Likelihood ridges and multimodality in population growth rate models.

A central problem in population ecology is to use time series data to estimate the form of density dependence in the per capita growth rate (pgr). This is often accomplished with phenomenological models such as the theta-Ricker or generalized Beverton-Holt. Using the theta-Ricker model as a simple but flexible description of density dependence, we apply theory and simulations to show how multimodality and ridges in the likelihood surface can emerge even in the absence of model misspecification or observation error. The message for model fitting of real data is to consider the likelihood surface in detail, check whether the best-fit model is located on a likelihood ridge and, if so, evaluate predictive differences of biologically plausible models along the ridge. We present a detailed analysis of a focal data set showing how multimodality and ridges emerge in practice for fits of several parametric models, including a state-space model with explicit accommodation of observation error. Best-fit models for these data are biologically dubious beyond the range of the data, and likelihood ratio confidence regions include a wide range of more biologically plausible models. We demonstrate the broad relevance of these findings by presenting analyses of 25 additional data sets spanning a wide range of taxa. The results here are relevant to information-theoretic and Bayesian methods, which also rely on likelihoods. Beyond presentation of best-fit models and confidence regions around individual parameters, effort toward understanding features of the likelihood surface will help ensure the most robust translation from statistical analysis to biological interpretation.