[R-sig-ME] Specifying a repeated-measure design with 3 fully crossed within-subject factors

[Michael Cone]

So, maybe I abstracted this a little bit too much. The following is an 
example I took from
Tamhane, A. C. and Hayter, A. J. (2004) Comparing Variances of Several 
Measurement Methods Using a Randomized Block Design with Repeat 
Measurements: A Case Study. In Advances in Ranking and Selection, 
Multiple Comparisons, and Reliability, Balakrishnan, N., Kannan, N. and 
Nagaraja, H. N. (Editors), Birkhauser, Boston, 165–178.

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The insertion gain of a hearing aid is defined as the difference 
between the sound pressure level (SPL) measured at the eardrum of the 
wearer with the hearing aid in place and the SPL at the eardrum with no 
aid in place, the stimulus being the same under both conditions.

The study measured the insertion gain of a hearing aid at different 
loudspeaker locations. The standard practice was to locate the 
loadspeaker in the ear-level horizontal plane of the subject. It was 
claimed that loudspeaker locations above the horizontal plane would 
yield more precise (less variable) results.

The study compared the following loudspeaker locations:
- Location 0: 0° azimuth, 0° elevation (Standard/Control)
- Location 1: 45° azimuth, 0° elevation (New)
- Location 2: 0° azimuth, 90° elevation (New)
- Location 3: 45° azimuth 45° elevation (new)
There were 10 subjects with five replicate measurements of insertion 
gain at each of the four loudspeaker locations.
The investigator was primarily interested in comparing the 
within-subject variances for different loudspeaker locations
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Imagine this study being conducted with 4 different kinds of auditory 
stimuli, each presented at the same combinations of azimuth and 
elevation, with replicate measurements, all presented to 5 subjects. A 
minimal reproducible example:

df <- expand.grid(meas.num = seq(1, 10),
                   elevation = c(0, 15, 30, 45, 60, 75, 90),
                   azimuth = c(0, 15, 30, 45),
                   stimuli = c("stim1", "stim2", "stim3", "stim4"),
                   subject = c("subj1", "subj2", "subj3", "subj4", 
"subj5",
                   "subj6"))
df$sex <- "m"
df[df$subject %in% c("subj1", "subj2", "subj3"), ]$sex <- "f"
df$val <- rnorm(nrow(df), mean=10, sd=5)  # dummy measurement values

Here, I would be interested in comparing
(1) the overall within-subject variances for each stimulus, as well as
(2) the within-subject variances between differing combinations of 
azimuth & elevation both within the same stimulus and between differing 
stimuli,
both within each sex and for both sexes.

I would greatly appreciate any comments, questions, or pointers in the 
right direction.

Michael