  Motivated by the goal of expanding currently existing genotype x environment
interaction (GxE) models to simultaneously include multiple genetic variants
and environmental exposures in a parsimonious way, we developed a novel method
to estimate the parameters in a GxE model, where G is a weighted sum of genetic
variants (genetic score) and E is a weighted sum of environments (environmental
score). The approach uses alternating optimization to estimate the parameters
of the GxE model. This is an iterative process where the genetic score weights,
the environmental score weights, and the main model parameters are estimated in
turn assuming the other parameters to be constant. This technique can be used
to construct relatively complex interaction models that are constrained to a
particular structure, and hence contain fewer parameters.
  We present the model as a two-way interaction longitudinal mixed model, for
which ordinary linear regression is a special case, but it can easily be
extended to be compatible with k-way interaction models and generalized linear
mixed models. The model is implemented in R (LEGIT package) and using SAS
macros (LEGIT_SAS). Here we present examples from the Maternal Adversity,
Vulnerability, and Neurodevelopment (MAVAN) study where we improve
significantly upon already existing models using alternating optimization.
Furthermore, through simulations, we demonstrate the power and validity of this
approach even with small sample sizes.
