  Standard logistic regression analysis of case-control data has low power to
detect gene-environment interactions, but until recently it was the only method
that could be used on complex polygenic data for which parametric
distributional models are not feasible. Under the assumption of
gene-environment independence in the underlying population, Stalder et al.
(2017, Biometrika, 104, 801-812) developed a retrospective method that treats
both genetic and environmental variables nonparametrically. However, the
mathematical symmetry of genetic and environmental variables is overlooked. We
propose an improvement to the method of Stalder et al. (2017) that increases
the efficiency of the estimates with no additional assumptions and modest
computational cost. This improvement is achieved by treating the genetic and
environmental variables symmetrically to generate two sets of parameter
estimates that are combined to generate a more efficient estimate. We employ a
semiparametric framework to develop the asymptotic theory of the estimator,
show its asymptotic efficiency gain, and evaluate its performance via
simulation studies. The method is illustrated using data from a case-control
study of breast cancer.
