Interpreting Significant Interaction Term Odds/Hazard Ratio with Binary Variables

Question

I am running a cox proportional hazards regression with Stata and am having trouble interpreting the interaction term between two binary categorical variables.

For simplicity, here's a summary of the output and the problem i'm struggling with:

stcox a##b gives me the following:

Hazard Ratio for A: 1.1 (p <0.01) Hazard Ratio for B: 1.1 (p=0.3) Hazard Ratio for A#B: 1.4 (p<0.01).

Now, my question is the following:

Do I interpret the Hazard Ratio for the interaction term as

-the interaction between A and B provides a hazard ratio of 1.4

or

-the interaction between A and B provides a hazard ratio that is 1.4 times greater than what would be expected by the sum of A and B if there was no true interaction present

The reason why I ask is because when I just include the interaction term without the main effects (stcox a#b), I get a hazard ratio of 1.80.

Thank you so much!

Answer 1

The correct answer (assuming that you are using treatment or dummy coding for each of A and B) is close to your second suggestion, with one modification:

the interaction between A and B provides a hazard ratio that is 1.4 times greater than what would be expected by the combination of A and B if there was no true interaction present

I replaced your "sum" of A and B with "combination" because someone else reading this page might interpret "sum of A and B" to mean the sum of their hazard ratios (HR). That would be wrong: hazard ratios multiply. To avoid confusion when combining coefficients, I work in the original coefficient scale of log-hazards (where sums are correct and standard errors are symmetric), and only exponentiate to get HR at the end.

In the original log-hazard coefficient scale, this is the same as for any interaction term in a regression model.

A few cautions:

First, the title might be taken to imply that "Odds Ratio" and "Hazard Ratio" are the same thing. They aren't. See this page among others for discussion.

Second, you mentioned modeling with only the interaction term and not the individual terms for A and B. That was OK to try to help figure out how to interpret the interaction HR, but that's almost never a good idea for formal modeling.

Third, your result shows why it's not a good idea to throw out an "insignificant" predictor from a model, particularly when there's an interaction. The apparently "insignificant" HR for B (p = 0.3) only means that its coefficient can't be distinguished from 0 (HR distinguished from 1) in the situation where A is at its reference level. Keeping B in the model helps illuminate the association of A with outcome and refines predictions from the model.