best regression model for a percentage of function variable I am trying to choose the best regression model for a dependent variable 'percentage of normal shoulder function'. This variable is non normal and clustered at the high end of the possible range, ie most scores were above 90% with several 100%s. The percentage was calculated from a scored survey. Independent variables include 2 continuous and 1 categorical.
I have read about using a couple different models, including tobit regression, beta regression, a gamma GLM, but I am unsure which is the best to use in this case. Given the non normal distribution and the inclusion of 100%s in the data set it appears that beta regression and tobit probably aren't the best choices here.
 A: The details will depend on your data, so it is hard to give general advice here. Personally, I would first try to use a tobit model that is also right-censored at 100%. Typically the fit of such models can be improved if regressors for the variance can also be included. In R such an implementation is available in our crch package. See https://doi.org/10.32614/RJ-2016-012
If you are working with a software environment that does not support right-censoring you could also use the trick of modeling 100 - y where y is your percentage variable. That would correspond to amount of reduction in shoulder function and consequently be clustered at zero.
A: The most appropriate solution will depend on several factors on the distribution.
If there is clear evidence of censoring, then some form of a censored regression needs to be used. You may even need to transform the dependent variable to make sure that the uncensored part resembles a part of a recognizable distribution.
If there is no evidence of censoring and concentration at 100% is a natural part of the distribution, then the beta regression or the fractional response regression (set family = 'quasibinomial' in GLM if using R) could be a good choice here. The choice between the two depends on your view of the likelihood function or distribution of the residuals.
