# Beta regression coefficient interpretation

I'm trying to use beta regression on a data set where the response variable is in percentage (i.e. between 0 and 1), so let's say it's the unemployment rate. I'm having trouble wrapping my head around the coefficients of the model summary. If I had predictor A which has a coefficient of -0.5, should I take a similar approach to what I would do for linear regression, so something like "an increase in A reduces the unemployment rate by 0.5"? Or should I say "decrease the odd ratio by exp(0.5)" like logistic regression?

I'm attaching a screenshot, in case this helps.

• It is a bit strange doing non-linear regression and then trying to describe the results in linear terms. Here your result suggests that adding $1$ to $A$ multiplies the odds of being unemployed by $e^{-0.5} \approx 0.6065$, but now you have to explain "odds". Commented Oct 18, 2023 at 21:14
• @Henry Can you clarify what's so strange about it? Commented Oct 19, 2023 at 6:23
• @dimitriy: If you want your conclusion to look like "adding $1$ to $A$ (other things staying the same) tends to subtract $0.3$ from $X$" then you are suggesting you have a linear model. Commented Oct 19, 2023 at 7:33
• @Henry I take derivatives of nonlinear functions all the time, and they have that exact incrementality interpretation. I even do that in OLS with interactions. There's nothing special about conditional expectations that precludes doing this. Commented Oct 19, 2023 at 17:20

The mean is usually logit-linked in beta regression, and so is yours as you can see in the first line of the summary. You'd interpret these like you would for a logistic regression. See also here.

The logit link function complicates things a bit a bit compared to OLS.

For a continuous variable $$x$$, the additive marginal effect of $$x$$ with a logit link function is

$$\Lambda(\alpha + \beta x)\cdot \left[1-\Lambda(\alpha + \beta x)\right]\cdot\beta = p \cdot (1 - p) \cdot \beta,$$ where the inverse logit function $$\Lambda$$ is $$\Lambda(z)=\frac{\exp{z}}{1+\exp{z}}.$$

You can get the average marginal effect by taking the mean of that quantity in your sample.

For dummies, you predict with the dummy set to 1, then with the dummy set to 0, and then average the difference.

In R, you can use the margins package or marginaleffects to do this for you. Unlike the manual approach, this should get the SEs right as well.

This is more accessible than the odds ratio multiplicative effects as I try avoiding odds outside gambling scenarios.