How to analyze proportions data: Does the proportion of spent time in option A differ from the proportion of time spent in option B? My research question is whether sheep prefer lying down on straw bedding (option A) or on a new type of bedding (option B). If sheep spent more time lying down on one option, we will say they have a preference for this bedding.
I have the data set up with two rows per sheep per day, one row for the proportion of time spent lying on each bedding type. For example:




Sheep
Day
Bedding Type
Proportion of Time




1
1
OptionA
0.72


1
1
OptionB
0.28


1
2
OptionA
0.66


1
2
OptionB
0.34




I am analyzing this in SAS using Proc MIXED. My model is Time = u + bedding type + sheep + e, with day as repeated measure (subject = sheep) and random effect of sheep.
However, I realize that two rows for every day are related - they always add up to 1 since it is proportions data.
Specific Questions:

*

*Do I need to add something in to account for the relatedness (?) of the two observations per day?

*Is using a general linear mixed model the appropriate way to analyze this? Would something like a paired t test or a Wilcoxon Signed Rank Test be more appropriate?
Edit: My concern with using an ANOVA is that the two "observations" per day are not independent, and independence is one of the assumptions of ANOVA.

Thank you in advance.

Experiment (using R):
set.seed(2021)
x.b = rbeta(6, 5,9)  # fictitious data
stripchart(x.b)


t.test(x.b, mu=.5, alt="less")

        One Sample t-test

data:  x.b
t = -3.7488, df = 5, p-value = 0.006655
alternative hypothesis: 
 true mean is less than 0.5
95 percent confidence interval:
      -Inf 0.4616541
sample estimates:
mean of x 
0.4170871 

wilcox.test(x.b, mu=.5, alt="less")

        Wilcoxon signed rank test

data:  x.b
V = 1, p-value = 0.03125
alternative hypothesis: 
 true location is less than 0.5

Comments: It is not obvious (to me anyhow) that the proportions should be normally distributed,
so it may be best to use the Wilcoxon signed-rank test. However, with only six days the Wilcoxon
test will have poor power. For my fake data
the Wilcoxon test does show that percentages
are significantly smaller than half.
