How do I interpret a betareg coefficient of 6.6970 for a categorical variable with only two categories, given that the response is a proportion?

Question

I cannot seem to find an exact answer to my question online. I used the betareg package in R to run a glm with a response variable that is a proportion, so it is between 0 and 1. One of my predictor variables is categorical, with two categories. The other two predictors are continuous, with an interaction included between the categorical and one of the continuous predictors. (I also included an offset to account for variable effort and area. Attached at the bottom is an image link of the R output for my model.

The coefficient for the categorical predictor "SIDE" is 6.6970. I think this should mean that switching from the reference category (disturbed) to the other category (undisturbed) with log.prey and log.rainfall held constant causes an increase of e^6.6970 for each unit increase in the proportion response variable. However, the proportion is never going to increase by 1 because it is a proportion. What is the unit increase in this case? Also, e^6.6970 is 809.97. I have no idea how to interpret that in terms of percent change. Can anyone help me out?

THANKS!

Answer 1

You need to take the inverse logit, because beta regression is modeling the logit of the mean. The inverse of $\text{logit}(x)$ is $\frac{e^x}{e^x + 1}$.