I have read many articles, but they are hard to understand. Please explain regression adjustment for propensity scores in a mathematical way. My understanding of regression adjustment is to include propensity score as an independent variable into the model (where Y is outcome variable):

Y = intercept + coefficent$_a\cdot$Propensity + coefficient$_b\cdot$treatment

When we fit the model, if coefficent$_a$ (which is the coefficient of propensity score term in the equation) has a value greater than 0.05, then we can say treatment effect is not significant. Am I correct?

However, some articles said treat effect = $Y_t-Y_c-B(X_t-X_c)$. What does that mean? I thought treatment effect is equal to $Y_t-Y_c$, what does the $B(X_t-X_c)$ do here? Does $X_t$ include all the covariances and treatment term, or only covariance terms?

• The size and direction of the coefficient on the propensity score is irrelevant for learning about the average treatment effect. That effect (if you have the true propensity scores) is what coefficient b estimates. Commented Sep 5, 2014 at 21:00