Bounded Continuous Outcomes

Abstract

This chapter presents a TMLE of the additive treatment effect on a bounded continuous outcome. A TMLE is based on a choice of loss function and a corresponding parametric submodel through an initial estimator, chosen so that the loss-functionspecific score of this parametric submodel at zero fluctuation equals or spans the efficient influence curve of the target parameter. Two such TMLEs are considered: one based on the squared error loss function with a linear regression model, and one based on a quasi-log-likelihood loss function with a logistic regression submodel. The problem with the first TMLE is highlighted: the linear regression model is not a submodel and thus does not respect global constraints implied by the statistical model. It is theoretically and practically demonstrated that the TMLE with the logistic regression submodel is more robust than a TMLE based on least squares linear regression. Some parts of this chapter assume familiarity with the core concepts, as presented in Chap. 5. The less theoretically trained reader should aim to navigate through these parts and focus on the practical implementation and importance of the presented TMLE procedure. This chapter is adapted from Gruber and van der Laan (2010b).