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I have a dataset where 50 Subjects were each exposed to 2 Testing Conditions over 5 Time Periods and they reported their Happiness Levels at every time point (i.e., I have 50 * 2 * 5 Happiness data points in total). Based on past work, I know I need to run a "linear mixed-effects model with a fully specified random-effects structure, with subjects as a non-nested random factor, and all intercepts and slopes included".

If I just had 50 Subjects with 5 Time Periods, I know that my lme4 formula would be:

Happiness ~ Time + (Time | Subjects)

But if I have 50 Subjects with 5 Time Periods AND 2 Testing Conditions, would this be:

Happiness ~ Time * TestingCondition + (Time * TestingCondition | Subjects)

I'm still learning more complicated LMM designs at the moment, so if I'm completely off the mark here, I'd appreciate any help or advice I can get!

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This is almost right, although it probably needs to be adjusted for practical/identifiability purposes. The rubric is that any condition or treatment that varies within groups should be allow to vary across groups; since you have measurements for both testing conditions and every time point for every subject, you can in principle measure the variation of all of these effects (for example: "how does the difference in testing conditions at time point 3 vary across subjects?" [this quantity isn't explicitly estimated by the model]).

There are a few important points to consider, however, in two cases.

Time considered as categorical

In this case you have two problems:

  • since you only have one measurement per time point $\times$ treatment combination per subject, the among-subject variance in the interaction is confounded with the residual variance term. There are a few different options for overriding this, but ...
  • in addition, your model will almost certainly be overparameterized/too complex. With two time points and 5 treatments, you have 10 parameters that vary across subjects, which implies a 10 $\times$ 10 covariance matrix with (10*11)/2 = 55 parameters to estimate. It's extremely unlikely that you will be able to do this with 50 subjects/500 data points ... you could:
    • use a Bayesian framework (e.g. blme, MCMCglmm, or brms) to put regularizing priors on the covariance matrix
    • simplify the model, at least to (Time + Treatment|Subject) (this will "only" require a 6 $\times$ 6 covariance matrix with 21 parameters ...)
    • simplify in some other way, e.g. use glmmTMB to fit a factor-analytic covariance matrix, or afex::mixed to fit a zero-covariance model

Time considered as continuous

This version is much easier, although it requires you to assume a linear trend across time points.

Now (Treatment*Time|Subject) only implies a 4 $\times$ 4 covariance matrix = 10 parameters. Still a little bit ambitious, but it might work.

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  • $\begingroup$ This is wonderful, thank you for the detailed answer Ben! So if I'm understanding this correctly: (1) if I consider Time to be categorical, my formula would be "Happiness ~ Time * Testing + (Time + Testing | Subject)" to account for the overcomplexity of my model OR (2) if I consider Time to be continuous, my formula would be "Happiness ~ Time * Testing + (Time * Testing | Subject)"? $\endgroup$
    – EPP
    Jun 17, 2022 at 2:30
  • $\begingroup$ Yes, although you might not be entirely out of the woods - either of these models might still be too complex for the data. $\endgroup$
    – Ben Bolker
    Jun 17, 2022 at 15:58
  • $\begingroup$ Okay good to know, thanks a bunch for the quick responses @BenBolker, and I'll definitely look into whether I'll need to simplify the model further! $\endgroup$
    – EPP
    Jun 17, 2022 at 17:02
  • $\begingroup$ If this solved your problem you are encouraged to click on the check-mark to accept it .. $\endgroup$
    – Ben Bolker
    Jun 17, 2022 at 17:09

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