betareg (version 3.2-1)

ReadingSkills: 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 (2024) propose to investigate the original unrestricted accuracy1 variable using their extended-support beta mixture regression.

References

Cribari-Neto F, Zeileis A (2010). Beta Regression in R. Journal of Statistical Software, 34(2), 1--24. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.18637/jss.v048.i11")

Kosmidis I, Zeileis A (2024). Extended-Support Beta Regression for [0, 1] Responses. 2409.07233, arXiv.org E-Print Archive. tools:::Rd_expr_doi("10.48550/arXiv.2409.07233")

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 Unpublished manuscript, The Australian National University, Canberra.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.

See Also

betareg, MockJurors, StressAnxiety

Examples

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)
## 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)

## Extended-support beta mixture regression (Kosmidis & Zeileis 2024)
rs_xbx <- betareg(accuracy1 ~ dyslexia * iq | dyslexia + iq, data = ReadingSkills)
summary(rs_xbx)

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