# Better to use wald test or likelihood ratio test to make pairwise comparisons after omnibus test in this scenario

I am testing the association between a gene and a binary disease. The gene has many different "versions". These versions are called alleles.I am also including covariates for sex, age, etc.

Right now I am doing a logistic regression-based omnibus test for the gene like this (pseudocode):

full<- "disease ~ sex + age + allele1 + allele2 + allele3"
null<- "disease ~ sex + age"
anova(null, full, test='Chisq')


I think I realize I could follow omnibus with the wald test to determine how significant each allele is, but I am wondering if this is best done with the LR test, which would allow me to account for the covariates on each allele comparisons, like this:

full1<- "disease ~ sex + age + allele1"
null<- "disease ~ sex + age"
anova(null, full1, test='Chisq')

full2<- "disease ~ sex + age + allele2"
null<- "disease ~ sex + age"
anova(null, full2, test='Chisq')


, etc.

My gut feeling is that the LR approach would be better for by allele comparisons, because of the inclusion of covariates. Is this the case?

The coefficients returned by commands (in R) like summary(full) for a model fit by maximum likelihood (e.g., logistic regression) represent Wald tests for each predictor coefficient based on the full model with covariates. The variances for the coefficient estimates are based on the full variance-covariance matrix calculated at the full model solution. So there's no problem doing Wald tests in a way that incorporates information about the covariates.