Which regression family will fit with a distribution ranging from 0 to 1?

Modeling
1 
     I have a response variable ranging from 0 to 1, so I tried beta regression (stan_reg in rstanarm and used the beta family in brms).  I chose beta because my variable represents a continuum from bad to good (more like a ratio). However, I have some observations with 0s and 1s, so when I fitted with the beta regression, it failed to fit since I should not have values either in 0s or 1s. So, which regression will better fit my analysis? I tried with normal regression, but there are just few values from 0 to 1. All the values are in 0.25 increments: 0, 0.25, 0.5, 0.75,  and 1.  I don't want to recode the variable since each value represent some condition which I don't want to miss.  Is there a distribution, other than the ordinal distribution (which of course requires recoding), that will fit better represent this distribution?
2

This may be of interest to you.

3

Thank you so much professor. It is exactly the kind of regression analysis I was looking for. Much appreciated. For a week, I tried everything and I was unconvinced about my functional specification of the distribution. I think this regression analysis will fit my response distribution much better.

4

I should be clear, this is not my method and I am not a professor. Glad to hear it answers your question

5

With discrete values, you can use ordinal regression. Here’s a discussion in the User’s Guide:

The function itself is described here:

and also fully encapsulated in a higher-level function for use in a GLM setting: