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?
 A: Your one-at-a-time approach for evaluating the alleles via likelihood-ratio (LR) tests runs a risk of omitted-variable bias. If any of your omitted alleles is associated with outcome, then your estimated coefficient for the included allele will tend to have a bias that will affect your ability to interpret the LR tests appropriately.
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.
Be sure that your coding of the alleles doesn't get you into trouble, however. If those are the only 3 possible alleles you might end up with a linear dependence among the predictors. Also, you aren't allowing for interactions among alleles, so you might be missing something important.
