

<!-- README.md is generated from README.qmd via: quarto render README.qmd --to gfm -->

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# Beta Regression

## Overview

The R package [betareg](https://zeileis.codeberg.page/betareg/)
provides:

- Beta regression for modeling beta-distributed dependent variables on
  the open unit interval (0, 1), e.g., rates and proportions, see
  Cribari-Neto and Zeileis (2010,
  [doi:10.18637/jss.v034.i02](https://doi.org/10.18637/jss.v034.i02)).

- Extended-support beta regression models for variables on the closed
  unit interval \[0, 1\] with boundary observations at 0 and/or 1, see
  Kosmidis and Zeileis (2026,
  [doi:10.1093/jrsssc/qlaf039](https://doi.org/10.1093/jrsssc/qlaf039)).

- Alternative specifications of the classical beta regression model:
  Bias-corrected and bias-reduced estimation, finite mixture models, and
  recursive partitioning for (0, 1) beta regression, see Grün, Kosmidis,
  and Zeileis (2012,
  [doi:10.18637/jss.v048.i11](https://doi.org/10.18637/jss.v048.i11)).

## Installation

The stable version of `betareg` is available on
[CRAN](https://CRAN.R-project.org/package=betareg):

``` r
install.packages("betareg")
```

The latest development version can be installed from
[R-universe](https://zeileis.R-universe.dev/betareg):

``` r
install.packages("betareg", repos = "https://zeileis.R-universe.dev")
```

## License

The package is available under the [General Public License version
3](https://www.gnu.org/licenses/gpl-3.0.html) or [version
2](https://www.gnu.org/licenses/old-licenses/gpl-2.0.html)

## Illustration

A nice first illustration of beta regression is the analysis of reading
accuracy scores from primary school children from Smithson & Verkuilen
(2006). Package and data can be loaded via:

``` r
library("betareg")
data("ReadingSkills", package = "betareg")
```

The reading `accuracy` was scaled to be within (0, 1). Its mean is
explained by verbal `iq` score with separate lines by `dyslexia`
(control vs. dyslexic). The precision parameter is explained by main
effects of the two explanatory variables. More details are provided in
`?ReadingSkills`.

``` r
br <- betareg(accuracy ~ dyslexia * iq | dyslexia + iq, data = ReadingSkills)
summary(br)
## 
## Call:
## betareg(formula = accuracy ~ dyslexia * iq | dyslexia + iq, data = ReadingSkills)
## 
## Quantile residuals:
##     Min      1Q  Median      3Q     Max 
## -2.3625 -0.5872  0.3026  0.9425  1.5874 
## 
## Coefficients (mean model with logit link):
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   1.1232     0.1428   7.864 3.73e-15 ***
## dyslexia     -0.7416     0.1428  -5.195 2.04e-07 ***
## iq            0.4864     0.1331   3.653 0.000259 ***
## dyslexia:iq  -0.5813     0.1327  -4.381 1.18e-05 ***
## 
## Phi coefficients (precision model with log link):
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   3.3044     0.2227  14.835  < 2e-16 ***
## dyslexia      1.7466     0.2623   6.658 2.77e-11 ***
## iq            1.2291     0.2672   4.600 4.23e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
## 
## Type of estimator: ML (maximum likelihood)
## Log-likelihood:  65.9 on 7 Df
## Pseudo R-squared: 0.5756
## Number of iterations: 25 (BFGS) + 1 (Fisher scoring)
```

The regression summary shows that `accuracy` increases with `iq` for the
control group but not for the dyslexic group (even slightly decreases).
This can be brought out more clearly graphically. This also highlights
that the model employs a logit link so that the fitted curves always
remain within (0, 1).

``` r
pal <- palette.colors()[c(4, 8)]
pch <- c(19, 17)
plot(accuracy ~ iq, data = ReadingSkills, col = pal[dyslexia], pch = pch[dyslexia])
iq <- -30:30/10
lines(iq, predict(br, newdata = data.frame(dyslexia = "no", iq = iq)), col = pal[1], lwd = 2)
lines(iq, predict(br, newdata = data.frame(dyslexia = "yes", iq = iq)), col = pal[2], lwd = 2)
legend("topleft", c("Control", "Dyslexic"), pch = pch, col = pal, bty = "n")
```

<img src="man/figures/README-plot-1.png" style="width:75.0%" />

## Extended models

For going beyond this basic analysis the following extensions can be
considered.

### Extended-support beta regression

To analyze the original accuracy scores in \[0, 1\] (without scaling the
perfect scores of `1` to `0.99`), use the variable `accuracy1` (instead
of `accuracy`) in the code above. The `betareg()` model then
automatically estimates an additional exceedence parameter that accounts
for the boundary probability of a perfect score. For this data set, most
coefficients shrink a bit, rendering some coefficients only weakly
significant but the qualitative interpretations still remain similar.

``` r
betareg(accuracy1 ~ dyslexia * iq | dyslexia + iq, data = ReadingSkills)
```

See Kosmidis and Zeileis (2026) and the documentation of `betareg()` for
more details.

### Bias reduction

Bias-reduced estimation (instead of the default maximum likelihood
estimation) can be used by adding the argument `type = "BR"` in
`betareg()`. This slightly shrinks all coefficient estimates but, on
this data, leads to qualitatively identical results.

``` r
betareg(accuracy ~ dyslexia * iq | dyslexia + iq, data = ReadingSkills, type = "BR")
```

See Grün et al. (2012) and the documentation of `betareg()` for more
details.

### Beta regression trees

To find subgroups in a beta regression by recursively splitting
subsamples, beta regression trees can be used. Here, this strategy can
be used to figure out the different `iq` effects by `dyslexia` rather
than fixing the variables’ interaction in advance.

``` r
betatree(accuracy ~ iq | iq, ~ dyslexia + ..., data = ReadingSkills, minsize = 10)
```

See Grün et al. (2012) and the documentation of `betatree()` for more
details.

### Finite mixtures of beta regressions

To find clusters in a beta regression finite mixtures of beta
regressions can be used. Here, this technique can be employed to find
the different `iq` effects in the data without even requiring the
`dyslexia` information.

``` r
betamix(accuracy ~ iq, data = ReadingSkills, k = 3, ...)
```

See Grün et al. (2012) and the documentation of `betamix()` for more
details.
