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We are conducting an individual participant level meta-analysis on a series of clustered randomised controlled trials, where we are mainly interested in an interaction effect with a characteristic (whether students have had contact with a particular kind of service). In essence, there are two things we need a multilevel model for:

  • Dealing with the nested clustering - in this case schools within trials
  • Producing an interaction effect with the random effects for the trials

The data is unfortunately protected from being shared but the structure is:

  • Outcome variable - test scores (score)
  • Trial - which trial a person appears in
  • School - this is the level the trials were randomised on, so it is nested below trial
  • Service - another individual-level characteristic that we want to interact trial with

If we had a fixed effects model then this would look something like:

lm(score ~ trial*service, ...)

if we just ignored the nested clustering (where trial is a factor variable for the treatment arm and which trial it is in).

If I could ignore the interaction then in lme4 it would be something like:

lmer(score ~ (1|school/trial), ...)

but what I would like to see is how to interact that with service, which maybe I could do by setting:

lmer(score ~ (1|school/trial/service), ...)

but feel this is wrong as service is an individual-level characteristic, so its weird to put it "above" school or trial in the clustering.

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When you write "score ~ (1|school/trial/service)" this treats service as a random effect, nested within trial, nested within school.

From your description, it sounds like service may actually be a fixed effect? If so, then I think you would want "score ~ service + (1|school/trial) + (1|school/trail:service)" - that gives you a fixed main effect for service, and a random effect for the interaction of service with trial-nested-within-school.

The answer above assumes service is categorical. If service is continuous, you will need a random slope for the interaction instead of random intercepts, eg. "score ~ service + (1|school/trial) + (service|school/trail)". This is not an interaction in the sense that you'll get an output value, but it does account for differing effects of service (as a continuous variable) in different trial-nested-within-school groups.

For more information on interactions with lmer, these slides may be helpful.

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