Title: Congenial Causal Inference with Binary Structural Nested Mean Models,
Abstract: Structural nested mean models (SNMMs) are among the fundamental tools for
inferring causal effects of time-dependent exposures from longitudinal studies.
With binary outcomes, however, current methods for estimating multiplicative
and additive SNMM parameters suffer from variation dependence between the
causal SNMM parameters and the non-causal nuisance parameters. Estimating
methods for logistic SNMMs do not suffer from this dependence. Unfortunately,
in contrast with the multiplicative and additive models, unbiased estimation of
the causal parameters of a logistic SNMM rely on additional modeling
assumptions even when the treatment probabilities are known. These difficulties
have hindered the uptake of SNMMs in epidemiological practice, where binary
outcomes are common. We solve the variation dependence problem for the binary
multiplicative SNMM by a reparametrization of the non-causal nuisance
parameters. Our novel nuisance parameters are variation independent of the
causal parameters, and hence allows the fitting of a multiplicative SNMM by
unconstrained maximum likelihood. It also allows one to construct true (i.e.
congenial) doubly robust estimators of the causal parameters. Along the way, we
prove that an additive SNMM with binary outcomes does not admit a variation
independent parametrization, thus explaining why we restrict ourselves to the
multiplicative SNMM.