Reliability for Dichotomous Item Response Data Using the Method of Green and Yang (2009)

This function estimates the model-based reliability of dichotomous data using the Green & Yang (2009) method. The underlying factor model is \(D\)-dimensional where the dimension \(D\) is specified by the argument nfactors. The factor solution is subject to the application of the Schmid-Leiman transformation (see Reise, 2012; Reise, Bonifay, & Haviland, 2013; Reise, Moore, & Haviland, 2010).

Usage

greenyang.reliability(object.tetra, nfactors)

Arguments

object.tetra: Object as the output of the function tetrachoric, the fa.parallel.poly from the psych package or the tetrachoric2 function (from sirt). This object can also be created as a list by the user where the tetrachoric correlation must must be in the list entry rho and the thresholds must be in the list entry thresh.
nfactors: Number of factors (dimensions)

Value

A data frame with columns:

coefficient: Name of the reliability measure. omega_1 (Omega) is the reliability estimate for the total score for dichotomous data based on a one-factor model, omega_t (Omega Total) is the estimate for a \(D\)-dimensional model. For the nested factor model, omega_h (Omega Asymptotic) is the reliability of the general factor model, omega_ha (Omega Hierarchical Asymptotic) eliminates item-specific variance. The explained common variance (ECV) explained by the common factor is based on the \(D\)-dimensional but does not take item thresholds into account. The amount of explained variance ExplVar is defined as the quotient of the first eigenvalue of the tetrachoric correlation matrix to the sum of all eigenvalues. The statistic EigenvalRatio is the ratio of the first and second eigenvalue.
dimensions: Number of dimensions
estimate: Reliability estimate

References

Green, S. B., & Yang, Y. (2009). Reliability of summed item scores using structural equation modeling: An alternative to coefficient alpha. Psychometrika, 74, 155-167.

Reise, S. P. (2012). The rediscovery of bifactor measurement models. Multivariate Behavioral Research, 47, 667-696.

Reise, S. P., Bonifay, W. E., & Haviland, M. G. (2013). Scoring and modeling psychological measures in the presence of multidimensionality. Journal of Personality Assessment, 95, 129-140.

Reise, S. P., Moore, T. M., & Haviland, M. G. (2010). Bifactor models and rotations: Exploring the extent to which multidimensional data yield univocal scale scores, Journal of Personality Assessment, 92, 544-559.

Note

This function needs the psych package.

Examples

if (FALSE) {
#############################################################################
# EXAMPLE 1: Reliability estimation of Reading dataset data.read
#############################################################################
miceadds::library_install("psych")
set.seed(789)
data( data.read )
dat <- data.read

# calculate matrix of tetrachoric correlations
dat.tetra <- psych::tetrachoric(dat)      # using tetrachoric from psych package
dat.tetra2 <- sirt::tetrachoric2(dat)       # using tetrachoric2 from sirt package

# perform parallel factor analysis
fap <- psych::fa.parallel.poly(dat, n.iter=1 )
  ##   Parallel analysis suggests that the number of factors=3
  ##   and the number of components=2

# parallel factor analysis based on tetrachoric correlation matrix
##       (tetrachoric2)
fap2 <- psych::fa.parallel(dat.tetra2$rho, n.obs=nrow(dat),  n.iter=1 )
  ## Parallel analysis suggests that the number of factors=6
  ## and the number of components=2
  ## Note that in this analysis, uncertainty with respect to thresholds is ignored.

# calculate reliability using a model with 4 factors
greenyang.reliability( object.tetra=dat.tetra, nfactors=4 )
  ##                                            coefficient dimensions estimate
  ## Omega Total (1D)                               omega_1          1    0.771
  ## Omega Total (4D)                               omega_t          4    0.844
  ## Omega Hierarchical (4D)                        omega_h          4    0.360
  ## Omega Hierarchical Asymptotic (4D)            omega_ha          4    0.427
  ## Explained Common Variance (4D)                     ECV          4    0.489
  ## Explained Variance (First Eigenvalue)          ExplVar         NA   35.145
  ## Eigenvalue Ratio (1st to 2nd Eigenvalue) EigenvalRatio         NA    2.121

# calculation of Green-Yang-Reliability based on tetrachoric correlations
#   obtained by tetrachoric2
greenyang.reliability( object.tetra=dat.tetra2, nfactors=4 )

# The same result will be obtained by using fap as the input
greenyang.reliability( object.tetra=fap, nfactors=4 ) }