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

Question

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

Answer 1

In the actual reference they say the following, suggesting this is not really applicable for proportions in the 'natural scale':

Our recommendation applies to a sample of scores from a continuous bounded scale that has already been linearly transformed to the [0,1] interval.

Other suggestions from the supplement include moving boundary values by a constant, possibly based on the sample size, which will introduce a small amount of bias. None of these will meaningfully change the way you'd interpret the model parameters though, they will just restrict the data to be within the support of the beta distribution (and thereby produce small numerical differences in the actual fit).

I think a better solution here might be to fit a zero- and/or one-inflated beta regression that actually handles the boundary cases.

As an aside, exponentiating the model parameters will not give you the ratio of change in proportion but in odds, these are not the same thing!