Omitting a moderator I'm wondering what the effect is if I don't include moderators in my model? Is this the same or different from an omitted variable bias? I am having a hard time grasping this conceptually.
More information: I did a GEE analysis for a binary outcome for a publication. The independent variables were the study group and several demographic variables along with a few other participant characteristics. The CI of the odds ratios were quite wide, but I think it is due to the small sample size. One reviewer indicated we should include moderators in the model to decrease the variability of the estimates. I have never heard of this, so I was wondering if this was correct. 
 A: For causal modeling, I think the popular idea is to adjust for precision variables (see Vittinghoff et al "Regression Methods in Biostatistics"). Precision variables are variables that are independent of the exposure of interest but highly prognostic of an outcome. Technically, the actual point estimate shouldn't change much, but the confidence intervals should "narrow" substantially. This is only feasible if you've collected very comprehensive data on the sample and you are able to justify why the variable's role in the multivariate analysis should serve that intended purpose. For this reason, Stephen Senn has advocated using adjustment for prognostic variables in analyses of randomized controlled clinical trial data to improve power and reduce NNT (a very important point!).
Suppose for instance I were looking at the relationship between mammographic breast density and breast cancer risk. Density is a physiologic aspect. Genetic risk factors, such as the presence or absence of the BRCA gene in extreme examples, or family history of breast cancer as a proxy for elevated genetic risk, would serve as a precision variable in such an analysis.
It's very important to ensure that prognostic variables or precision variables are not in the causal pathway (e.g. a direct consequence of the exposure of interest, or a "mediator"). This will eliminate your power to detect an association between the outcome and exposure of interest. Returning to the breast density example, adjusting for prior benign findings or recalled mammographic screens would serve this purpose. Dense breasts are more likely to confuse radiologists and cause them to recall patients for additional screening or even a biopsy. This is one of the ways by which dense breasts are indeed actually associated with greater breast cancer risk, however adjusting for this downstream effect will cause you to infer otherwise.
As a side note, for most categorical analyses (esp logistic regression), the idea that point estimates "do not change" must be relaxed somewhat. This is due to issues of non-collapsibility of the odds ratio. So, while the odds ratio and confidence interval may be compared between models which do and do not adjust for prognostic factors, you may expect small changes in the odds ratio due to this. Contrast that with confounding. I personally never compare adjusted and unadjusted results because I expect point estimates to be substantially different after adjustment for confounding variables, and it's only the latter (confounding adjusted) that I would believe.
