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

In semTools: Useful Tools for Structural Equation Modeling

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.

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 common-factor variance, the residual variance of the first-order common factors (i.e., not accounted for by the second-order factor), and the measurement error of observed 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 contains first-order factor loadings, \bold{B} contains second-order factor loadings, Φ_2 is the covariance matrix of the second-order factor(s), Ψ_{u} is the covariance matrix of residuals from first-order factors, and Θ is the covariance matrix of the measurement errors from observed indicators. Thus, we can calculate the proportion of variance of a composite score calculated from the observed indicators (e.g., a total score or scale mean) that is attributable to the second-order factor, i.e. coefficient omega at Level 1:

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.

The model-implied covariance matrix among first-order factors (Φ_1) can be calculated as:

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

Thus, the proportion of variance among first-order common factors that is attributable to the second-order factor (i.e., coefficient omega at Level 2) can be calculated as:

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. This Level-2 omega can be interpreted as an estimate of the reliability of a hypothetical composite calculated from error-free observable variables representing the first-order common factors. This might only be meaningful as a thought experiment.

An additional thought experiment is possible: If the observed indicators contained only the second-order common-factor variance and unsystematic measurement error, then there would be no first-order common factors because their unique variances would be excluded from the observed measures. An estimate of this hypothetical composite reliability can be calculated as the partial coefficient omega at Level 1, or the proportion of observed variance explained by the second-order factor after partialling out the uniqueness from the first-order factors:

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, which might not be desirable.

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 ([e-mail])

See Also

semTools-deprecated

Examples

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")

semTools documentation built on May 10, 2022, 9:05 a.m.