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semTools (version 0.5-1)

maximalRelia: Calculate maximal reliability

Description

Calculate maximal reliability of a scale

Usage

maximalRelia(object)

Arguments

object

The lavaan model object provided after running the cfa, sem, growth, or lavaan functions.

Value

Maximal reliability values of each group. The maximal-reliability weights are also provided. Users may extracted the weighted by the attr function (see example below).

Details

Given that a composite score (W) is a weighted sum of item scores:

W=\boldw\boldx,

where \boldx is a k×1 vector of the scores of each item, \boldw is a k×1 weight vector of each item, and k represents the number of items. Then, maximal reliability is obtained by finding \boldw such that reliability attains its maximum (Li, 1997; Raykov, 2012). Note that the reliability can be obtained by

ρ=\boldw\boldST\boldw\boldw\boldSX\boldw

where \boldST is the covariance matrix explained by true scores and \boldSX is the observed covariance matrix. Numerical method is used to find \boldw in this function.

For continuous items, \boldST can be calculated by

\boldST=ΛΨΛ,

where Λ is the factor loading matrix and Ψ is the covariance matrix among factors. \boldSX is directly obtained by covariance among items.

For categorical items, Green and Yang's (2009) method is used for calculating \boldST and \boldSX. The element i and j of \boldST can be calculated by

[\boldST]ij=ci=1Ci1cj1Cj1Φ2(τxci,τxcj,[ΛΨΛ]ij)ci=1Ci1Φ1(τxci)cj1Cj1Φ1(τxcj),

where Ci and Cj represents the number of thresholds in Items i and j, τxci represents the threshold ci of Item i, τxcj represents the threshold ci of Item j, Φ1(τxci) is the cumulative probability of τxci given a univariate standard normal cumulative distribution and Φ2(τxci,τxcj,ρ) is the joint cumulative probability of τxci and τxcj given a bivariate standard normal cumulative distribution with a correlation of ρ

Each element of \boldSX can be calculated by

[\boldST]ij=ci=1Ci1cj1Cj1Φ2(τVci,τVcj,ρij)ci=1Ci1Φ1(τVci)cj1Cj1Φ1(τVcj),

where ρij is a polychoric correlation between Items i and j.

References

Li, H. (1997). A unifying expression for the maximal reliability of a linear composite. Psychometrika, 62(2), 245--249. doi:10.1007/BF02295278

Raykov, T. (2012). Scale construction and development using structural equation modeling. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (pp. 472--494). New York, NY: Guilford.

See Also

reliability for reliability of an unweighted composite score

Examples

Run this code
# NOT RUN {
total <- 'f =~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 '
fit <- cfa(total, data = HolzingerSwineford1939)
maximalRelia(fit)

# Extract the weight
mr <- maximalRelia(fit)
attr(mr, "weight")

# }

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