Abstract
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.
Highlights
Biologists and ecologists routinely study quantities that are not di‐ rectly measurable, espe‐ cially in wild populations, and need to use models to calculate estimates of these quantities
We propose a second option, that is, applying a generalized least square (GLS) approach to the estimates produced by the unconstrained multisite model
The two‐step approach presented in this paper works well in the context of CR analyses and has the additional advantage of allowing manipulating the biologi‐ cal quantities themselves rather than the compounded parameters of multinomial logits
Summary
Biologists and ecologists routinely study quantities that are not di‐ rectly measurable (probabilities of survival, capture, etc....), espe‐ cially in wild populations, and need to use models to calculate estimates of these quantities. The same problem will arise every time that the quantities to be constrained are parameters with a multinomial distribution, that is, variables with more than two mo‐ dalities such as the breeding status based on the number of offspring produced or the settlement probabilities in different areas To circumvent this difficulty, we propose a second option (op‐ tion 2, Figure 1), that is, applying a generalized least square (GLS) approach to the estimates produced by the unconstrained multisite model. The one‐to‐one individual transformed estimates (γ) and their VC matrix (V(γ)) can be used to perform a GLS linear re‐ gression between the movement probability estimates and a co‐ variate (e.g., distance between sites A, B, C, and D) R routines, numerical steps, variance–covariance matrices, and results are provided in sup‐ porting information (Appendix S1)
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