marginal_rxx: Function to calculate the marginal reliability

In mirt: Multidimensional Item Response Theory

Function to calculate the marginal reliability

Description

Given an estimated model and a prior density function, compute the marginal reliability (Thissen and Wainer, 2001). This is only available for unidimensional tests.

Usage

marginal_rxx(mod, density = dnorm, ...)

Arguments

mod	an object of class 'SingleGroupClass'
density	a density function to use for integration. Default assumes the latent traits are from a normal (Gaussian) distribution
...	additional arguments passed to the density function

Author(s)

Phil Chalmers [e-mail]

References

Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v048.i06")}

Thissen, D. and Wainer, H. (2001). Test Scoring. Lawrence Erlbaum Associates.

See Also

empirical_rxx, extract.group, testinfo

Examples

dat <- expand.table(deAyala)
mod <- mirt(dat)

# marginal estimate treating item parameters as known
marginal_rxx(mod)

# compare to alpha
itemstats(dat)$overall$alpha



# empirical estimate (assuming the same prior)
fscores(mod, returnER = TRUE)

# empirical rxx the alternative way, given theta scores and SEs
fs <- fscores(mod, full.scores.SE=TRUE)
head(fs)
empirical_rxx(fs)

#############
# example demonstrating correlation attenuation

theta <- rnorm(1000)
X <- theta + rnorm(1000, sd=2)
cor(X, theta)    # correlation without measurement error (what you want)

# measured with a 10 item GRM test
nitems <- 10
a <- matrix(rlnorm(nitems,.2,.3))
diffs <- t(apply(matrix(runif(nitems*4, .3, 1), nitems), 1, cumsum))
diffs <- -(diffs - rowMeans(diffs))
d <- diffs + rnorm(nitems)
dat <- simdata(a, d, itemtype = 'graded', Theta=matrix(theta))

# correlation with total score (attenuated)
cor(rowSums(dat), X)

# fit single group model
mod <- mirt(dat)

# EAP correlation (also attenuated)
fs <- fscores(mod)
cor(fs, X)

# correction for attenuation, r_x.theta = r_x.theta.hat / sqrt(rxx_theta.hat)
(rxx <- marginal_rxx(mod))  # alternatively, could use empirical_rxx()
cor(fs, X) / sqrt(rxx)  # correction estimate
cor(X, theta)           # compare to true correlation

mirt documentation built on Sept. 10, 2025, 1:07 a.m.