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I am using GENESIS R package to fit GLMM models to analyze genome-wide associations of SNPs with binary traits in a dataset with familial design. Running the "fitNullModel" and "assocTestSingle" functions (with family = "binomial" and test="Score") the results return Score/Score.SE for each SNP and, in addition, contain Est/Est.SE which are (according to user manual) approximations of effect sizes and their standard errors.

I appreciate if you can clarify if these Est/Est.SE approximations are based on a Wald's test? and if the family is kept as binomial or internally changed to gaussian to decrease the computational burden when Est/Est.SE are computed?

Also, is my understanding correct that when a GXE term is included, the effect sizes (Est.G, Est.G:env) are based on a Wald’s test and a binomial family?

Est is not recalculated, rather it is approximated based on the score test results, as described in this paper. Est = Score/(Score.SE^2) and Est.SE = 1/Score.SE. When a GxE term is included, the effect sizes are similarly approximated based on a multi-parameter score test.