# Propensity score weighting in Cox PH analysis and covariate selection

Regarding propensity score weighting (IPTW) when doing Cox proportional hazard modeling of time-to-event survival data:

I have prospective registry data where we're interested in looking at treatment effect of a medication that in most cases the patients were already taking at baseline. I'm therefore not sure how to best analyze the data. Potentially, some of the baseline variables are to a substantial degree influenced by the treatment and not the other way around (e.g. certain biomarkers). I'm a bit lost as to which covariates I should include in the propensity score model for estimating weights and which covariates I should include as covariates in the coxph model (if any at all). Any hints in the right direction would be helpful! I haven't been able to find any literature regarding this in CoxPh modeling as of yet.

I'm thinking that covariates that represent treatments instituted at baseline that (might) influence the outcome should be included as Cox PH covariates, but I'm not sure of this.

How do I determine which variables should be included as covariates in the Cox model instead of being used in calculating the propensity score weights?

Follow-up question:

I understand the inherit problem of evaluating a treatment effect of a certain intervention that has already begun - i.e. is prevalent among the patients, prior to start of observation. Both in regards to introducing bias related to time-variation of the risk (e.g. adverse side effects more common the first year of therapy) and covariates being affected by treatment. If I'm not mistaken - this has been proposed as a cause of discrepancy between observational and randomized that in regards to cardiovascular endpoints and hormon replacement therapy. In my dataset on the other hand, we're interested in looking at an possible adverse effect of the treatment.

If I use propensity score adjustment to investigate the treatment effect among prevalent users, i.e. already using the medication before observation starts, in cohort data and we observe an adverse effect of a pharmacological therapy (and this is what we were looking for). Can I rule out the possibilty of overestimating the risk associated with the treatment? I.e. as long as the risk is significantly elevated, it's most "definitely" not protective?

I can't quite picture an example where this kind of bias can introduce an overestimation of risk of falsy risk association in this context.

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In theory, every variable you select as part of the propensity score weight need not be included as covariates in the model, because the weighting has already controlled for their potential confounding. With a proper weighting model you can, quite literally, just model the effect of the exposure.

That being said, there are reasons you may wish to include terms in the model:

• "Doubly robust" estimates. There is no reason, save for a loss in precision, that you cannot use variables both in the weighting model and as covariates. In theory, you are protecting yourself against confounding two ways (hence this technique being referred to as "doubly robust"). Keep in mind this only protects you against either the PS model or the covariate model being misspecified by giving you a "second chance" to specify the correct model, it isn't a magic fix-all.
• Multiple estimates of interest. Weighting makes the effect estimates from the covariates disappear - if you want a regression coefficient for the variable, you're going to want to include it as a covariate in the CoxPH step and not in the PS model.

Try searching for "Doubly robust" and similar terms in journals like Epidemiology or The American Journal of Epidemiology as well as the biostatistical literature and you should uncover some useful sources.

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Thank you for a (quick) and clear answer! I've seen doubly robust being mentioned, but did look to much into. I definently will now. Would you say that using doubly robust estimates is warranted when covariates are not adequatly adjusted after weighting (e.g.) still significant differences between treatment groups? – Kjetil Loland Jan 11 '13 at 14:23
@KjetilLoland That may be a reason to use doubly robust estimation - generally, it's something to look at whenever you're concerned that one method for controlling for variables is suffering from misspecificication. I'd also check to make sure your PS model isn't acting up and is giving you nice, overlapping propensity scores between the two groups. – Fomite Jan 11 '13 at 23:46
Due to non-collapsibility of the hazard ratio it is not adequate to just include the variables in the PS. It is common to let the PS include the "kitchen sink" and for the known most important predictors to be included again as covariates. This will prevent underestimation of the hazards ratio of the exposure. – Frank Harrell Jan 16 '13 at 17:22
Again, thank you both @EpiGrad and Frank for your answers. I can't exactly say that the treatment groups have nice, overlapping propensity scores. So I would probably end up using extensive covariate adjustment. On a side note, i noticed i wrote IPTW, when I'm in fact is using the twang package - which utilizes generalized boosted regression to estimates weights (if I'm right) - but I guess that doesn't change the general approach much. – Kjetil Loland Jan 17 '13 at 13:44
@KjetilLoland You can at least visually inspect whether or not your PS scores overlap by looking at overlapping plots of their distribution by treatment group. – Fomite Jan 17 '13 at 23:11

It is important to distinguish "affected by treatment" and "related to treatment". The latter can include treatment selection factors such as the ones we are trying to adjust for with propensity and/or covariate adjustment. "Affected by treatment" implies that the covariates are measured after time zero (e.g., after randomization or after treatment start), which means they should seldom be used.

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Again thank you Dr. Harrell. Some of our variables are definitely "affected by treatment". The treatment we're trying to investigate was initiated prior to baseline, which of course is not ideal. Whether or not propensity score adjustment is suitable for this kind of analysis at all is maybe a better question. I'm however not aware of any other way to investigate this. – Kjetil Loland Jan 20 '13 at 20:11
The study design may not be suitable for what you want to do. The study will be very difficult to interpret. You might get subject matter experts to try to come up with a subset of variables that are highly likely to not change with treatment, but adjustment for confounding by implication may be incomplete. – Frank Harrell Jan 22 '13 at 0:07
I see. I guess this starts to look a bit like the old observational vs randomized HRT studies on CVD. Correct me if I'm wrong, but isn't all I "risk" to underestimate the risk of a possible adverse treatment effect (which is what we're looking for) - i.e. as long as we show the treatment to be adverse, those kind of confounders could only weaken the finding? I've updated the question accordingly. – Kjetil Loland Jan 22 '13 at 13:43
This is more than an observational vs. randomized issue but there is a relationship to HRT studies. You might be right that some careful reasoning might justify treating the results as providing a lower bound. – Frank Harrell Jan 22 '13 at 14:09