Incorporating covariates like age into your model should be based primarily on how they are expected to be related to outcome. For example, if knowledge of the subject matter suggests that age is related to outcome on its own but that its relation to outcome doesn't depend on the other variables, then it should be incorporated on its own without any interactions. If, however, you expect for example that the effect of age on outcome will differ with treatment, then you should include an age-treatment interaction. Unless you think that the relation of age to outcome depends on
Location then you shouldn't be including an interaction of
If you are primarily interested in the effect of
Treatment and you are still concerned about lack of balance in baseline covariate differences, you could consider analysis that incorporates propensity scores. The propensity score relates the probability of treatment-group membership to the combination of covariate values, for example by logistic regression or by tree-based approaches as provided by the R package
twang. Provided that there is overlap in the baseline characteristics for the two treatment groups, then you can get balance by selecting cases that received the 2 treatments matched on their propensities, or you can use the inverses of the propensity scores as weights in your regression. Regressions with mixed models can incorporate weights.
Inverse propensity score weighting can be done along with incorporating age and other covariates into the model proper, to give a doubly robust model.
With respect to specifics of your data set, you say
ages are significantly different by t-test between people in group
Var2=TreatmentA and people in group Var2=TreatmentB, but only when
Var1=LocationX (and not when Var2=LocationY)
and in your last model you propose to adjust for this by including an
age:Location interaction without a main term for
age. There are a couple of issues with this.
First, including an interaction without a main effect for each of the variables is generally not good practice and at best makes interpretation of the coefficients troublesome; see this page for extensive discussion. You could, however, add a main effect of age along with this interaction, and that would be OK if justified by your understanding of the subject matter.
Second, it's not clear that this
age:Location interaction really controls for an important issue at hand. If a linear model is valid (e.g., outcome linearly related to age), a model with that interaction and with age itself would correct for overall differences in outcome due to age, and differences between locations with respect to the influence of age, but it wouldn't correct for any influence of treatment on outcome that depends on age. I suspect that an
age:treatment interaction would be even more of an issue than the
age:Location interaction, particularly if clinicians at one location seem to be preferring one treatment over another based on age. So think carefully about what interactions make sense to include based on your understanding of factors that are contributing to outcome overall, and don't be focused in your model solely on the particular age mismatch between treatments at one location. Again, propensity score methods can be useful in dealing with this type of covariate imbalance to provide a model that can predicts how outcomes would differ between the two treatments if applied to the entire population.
Subsetting typically loses information by leaving informative cases out of the analysis. If a linear model including clinically relevant interactions (e.g., with
Treatment as needed) is well specified and calibrated then a combined model will take advantage of the greater number of cases to provide more precise estimates of the coefficients than will subsetting.