Solved – How to interpret the odds ratio of an interaction term in Conditional Logistic Regression

interactioninterpretationlogisticodds-ratioregression

I am wondering what the correct interpretation of the odds ratio of an interaction term in conditional logistic regression is.

Let's say there are two independent variables A and B, as well as an interaction term (AxB).
Coefficient of A=𝛽1, Coefficient of B=𝛽2 and Coefficient of (AxB)=𝛽3.

The odds ratio for the independent variable A would be exp(𝛽1).
The odds ratio for the independent variable B would be exp(𝛽2).

Questions:

  1. Is it correct to say that the odds ratio of (AxB) is exp(𝛽3)?

  2. If exp(𝛽3) is 1.5, would it be correct to interpret the odds ratio of (AxB) as: an increase in the interaction term (AxB) by one unit of measure increases the odds of "success" by a factor of 1.5?

  3. Or is the interpretation in question two wrong and it would be correct to say the following: A possible alternative interpretation would be that the odds ratio of the interaction term (AxB) is exp(𝛽1+𝛽3B). This could be interpreted as: if A increases by one unit, then the odds increase by a factor of exp(𝛽1+𝛽3B), which would be dependent on values of B.

Thank you.

Best Answer

None of those interpretations are quite right. I think you have to connect a few concepts first. (Numbering ideas here that don't really relate to your own numbers there).

  1. Conditional logistic regression only differs from "ordinary" logistic regression in that the analysis is based on matches sets, so in interpreting the effects you must state what you are controlling for, or the matching in some regard. For instance, if this were a twin's analysis, you would say something like "Smoking was associated with a 2-fold difference in the odds of psychiatric disorder among twins".

  2. The (exponentiated) coefficient for an interaction (or product) term in a logistic regression is not an odds ratio, it is a ratio of odds ratios or an odds ratio ratio (ORR). The point is that you never observe a "difference" or "increase" in the product term without a difference in the lower level terms... so the standard interpretation doesn't apply.

  3. In a logistic regression model, the interpretation of an (exponentiated) coefficient term for an interaction (say between X and W) is like the following. "For a unit difference in W, the ratio of odds ratio of Y and X is $\exp(\gamma)$".

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