INTRODUCTION

One of the more common ways of estimating causal effects with experimental, as well as observational, data in many disciplines is based on regression methods. Typically an additive linear regression function is specified for the observed outcome as a function of a set of predictor variables. This set of predictor variables includes the indicator variable for the receipt of treatment and usually additional pre-treatment variables. The parameters of the regression equation are estimated by least squares, with the primary focus on the coefficient for the treatment indicator. Inferences, including point estimates, standard errors, tests, and confidence intervals, are based on standard least squares methods. Although popular, the use of these methods in this context is not without controversy, with some researchers arguing that experimental data should be analyzed based on randomization inference. As Freedman writes bluntly, “Experiments should be analyzed as experiments, not as observational studies” (Freedman, 2006, p. 691). It has also been pointed out that the justification for least squares methods does not follow from randomization. Again Freedman: “randomization does not justify the assumptions behind the ols [ordinary least squares] model” (Freedman, 2008a, p. 181). In this chapter we discuss in some detail the rationale for, and the interpretation and implementation of, regression methods in the setting with completely randomized experiments. This chapter can be viewed as providing a bridge between the previous chapter, which was largely focused on exact finite-sample results based on randomization, and the next chapter, which is based on fully parametric models for imputation of the unobserved potential outcomes.

The most important difference between the methods discussed in Chapters 5 and 6 and the ones discussed here is that they rely on different sampling perspectives. Both the Fisher approach discussed in Chapter 5 and the Neyman methods discussed in Chapter 6 view the potential outcomes as fixed and the treatment assignments as the sole source of randomness. In the regression analysis discussed in this chapter, the starting point is an infinite super-population of units.