# Interpretation of betareg coefficients where observations transformed to account for y=0 or y=1

I am running a beta-regression using betareg in R (with default logit link function). My response variable is a proportion, and may include 0 and/or 1. I've transformed the data following the betareg manual:"if y also assumes the extremes 0 and 1, a useful transformation in practice is (y · (n − 1) + 0.5)/n where n is the sample size (Smithson and Verkuilen 2006)."

Normally, I would interpret the regression coefficient in terms of exp(b) - e.g. a unit change in x changes the proportion by (exp(b)*100-100)%,
but how do I account for the transformation carried out?

An example including y=1

> yb<-seq(32)
> a<-c(23,25,36,27,21,18,18,8,15,11,8,14,15,11,6,14,11,14,8,10,9,7,17,14,12,17,8,10,8,12,10,12)
> b<-c(18,19,8,12,8,5,8,7,5,5,4,0,2,0,4,2,1,0,1,5,3,3,1,5,10,6,17,20,18,20,26,28)
>
> df<-data.frame(x=yb, A=a, B=b)
>
> ## Calculate total
> df$$n<-df$$A+df$$B > ## Calculate proportion of A > df$$prop<-df$$A/df$$n
> ## Transform
> df$$prop_trans<-(df$$prop*(df$$n-1)+0.5)/df$$n
>
> ## Betaregression
> bregt<-betareg(prop_trans ~ x, data=df)
> summary(bregt)

Call:
betareg(formula = prop_trans ~ x, data = df)

Standardized weighted residuals 2:
Min      1Q  Median      3Q     Max
-1.2435 -0.8567 -0.2671  0.6495  2.4999

Coefficients (mean model with logit link):
Estimate Std. Error z value Pr(>|z|)
(Intercept)  1.39672    0.31237   4.471 7.77e-06 ***
x           -0.04077    0.01581  -2.578  0.00993 **

Phi coefficients (precision model with identity link):
Estimate Std. Error z value Pr(>|z|)
(phi)    5.486      1.289   4.255 2.09e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Type of estimator: ML (maximum likelihood)
Log-likelihood: 11.44 on 3 Df
Pseudo R-squared: 0.1245
Number of iterations: 12 (BFGS) + 1 (Fisher scoring)
>
> ## Interpretation - unit change in x changes the proportion by ...%
> b=summary(bregt)$$coef$$mean[2,1]
> round(exp(b)*100-100,digits=1)
[1] -4