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

reliabilityL2 {semTools} | R Documentation

## 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

### 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{Σ} = Λ \bold{B} Φ_2 \bold{B}^{\prime} Λ^{\prime} + Λ Ψ_{u} Λ^{\prime} + Θ,

where \hat{Σ} is the model-implied covariance matrix, Λ is the first-order factor loading, \bold{B} is the second-order factor loading, Φ_2 is the covariance matrix of the second-order factors, Ψ_{u} is the covariance matrix of the unique scores from first-order factors, and Θ 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:

ω_{L1} = \frac{\bold{1}^{\prime} Λ \bold{B} Φ_2 \bold{B}^{\prime} Λ^{\prime} \bold{1}}{\bold{1}^{\prime} Λ \bold{B} Φ_2 \bold{B} ^{\prime} Λ^{\prime} \bold{1} + \bold{1}^{\prime} Λ Ψ_{u} Λ^{\prime} \bold{1} + \bold{1}^{\prime} Θ \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 (Φ_1) can be calculated:

Φ_1 = \bold{B} Φ_2 \bold{B}^{\prime} + Ψ_{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:

ω_{L2} = \frac{\bold{1_F}^{\prime} \bold{B} Φ_2 \bold{B}^{\prime} \bold{1_F}}{\bold{1_F}^{\prime} \bold{B} Φ_2 \bold{B}^{\prime} \bold{1_F} + \bold{1_F}^{\prime} Ψ_{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:

ω_{L1} = \frac{\bold{1}^{\prime} Λ \bold{B} Φ_2 \bold{B}^{\prime} Λ^{\prime} \bold{1}}{\bold{1}^{\prime} Λ \bold{B} Φ_2 \bold{B}^{\prime} Λ^{\prime} \bold{1} + \bold{1}^{\prime} Θ \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.

### Value

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

### Author(s)

Sunthud Pornprasertmanit (psunthud@gmail.com)

### See Also

`reliability` for the reliability of the first-order factors.

### Examples

    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")

[Package semTools version 0.5-1 Index]