Covariate and multinomial: Accounting for distance in movement in capture-recapture analyses

Many biological quantities cannot be measured directly but rather need to be estimated from models. Estimates from models are statistical objects with variance and, when derived simultaneously, covariance. It is well known that their variance-covariance (VC) matrix must be considered in subsequent analyses. Although it is always preferable to carry out the proposed analyses on the raw data themselves, a two-step approach cannot always be avoided. This situation arises when the parameters of a multinomial must be regressed against a covariate. The Delta method is an appropriate and frequently recommended way of deriving variance approximations of transformed and correlated variables. Implementing the Delta method is not trivial, and there is a lack of a detailed information on the procedure in the literature for complex situations such as those involved in constraining the parameters of a multinomial distribution. This paper proposes a how-to guide for calculating the correct VC matrices of dependant estimates involved in multinomial distributions and how to use them for testing the effects of covariates in post hoc analyses when the integration of these analyses directly into a model is not possible. For illustrative purpose, we focus on variables calculated in capture-recapture models, but the same procedure can be applied to all analyses dealing with correlated estimates with multinomial distribution and their variances and covariances.