Targeted maximum likelihood estimation for integral projection models in population ecology

In population ecology, integral projection models (IPMs) are widely used to study population growth and the dynamics of population structure (e.g., age and size distributions). These models typically use data on the growth, survival, and reproduction of marked individuals to parameterize models for each demographic rate. The resulting models can be used to predict changes in the population from one time point to the next and to predict long-term properties such as long-term population growth rate, the sensitivity of that growth rate to environmental factors and to changes in demographic rates, and the variation in lifetime outcomes among individuals. These quantities must be inferred from the model, and we often lack any way to ground-truth their plausibility directly from the available data. They nonetheless reflect key ecological processes with significant management and policy implications. Building IPMs requires us to develop sub-models for individual fates over the next time step—Did they survive? How much did they grow or shrink? Did they reproduce?—conditional on their initial state as well as on environmental covariates. This must be done in a manner that gives the most precise possible estimates, and most accurate uncertainty quantification, for the long-term model properties that we are interested in predicting. These models include three core demographic submodels—for growth, survival, and reproduction rates—to describe how individuals change from one time point to the next. Targeted maximum likelihood estimation (TMLE) methods are particularly well suited to a situation in which we are largely interested in estimation and inference on quantities derived from models. These methods build machine learning-based models that estimate the probability distribution from which the empirical data were drawn, and then the user specifies a target of inference as a function of this distribution. An initial estimate for the distribution is then modified by tilting in the direction of the influence function to both de-bias the parameter estimate and provide more accurate inference. In this paper we employ TMLE to develop robust and efficient estimators for quantities derived from a fitted IPM as targets of interest, with a particular focus on long-term stable population growth, its elasticity to fecundity, and the expected growth rate under year-specific covariates. Mathematically, we derive the influence functions for these targets of influence and formulate and use these to update our model so as to remove bias from our estimates. Empirically, we conduct extensive simulations and demonstrate our method’s efficacy using empirical data from a long-term study of plant communities on the Idaho steppe and from experimental rotifer populations.