Causal Logistic Models for Non-compliance Under Randomized Treatment with Univariate Binary Response

We propose a method for estimating the marginal causal log-odds ratio for binary outcomes under treatment non-compliance in placebo-randomized trials. This estimation method is a marginal alternative to the causal logistic approach by Nagelkerke et al. (2000) that conditions on partially unknown compliance (that is, adherence to treatment) status, and also differs from previous approaches that estimate risk differences or ratios in subgroups defined by compliance status. The marginal causal method proposed in this paper is based on an extension of Robins' G-estimation approach for fitting linear or log-linear structural nested models to a logistic model. Comparing the marginal and conditional causal log-odds ratio estimates provides a way of assessing the magnitude of unmeasured confounding of the treatment effect due to treatment non-adherence. More specifically, we show through simulations that under weak confounding, the conditional and marginal procedures yield similar estimates, whereas under stronger confounding, they behave differently in terms of bias and confidence interval coverage. The parametric structures that represent such confounding are not identifiable. Hence, the proof of consistency of causal estimators and corresponding simulations are based on two different models that fully identify the causal effects being estimated. These models differ in the way that compliance is related to potential outcomes, and thus differ in the way that the causal effect is identified. The simulations also show that the proposed marginal causal estimation approach performs well in terms of bias under the different levels of confounding due to non-adherence and under different causal logistic models. We also provide results from the analyses of two data sets further showing how a comparison of the marginal and conditional estimators can help evaluate the magnitude of confounding due to non-adherence.