Overview
I have an outcome variable that is a percentage with some 0% and 100% included. My main goal is to test the difference between two groups on this percentage. I am using brms and zero_one_inflated_beta()
.
brms code
fit < brm(
formula = bf(
pct ~ 1 + group + (1  participant) + (1  target),
phi ~ 1 + group + (1  participant) + (1  target),
zoi ~ 1 + group + (1  participant) + (1  target),
coi ~ 1 + group + (1  participant) + (1  target)
),
family = zero_one_inflated_beta(),
...
)
Results
Here are my results (simplified to just median and pd for your readability):
Parameter Median pd
(Intercept) 1.19 100%
phi_Intercept 1.74 100%
zoi_Intercept 3.44 100%
coi_Intercept 7.60 100%
group1 0.38 99.36%
phi_group1 0.39 99.74%
zoi_group1 0.54 76.92%
coi_group1 3.72 99.05%
Density Definitions
Interpretation
Here is my interpretation. Iâ€™d greatly appreciate a confirmation that this is right or a correction.
(Intercept)
= \mu, the beta distributionâ€™s mean parameter (in logit units), for group0
phi_Intercept
= \phi, the beta distributionâ€™s precision parameter (in log units), for group0
zoi_Intercept
= \alpha, the probability that an observation is either a zero or a one (in logit units), for group0
coi_Intercept
= \gamma, the probability that an observation is a one given that it is either a zero or a one (in logit units), for group0
group1
= the difference between group1 and group0 in \mu (in logit units), the beta distributionâ€™s mean parameter
phi_group1
= the difference between group1 and group0 in \phi (in log units), the beta distributionâ€™s precision parameter
zoi_group1
= the difference between group1 and group0 in \alpha, the probability that an observation is either a zero or a one (in logit units)
coi_group1
= the difference between group1 and group0 in \gamma, the probability that an observation is a one given that it is either a zero or a one (in logit units)
Questions
In addition to a confirmation that the above interpretations are right, please help with me this question:
Can I (and how do I) test the following hypotheses?

The groups differ on the overall probability of a one occurring

The groups differ on the overall probability of a zero occurring

The groups differ on the overall mean score, inclusive of the zero, one, and beta components
I assume that I will use hypothesis()
and do some parameter algebra, perhaps converting back from logit units to probabilities?