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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?

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  • $\begingroup$ 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. $\endgroup$ Commented Sep 5, 2014 at 21:00

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"When we fit the model, if coefficenta (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?"

You need to look at the coefficient of the treatment indicator to decide whether the treatment effect is significant or not. Remember that propensity score is a scalar summary of a number of covariates. The coefficient for the propensity scores shows whether there are significant differences between your treatment and control groups regarding one or more of these covariates. By adding the propensity scores you "control" for these differences to obtain the net treatment effect which is given by the coefficient b in your equation.

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