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What is the most appropriate method for analyzing clustered data (i.e., repeated measurements nested within subjects) when there are very few clusters (e.g., 3 subjects)?

Here's a brief description of the study:

  • There are 3 subjects, each experiencing 5 conditions.
  • Within each condition, there are 12-15 repeated measures.
  • The measures are: the number of responses on two alternatives (B1 and B2) and number of reinforcers earned for each type of response (R1 and R2).

The question is how subjects allocate responses to the two alternatives (B1 and B2) based on the number of reinforcers earned under those alternatives (R1 and R2).

The image below provides examples of some possibilities.

enter image description here

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  • $\begingroup$ I don't understand your research question. Please can you explain the study design and describe all of the variables. What is the outcome/response/dependent variable, and what are the independent variables, and how are they all measured ? $\endgroup$
    – Robert Long
    Nov 13, 2020 at 20:10
  • $\begingroup$ The dependent variable is the number of responses (B1=number of responses on option 1, B2=number of responses on option 2). The independent variable is the number of reinforcers delivered for each response option. In the five conditions, reinforcement (food) is concurrently available on the two options, but the schedule of reinforcement for each response differs across conditions. In the control condition, the schedule of reinforcement is the same for both options. In the other conditions, reinforcement is available more frequently for one option (e.g., every 12s for B1 vs. every 60s for B2). $\endgroup$
    – Carolyn
    Nov 13, 2020 at 20:37
  • $\begingroup$ How are B1 and B2 related ? Are these seperate outcomes ? If you had more subjects what kind of model would you fit ? $\endgroup$
    – Robert Long
    Nov 13, 2020 at 20:41
  • $\begingroup$ B1 and B2 are separate outcomes. B1=moving to one location and B2=moving to a second location. I think a linear mixed-effects model might be suitable if there were more subjects. $\endgroup$
    – Carolyn
    Nov 13, 2020 at 20:55

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From the description in the question and the comments, it appears that a mixed effects model with random intercepts for subjects would be the preferred analysis model for these data. However, given the small number of subjects, this is not appropriate within the usual frequentist paradigm. So, I would suggest fitting fixed effects for subjects (ie just a linear model, within random effects), which will also control for the non-independence of observations within subjects.

I might also consider a Bayesian mixed model approach with random effects for subjects if I knew what intra-class correlation was likely, in order to set a reasonable prior on the variance of the random intercepts for subject (this might be available from prior studies).

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  • $\begingroup$ Thank you, Robert. Regarding your suggestion to fit fixed effects for subjects, are you suggesting fitting a separate model for each subject? How does that control for the non-independence of observations within subjects? $\endgroup$
    – Carolyn
    Nov 13, 2020 at 21:33
  • $\begingroup$ No. I mean including subject as a fixed effect instead of a random effect. Both methods control for non independence but when the number of subjects is high it is preferable to use random effects. $\endgroup$
    – Robert Long
    Nov 13, 2020 at 22:51

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