Estimate
4.3 Example: Estimate Propensity Scores Using the Simulated REFLECTIONS Data
4.3.2 Automatic Logistic Model Selection
4.1 Introduction
This chapter will introduce the basics of the propensity score and focus on the process for estimating the propensity score using real world data. It is organized as follows. First, we will introduce the theoretical properties of the propensity score. Second, we will discuss best practice guidance for estimating the propensity score and provide associated SAS code. This guidance includes the selection of an appropriate statistical model for propensity score estimation, the covariates included in the estimation model, the methods to address missing covariate values, and the assessment of quality of the estimated propensity score. Based on the guidance, propensity score will be estimated for the simulated REFLECTIONS data (described in Chapter 3). The estimated propensity scores will be further used to adjust for confounding in analyzing simulated REFLECTIONS data via matching (Chapter 6), stratification (Chapter 7) and weighting (Chapter 8). Those chapters focus on the scenario of comparing two interventions and we leave the discussion on comparing multiple (>2) interventions using propensity scores to Chapter 10. For simplicity, the term “treatment” refers to the intervention whose causal effect is of research interest and the term “control” indicates the intervention that is compared to the treatment. Note also throughout this book, the terms “treatment” and “cohort” and “interventions” are used interchangeably to denote general groups of patients identified by their treatment selection or other patient factors.
In Chapter 2, we discussed the concept of using randomized experiments to assess causal treatment effects and the difficulties in estimating such effects without randomization. The existence of confounders can bias the causal treatment effect estimates in observational studies. Thus, to analyze observational data for causal treatment effects, the most important methodological challenge is to control bias due to lack of randomization. Cochran (1972) summarized three basic methods – matching, standardization and covariance adjustments via modeling – that attempt to reduce the bias due to confounders (which he termed as “extraneous variables”) in non-randomized settings and these methods set the stage for developing bias control methods in observational studies. Over the past decades, new methods have been proposed to deal with the rising challenge of analyzing more complex observational data, and the propensity score has been the foundation for many of these approaches.
In 1983, Rubin and Rosenbaum proposed the use of the propensity score in analyzing observational data to obtain unbiased causal treatment effect estimates. Since then, bias control methods based on propensity score have become widely accepted. They have been used in many research fields such as economics, epidemiology, health care and the social sciences. To define the propensity score, we introduce the following notation: let
Intuitively, conditioning on the propensity score, each subject has the same chance of receiving treatment. Thus, propensity score is a tool to mimic randomization when randomization is not available.
Like other statistical methods, the validity of the propensity scoring methods is not without assumptions. For causal inference using the propensity score, the following assumptions are necessary:
● Stable Unit Treatment Value Assumption (SUTVA): the potential outcomes (see Chapter 2) for any subject do not vary with the intervention assigned to other subjects, and, for each subject, there are no different forms or versions of each intervention level, which lead to different potential outcomes.
● Positivity: the probability of assignment to either intervention for each subject is strictly between 0 and 1.
● Unconfoundedness: the assignment to treatment for each subject is independent of the p otential outcomes, given a set of pre-intervention covariates.
If these assumptions hold, then the propensity score is a balancing score, which means the treatment assignment is independent of the potential outcome, given the propensity score. Conditioning on the propensity score, the distributions of measured baseline confounders are similar between treatment and control groups. However, unless in a randomized clinical trial, the true propensity score of a subject is unknown. Thus, if the researchers plan to use propensity score to control for bias when estimating causal treatment effects, proper estimation of propensity score is critical. For the remainder of this chapter, we will discuss the key considerations in estimating propensity scores, along with SAS code for implementation.
4.2 Estimate Propensity Score
In this section, we discuss four important issues in estimating propensity scores, (1) selection of covariates in the estimation model; (2) addressing missing covariates values; (3) selection of an appropriate modeling approach; (4) assessment of quality of the estimated propensity score. Keep in mind that the purpose of using a propensity score in observational studies is to create the balance in distributions of the baseline confounders between interventions, so that estimating the causal treatment effect can be similar to randomized clinical trials. A “good” propensity score estimate should always induce balance in baseline confounders between treatment and control groups. In section 4.2.4, we will discuss the standard of “good” propensity scores in a more formal way by introducing statistical approaches for assessing the quality of propensity scores.
4.2.1 Selection of Covariates
As the true propensity score of each subject is unknown in any observational study, in practice, models are always used to estimate the propensity score. The selection of which covariates to include in the estimation models is an important step. First, the unconfoundedness assumption requires all baseline confounders are identified and included appropriately. Thus, failure to include a confounder in the estimation model will most likely result in a biased estimate of the causal treatment effect. However, blindly including every possible covariate into the model might also not be a good strategy. If certain type of covariates, for instance, “colliders” (Pearl, 2000), are included, it might exacerbate the bias of the treatment effect estimate, which is contrary to the purpose of using propensity score. Ding et al. (2017) also found that instrumental variables should not be included in the propensity score estimation