betareg

Dyslexia and IQ Predicting Reading Accuracy

Description

Data for assessing the contribution of non-verbal IQ to children’s reading skills in dyslexic and non-dyslexic children.

Usage

data("ReadingSkills", package = "betareg")

Format

A data frame containing 44 observations on 3 variables.

accuracy: numeric. Reading score with maximum restricted to be 0.99 rather than 1 (see below).
dyslexia: factor. Is the child dyslexic? (A sum contrast rather than treatment contrast is employed.)
iq: numeric. Non-verbal intelligence quotient transformed to z-scores.
accuracy1: numeric. Unrestricted reading score with a maximum of 1 (see below).

Details

The data were collected by Pammer and Kevan (2004) and employed by Smithson and Verkuilen (2006). The original reading accuracy score was transformed by Smithson and Verkuilen (2006) so that accuracy is in the open unit interval (0, 1) and beta regression can be employed. First, the original accuracy was scaled using the minimal and maximal score (a and b, respectively) that can be obtained in the test: accuracy1 = (original_accuracy - a) / (b - a) (a and b are not provided). Subsequently, accuracy was obtained from accuracy1 by replacing all observations with a value of 1 with 0.99.

Kosmidis and Zeileis (2026) propose to investigate the original unrestricted accuracy1 variable using their extended-support beta mixture regression.

Source

Example 3 from Smithson and Verkuilen (2006) supplements.

References

Cribari-Neto F, Zeileis A (2010). Beta Regression in R. Journal of Statistical Software, 34(2), 1–24. doi:10.18637/jss.v034.i02

Grün B, Kosmidis I, Zeileis A (2012). Extended Beta Regression in R: Shaken, Stirred, Mixed, and Partitioned. Journal of Statistical Software, 48(11), 1–25. doi:10.18637/jss.v048.i11

Kosmidis I, Zeileis A (2026). Extended-Support Beta Regression for [0, 1] Responses. Journal of the Royal Statistical Society C, 75(1), 139–157. doi:10.1093/jrsssc/qlaf039

Pammer K, Kevan A (2004). The Contribution of Visual Sensitivity, Phonological Processing and Non-Verbal IQ to Children’s Reading. Unpublished manuscript, The Australian National University, Canberra.

Smithson M, Verkuilen J (2006). A Better Lemon Squeezer? Maximum-Likelihood Regression with Beta-Distributed Dependent Variables. Psychological Methods, 11(7), 54–71.

Examples

library("betareg")

options(digits = 4)
data("ReadingSkills", package = "betareg")

## Smithson & Verkuilen (2006, Table 5)
## OLS regression
## (Note: typo in iq coefficient: 0.3954 instead of 0.3594)
rs_ols <- lm(qlogis(accuracy) ~ dyslexia * iq, data = ReadingSkills)
summary(rs_ols)


Call:
lm(formula = qlogis(accuracy) ~ dyslexia * iq, data = ReadingSkills)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.6640 -0.3797  0.0369  0.4089  2.5035 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    1.601      0.226    7.09  1.4e-08 ***
dyslexia      -1.206      0.226   -5.34  4.0e-06 ***
iq             0.359      0.225    1.59    0.119    
dyslexia:iq   -0.423      0.225   -1.88    0.068 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.2 on 40 degrees of freedom
Multiple R-squared:  0.615, Adjusted R-squared:  0.586 
F-statistic: 21.3 on 3 and 40 DF,  p-value: 2.08e-08

## Beta regression (with numerical rather than analytic standard errors)
## (Note: Smithson & Verkuilen erroneously compute one-sided p-values)
rs_beta <- betareg(accuracy ~ dyslexia * iq | dyslexia + iq,
  data = ReadingSkills, hessian = TRUE)
summary(rs_beta)


Call:
betareg(formula = accuracy ~ dyslexia * iq | dyslexia + iq, data = ReadingSkills, 
    hessian = TRUE)

Quantile residuals:
   Min     1Q Median     3Q    Max 
-2.362 -0.587  0.303  0.942  1.587 

Coefficients (mean model with logit link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)    1.123      0.151    7.44  9.8e-14 ***
dyslexia      -0.742      0.151   -4.90  9.7e-07 ***
iq             0.486      0.167    2.91  0.00360 ** 
dyslexia:iq   -0.581      0.173   -3.37  0.00076 ***

Phi coefficients (precision model with log link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)    3.304      0.227   14.59  < 2e-16 ***
dyslexia       1.747      0.294    5.94  2.8e-09 ***
iq             1.229      0.460    2.67   0.0075 ** 
---
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: 0.576
Number of iterations in BFGS optimization: 25

## Extended-support beta mixture regression (Kosmidis & Zeileis 2026)
rs_xbx <- betareg(accuracy1 ~ dyslexia * iq | dyslexia + iq, data = ReadingSkills)
summary(rs_xbx)


Call:
betareg(formula = accuracy1 ~ dyslexia * iq | dyslexia + iq, data = ReadingSkills)

Randomized quantile residuals:
   Min     1Q Median     3Q    Max 
-2.418 -0.598  0.065  0.730  2.236 

Coefficients (mu model with logit link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)    0.903      0.217    4.17  3.1e-05 ***
dyslexia      -0.606      0.182   -3.34  0.00084 ***
iq             0.329      0.188    1.75  0.07989 .  
dyslexia:iq   -0.388      0.199   -1.94  0.05205 .  

Phi coefficients (phi model with log link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)    3.499      0.530    6.60  4.1e-11 ***
dyslexia       1.736      0.449    3.86  0.00011 ***
iq             0.697      0.571    1.22  0.22243    

Exceedence parameter (extended-support xbetax model):
        Estimate Std. Error z value Pr(>|z|)  
Log(nu)   -1.790      0.854    -2.1    0.036 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Exceedence parameter nu: 0.167
Type of estimator: ML (maximum likelihood)
Log-likelihood: 18.5 on 8 Df
Number of iterations in BFGS optimization: 29

## Coefficients in XBX are typically somewhat shrunken compared to beta
cbind(XBX = coef(rs_xbx), Beta = c(coef(rs_beta), NA))

                      XBX    Beta
(Intercept)        0.9030  1.1232
dyslexia          -0.6065 -0.7416
iq                 0.3288  0.4864
dyslexia:iq       -0.3875 -0.5813
(phi)_(Intercept)  3.4990  3.3044
(phi)_dyslexia     1.7360  1.7466
(phi)_iq           0.6965  1.2291
Log(nu)           -1.7905      NA

## Visualization
plot(accuracy1 ~ iq, data = ReadingSkills, col = c(4, 2)[dyslexia], pch = 19)
nd <- data.frame(dyslexia = "no", iq = -30:30/10)
lines(nd$iq, predict(rs_xbx, nd), col = 4)
lines(nd$iq, predict(rs_beta, nd), col = 4, lty = 5)
lines(nd$iq, plogis(predict(rs_ols, nd)), col = 4, lty = 3)
nd <- data.frame(dyslexia = "yes", iq = -30:30/10)
lines(nd$iq, predict(rs_xbx, nd), col = 2)
lines(nd$iq, predict(rs_beta, nd), col = 2, lty = 5)
lines(nd$iq, plogis(predict(rs_ols, nd)), col = 2, lty = 3)
legend("topleft", c("Dyslexia: no", "Dyslexia: yes", "OLS", "XBX", "Beta"),
  lty = c(0, 0, 3, 1, 5), pch = c(19, 19, NA, NA, NA), col = c(4, 2, 1, 1, 1), bty = "n")

## see demo("SmithsonVerkuilen2006", package = "betareg") for further details