reliabilityL2: Calculate the reliability values of a second-order factor

Description

Calculate the reliability values (coefficient omega) of a second-order factor

Usage

reliabilityL2(object, secondFactor, omit.imps = c("no.conv", "no.se"))

Arguments

object

A '>lavaan or '>lavaan.mi object, expected to contain a least one exogenous higher-order common factor.

secondFactor

The name of a single second-order factor in the model fitted in object. The function must be called multiple times to estimate reliability for each higher-order factor.

omit.imps

character vector specifying criteria for omitting imputations from pooled results. Can include any of c("no.conv", "no.se", "no.npd"), the first 2 of which are the default setting, which excludes any imputations that did not converge or for which standard errors could not be computed. The last option ("no.npd") would exclude any imputations which yielded a nonpositive definite covariance matrix for observed or latent variables, which would include any "improper solutions" such as Heywood cases. NPD solutions are not excluded by default because they are likely to occur due to sampling error, especially in small samples. However, gross model misspecification could also cause NPD solutions, users can compare pooled results with and without this setting as a sensitivity analysis to see whether some imputations warrant further investigation.

Value

Reliability values at Levels 1 and 2 of the second-order factor, as well as the partial reliability value at Level 1

Details

The first formula of the coefficient omega (in the reliability) will be mainly used in the calculation. The model-implied covariance matrix of a second-order factor model can be separated into three sources: the second-order factor, the uniqueness of the first-order factor, and the measurement error of indicators:

$$ \hat{\Sigma} = \Lambda \bold{B} \Phi_2 \bold{B}^{\prime} \Lambda^{\prime} + \Lambda \Psi_{u} \Lambda^{\prime} + \Theta, $$

where $\hat{\Sigma}$ is the model-implied covariance matrix, $\Lambda$ is the first-order factor loading, $\bold{B}$ is the second-order factor loading, $\Phi_2$ is the covariance matrix of the second-order factors, $\Psi_{u}$ is the covariance matrix of the unique scores from first-order factors, and $\Theta$ is the covariance matrix of the measurement errors from indicators. Thus, the proportion of the second-order factor explaining the total score, or the coefficient omega at Level 1, can be calculated:

$$ \omega_{L1} = \frac{\bold{1}^{\prime} \Lambda \bold{B} \Phi_2 \bold{B}^{\prime} \Lambda^{\prime} \bold{1}}{\bold{1}^{\prime} \Lambda \bold{B} \Phi_2 \bold{B} ^{\prime} \Lambda^{\prime} \bold{1} + \bold{1}^{\prime} \Lambda \Psi_{u} \Lambda^{\prime} \bold{1} + \bold{1}^{\prime} \Theta \bold{1}}, $$

where $\bold{1}$ is the k-dimensional vector of 1 and k is the number of observed variables. When model-implied covariance matrix among first-order factors ($\Phi_1$) can be calculated:

$$ \Phi_1 = \bold{B} \Phi_2 \bold{B}^{\prime} + \Psi_{u}, $$

Thus, the proportion of the second-order factor explaining the varaince at first-order factor level, or the coefficient omega at Level 2, can be calculated:

$$ \omega_{L2} = \frac{\bold{1_F}^{\prime} \bold{B} \Phi_2 \bold{B}^{\prime} \bold{1_F}}{\bold{1_F}^{\prime} \bold{B} \Phi_2 \bold{B}^{\prime} \bold{1_F} + \bold{1_F}^{\prime} \Psi_{u} \bold{1_F}}, $$

where $\bold{1_F}$ is the F-dimensional vector of 1 and F is the number of first-order factors.

The partial coefficient omega at Level 1, or the proportion of observed variance explained by the second-order factor after partialling the uniqueness from the first-order factor, can be calculated:

$$ \omega_{L1} = \frac{\bold{1}^{\prime} \Lambda \bold{B} \Phi_2 \bold{B}^{\prime} \Lambda^{\prime} \bold{1}}{\bold{1}^{\prime} \Lambda \bold{B} \Phi_2 \bold{B}^{\prime} \Lambda^{\prime} \bold{1} + \bold{1}^{\prime} \Theta \bold{1}}, $$

Note that if the second-order factor has a direct factor loading on some observed variables, the observed variables will be counted as first-order factors.

Examples

Run this code

# NOT RUN {
library(lavaan)

HS.model3 <- ' visual  =~ x1 + x2 + x3
               textual =~ x4 + x5 + x6
               speed   =~ x7 + x8 + x9
			          higher =~ visual + textual + speed'

fit6 <- cfa(HS.model3, data = HolzingerSwineford1939)
reliability(fit6) # Should provide a warning for the endogenous variables
reliabilityL2(fit6, "higher")

# }