# With multiple imputed data, how do you probe a categorical-by-categorical interaction in logistic regression?

I am working with 10 multiple imputed datasets in SAS. I used the command PROC SURVEYLOGISTIC to fit a multivariate model with 6 predictors (3 dichotomous and 3 categorical) and their interaction terms, controlling for a numbers of relevant covariates.

The predictor variables are as follow:

• CVD (0 = No; 1 =Yes)
• HCA (0 = Low; 1= High)
• EDU (1=High School Dropout; 2=Graduated High School; 3=Some College; 4=Graduated College)
• POV (1=Low; 2=Medium; 3=High)
• LAN (1=Spanish; 2=English)
• STA (0=Old, 1=New, 2=None)

My outcome variable is diabetes medication use and is coded as 0 = no use and 1 = current use I found the following:

1. One of the levels of the interaction of EDU*STA is marginally significant.
2. Only one of the levels of the interaction of POV*STA is marginally significant.

My questions are:

1. When working with multiple imputed data, how do you probe a categorical-by-categorical interaction in logistic regression?
2. What would be the appropriate approach to graphically represent the interaction?

If you have followed Rubin's rules for combining the results from the models on the multiply imputed (MI) data sets (which I presume SAS has done for you), there is no difference from standard multiple regression (whether logistic or otherwise) in how to evaluate interaction terms. The variance-covariance matrix of regression coefficients following MI and Rubin's rules takes into account both the within-imputation and across-imputation variances.

Whether MI is involved or not, you need to be very careful in evaluating individual coefficient estimates in interactions involving multiple-level categorical predictors. With the usual coding, individual pairwise interaction coefficients represent differences in outcome (log-odds here) for combinations of 2 predictor levels other than reference levels of those predictors. Thus changing the reference level of one categorical predictor can change both the values and apparent "significance" of interaction coefficients with other predictors (in addition to the coefficients of its own individual levels, which are usually expressed as differences in outcome from the reference). For example, if my recollection is correct that SAS uses the last category as the default reference level, then just doing the same analysis in SAS versus R (which by default uses the first category) could give different coefficients and p-values with multi-level categorical predictors because of the different choice of reference levels.

Unless you have specific pre-specified hypotheses about interaction terms, it's usually safest first to examine whether a group of interaction terms all together are significantly different from 0 before you evaluate any particular interaction terms. That's similar to the advice to make sure an ANOVA has overall significance before proceeding to individual comparisons. If you included all pairwise interactions in your model you have almost 40 pairwise interaction coefficients, so there's a good chance that you will find a couple of them apparently "significant" at p < 0.05 just by chance. You want to rule that problem out first.

For example, the EDU*STA interaction provides 6 separate coefficients. Examine whether the whole set of those coefficients is different from 0. That's sometimes called a "chunk test." Analyzing a set of interaction coefficients all together gets around the problems arising from which particular levels were specified as references. That can be done with a Wald test; I think you can do that with the CONTRAST statement in SAS, specifying all of a set of interaction coefficients = 0 as a combined null hypothesis (although I don't use SAS and might be mistaken). You also could do a likelihood-ratio test between the full model and the model without that particular set of interactions.

If you do find that an overall test on a set of interactions significantly rules out the null hypothesis, then you can proceed to evaluate particular circumstances. For illustration you should display model predictions for particular scenarios involving the interaction terms of interest, as the interaction terms alone can be hard to interpret without the rest of the model. For example, if you find a particular EDU*STA interaction to be interesting, fix all the other covariate/predictor values at reasonable levels and show how particular combinations of EDU and STA values lead to different outcomes than you would have expected based on each individually.