How to interpret the coefficients from a beta regression?

Question

I have some data that is bounded between 0 and 1. I have used the betareg package in R to fit a regression model with the bounded data as the dependent variable. My question is: how do I interpret the coefficients from the regression?

Answer 1

So you need to figure out what scale you are modeling the response on. In the case of the betareg function in R we have the following model

$$\text{logit}(y_i)=\beta_0+\sum_{i=1}^p\beta_i$$

where the $\text{logit}(y_i)$ is the usual log-odds we are used to when using the logit link in the glm function (i.e., family binomial) in R. Thus the beta coefficients that betareg returns are the additional increase (or decrease if the beta is negative) in the log-odds of your response. I am assuming you want to be able to interpret the betas on the probability scale (i.e., on the interval (0,1)) thus once you have you beta coefficients all you need to do is simply change the response, i.e.,

$$\text{logit}(y_i)=\beta_0+\sum_{i=1}^p\beta_i\Rightarrow y_i=\frac{e^{\beta_0+\sum_{i=1}^p\beta_i}}{1+e^{\beta_0+\sum_{i=1}^p\beta_i}}$$

Thus you should realize that we are basically using the same results and interpretations from standard generalized linear modeling (under the logit link). One of the main differences between logistic regression and beta regression is that you are allowing the variance of your response to be much larger than it could be in logistic regression in order to deal with the typical problem of over-dispersion.