Coherent modeling of longitudinal causal effects on binary outcomes

Analyses of biomedical studies often necessitate modeling longitudinal causal effects. The current focus on personalized medicine and effect heterogeneity makes this task even more challenging. Toward this end, structural nested mean models (SNMMs) are fundamental tools for studying heterogeneous treatment effects in longitudinal studies. However, when outcomes are binary, current methods for estimating multiplicative and additive SNMM parameters suffer from variation dependence between the causal parameters and the noncausal nuisance parameters. This leads to a series of difficulties in interpretation, estimation, and computation. These difficulties have hindered the uptake of SNMMs in biomedical practice, where binary outcomes are very common. We solve the variation dependence problem for the binary multiplicative SNMM via a reparameterization of the noncausal nuisance parameters. Our novel nuisance parameters are variation independent of the causal parameters, and hence allow for coherent modeling of heterogeneous effects from longitudinal studies with binary outcomes. Our parameterization also provides a key building block for flexible doubly robust estimation of the causal parameters. Along the way, we prove that an additive SNMM with binary outcomes does not admit a variation independent parameterization, thereby justifying the restriction to multiplicative SNMMs.