Estimating treatment effects of longitudinal designs using regression models on propensity scores

Abstract

We derive regression estimators that can compare longitudinal treatments using only the longitudinal propensity scores as regressors. These estimators, which assume knowledge of the variables used in the treatment assignment, are important for reducing the large dimension of covariates for two reasons. First, if the regression models on the longitudinal propensity scores are correct, then our estimators share advantages of correctly specified model-based estimators, a benefit not shared by estimators based on weights alone. Second, if the models are incorrect, the misspecification can be more easily limited through model checking than with models based on the full covariates. Thus, our estimators can also be better when used in place of the regression on the full covariates. We use our methods to compare longitudinal treatments for type II diabetes mellitus.

Figure 1

Comparing estimators of the relative risk of hospitalization and difference in average monthly total health care charge in a longitudinal causal framework (Jul–Nov05). RLP MC = regression on longitudinal propensity score with AIC based model checking, RLP SAT = regression on longitudinal propensity score with AIC based model checking with $E(Y^{\mathit{\text{obs}}}|1,e_1^{str},\cdots 1,e_T^{str})$ fitted using only patients with (Z₁, Z₂, Z₃) = (1, 1, 1) and IPW = inverse propensity score weighting. OOO/III = Other-Other-Other versus Insulin-Insulin-Insulin, EEE/III = Exenatide-Exenatide-Exenatide versus Insulin-Insulin-Insulin. The circle represent the point estimates and the bar represents the 95% confidence intervals. The vertical axis is both for point estimates on the left and right of the vertical line in the middle (demarcation line).

Figure 2

Checking the prediction accuracy of the model selected from the second set described in Sec. 5.3. Here, the models for $E(Y^{\mathit{\text{obs}}}|1,e_1^{str},\cdots 1,e_T^{str})$ were fitted using only patients with (Z₁, Z₂, Z₃) = (1, 1, 1). Here pek observed is $\text{pr}(e_k^{str}|1,e_1^{str},\cdots 1,e_{k-1}^{str})$ while pek estimate is $\text{pr}(e_k^{str}|1,e_1^{str},\cdots 1,e_{k-1}^{str};{\hat{\beta }}_k)$. Here, n is the corresponding number of patients and e1 is $e_1^{str}$. We can easily check the models by verifying that most circles are close to the line with a special emphasis on large circles.