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I am decently familiar with basic random effect structures for mixed effect models (i.e., random intercepts per id) but struggle to identify the most justifiable structures for more complex study designs (i.e., crossed and/or nested random effects). I have examined most of the other questions/topics here and SO that discuss this concept but it’s’ still not clicking for me. Is there a series of questions / processes one can follow when looking at a naive dataset to understand what random effects may be the best place to start? I am aware that there is no black/white answer here and much of these depends on your research question / what seems to be the most appropriate for a given dataset (i.e., model comparisons)

Here is a theoretical study design I’m struggling with:

  • 20 participants (i.e., id)
  • response variable is continuous (i.e., response)
  • Within each participant, their upper limbs (i.e., limb = dominant/nondominant) are randomized into one of two conditions (i.e., condition = A or B.)
  • Each limb is measured on two occasions (i.e., time = pre / post) after receiving the condition
  • Finally, each participant completes two “phases”, where after a washout period, they complete the same intervention again, re-randomizing their limbs to the two conditions
  • In terms of the research question, I am the most interested in comparing the change scores (after adjusting for phase) between the two conditions. Here is my current thinking on the model that best approaches this study design and research question:
lme4::lmer(response ~ condition * time + phase + (1 | id / limb))
zr2015
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  • I am inclined to say that the best practice is a structured literature review. The random effects structure is supposed to be reflective of our research question. I commented on this more extensively in the CV.SE thread: https://stats.stackexchange.com/questions/189021/model-selection-for-random-effects-can-unselected-random-effects-be-used-as-fix/189160#189160 which handles a particular subcase of this question. (cont.) – usεr11852 Apr 18 '23 at 14:58
  • I suggest deviating from the experimentally relevant random effects structure only if it is clear that we cannot correctly estimate it given the available data. Particularly for this post, why is limb random? Don't participants have usually two? – usεr11852 Apr 18 '23 at 14:59
  • @usεr11852 I suppose I was thinking it more from the perspective of handling correlated observations that violate independence. We include random intercepts for participants to account for the dependency of their scores from the repeated measures. Because limbs are nested within participants, this seems to be an extension of the same concept in my mind - but I very well could be mistaken – zr2015 Apr 18 '23 at 16:15
  • I would assume that limb then should be included as a fixed term too. – usεr11852 Apr 18 '23 at 16:40
  • @usεr11852 I sincerely appreciate the feedback. Can I ask what is leading you to that conclusion? That is where I'm getting confused, I don't quite understand the philosophical difference that is leading you to make that determination. What separates participant id from limb in that only the latter should be included as a fixed term? I am aware that id is very rarely (if ever) included as a fixed term, but I'm trying to understand it from a conceptual basis on why that is the case. Thanks again for the feedback – zr2015 Apr 18 '23 at 16:45
  • I am suggested using limb as a fixed effect too (not id) because (simplistically) I do believe that there might be some differences in how people perform a task with their right instead of their left limb. In that sense, it might even make sense to have information for their dominant arm. – usεr11852 Apr 18 '23 at 20:42
  • @usεr11852 yep, you're suggesting response ~ condition * time + phase + limb + (1 | id / limb) - I still don't quite understand how you're arriving at that conclusion though. What question(s) are you asking yourself to determine whether the effect should also be included as a fixed effect? I apologize if I'm just missing the point here, this distinction is challenging for me to grasp – zr2015 Apr 21 '23 at 23:37

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