Rank and Redundancy of Multistate Mark-Recapture Models for Seabird Populations with Unobservable States

Unobservable stages are common in many life cycles. Estimates of the vital rates, such as survival and breeding probabilities, of these stages are essential for demographic analysis but difficult to obtain. Explicit modeling of these states in multi-state mark-recapture methods can provide such estimates. However, models can be rank-deficient, meaning that not all parameters can be estimated. Determining whether a model is full rank is essential for interpretation of model selection and estimation results. Full rank models can be obtained by imposing biologically reasonable constraints on parameters. Developing such models requires an efficient way to assess model rank and determine which parameters, if any, are redundant. We introduce the use of automatic differentiation (AD) for this purpose. It generates the Jacobian matrix of the likelihood function in a way that is numerically stable, can accommodate large complicated models, and produces rank estimates accurate to machine precision. It reveals whether a model is full rank or rank-deficient (either intrinsically or for a particular data set), how many parameters or parameter combinations can be estimated, and which parameters are confounded. We use the method to explore three examples relevant to seabirds: a model with multiple breeding sites, a model distinguishing successful and failed breeders, and a model for pre-breeder survival and recruitment. We find a surprisingly large number of time-invariant and time-varying models to be of full rank, thus allowing estimation of all parameters, despite the unobservable states. We present a biological example for the Wandering Albatross (Diomedea exulans). Reliable assessment of model rank for multi-state mark-recapture models with unobservable stages will make it possible to use these methods in demographic applications.