Calculate maximal reliability

Description

Calculate maximal reliability of a scale

Usage

maximalRelia(object)

Arguments

Details

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

W = old{w}^prime old{x} ,

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

ρ = frac{old{w}^prime old{S}_T old{w}}{old{w}^prime old{S}_X old{w}}

where old{S}_T is the covariance matrix explained by true scores and old{S}_X is the observed covariance matrix. Numerical method is used to find old{w} in this function.

For continuous items, old{S}_T can be calculated by

old{S}_T = Λ Ψ Λ^prime,

where Λ is the factor loading matrix and Ψ is the covariance matrix among factors. old{S}_X is directly obtained by covariance among items.

For categorical items, Green and Yang's (2009) method is used for calculating old{S}_T and old{S}_X. The element i and j of old{S}_T can be calculated by

≤ft[old{S}_T ight]_{ij} = ∑^{C_i - 1}_{c_i = 1} ∑^{C_j - 1}_{c_j - 1} Φ_2≤ft( τ_{x_{c_i}}, τ_{x_{c_j}}, ≤ft[ Λ Ψ Λ^prime ight]_{ij} ight) - ∑^{C_i - 1}_{c_i = 1} Φ_1(τ_{x_{c_i}}) ∑^{C_j - 1}_{c_j - 1} Φ_1(τ_{x_{c_j}}),

where C_i and C_j represents the number of thresholds in Items i and j, τ_{x_{c_i}} represents the threshold c_i of Item i, τ_{x_{c_j}} represents the threshold c_i of Item j, Φ_1(τ_{x_{c_i}}) is the cumulative probability of τ_{x_{c_i}} given a univariate standard normal cumulative distribution and Φ_2≤ft( τ_{x_{c_i}}, τ_{x_{c_j}}, ρ ight) is the joint cumulative probability of τ_{x_{c_i}} and τ_{x_{c_j}} given a bivariate standard normal cumulative distribution with a correlation of ρ

Each element of old{S}_X can be calculated by

≤ft[old{S}_T ight]_{ij} = ∑^{C_i - 1}_{c_i = 1} ∑^{C_j - 1}_{c_j - 1} Φ_2≤ft( τ_{V_{c_i}}, τ_{V_{c_j}}, ρ^*_{ij} ight) - ∑^{C_i - 1}_{c_i = 1} Φ_1(τ_{V_{c_i}}) ∑^{C_j - 1}_{c_j - 1} Φ_1(τ_{V_{c_j}}),

where ρ^*_{ij} is a polychoric correlation between Items i and j.

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).

Author(s)

Sunthud Pornprasertmanit (psunthud@gmail.com)

References

Li, H. (1997). A unifying expression for the maximal reliability of a linear composite. Psychometrika, 62, 245-249.

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: Guilford.

See Also

reliability for reliability of an unweighted composite score

Examples

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")

Results