This morning, I read a paper by Roland Langrock and Ruth King in a 2013 issue of Annals of Applied Statistics that had gone too far under my desk to be noticed… This problem of using continuous variates in capture-recapture models is a frustrating one as it is not clear what one should do at times the subject and therefore its covariates are not observed. This is why I was quite excited by the [trinomial] paper of Catchpole, Morgan, and Tavecchia when they submitted it to JRSS Series B and I was the editor handling it. In the current paper Langrock and King build a hidden Markov model on the capture history (as in Jérôme Dupui’s main thesis paper, 1995), as well as a discretised Markov chain model on the covariates and a logit connection between those covariates and the probability of capture. (At first, I thought the Markov model was a sheer unconstrained Markov chain on the discretised space and found curious that increasing the number of states had a positive impact on the estimation but, blame my Métro environment!, I had not read the paper carefully.)
“The accuracy of the likelihood approximation increases with increasing m.” (p.1719)
While I acknowledge that something has to be done about the missing covariates, and that this approach may be the best one can expect in such circumstances, I nonetheless disagree with the above notion that increasing the discretisation step m will improve the likelihood approximation, simply because the model on the covariates that was chosen ex nihilo has no reason to fit the real phenomenon, especially since the value of the covariates impact the probability of capture: the individuals are not (likely to get) missing at random, i.e., independently from the covariates. For instance, in a lizard study on which Jérôme Dupuis worked in the early 1990’s, weight and survival were unsurprisingly connected, with a higher mortality during the cold months where food was sparse. Using autoregressive-like models on the covariates is missing the possibility of sudden changes in the covariates that could impact the capture patterns. I do not know whether or not this has been attempted in this area, but connecting the covariates between individuals at a specific time, so that missing covariates can be inferred from observed covariates, possibly with spatial patterns, would also make sense.
In fine, I fear there is a strong and almost damning limitation to the notion of incorporating covariates into capture-recapture models, namely, if a covariate is determinantal in deciding of a capture or non-capture, the non-capture range of the covariate will never be observed and hence cannot be derived from the observed values.