dead before the next occasion or not observed. The authors termed this the trinomial model and the resulting likelihood, which depends only on the observed values of the covariate, can be constructed without modelling the missing covariate values. The resulting likelihood is simple to evaluate and provides consistent ML estimates of the effect of the covariate on survival (Catchpole, Morgan & Tavecchia 2008), but the model considers only part of the data and is less efficient than methods that model the complete capture histories. Bonner, Morgan & King (2010) used simulation studies to compare estimators of the survival probabilities produced by the trinomial model and a Bayesian mark–recapture–recovery model based on the complete data likelihood implementation of the Cormack–Jolly–Seber developed by Bonner & Schwarz (2006). We found that inferences from the trinomial model were generally less precise but that the trinomial model could provide more accurate estimates of the capture and survival probabilities if the distribution of the covariate imposed by the Bayesian model was far from the truth.

In Bonner, Morgan & King (2010), we noted that the trinomial model can be implemented in the existing software package program mark (White & Burnham 1999) by recasting the observed capture histories into one of the existing data types. This report describes the equivalence between the trinomial model and the existing mark–recapture–recovery model that allows the trinomial model to be implemented in program mark, and a detailed example is included in Appendix S1. In short, the trinomial model is implemented by breaking each capture history into a series of individual events that are separately entered into the mark–recapture–recovery data set with group variables modelling differences over time. I hope that implementing the model in program mark will allow users to fit the trinomial model more easily and to take advantage of program mark's existing features, including its powerful optimization routines and advanced model selection tools. To further this goal, I have created an r package, trimark, which provides functions to assist in fitting the trinomial model both in program mark and through the rmark interface (Laake *et al*. 2012). This package is included in Attachment S1 and updated versions will be available from the author.