Calculate maximal reliability of a scale
maximalRelia(object)The lavaan model object provided after running the cfa, sem, growth, or lavaan functions.
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).
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
$$ \rho = \frac{\bold{w}^\prime \bold{S}_T \bold{w}}{\bold{w}^\prime \bold{S}_X \bold{w}}$$
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
$$ \bold{S}_T = \Lambda \Psi \Lambda^\prime,$$
where \(\Lambda\) is the factor loading matrix and \(\Psi\) 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
$$ \left[\bold{S}_T\right]_{ij} = \sum^{C_i - 1}_{c_i = 1} \sum^{C_j - 1}_{c_j - 1} \Phi_2\left( \tau_{x_{c_i}}, \tau_{x_{c_j}}, \left[ \Lambda \Psi \Lambda^\prime \right]_{ij} \right) - \sum^{C_i - 1}_{c_i = 1} \Phi_1(\tau_{x_{c_i}}) \sum^{C_j - 1}_{c_j - 1} \Phi_1(\tau_{x_{c_j}}),$$
where \(C_i\) and \(C_j\) represents the number of thresholds in Items \(i\) and \(j\), \(\tau_{x_{c_i}}\) represents the threshold \(c_i\) of Item \(i\), \(\tau_{x_{c_j}}\) represents the threshold \(c_i\) of Item \(j\), \( \Phi_1(\tau_{x_{c_i}})\) is the cumulative probability of \(\tau_{x_{c_i}}\) given a univariate standard normal cumulative distribution and \(\Phi_2\left( \tau_{x_{c_i}}, \tau_{x_{c_j}}, \rho \right)\) is the joint cumulative probability of \(\tau_{x_{c_i}}\) and \(\tau_{x_{c_j}}\) given a bivariate standard normal cumulative distribution with a correlation of \(\rho\)
Each element of \(\bold{S}_X\) can be calculated by
$$ \left[\bold{S}_T\right]_{ij} = \sum^{C_i - 1}_{c_i = 1} \sum^{C_j - 1}_{c_j - 1} \Phi_2\left( \tau_{V_{c_i}}, \tau_{V_{c_j}}, \rho^*_{ij} \right) - \sum^{C_i - 1}_{c_i = 1} \Phi_1(\tau_{V_{c_i}}) \sum^{C_j - 1}_{c_j - 1} \Phi_1(\tau_{V_{c_j}}),$$
where \(\rho^*_{ij}\) is a polychoric correlation between Items \(i\) and \(j\).
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.
reliability for reliability of an unweighted composite score
# 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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