In the Women’s Health Initiative study protocol, participants were required to permanently stop their assigned hormone therapy if they developed the following events: deep vein thrombosis, pulmonary embolus, endometrial hyperplasia with atypia, malignant melanoma, endometrial cancer, breast cancer, triglycerides above 1,000 mg/dL, or starting anticoagulant medications, estrogen, progesterone, testosterone, tamoxifen, or other selective estrogen-receptor modulators. We performed a secondary analysis to estimate the survival curve that would have been observed if all women in the hormone therapy arm had fully complied with this protocol. More specifically, we estimated survival under the dynamic regime “take hormone therapy until one of the above events occurs, then stop taking hormone therapy.” To do so, we artificially censored participants in the hormone arm at the time they deviated from the protocol (i.e., did not stop taking their assigned study hormone after they had one of these events). This artificial censoring may result in selection bias because the distribution of risk factors of breast cancer may differ between the censored and the uncensored.
To adjust for such potential selection bias, one would estimate time-varying, subject-specific inverse probability weights whose denominator is the subject’s estimated probability of remaining uncensored at each time, conditional on past joint predictors of censoring and the outcome. Note, however, that the predictors of censoring at time t are in fact the predictors of hormone therapy continuation at t because those who continue to take their study pills are precisely those who are censored. Therefore there is no need to estimate separate inverse probability weights to adjust for selection bias due to artificial censoring because the treatment weights estimated in the primary analysis already adjust for the potential time-varying selection bias due to artificial censoring. Also note that most protocol-mandated reasons for stopping treatment were not risk factors for breast cancer and thus need not be used to estimate the weights.
The specification of a marginal structural model for dynamic regimes is straightforward when only two regimes are compared23 (link) or when the goal of the structural model is to smooth over a set of regimes that can be placed in 1-to-1 correspondence with an indexing continuous variable.8 ,24 (link) The situation is not so straightforward when, as is required in this example, the dose-response function depends on a summary dose measure (such as cumulative use) that can take the same value for many different regimes. SeeAppendix 2 for further discussion and caveats. For simplicity, we used the same specification as in the primary analysis; specifically we assumed the log discrete hazard is a linear function of cumulative dose.
To adjust for such potential selection bias, one would estimate time-varying, subject-specific inverse probability weights whose denominator is the subject’s estimated probability of remaining uncensored at each time, conditional on past joint predictors of censoring and the outcome. Note, however, that the predictors of censoring at time t are in fact the predictors of hormone therapy continuation at t because those who continue to take their study pills are precisely those who are censored. Therefore there is no need to estimate separate inverse probability weights to adjust for selection bias due to artificial censoring because the treatment weights estimated in the primary analysis already adjust for the potential time-varying selection bias due to artificial censoring. Also note that most protocol-mandated reasons for stopping treatment were not risk factors for breast cancer and thus need not be used to estimate the weights.
The specification of a marginal structural model for dynamic regimes is straightforward when only two regimes are compared23 (link) or when the goal of the structural model is to smooth over a set of regimes that can be placed in 1-to-1 correspondence with an indexing continuous variable.8 ,24 (link) The situation is not so straightforward when, as is required in this example, the dose-response function depends on a summary dose measure (such as cumulative use) that can take the same value for many different regimes. See
