maximalRelia  R Documentation 
Calculate maximal reliability of a scale
maximalRelia(object, omit.imps = c("no.conv", "no.se"))
object 
A 
omit.imps 

Given that a composite score (W) is a weighted sum of item scores:
W = \bold{w}^\prime \bold{x} ,
where \bold{x} is a k \times 1 vector of the scores of each item, \bold{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 \bold{w} such that reliability attains its maximum (Li, 1997; Raykov, 2012). Note that the reliability can be obtained by
ρ = \frac{\bold{w}^\prime \bold{S}_T \bold{w}}{\bold{w}^\prime \bold{S}_X \bold{w}}
where \bold{S}_T is the covariance matrix explained by true scores and \bold{S}_X is the observed covariance matrix. Numerical method is used to find \bold{w} in this function.
For continuous items, \bold{S}_T can be calculated by
\bold{S}_T = Λ Ψ Λ^\prime,
where Λ is the factor loading matrix and Ψ is the covariance matrix among factors. \bold{S}_X is directly obtained by covariance among items.
For categorical items, Green and Yang's (2009) method is used for calculating \bold{S}_T and \bold{S}_X. The element i and j of \bold{S}_T can be calculated by
≤ft[\bold{S}_T\right]_{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 \right]_{ij} \right)  ∑^{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 \right) 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 \bold{S}_X can be calculated by
≤ft[\bold{S}_T\right]_{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} \right)  ∑^{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.
Maximal reliability values of each group. The maximalreliability
weights are also provided. Users may extracted the weighted by the
attr
function (see example below).
Sunthud Pornprasertmanit (psunthud@gmail.com)
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.
reliability
for reliability of an unweighted
composite score
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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