# Regression with frequency rating in percent as dv

I seem to have a similar issue as asked in this thread, nevertheless I'm still clueless about what model I have to fit to my data.

I conducted a repeated measure experiment (consisting of 2 sessions) where participants were assigned to either Placebo or Drug - this is the repeated measure variable as in the second session drug-condition was switched for each pp. They then had to compete with 3 opponents in a simple game. Unbeknownst to the participants the probabilities to win against the different opponents were rigged, resulting in an inferior, intermediate and superior opponent.

After the competition participants were asked to answer the question "How often did this opponent win against you?", which they did by rating on a scale from 0 - 100 % (in steps of 10).

I need to analyse whether participants were able to correctly rate the probabilities of each opponent to win, grouped by drug-condition. Or, rather than comparing their ratings to these values, if they managed to gain implicit knowledge about the hierarchical structure of the group.

As far as I know it is not appropriate to use a repeated measure ANOVA here because the values of my dv are percentages. To my knowledge, beta regression is too not applicable as my dv contains the values 0 & 100.

Please see below my example data:

structure(list(ID = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 10L,
10L, 10L, 10L), .Label = c("1", "2", "3", "4", "5", "6", "7",
"8", "9", "10", "11", "12", "14", "15", "16", "17", "18", "19",
"20", "21", "22", "23", "24", "25", "26", "27", "28", "29", "30",
"31", "33", "34", "35", "36", "37", "38", "39", "40", "41", "42",
"43", "44", "45", "46", "47"), class = "factor"), Opp = structure(c(2L,
1L, 3L, 2L, 1L, 3L, 3L, 1L, 1L, 2L), .Label = c("A", "B",
"C"), class = "factor"), Drug = structure(c(1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L), .Label = c("Placebo", "Drug"), class = "factor"),
win.frequ = c(40, 50, 70, 70, 40, 40, 70, 70, 60, 60)), .Names = c("ID",
"Opp", "Drug", "win.frequ"), row.names = c(NA, 10L), class = "data.frame")


My questions now:

• Do I have to transform my data and if yes, how?
• Is using GLMM the right approach here and if so, can I enter my data just as it is? Do I use glmer then?
• Is it possible to use a logistic regression?
• As I could also consider my data as a rating from 0 to 10, would it be appropriate to conduct an ordinal logistic regression? Or a fractional logit regression? I'm completely lost...

Any explanation on what I need to do and why which method is correct/incorrect would be greatly appreciated!

• your win frequencies in this example are always multiples of 10. Is this true in the full data set (i.e., the only reported outcomes are {0,10,20,30,40,50,60,70,80,90,100}) ? – Ben Bolker Nov 25 '18 at 18:04
• @Ben Bolker, yes, that's correct! – joy2709 Nov 25 '18 at 19:03

I would say a Beta mixed effects models is still the way to go. You’re right that the Beta distribution is defined in the $$(0, 1)$$ interval, and the fact that you have 0 and 1 is problematic. But depending on the frequency of the 0’s and 1’s, you could either transform back to the $$(0, 1)$$ interval or considered a zero- and one-inflated Beta distribution. For the former option, you can transform your outcome as $$Y^* = \{Y \times (n - 1) + 0.5\} / n$$, where $$Y$$ Is your original outcome, and $$n$$ the sample size. For th latter option, you could consider the brms package in R that provides the zero- and one-inflated Beta distribution.
• The transformation I mentioned will put your new outcome $Y^*$ into the (0, 1), and is mentioned for example in the Journal of Statistical Software paper for the betareg package. And AFAIK brms should be able to handle random effects using lme4-type of syntax. – Dimitris Rizopoulos Nov 26 '18 at 12:34