# Repeated measures with population-level effects

Hello,

I’m trying to analyse some repeat measurements with an intercept varying over participants. I also want to use the some features on a participant level to predict their intercept. This is similar to equations 11.1 and 11.2 on p. 241 in Gelman and Hill’s ARM. Unfortunately, there seems to be no further discussion of this model in the book.

Let’s say I have 1 measurement for 12 categories A through L for each participant, so I have 12 trials per participant and want to model the probability of a correct answer depending on the category and allowing for a varying intercept per participant:

`correct ~ 1 + category + (1 | respid)`

So this will give me an estimate of

• overall difficulty of the task
• the effect of each category on the task
• an estimate of each participant’s skill

The model that interests me tries to model the effect of participant features on participant skill, something like:

`correct ~ 1 + category + alpha[respid]`

Where `alpha[respid]` might be something like

`alpha[respid] ~ 1 + education[respid] + age[respid]`

Is it possible to specify this model in brms?

I have tried fitting

`correct ~ 1 + category + education + age + (1 | respid)`

Which works fine and recovers the parameters from my simulated data, but I think this treats every participant education and age variable as occurring 12 times. This would overestimate the relation between education and age on the dependent variable.

I have also tried

`correct ~ 1 + category + ( 1 | respid / 1 + education + age )`

But if I understand correctly this estimates an effect of education and age on each participant’s intercept separately, so instead of one population wide effect of education on the intercept, this estimates one effect for each participant.

How can I specify a model to estimate the effect of education on participants intercept?

• Operating System: Windows 10
• brms Version: 2.9.0

This is the model you want; it does not, as you fear, imply that you have more measurements than you actually obtained, merely that each outcome is influenced by the participant’s characteristics.

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Excellent, thanks so much for your help!