Two-stage Regression for Treatment Effect Estimation.

Abstract

Two-stage regression is a common tool for instrumental variable analysis in applied research. This dissertation introduces additional uses of two-stage models that enable researchers to make inferences that are unavailable in one-stage models.

The first part of the dissertation explores recent methodology that examines whether the predicted response to control affects the magnitude of the treatment effect. The method is a two-stage variation of the Peters-Belson method, studying the interaction between the treatment effect and a prognostic score, i.e. an outcome regression that was fitted to control observations and then extrapolated into the treatment group. The dissertation expands this method into a two-stage regression method termed “Peters-Belson with Prognostic Heterogeneity” (PBPH), which addresses propagation of error from the first stage to the second. A non-standard construction based on stacked estimating equations tests hypotheses about the treatment-prognostic score interaction, deriving confidence intervals by inverting families of tests. These tests combine characteristics of Wald and generalized score tests, improving the small-sample coverage of a similar Wald method and the power of comparable generalized score tests.

Following this, the dissertation enhances the PBPH methodology, addressing complications that the applied researcher may encounter. The method is adapted to accommodate generalized linear models (as opposed to linear models only) at the first stage. Further adaptations accommodate designs assigning study subjects to treatment conditions by cluster, such as cluster randomized trials. A simulation study clarifies sample size requirements’ dependency on the complexity of the first stage model, culminating in a theoretically motivated rule of thumb for the maximum number of first stage regressors as a function of n.

The final chapter examines treatment effect estimation with a binary response. Simulations reveal that using two-stage regression models sheds light on whether a treatment effect is linear on the logit scale (logistic regression) or linear on the probability scale (linear regression). Often, matching is performed on observational data to aid in treatment effect estimation. Conditional logistic regression is a typical approach to matched data with binary response; however, we show that two-stage regression in this setting offers benefits not available to a one-stage conditional logistic model.