  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. Towards 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 non-causal 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
reparametrization of the non-causal 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 parametrization 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 parametrization, thereby justifying the restriction to
multiplicative SNMMs.
