I have been studying equation 17 in this paper. This is their recommended estimator of treatment effect in a randomized trial. I just noticed something that I think is by no means profound but still sort of neat. I wonder if anyone else has noticed this.
They advocate fitting separate outcome models on each arm of a trial, applying both models to the full dataset, and then using a function of the unconditional estimator of treatment effect and an odd looking function of the two sets of predictions. Based on earlier work by Tsiatis and allies, this estimator is optimal in a well-defined sense among a broad and useful class of estimators. Let R1bar be the average residual on just the treatment sample from the outcome model fit on that sample. Similarly, let R0bar be the average residual on just the control sample from the outcome model fit on that sample. Let Pred1bar be the average prediction across the entire sample of the predictions formed using the outcome model fit on just the treatment sample. Similarly, let Pred2bar be the average prediction across the entire sample of the predictions formed using the outcome model fit on just the treatment sample. Then their recommended treatment effect is Betahat= (R1bar-R0bar)+(Pred1bar-Pred0bar). (17) If it happens (as it almost always does) that R1bar=R0bar=0, then this simplifies to just Pred1bar-Pred0bar. This is a very simple and intuitive description of the estimator. It also suggests using imputation to form these predictions, and Rubin's formula for combining multiple imputations to estimate its variance. Steff van Buuren just presented a paper with this theme at JSM in Boston. Andrea Piesse presented a very similar paper back in 2010 at JSM in Vancouver. I did not make the connection to Tsiatis et al when Andrea and I were working on this back in 2010, but this suggests that the multiple imputation procedure for analysis of RCTs is the optimal approach. --Dave https://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=2&cad=rja&uact=8&ved=0CCgQFjAB&url=https%3A%2F%2Fwww.amstat.org%2Fsections%2Fsrms%2Fproceedings%2Fy2010%2FFiles%2F306552_56705.pdf&ei=kwDtU_j6L8PesAScyYHQAQ&usg=AFQjCNGQoUCVYG8vYP4mrsligw_koskW7A&sig2=uZuf0GolkvfLhgJRj6izKA David Judkins | Principal Scientist | Abt Associates 4550 Montgomery Ave. Suite 800 North Bethesda, MD 20814-3343 O: 301.347.5952 | Fax.301.828.9672 | [email protected]<mailto:[email protected]> ________________________________ This message may contain privileged and confidential information intended solely for the addressee. Please do not read, disseminate or copy it unless you are the intended recipient. If this message has been received in error, we kindly ask that you notify the sender immediately by return email and delete all copies of the message from your system.
