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I have two studies:
STUDY1:
Independent variables: 
 1. between-subject factor ABILITY (typical, deficit) 
 2. within-subject categorical variable TEST_LANGUAGE (lang1, lang2) 
 3. within-subject categorical variable YEAR (year1, year2).
Dependent variable:
 1. VSCORE, a percentage value, in effect a continuous variable with min=0.0 max=100.0

STUDY2:
Independent variables: 
 1. between-subject a percentage value, in effect a continuous variable with min=20.0 max=100.0 
 2. within-subject categorical variable TEST_LANGUAGE (lang1, lang2) 
 3. within-subject categorical variable YEAR (year1, year2).
Dependent variable:
 1. VSCORE, a percentage value, in effect a continuous variable with min=0.0 max=100.0

 Should I be running generalized linear mixed-effect models with family=binomial, link=logit? I am not sure if this is the right test.
I am using R.
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  • $\begingroup$ In addition to Dimitri's suggestion you can check the R packages glmmTMB and glmmADMB. The package mgcv also allows for a beta response. $\endgroup$ – usεr11852 Aug 28 '18 at 21:36
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You could consider a Beta mixed effects model that can be fitted under maximum likelihood in R by package GLMMadaptive. For more info, check the vignette: https://drizopoulos.github.io/GLMMadaptive/articles/Custom_Models.html#beta-mixed-effects-model

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