Came across a current online review piece on 'Zero-One Inflated Beta Models', by Karen Grace-Martin in "The Analysis Factor", outlining the proposed solution (noted above by Matze O in 2013) to address the 0/1 occurrence issue. To quote parts from the non-technical review:
So if a client takes their medication 30 out of 30 days, a beta regression won’t run. You can’t have any 0s or 1s in the data set.
Zero-One Inflated Beta Models
There is, however, a version of beta regression model that can work in this situation. It’s one of those models that has been around in theory for a while, but is only in the past few years become available in (some) mainstream statistical software.
It’s called a Zero-One-Inflated Beta and it works very much like a Zero-Inflated Poisson model.
It’s a type of mixture model that says there are really three processes going on.
One is a process that distinguishes between zeros and non-zeros. The idea is there is something qualitatively different about people who never take their medication than those who do, at least sometimes.
Likewise, there is a process that distinguishes between ones and non-ones. Again, there is something qualitatively different about people who always take their medication than those who do sometimes or never.
And then there is a third process that determines how much someone takes their medication if they do some of the time.
The first and second processes are run through a logistic regression and the third through a beta regression.
These three models are run simultaneously. They can each have their own set of predictors and their own set of coefficients...
Depending on the shape of the distribution, you may not need all three processes. If there are no zeros in the data set, you may only need to accommodate inflation at 1.
It’s highly flexible and adds important options to your data analysis toolbox."
Here is also a more recent December 2015 technical paper source for 'zoib: An R Package for Bayesian Inference for Beta Regression and Zero/One Inflated Beta Regression'. The authors note that the y variable, in a Zero/one inflated beta (ZOIB) regression model(s), can be applied when y takes values from closed unit interval [0, 1]. Apparently, the zoib model assumes that Yij follows a piecewise distribution (see system depicted in (1) on p.36).