marginal.truescore.reliability: 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

Vector of item difficulties

Vector of item discriminations

Vector of guessing parameters

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.testReliability of the test
itemTrue score variance (sig2.true, error variance (sig2.error) and item reliability (rel.item). Expected proportions correct are in the column pi.
piAverage proportion correct for all items and persons
sig2.tauTrue score variance $\sigma^2_{\tau}$ (calculated by the formula in May & Nicewander, 1994)
sig2.errorError 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 Mesaurement, 31, 313-325.

Examples

Run this code

#############################################################################
# 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 <- 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 <- rasch.mml2( data.read )
marginal.truescore.reliability( b=mod$item$b )
  ##   Reliability= 0.653 

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

# 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.