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:

$$ W = \bold{w}^\prime \bold{x} ,$$

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 $j$.

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.

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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