[book] Analysis of Observational Health Care Data Using SAS#

Chapter 2-3 Propensity Score Stratification and Regression / Propensity Score Matching for Estimating Treatment Effects#

1. Estimating propensity score (PS)#

  • Logistic regression

  • Probit regression: \(P(Y=1|X) = \Phi(X^T\beta)\), where \(\Phi(\cdot)\) is the CDF of standard normal distribution

  • discriminant analysis

  • CART

  • Neural nets

  • GAM(?)

2. Using PS#

  • Stratification

  • Matching: with caliper (0.2 SD of logit of PS [1])

  • Regression covariate

  • IPW (Chapter 5): \(\frac{1}{p_i}\) for treatment arm, \(\frac{1}{1-p_i}\) for control arm, where \(p_i\) is the PS for subject \(i\)

3. Assess the balance after applying PS method#

  • standardized difference of each covariate

  • PS distribution

  • distribution of the covariates

  • goodness of fit statistics

4. Estimating treatment effect#

  • treatment effect estimate

  • S.E. of treatment effect

5. Sensitivity analysis for PS#

Chapter 4 Doubly Robust Estimation of Treatment Effects#

Conceptual overview#

  • Doubly robust (DR) estimation builds on the propensity score approach of Rosenbaum and Rubin (1983) and the inverse probability of weighting (IPW) approach of Robins and colleagues (Robins, 1998; Robins, 1999a; Robins, 1999b; Robins et al., 2000).

  • DR estimation combines inverse probability weighting by a propensity score with regression modeling of the relationship between covariates and outcome for each treatment.

    • It combines it in such a way that, as long as either the propensity score model or the outcome regression models are correctly specified, the effect of the exposure on the outcome will be correctly estimated, assuming that there are no unmeasured confounders (Robins et al., 1994; Robins, 2000; van der Laan and Robins, 2003; Bang and Robins, 2005).

  • Procedure

    • one builds and fits a (binary) regression model for the probability that a particular patient received a given treatment as a function of that individual’s covariates (the propensity score).

    • Maximum likelihood regression is conducted separately within the exposed and unexposed populations to predict the mean response (outcome) as a function of confounders and risk factors.

    • each individual observation is given a weight equal to the inverse of the probability of the treatment he/she received based on baseline covariates (as in IPW analysis) to create two pseudopopulations of subjects that represent the expected response in the entire population under those two treatment conditions.

  • one trades a possible loss of precision in using the DR estimator for this additional protection.

Statistical model#

  • The estimator of the causal effect

    (1)#\[\begin{align} \hat{\Delta}_{DR} \\ & = n^{-1}\sum_{i=1}^n\left[\frac{Z_iY_i}{e(X_i, \hat{\beta})} - \frac{Z_i-e(X_i, \hat{\beta})}{e(X_i, \hat{\beta})}m_1(X_i, \hat{\alpha}_1)\right] \\ & - n^{-1}\sum_{i=1}^n\left[\frac{(1-Z_i)Y_i}{1-e(X_i, \hat{\beta})} - \frac{Z_i-e(X_i, \hat{\beta})}{1-e(X_i, \hat{\beta})}m_0(X_i, \hat{\alpha}_0)\right] \\ & = \hat{\mu_{1, Dr}} - \hat{\mu}_{0, DR} \end{align}\]

  • The standard error for the DR estimator is estimated using the delta method

    \[ SE_{\hat{\Delta}_{DR}} = \sqrt{n^{-2}\sum_{i=1}^n \hat{I}^2_i}, \]

    where \(\hat{I}_i\) is defined as

    (2)#\[\begin{align} \hat{I}_i \\ & = \left[\frac{Z_iY_i}{e(X_i, \hat{\beta})} - \frac{Z_i-e(X_i, \hat{\beta})}{e(X_i, \hat{\beta})}m_1(X_i, \hat{\alpha}_1)\right]\\ & - \left[\frac{(1-Z_i)Y_i}{1-e(X_i, \hat{\beta})} - \frac{Z_i-e(X_i, \hat{\beta})}{1-e(X_i, \hat{\beta})}m_0(X_i, \hat{\alpha}_0)\right] - \hat{\Delta}_{DR} \end{align}\]