Fortunes and Misadventures With Parametric Models: They Can Be Confounding, Burdensome and Unstable, Yet Insightful, Powerful and Flexible

Abstract

Statistical models allow us to represent latent structure in data, giving us the ability to wield the power of the unobserved. At the same time, statistical models can confound and trouble us in at least three ways. First, the correspondence between model parameters and physical quantities of interest is not always clear. In the pursuit of causal inference of zero-inflated outcomes, for example, we can employ zero-inflated generalized linear models to harness covariate information and gain precision. However, matching model parameters to causal quantities of interest is not as straight-forward as one would think, especially when involving covariates. I use analysis and simulation to investigate the appropriate use of models here. Secondly, while complex models can capture equally complex structure in data, fitting these models can be a burdensome task. For example, to describe the dependence between time-varying covariates and time-varying outcomes, we might employ latent variable models. The description of these models, however, involve a large number of parameters. I enlist a marginalization strategy that induces a two-stage procedure which greatly simplifies model fitting. Finally, we are not guaranteed to learn about all model parameters when combining models with data. Indeed, it is possible that some parameters are not informed by the data directly at all, but only indirectly through their relationship with other model parameters. I develop strategies for understanding this as the flow of information from data to model parameters using unidentifiable and orthogonal parameters as building blocks. In this thesis, I describe these situations to highlight the difficulties of using parametric models to gain scientific knowledge from data.