marginal.truescore.reliability: True-Score Reliability for Dichotomous Data

In sirt: Supplementary Item Response Theory Models

True-Score Reliability for Dichotomous Data

Description

This function computes the marginal true-score reliability for dichotomous data (Dimitrov, 2003; May & Nicewander, 1994) for the four-parameter logistic item response model (see rasch.mml2 for details regarding this IRT model).

Usage

marginal.truescore.reliability(b, a=1+0*b,c=0*b,d=1+0*b,
    mean.trait=0, sd.trait=1, theta.k=seq(-6,6,len=200) )

Arguments

b	Vector of item difficulties
a	Vector of item discriminations
c	Vector of guessing parameters
d	Vector of upper asymptotes
mean.trait	Mean of trait distribution
sd.trait	Standard deviation of trait distribution
theta.k	Grid at which the trait distribution should be evaluated

Value

A list with following entries:

rel.test	Reliability of the test
item	True score variance (sig2.true, error variance (sig2.error) and item reliability (rel.item). Expected proportions correct are in the column pi.
pi	Average proportion correct for all items and persons
sig2.tau	True score variance \sigma^2_{\tau} (calculated by the formula in May & Nicewander, 1994)
sig2.error	Error variance \sigma^2_{e}

References

Dimitrov, D. (2003). Marginal true-score measures and reliability for binary items as a function of their IRT parameters. Applied Psychological Measurement, 27, 440-458.

May, K., & Nicewander, W. A. (1994). Reliability and information functions for percentile ranks. Journal of Educational Measurement, 31, 313-325.

See Also

See greenyang.reliability for calculating the reliability for multidimensional measures.

Examples

#############################################################################
# EXAMPLE 1: Dimitrov (2003) Table 1 - 2PL model
#############################################################################

# item discriminations
a <- 1.7*c(0.449,0.402,0.232,0.240,0.610,0.551,0.371,0.321,0.403,0.434,0.459,
    0.410,0.302,0.343,0.225,0.215,0.487,0.608,0.341,0.465)
# item difficulties
b <- c( -2.554,-2.161,-1.551,-1.226,-0.127,-0.855,-0.568,-0.277,-0.017,
    0.294,0.532,0.773,1.004,1.250,1.562,1.385,2.312,2.650,2.712,3.000 )

marginal.truescore.reliability( b=b, a=a )
  ##   Reliability=0.606

#############################################################################
# EXAMPLE 2: Dimitrov (2003) Table 2
#  3PL model: Poetry items (4 items)
#############################################################################

# slopes, difficulties and guessing parameters
a <- 1.7*c(1.169,0.724,0.554,0.706 )
b <- c(0.468,-1.541,-0.042,0.698 )
c <- c(0.159,0.211,0.197,0.177 )

res <- sirt::marginal.truescore.reliability( b=b, a=a, c=c)
  ##   Reliability=0.403
  ##   > round( res$item, 3 )
  ##     item    pi sig2.tau sig2.error rel.item
  ##   1    1 0.463    0.063      0.186    0.252
  ##   2    2 0.855    0.017      0.107    0.135
  ##   3    3 0.605    0.026      0.213    0.107
  ##   4    4 0.459    0.032      0.216    0.130

#############################################################################
# EXAMPLE 3: Reading Data
#############################################################################
data( data.read)

#***
# Model 1: 1PL
mod <- sirt::rasch.mml2( data.read )
marginal.truescore.reliability( b=mod$item$b )
  ##   Reliability=0.653

#***
# Model 2: 2PL
mod <- sirt::rasch.mml2( data.read, est.a=1:12 )
marginal.truescore.reliability( b=mod$item$b, a=mod$item$a)
  ##   Reliability=0.696

## Not run: 
# compare results with Cronbach's alpha and McDonald's omega
# posing a 'wrong model' for normally distributed data
library(psych)
psych::omega(dat, nfactors=1)     # 1 factor
  ##  Omega_h for 1 factor is not meaningful, just omega_t
  ##   Omega
  ##   Call: omega(m=dat, nfactors=1)
  ##   Alpha:                 0.69
  ##   G.6:                   0.7
  ##   Omega Hierarchical:    0.66
  ##   Omega H asymptotic:    0.95
  ##   Omega Total            0.69

##! Note that alpha in psych is the standardized one.

## End(Not run)

sirt documentation built on Aug. 11, 2023, 5:07 p.m.