ReadingSkills: Dyslexia and IQ Predicting Reading Accuracy

In betareg: Beta Regression

Description

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

Usage

data("ReadingSkills")

Format

A data frame containing 44 observations on 3 variables.

accuracy

reading score scaled to the open unit interval (see below).

dyslexia

factor. Is the child dyslexic? (A sum contrast rather than treatment contrast is employed.)

iq

non-verbal intelligence quotient transformed to z-scores.

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: (original_accuracy - a) / (b - a) (a and b are not provided). Subsequently, the scaled score is transformed to the unit interval using a continuity correction: (scaled_accuracy * (n-1) - 0.5) / n (either with some rounding or using n = 50 rather than 44).

Source

Example 3 from Smithson and Verkuilen (2006) supplements.

References

Cribari-Neto, F., and 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., and 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

Pammer, K., and 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 Unpublished manuscript, The Australian National University, Canberra.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., and Verkuilen, J. (2006). A Better Lemon Squeezer? Maximum-Likelihood Regression with Beta-Distributed Dependent Variables. Psychological Methods, 11(7), 54–71.

See Also

betareg, MockJurors, StressAnxiety

Examples

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)
## 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)

## visualization
plot(accuracy ~ iq, data = ReadingSkills, col = as.numeric(dyslexia), pch = 19)
nd <- data.frame(dyslexia = "no", iq = -30:30/10)
lines(nd$iq, predict(rs_beta, nd))
lines(nd$iq, plogis(predict(rs_ols, nd)), lty = 2)
nd <- data.frame(dyslexia = "yes", iq = -30:30/10)
lines(nd$iq, predict(rs_beta, nd), col = 2)
lines(nd$iq, plogis(predict(rs_ols, nd)), col = 2, lty = 2)

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

Example output

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

Residuals:
     Min       1Q   Median       3Q      Max 
-2.66405 -0.37966  0.03687  0.40887  2.50345 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   1.6011     0.2259   7.089 1.41e-08 ***
dyslexia     -1.2056     0.2259  -5.338 4.01e-06 ***
iq            0.3594     0.2255   1.594   0.1188    
dyslexia:iq  -0.4229     0.2255  -1.875   0.0681 .  
---
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:  0.6151,	Adjusted R-squared:  0.5862 
F-statistic: 21.31 on 3 and 40 DF,  p-value: 2.083e-08


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

Standardized weighted residuals 2:
    Min      1Q  Median      3Q     Max 
-2.3900 -0.6416  0.1572  0.8524  1.6446 

Coefficients (mean model with logit link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)   1.1232     0.1509   7.444 9.76e-14 ***
dyslexia     -0.7416     0.1515  -4.897 9.74e-07 ***
iq            0.4864     0.1671   2.911 0.003603 ** 
dyslexia:iq  -0.5813     0.1726  -3.368 0.000757 ***

Phi coefficients (precision model with log link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)   3.3044     0.2265  14.589  < 2e-16 ***
dyslexia      1.7466     0.2940   5.941 2.83e-09 ***
iq            1.2291     0.4596   2.674  0.00749 ** 
---
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 in BFGS optimization: 25 

betareg documentation built on Feb. 10, 2021, 1:07 a.m.