Below are histograms representing estimated propensity scores (on the linear predictor scale from a logistic regression) for units in an observational study, depicted separately for treated and untreated units. These units are a random sample from a population of interest. One of your colleagues approaches you for guidance on how best to use these data to estimate the average treatment effect on the treated (ATT) in this population. Assume throughout the following questions that your colleague will implement the stated approach correctly (e.g. with any necessary adjustments for uncertainty intervals, appropriate prior distributions, etc.). Untreated Units Treated Units q o -4 -2 0 6 -4 -2 0 6 logit(ps) logit(ps) Part 1: First, your colleague says: "I think I'll analyze these data with 1:1 nearest neighbor propensity score matching without replacement, followed by a difference in means in matched treated and control groups. I'll use the logit of the estimated propensity score as the distance metric." Based on the histograms above, what might you say to your colleague about the expected quality of the matches (in terms of covariate balance)? Limit your answer to no more than 2 sentences. Part 2: Next, your colleague says: "On second thought, I'll instead implement 1:1 nearest neighbor propensity score matching with replacement and a caliper of 0.25 (on the logit(ps) scale), followed by a difference in means in matched treated and control groups." You see that this might change the quality of matches and the covariate balance, but what tradeoff should you warn your colleague about? Limit your answer to no more than 2 sentences. Part 3: Then, your colleague says: "Ok, forget about matching. I'll just fit a linear regression for the outcome, but be sure to adjust for the covariates that were used to estimate the propensity score with a main effect term for each. I won't even use the propensity score estimates. I'll use posterior predictions from the regression model to estimate the ATT." What would you warn your colleague about with this approach in light of the propensity score histograms above? Limit your answer to no more than 2 sentences. Part 4: Your colleague responds: "Ok, instead I'll fit a Bayesian Additive Regression Tree specified to include the covariates used to estimate the propensity score. I still won't use the propensity score. I'll use posterior predictions from this model to estimate the ATT." You agree that this is an improvement over the linear regression. Does the use of BART in this way mean that your colleague doesn't have to worry about the issue highlighted by the propensity score histograms? Limit your answer to no more than 2 sentences.