reliabilityL2-deprecated: Calculate the reliability values of a second-order factor
| reliabilityL2-deprecated | 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, omit.imps = c("no.conv", "no.se"))
Arguments
object |
A lavaan::lavaan or lavaan.mi::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 |
omit.imps |
|
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{\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 contains first-order factor loadings, \bold{B} contains
second-order factor loadings, \Phi_2 is the covariance matrix of the
second-order factor(s), \Psi_{u} is the covariance matrix of residuals
from first-order factors, and \Theta 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:
\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.
The model-implied covariance matrix among first-order factors (\Phi_1)
can be calculated as:
\Phi_1 = \bold{B} \Phi_2 \bold{B}^{\prime} + \Psi_{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:
\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. 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:
\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, 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 (psunthud@gmail.com)
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")
