Mixed Models – Controlling for Time Elapsed Between Baseline and Treatment

Question

I'm having a mixed model with 5 repeated measurements (time is categorical because the timepoints represent discrete events).

Edit: I have 5 meaningful timepoints: t0 baseline (neutral timepoint); t1 invitation to intervention; t2 shortly before intervention; t3 shortly after intervention; t4 participants receive the result (good or bad news).

I also have 2 treatment groups (randomized trial).The outcome variable is self-reported quality of life (e.g. mental health measured by questionnaire). My plan is to first build a mixed model to test for the significance of the interaction between treatment and time, and then look at specific contrasts (e.g. difference in change in QoL between groups at t0 vs t1)

While the measurement points are equally spaced between the last 4 measurement points for both groups, the average time elapsed between baseline (t0) and the intervention appointment (and thus also t1, t2, t3, t4) differs considerably between the two treatment groups (around 43 days), see image below for illustration.

The reason for this difference is not random sampling error, but a a logistical one, i.e. it was easier to plan in the treatment appointment for treatment group 1 than treatment group 2, as less technologically advanced equipment was used.

I do think that this time difference could impact the outcome variable (e.g. the longer the time elapsed, the more likely it is that a significant event has happened that impacts quality of life).
Therefore I would like to control for it, for example when comparing the change in QoL between the two treatment groups for t0 vs. t1.

Does it make sense to use the time between baseline and intervention as a covariate in the model?

So, for example the R code for the model would be:

lmerFitQoL <- lmer(QoL ~ time * treatmentgroup+ timedifference + 
                   (1| id), data = datQoL)

where timedifference= difference in time between baseline and treatment appointment

Would that be a suitable solution in your opinion? I'm unsure if I'm on the right path here or if there is a better way.

Answer 1

It would seem most important to ensure that the groups had similar QoL outcome measures at t1 or t2, as those time points are fixed with respect to the intervention time. Your question says that you might include a t1 versus t0 comparison. That doesn't seem to make much sense in terms of evaluating the intervention, however, as the intervention doesn't happen until after t2.

You might want to examine changes from t0 to t1 in a separate analysis with a continuous measure of time in days. That would help you evaluate whether the two treatment groups were adequately well matched both at enrollment into the study and at the time point t1 that (unlike t0) occurs at a fixed time prior to the intervention. It would also let you see if there is any systematic change over time in the outcome measure absent the intervention.

If the groups are adequately well matched at t1, however, I don't see any need to use values at t0 as part of evaluating the intervention itself. You might, however, need to evaluate them as part of your quality control.

In response to comments

I think it's important to distinguish the direct effects of the intervention from possible changes in QoL values associated with the treatment-group assignment, presumably done at t0, which might lead to systematic differences between t0 and t1.

With similar distributions of QoL values between the 2 treatment groups at t1, the specific effects of interventions per se can probably be described as differences between pre-intervention (t1, t2) and post-intervention (t3, t4) QoL values. Think carefully how you want to do that, as the more coefficients you have to estimate the lower power you might have.

For example, might the QoL values at t1 and t2 be considered replicates rather than separate values? Might it make sense to model QoL differences between t1 and t2 against corresponding differences between t3 and t4, both representing 13-day periods? You need to apply your knowledge of the subject matter to make those decisions.

You certainly should examine potential changes between t0 and t1, but such changes would have to do with either the time interval or the group assignment (e.g., due to the potential psychological effects you mention) rather than with the intervention per se. They thus would require a type of explanation other than a direct effect of the intervention.

Don't overthink the t0 to t1 differences. What you presumably want to do is to assure yourself and your audience that any such differences between the 2 assignment groups are small enough not to affect your interpretation of the direct intervention effect. Don't worry so much about whether you have the "best" model for the t0 to t1 difference. Just develop one that's adequate to address that potential concern.

A simple analysis of the paired t1-t0 differences within individuals should be adequate and accomplish more simply what you propose in a comment to do with a mixed model. If you are only examining paired t1-t0 differences you don't need the time*treatmentgroup interaction, just the treatmentgroup assignment itself. Flexible inclusion of timeddifference in the model of the t1-t0 QoL paired differences with a regression spline makes sense. You will need more than the 2 degrees of freedom you propose in the model in your comment, however, as that doesn't allow any knots at all. I prefer to model splines with the rcs() function in the R rms() package, in part because (unlike ns()) it provides reasonable default parameter settings.