How to define interaction between 2 categorical covariates and Time and Intervention in a longitudinal model? (in R)

Question

Dears,

Let's assume, that I have a study like this:

1. longitudinal, 3 time points, T1, T2, T3. Let's assume T1 is post-intervention.
2. 2 interventions, A and B
3. 2 categorical covariates: Cov1 (2 levels: Cov1.1, Cov1.2) and Cov2 (2 levels: Cov2.1, Cov2.2) The covariates DO NOT interact with each other.

I want the interaction between Time and Intervention and the covariates to study the impact of the covariates on the intervention at each time point:

1. Whether there are differences between interventions at each time point
2. Whether these differences at all time points are affected by the baseline values of Cov1 and Cov2 (the baseline covariates may have different impact on subsequent time points, but itself they don't change over time).

I thought about something like:

Response ~ Intervention * Time * Cov1 + Intervention * Time * Cov2 I also saw somewhere also this shortcut: Response ~ (Intervention * Time) * (Cov1 + Cov2)

Or maybe it should be Response ~ Intervention * Time + Intervention * Time * Cov1... ? or Response ~~ Intervention * Time + Intervention : Time : Cov1 + Cov1 + Intervention : Time : Cov2 + Cov2 ?

This is exploratory analysis I will be doing with GEE followed by the analysis of contrasts (in R: emmeans). My data are complete.

My output should look like (using emmeans):

Strata: Time T1:
Cov1.1-TrtA vs. Cov1.1-TrtB 
Cov1.2-TrtA vs. Cov1.2-TrtB
Cov2.1-TrtA vs. Cov2.1-TrtB 
Cov2.2-TrtA vs. Cov2.2-TrtB 

PS: later the pairs for each covariate Cov1 (and Cov2) will be compared to see if in overall Cov1 (and Cov2) affects the effect TrtA vs. TrtB), I mean: [Cov1.1-TrtA vs. Cov1.1-TrtB] vs. [Cov1.2-TrtA vs. Cov1.2-TrtB ] and same for Cov2.

Strata: Time T2:
Cov1.1-TrtA vs. Cov1.1-TrtB 
Cov1.2-TrtA vs. Cov1.2-TrtB
Cov2.1-TrtA vs. Cov2.1-TrtB 
Cov2.2-TrtA vs. Cov2.2-TrtB
Strata: Time T3:
Cov1.1-TrtA vs. Cov1.1-TrtB 
Cov1.2-TrtA vs. Cov1.2-TrtB
Cov2.1-TrtA vs. Cov2.1-TrtB 
Cov2.2-TrtA vs. Cov2.2-TrtB 

You know my goals, now I need your advice of which formula best reflects my needs. I cannot provide the data, it's just about general idea. I used R syntax, because it's easiest for me to understand the syntax. But if you use any other software, SAS, SPSS, you can provide your own. I need the general idea on how to set up the interactions properly.

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EDIT: I just checked, that Response ~ Intervention*Time*(Cov1 + Cov2) gives me sensible model coefficients and it's equivalent to Response ~ Intervention + Time + Intervention:Time + Cov1 + Cov1:Time + Cov1:Intervention + Cov1:Time:Intervention + (same for Cov2)

The second options with Response ~ Intervention*Time + Cov1 + Intervention:Time:Cov1 + Cov2 + Intervention:Time:Cov2 gives me weird coefficients, I missed certain higher-level interactions.

Answer 1

As you note, your "second option" omits almost all of the two-way interactions among predictors (except for the Intervention:Time interaction) while maintaining the 3-way interactions. Omitting lower-level terms or interactions while including higher-level interactions is generally not a good idea, as explained on this page.

What might be missing from your model is the baseline Response value as a predictor. That depends on the nature of Response. It often helps to include the pre-intervention values of a Response variable as a predictor, to control for differences between treatment and control groups in the post-intervention Response values. If the Response is a change from a baseline, however, then you don't want include the baseline level as a predictor. This page and this page might help guide your design.

Finally, note that your model with 3-way interactions involving 3 time points requires estimating a large number of coefficients and a correspondingly large sample size. You need to estimate: for individual coefficients, 1 each for Intervention, Cov1, and Cov2 and 2 for Time (5 total individual coefficients); for 2-way interactions, 1 each for Intervention:Cov1 and Intervention:Cov2 and 2 each for the 3 two-way interactions involving Time (8 total 2-way interaction cofficients); for 3-way interactions, 2 for each of them. That's 15 coefficients to estimate beyond an intercept.

Based on a rule of thumb of 15 observations per coefficient to avoid overfitting, you will need on the order of 225 total observations. With 3 observations per individual and 2 treatment groups, that suggests at least 35 to 40 individuals per intervention group, with more possibly needed depending on the magnitude of the intervention effect you are investigating.