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I ran an experiment where I moved subjects arm about the elbow to a reference position, returned it to a home position and then asked them to try and replicate the position, all without vision of their arm. I then measured the error in their position matching as an estimate of their upper limb position sense acuity. This experiment aimed to compare position sense acuity between a group of older and young adults.

The experiment was designed such that subjects performed 4 repeats at each of 3 reference positions for both extension and flexion movements of the elbow. To make best use of the data (avoid averaging and potential loss of data across repeated measure levels with a mixed ANOVA), I would like to analyse the effect of age group (2 levels) on matching error whilst controlling for reference position (3 levels) and movement direction (2 levels) in a linear mixed effects model, but I’m having some issues working out how to model the random effects.

On the one hand, I have fairly consistently read that that random effects factors typically need a minimum of 5-6 levels to achieve a robust estimate of variance (e.g. pg.33, https://lme4.r-forge.r-project.org/book/Ch2.pdf and https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5970551/) which makes me think that it would not be possible to model reference position (3 levels) and movement direction (2 levels) with random intercepts in this way…

Err_model_1 <- lmer(error ~ age_group + (1|subjects) + (1|move_direct) + (1|ref_pos))

In which case I was considering including them as fixed factors instead, however, I have also read that for designs using within-subjects measures as fixed factors, they should also be modelled with a random slope for a maximal model that minimizes Type I error rates (Barr 2013; https://www.frontiersin.org/articles/10.3389/fpsyg.2013.00328/full) which somewhat contradicts the minimum 5-6 number of levels rule, but I think would look as follows…

Err_model_2 <- lmer(error ~ age_group * move_direct * ref_pos + (1 + move_direct + ref_pos | subjects)

  • Is there something specific to using the within-subjects measures as fixed factors in Err_model_2 that allows you to model them as a random effect validly?
  • Is there another way I would be able to model movement direction and reference position as random effects in this kind of model?

Any other help or comments would be appreciated, thanks!

Nkitchen1
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  • You might find the answers to [this CrossValidated question](https://stats.stackexchange.com/questions/37647/what-is-the-minimum-recommended-number-of-groups-for-a-random-effects-factor) helpful, though they give conflicting advice. – eipi10 Sep 04 '20 at 18:24
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    From simulation studies it is often quite clear, that random effects should contain 5+ levels to be sensible. Random slope makes somewhat sense in the case that effects are somewhat crossed (eg. `direction` is crossed in subject, similar to `ref_pos`). Lacking my thesis sources at hand, but using a (very) simplified example it isn't too hard to see the logic behind this. A **mixed effect** model is also known as a **variance component** model. We are estimating the **variance** between groups, not the mean of the groups. You wouldn't use 2 observations to estimate the variance either. – Oliver Sep 04 '20 at 20:04
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    why do you want to model movement direction and reference position as random effects, anyway? Are you clear on the distinction between random effects and nuisance variables? https://stats.stackexchange.com/questions/414460/where-to-specify-covariates-in-a-linear-mixed-effect-model – Ben Bolker Sep 08 '20 at 03:29
  • Thanks for responses! @BenBolker I had chosen to model them as random effects since I wanted to generalize across movement amplitude and directions, with the main focus being whether there is an effect of age on matching errors. However, looking at the link you posted (and comments from others) I think it may be better to model them as fixed effects. So I'm assuming my best option would still be a maximal model which would essentially look like Err_Model_2 from my original post. Do you know any papers with explicit "nuisance" factors? Are they reported in the same detail as "main" factors? – Nkitchen1 Sep 08 '20 at 16:24

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