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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.

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    $\begingroup$ I usually interpret the effect of variables and their interactions in a different way when the variables are categorical than when the variables are continuous. So this is not easy. Tentatively, I would say no to Q1, yes to Q2 (if it made sense for the variables in question) and yes to Q3 (if it made sense for the variables in question). See a related question. $\endgroup$ – Ertxiem Mar 24 at 23:33
  • $\begingroup$ I am looking at an interaction effect with continuous variables. You mentioned that you would interpret them differently - could you elaborate? $\endgroup$ – Statphil Mar 25 at 20:57
  • $\begingroup$ If you are interested in interaction effects in non-linear (logit/probit) models, then I would strongly advise you to read the excellent Ai & Norton paper (sciencedirect.com/science/article/pii/S0165176503000326) - They basically explain how tricky interaction effects in non-linear models can be (There is also a follow-up paper from Greene (people.stern.nyu.edu/wgreene/Lugano2013/…) who proposed a visual method to investigate such effects. $\endgroup$ – Umka Mar 25 at 21:04
  • $\begingroup$ I've added below an answer where I elaborate more on what I said and where I corrected the terminology I used. $\endgroup$ – Ertxiem Mar 25 at 21:54
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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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I will expand what I wrote on my comment and I'll correct my terminology, following AdamO.

Regarding the first question, I would answer no for two reasons: a) if $A$ or $B$ are continuous, you have to look in terms of increase in each variable if you want to talk about ratio of odds ratios; b) if $A$ and $B$ are categorical, it would only make sense if you have multiple categories (at least 4 if $A$ and $B$ are ate least dichotomous) and the ratio of odds ratio have to be interpreted in terms of the reference categories (I will talk about the categorical variables later).

With respect to the second question, if both $A$ and $B$ are continuous (as your comment indicates), I agree with your interpretation:

an increase in the interaction term ($A \times B$) by one unit of measure increases the odds ratio of "success" by a factor of $1.5$.

The third question says more or less the same thing as the second one but with a reorganization of the terms. Since $A$ and $B$ are continuous I agree with you on:

if $A$ increases by one unit, then the odds ratio increase by a factor of $\exp(𝛽_1+𝛽_3 B)$.

Regarding on the way I interpret the interaction terms, I usually begin by looking at the isolated terms to understand the effects of each variable, then I look at the interaction terms.

If both variables are continuous, I would look at sign of $𝛽_1+𝛽_3 B$ for low $B$ and for high $B$. If the signal changes it means that for low $B$ the effect of $A$ is in one direction, while for high $B$ the effect is on the opposite direction. If the sign does not change, the value of $B$ that makes $𝛽_1+𝛽_3 B$ closer to zero would mean that for that value of $B$ the effect of $A$ would be less visible, while for the value of $B$ on the "other side" of its mean, the effect of $A$ is "stronger".

If one or both variables are categorical, the interaction term has to be interpreted having in consideration the reference category of the variables, which usually makes our life hard because we will have to take in consideration the degrees of freedom of the interaction term. In general terms, if both are categorical, we would say something like the effect in the category $a_j$ of $A$ is smaller or larger among the individuals that belong to category $b_k$ of $B$.

See also other questions on interaction terms in conditional logistic regression and in logistic regression and a related question.

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  • $\begingroup$ This was very helpful. In order to interpret the interaction effect 𝛽1+𝛽3𝐵, should both coefficients 𝛽1 and 𝛽3 be statistically significant? What would be the implication for the interpretation if e.g. 𝛽1 is insignificant? $\endgroup$ – Statphil Mar 29 at 22:47
  • $\begingroup$ In that case, $A$ alone would not have a significant effect, but the effect of $B$ would change with the presence of $A$. Note that in this case sometimes it easier to think instead the effect of $B$ looking at $\beta_2 + \beta_3 A$. $\endgroup$ – Ertxiem Mar 30 at 1:49

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