semTools (version 0.5-1)

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)

Arguments

object

The lavaan model object provided after running the cfa, sem, growth, or lavaan functions that has a second-order factor

secondFactor

The name of the second-order factor

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:

$$\hat{\mathrm{\Sigma }}=\mathrm{\Lambda }\text{bold}B{\mathrm{\Phi }}_2\text{bold}{B}^{\prime }{\mathrm{\Lambda }}^{\prime }+\mathrm{\Lambda }{\mathrm{\Psi }}_u{\mathrm{\Lambda }}^{\prime }+\mathrm{\Theta },$$

where $\hat{\mathrm{\Sigma }}$ is the model-implied covariance matrix, $\mathrm{\Lambda }$ is the first-order factor loading, $\text{bold}B$ is the second-order factor loading, ${\mathrm{\Phi }}_2$ is the covariance matrix of the second-order factors, ${\mathrm{\Psi }}_u$ is the covariance matrix of the unique scores from first-order factors, and $\mathrm{\Theta }$ 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:

$${\omega }_{L1}=\frac{\text{bold}{1}^{\prime }\mathrm{\Lambda }\text{bold}B{\mathrm{\Phi }}_2\text{bold}{B}^{\prime }{\mathrm{\Lambda }}^{\prime }\text{bold}1}{\text{bold}{1}^{\prime }\mathrm{\Lambda }\text{bold}B{\mathrm{\Phi }}_2\text{bold}{B}^{\prime }{\mathrm{\Lambda }}^{\prime }\text{bold}1+\text{bold}{1}^{\prime }\mathrm{\Lambda }{\mathrm{\Psi }}_u{\mathrm{\Lambda }}^{\prime }\text{bold}1+\text{bold}{1}^{\prime }\mathrm{\Theta }\text{bold}1},$$

where $\text{bold}1$ is the k-dimensional vector of 1 and k is the number of observed variables. When model-implied covariance matrix among first-order factors (${\mathrm{\Phi }}_1$) can be calculated:

$${\mathrm{\Phi }}_1=\text{bold}B{\mathrm{\Phi }}_2\text{bold}{B}^{\prime }+{\mathrm{\Psi }}_u,$$

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:

$${\omega }_{L2}=\frac{\text{bold}{1_F}^{\prime }\text{bold}B{\mathrm{\Phi }}_2\text{bold}{B}^{\prime }\text{bold}{1}_F}{\text{bold}{1_F}^{\prime }\text{bold}B{\mathrm{\Phi }}_2\text{bold}{B}^{\prime }\text{bold}{1}_F+\text{bold}{1_F}^{\prime }{\mathrm{\Psi }}_u\text{bold}{1}_F},$$

where $\text{bold}{1}_F$ 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:

$${\omega }_{L1}=\frac{\text{bold}{1}^{\prime }\mathrm{\Lambda }\text{bold}B{\mathrm{\Phi }}_2\text{bold}{B}^{\prime }{\mathrm{\Lambda }}^{\prime }\text{bold}1}{\text{bold}{1}^{\prime }\mathrm{\Lambda }\text{bold}B{\mathrm{\Phi }}_2\text{bold}{B}^{\prime }{\mathrm{\Lambda }}^{\prime }\text{bold}1+\text{bold}{1}^{\prime }\mathrm{\Theta }\text{bold}1},$$

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.

See Also

reliability for the reliability of the first-order factors.

Examples

# 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")

# }