Interpreting results for interaction term continuous continuous and presenting them in tables

Question

I have run a series of analyses aiming to explore the association between a clinical score (it is a scale that goes from 0 to 15) and selected study outcomes. It is an imbalanced individual level panel dataset but not all individuals are present at every time point, therefore, I fitted mixed-effect regression models.

Because the hypothesis is that over time the burden of the clinical condition measured by this score is reducing, I fitted an interaction term continuous##continuous with the time variable. I have also calculated marginal effect for time at each value of the clinical score to investigate in details the interaction. Therefore, I have three variables of interest, time, clinical score, and the interaction term. But I am not considering time as single variable when interpreting the results because I am not interested in the overall effect of time on the outcomes. As for the other two variables I have interpreted them this way:

clinical score: this should be interpreted as the overall association (considering all time points) of the clinical condition and the study outcomes. If I am not mistaken if should be interpreted this way: for each additional point in the clinical score there is an increase of xx in the likelihood of having this outcome (in case of logistic model) over time

interaction term continuous continuous (slope?): I have interpreted it as the effect modification of increasing time on the increasing in the clinical score. in the following example I have clinical score: AOR 1.18 (p<0.001); interaction term AOR 0.98 (p<0.01); time AOR 1.00 (p<0.001). I have interpreted this way: Over time for each 1-point increase in the clinical score there is a 18% increase in the likelihood of the outcome. However, there is a 2% reduction of the effect for each additional year. Am I correct?

I was also trying to find a good example of how to present these results in tables. Online I could only find results for categorical categorical or categorical continuous interaction term. Any suggestion?

Thanks

Answer 1

This sort of interaction is complex to describe as you have found out.

When the clinical score is zero the effect of time is flat (as your adjusted odds ratio is 1)

At time zero the effect of clinical score is to multiply the odds by 1.18 for each one point increase in clinical score.

Anywhere else the effect of a move from time $i$ to time $j$ and a move in clinical score from $l$ to $l$ is found by multiplying out all the coefficients. So it is (j - i) * 1.0 + (l - k) * 1.18 + (j - i) * (k - l) * 0.98. It is easiest to try to explain this if you hold one of them constant of course. Since you are trying to describe a plane which tilts in two directions words are hard to find. You might experiment with plotting a 3-dimensional graph of that plane.

I suggest for publication you present the coefficients as you did in your question. Readers who understand 3-d geometry will understand but you cannot explain 3-d geometry in your paper for the other, that is for a different tutorial article.