sirt (version 0.31-20)

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

Description

This function computes the marginal true-score reliability for dichotous data (Dimitrov, 2003; May & Nicewander, 1994) for the four-parameter logistic item response 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 and 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 Mesaurement, 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)

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