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I have a regression model analyzing an continuous acoustic value, F1, based on three categorical predictors: vowel pair, tongue root position, and syllable number. My model includes all possible interactions and a random intercept for speaker. So, F1 ~ pair * tongue root position * syllable + (1|speaker).

Vowel pairs are defined thusly (approximations based on American English):
Pair 1: /i/ (the vowel in "beet") vs /ɪ/ (the vowel in "bit")
Pair 2: /e/ (approximately the vowel in "bait") vs /ɛ/ the vowel in "bet")
Pair 3: /u/ (the vowel in "boot") vs /ʊ/ (the vowel in "book")
Pair 4: /o/ (the vowel in "boat") vs /ɔ/ (the vowel in "bought")

From this I can generate pairwise comparisons in emmeans, comparing for instance the difference between F1 of /ɪ/ and /i/ in syllable 1 (comparison: pair = 1; tongue root position = 0 vs 1; syllable = 1) and every other pairwise comparison. I'm fine up to this point.

However, what I'm really interested in is comparing different pairwise comparisons to one another. I want to compare every other within-pairing difference to /ɪ/ vs /i/ in syllable 1. So, I want to see if the difference between F1 of /e/ and /ɛ/ in syllable 2 is larger than the difference of F1 in /ɪ/ and /i/ in syllable 1, if the difference in F1 of /u/ and /ʊ/ in syllable 2 is larger than the difference of F1 in /ɪ/ and /i/ in syllable 1 and so on and so forth.

I can take the output of emmeans and run t-tests over these. For instance, if the difference between F1 of /ɪ/ vs /i/ in syllable 1 is 0.32z, SE of 0.11, with n=150; the difference between F1 of /e/ and /ɛ/ in syllable 2 is 0.54z, SE of 0.08, n=150, I can plug these into a t-test, but that seems inappropriate.

Alternatively, I could take the data I have and derive a direct Delta-F1 by pairing tokens of /ɪ/ and /i/, /e/ and /ɛ/, etc to generate Delta-F1 values for each pairing. For instance, if the measurement for a given token of /i/ is -1.81z and the measurement for a given token of /ɪ/ is -1.46z, I would take the difference, 0.35z, as an input data point for the regression, with n' = original n/2. This would, in turn, be the input to the regressionː Delta-F1 ~ pair * syllable + (1|speaker), and then pairwise comparisons would be handled the usual way.

Which of these is most appropriate? Are there alternative ways to analyze this that I'm just unaware of?

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What you are asking for is interaction contrasts, or contrasts of contrasts. You can do something like:

emm <- emmeans(model, ~ tongue * syllable)
contrasts(emm, interaction = "pairwise")

Or equivalently,

emm <- emmeans(model, ~ tongue * syllable)
con <- contrast(emm, "pairwise", by = "syllable", name = "tongue_diff")
contrast(con, "pairwise", by = "tongue_diff")

You can do analogous things with different combinations of the factors, or with all three factors (in which case the second method requires 4 steps, and two factors in each by spec).

See vignette("interactions", "emmeans") and the section on "interaction contrasts" for more discussion.

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