I have a methodological question, and therefore no sample dataset is attached.
I'm planning to do a propensity score adjusted Cox regression that aims to examine whether a certain drug will reduce the risk of an outcome. The study is observational, comprising 10,000 individuals.
The data set contains 60 variables. I judge that 25 of these might affect treatment allocation. I would never adjust for all 25 of these in a Cox regression, but I've heard that you can include that many variables as predictors in a propensity score and then only include the propensity score subclass and treatment variable in the Cox regression.
(covariates that will not be equal after prop score adjustment would of course have to be included in the Cox regression).
Bottom line, is it really smart to include that many predictors in the prop score?
@Dimitriy V. Masterov Thank you for sharing these important facts. On the contrary to books and articles considering other regression frameworks, I don't see any (reading Rosenbaums book) guidelines on model selection in propensity score analyses. While standard textbooks / review articles seem to always recommend stringent variable selection and keeping the number of predictors low, I haven't seen much of this discussion in prop score analyses. You write: (1) "Theoretical insight, institutional knowledge, and good research should guide selection of Xs". I agree but there are circumstances where we have a variable at hand and don't really know (but it might be possible) if the variable effects either treatment allocation or outcome. For example: should I include kidney function, as measure by filtration rate, in a prop score aiming to adjust for statin treatment. Statin treatment has nothing to do with kidney function and I have already included an array of variables that will effect statin treatment. But it is still tempting to include kidney function; it might adjust even more. Now some would say that it should be included because it effects outcome, but I could give you another example (such as the binary variable urban / rural living) of a variable that don't effect treatment nor outcome, as far as we know. But I would like to include it, as long as it don't effect the prop score precision. (2) "Including Xs affected by the treatment, either ex post or ex ante in anticipation of treatment, will invalidate the assumption". I'm not sure what you mean here. But if I study the effect of statins on cardiovascular outcome, I will include various measurements of blood lipids in the propensity score. Blood lipids are effected by the treatment. I guess I misunderstood this statement.
@statsRus thank you for sharing the facts, particularly what you call "a note on selecting inputs". I think I reasons much the same way you do.
Unfortunately prop score methods discuss various adjustment strategies instead of model selection strategies. Perhaps model fit is not important. If that is the case, I would adjust for every available variable that might effect outcome and treatment allocation the slightest. I am not a statician, but if model fit is of no importance then I would like to adjust for all variables that might affect treatment allocation and outcome. This would in many cases mean including variables that will be effected by the treatment.
Furthermore, some people suggest that the subsequent Cox regression should only include the treatment variable and prop score subclass. While others suggest that the cox adjustment should include the prop score additionally to all other variables that you would adjust for.