Multilevel model with two timepoints?

I am conducting a study with a large population-based sample. We have measurements at two time-points spaced 8 years apart. I want to see if a variable at time 1 predicts changes in the DV over time. I have a set of covariates (both time-invariant and varying, e.g., race, age, gender). For our second aim, I am also looking at whether another time 1 variable moderates the association between the IV and DV. Because this is a population-based cohort study with missing data, we incorporate sampling weights into the analyses.

I conducted mixed models where I nested scores on the DV within the individual. Participant ID is a random effect and because we only have two measurements, time is a fixed effect. One of my co-authors is concerned that mixed models aren't appropriate for longitudinal designs with only two measurements. I also conducted linear regressions to see if there were differences in results between the two approaches. For these, I looked at whether the time 1 variable predicted the DV at time 2, adjusting for the DV at time 1. Generally, more of the effects were significant and larger with LMMs.

I can't find any guidance online that suggests you can't use mixed models for longitudinal data with two time points. Can anyone point me in the direction of literature I can read on this issue? Or, can anyone weigh in on whether LMMs make sense here or if I should just use regression? I have conducted LMMs before with clinical trial data, but these all had 3 measurements.

Thank you!

• Is there are reason why you aren't just defining the outcome as the difference between post and pre DV scores? Nov 15, 2022 at 15:58
• I suppose I wasn't sure how reliable a simple change score would be. I certainly can run them that way-I know sometimes we use the residualized change vs. just post-pre. Do you have a suggestion on which is better? Nov 15, 2022 at 16:06

The classic and most accepted analysis of this type of data is the ANCOVA model where you fit

$$E[Y_{t_2} | Y_{t_1}, X_{t_1} ] = \alpha + \beta Y_{t_1} + \gamma X_{t_1}$$

That is $$Y_{t_2}$$ is the response at follow-up, and $$Y_{t_1}$$ is the baseline assessment of response, and $$X_{t_1}$$ is the covariate of interest. For instance, in a randomized study of a novel TK2 inhibitor for psoriasis, the psoriasis area severity index (PASI) might be the response. The baseline PASI is an adjustment variable, and the randomization allocation is the $$X_{t_1}$$.

Formulated this way, the observations are mutually independent and models for independent data (i.e. linear regression) may be used. This is also the natural extension of the paired T test to allow covariate adjustment.

One alternative approach that I like with two time points when the topic of interest is the change between time points is latent difference score model (also known as latent change score model), where you can model change t1->t2 as a latent "entity".

See, for instance, https://www.frontiersin.org/articles/10.3389/fpsyg.2021.696419/full

You can easily (well, there's a bit of coding) do this in R lavaan. You can have your model estimate the regression coefficient of your T1 variable predicting the latent change variable to investigate whether it is related to change.

• Oooh this is intriguing, I am unfamiliar with these models. Thanks for linking that paper, I will read up on this! Nov 16, 2022 at 14:51

Same issues as his. I found two time points measurements not suited to the linear mixed model as your coauthor concerned. There are at least 3 measurements for each person, but it allows some missing data in your measurements. Hope this is helpful.

• Hi Simon110 and Welcome to CrossValidated! Your answer could be potentially useful but needs some further details. Could elaborate further on I found two time points measurements not suited to the linear mixed model as your coauthor concerned ? Jan 6, 2023 at 9:36
• Great question! Two time points aren't enough for a random intercept. Actually, a random effect is essentially applying a probability distribution to the given variable. That means you should make a reasonable prediction about other time points that were not in this study. So it has to need at least three time points to fit such a model. Jan 6, 2023 at 15:05