gkwreg (version 2.1.6)

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. This is a classic dataset demonstrating beta regression with interaction effects and heteroscedasticity.

Usage

ReadingSkills

Format

A data frame with 44 observations on 4 variables:

accuracy

numeric. Reading accuracy score scaled to the open unit interval (0, 1). Perfect scores of 1 were replaced with 0.99.

accuracy1

numeric. Unrestricted reading accuracy score in (0, 1), including boundary observations.

dyslexia

factor. Is the child dyslexic? Levels: no (control group) and yes (dyslexic group). Sum contrast coding is employed.

iq

numeric. Non-verbal intelligence quotient transformed to z-scores (mean = 0, SD = 1).

Details

The data were collected by Pammer and Kevan (2004) and employed by Smithson and Verkuilen (2006) in their seminal beta regression paper. The sample includes 19 dyslexic children and 25 controls recruited from primary schools in the Australian Capital Territory. Children's ages ranged from 8 years 5 months to 12 years 3 months.

Mean reading accuracy was 0.606 for dyslexic readers and 0.900 for controls. The study investigates whether dyslexia contributes to reading accuracy even when controlling for IQ (which is on average lower for dyslexics).

Transformation details: The original reading accuracy score was transformed by Smithson and Verkuilen (2006) to fit beta regression requirements:

1. First, the original accuracy was scaled using the minimal and maximal scores (a and b) that can be obtained in the test: accuracy1 = (original - a)/(b - a) (a and b values are not provided).

2. Subsequently, accuracy was obtained from accuracy1 by replacing all observations with a value of 1 with 0.99 to fit the open interval (0, 1).

The data clearly show asymmetry and heteroscedasticity (especially in the control group), making beta regression more appropriate than standard linear regression.

References

Cribari-Neto, F., and 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., and 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., and Zeileis, A. (2024). Extended-Support Beta Regression for (0, 1) Responses. arXiv:2409.07233. tools:::Rd_expr_doi("10.48550/arXiv.2409.07233")

Pammer, K., and Kevan, A. (2004). The Contribution of Visual Sensitivity, Phonological Processing and Nonverbal 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(1), 54--71.

Examples

# \donttest{
require(gkwreg)
require(gkwdist)

data(ReadingSkills)

# Example 1: Standard Kumaraswamy with interaction and heteroscedasticity
# Mean: Dyslexia × IQ interaction (do groups differ in IQ effect?)
# Precision: Main effects (variability differs by group and IQ level)
fit_kw <- gkwreg(
  accuracy ~ dyslexia * iq |
    dyslexia + iq,
  data = ReadingSkills,
  family = "kw",
  control = gkw_control(method = "L-BFGS-B", maxit = 2000)
)
summary(fit_kw)

# Interpretation:
# - Alpha (mean): Interaction shows dyslexic children benefit less from
#   higher IQ compared to controls
# - Beta (precision): Controls show more variable accuracy (higher precision)
#   IQ increases consistency of performance

# Example 2: Simpler model without interaction
fit_kw_simple <- gkwreg(
  accuracy ~ dyslexia + iq |
    dyslexia + iq,
  data = ReadingSkills,
  family = "kw",
  control = gkw_control(method = "L-BFGS-B", maxit = 2000)
)

# Test if interaction is significant
anova(fit_kw_simple, fit_kw)

# Example 3: Exponentiated Kumaraswamy for ceiling effects
# Reading accuracy often shows ceiling effects (many perfect/near-perfect scores)
# Lambda parameter can model this right-skewed asymmetry
fit_ekw <- gkwreg(
  accuracy ~ dyslexia * iq | # alpha
    dyslexia + iq | # beta
    dyslexia, # lambda: ceiling effect by group
  data = ReadingSkills,
  family = "ekw",
  control = gkw_control(method = "L-BFGS-B", maxit = 2000)
)
summary(fit_ekw)

# Interpretation:
# - Lambda varies by dyslexia status: Controls have stronger ceiling effect
#   (more compression at high accuracy) than dyslexic children

# Test if ceiling effect modeling improves fit
anova(fit_kw, fit_ekw)

# Example 4: McDonald distribution alternative
# Provides different parameterization for extreme values
fit_mc <- gkwreg(
  accuracy ~ dyslexia * iq | # gamma
    dyslexia + iq | # delta
    dyslexia * iq, # lambda: interaction affects tails
  data = ReadingSkills,
  family = "mc",
  control = gkw_control(method = "L-BFGS-B", maxit = 2000)
)
summary(fit_mc)

# Compare 3-parameter models
AIC(fit_ekw, fit_mc)
# }