I have the following study design:

  • 3 factors ($F_1,F_2,F_3$), where $F_1$ has 5 levels and both $F_2$ and $F_3$ have 6 levels (so 180 "conditions" in total)
  • $F_2$ and $F_3$ are nested in $F_1$ and are crossed (so this is a split-plot design with 5 plots, each containing a 6*6 matrix of subplots)
  • The levels of each factor are categorical, and the response is measured on a continuous but bounded 100 point scale (0-100).

51 participants were assigned 30 random levels of $F_2$ (from any level of $F_1$), where for a given level ($l$), the participant provided 6 responses: one for each level of $F_3$ that is crossed with $F_{1F2_l}$. Therefore each participant provided 180 responses in total.

I have 9126 responses in total, averaging approximately 50 for each condition (it is slightly unbalanced across both participant and conditions due to measurement errors).

I am trying to model this data for inference purposes using lme4, however I have a few questions:

  1. How to specify the model? My understanding is a bit shaky and I'm not sure from reading Bates (2014, Fitting linear mixed-effects models using lme4). Is this correct? resp ~ F1/F2 + F1/F3 + (1|participant)

  2. Using the model above, I see significant interactions between $F_1$-$F_2$ and $F_1$-$F_3$, as expected, but I want to know about the effect of each $F_2$ and $F_3$ within each level of $F_1$, and the above specified model doesn't give me that...

  3. An aside - is it ok to use a linear model with a bounded continuous DV? Looking at the data, participants tend to use the extremes of the range quite a lot. The range of the residuals is the same across all fitted values, but the mean residual decreases as the fitted value increases.


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