The coefficient alpha (Cronbach, 1951) can be calculated by
$$ \alpha = \frac{k}{k - 1}\left[ 1 - \frac{\sum^{k}_{i = 1}
\sigma_{ii}}{\sum^{k}_{i = 1} \sigma_{ii} + 2\sum_{i < j} \sigma_{ij}}
\right],$$
where \(k\) is the number of items in a factor, \(\sigma_{ii}\) is the
item i observed variances, \(\sigma_{ij}\) is the observed
covariance of items i and j.
The coefficient omega (Bollen, 1980; see also Raykov, 2001) can be
calculated by
$$ \omega_1 =\frac{\left( \sum^{k}_{i = 1} \lambda_i \right)^{2}
Var\left( \psi \right)}{\left( \sum^{k}_{i = 1} \lambda_i \right)^{2}
Var\left( \psi \right) + \sum^{k}_{i = 1} \theta_{ii} + 2\sum_{i < j}
\theta_{ij} }, $$
where \(\lambda_i\) is the factor loading of item i, \(\psi\) is
the factor variance, \(\theta_{ii}\) is the variance of measurement errors
of item i, and \(\theta_{ij}\) is the covariance of measurement
errors from item i and j.
The second coefficient omega (Bentler, 1972, 2009) can be calculated by
$$ \omega_2 = \frac{\left( \sum^{k}_{i = 1} \lambda_i \right)^{2}
Var\left( \psi \right)}{\bold{1}^\prime \hat{\Sigma} \bold{1}}, $$
where \(\hat{\Sigma}\) is the model-implied covariance matrix, and
\(\bold{1}\) is the \(k\)-dimensional vector of 1. The first and the
second coefficients omega will have the same value when the model has simple
structure, but different values when there are (for example) cross-loadings
or method factors. The first coefficient omega can be viewed as the
reliability controlling for the other factors (like \(\eta^2_{partial}\) in
ANOVA). The second coefficient omega can be viewed as the unconditional
reliability (like \(\eta^2\) in ANOVA).
The third coefficient omega (McDonald, 1999), which is sometimes referred to
hierarchical omega, can be calculated by
$$ \omega_3 =\frac{\left( \sum^{k}_{i = 1} \lambda_i \right)^{2}
Var\left( \psi \right)}{\bold{1}^\prime \Sigma \bold{1}}, $$
where \(\Sigma\) is the observed covariance matrix. If the model fits the
data well, the third coefficient omega will be similar to the
\(\omega_2\). Note that if there is a directional effect in the model, all
coefficients omega will use the total factor variances, which is calculated
by lavInspect(object, "cov.lv").
In conclusion, \(\omega_1\), \(\omega_2\), and \(\omega_3\) are
different in the denominator. The denominator of the first formula assumes
that a model is congeneric factor model where measurement errors are not
correlated. The second formula accounts for correlated measurement errors.
However, these two formulas assume that the model-implied covariance matrix
explains item relationships perfectly. The residuals are subject to sampling
error. The third formula use observed covariance matrix instead of
model-implied covariance matrix to calculate the observed total variance.
This formula is the most conservative method in calculating coefficient
omega.
The average variance extracted (AVE) can be calculated by
$$ AVE = \frac{\bold{1}^\prime
\textrm{diag}\left(\Lambda\Psi\Lambda^\prime\right)\bold{1}}{\bold{1}^\prime
\textrm{diag}\left(\hat{\Sigma}\right) \bold{1}}, $$
Note that this formula is modified from Fornell & Larcker (1981) in the case
that factor variances are not 1. The proposed formula from Fornell & Larcker
(1981) assumes that the factor variances are 1. Note that AVE will not be
provided for factors consisting of items with dual loadings. AVE is the
property of items but not the property of factors. AVE is calculated with
polychoric correlations when ordinal indicators are used.
Coefficient alpha is by definition applied by treating indicators as numeric
(see Chalmers, 2018), which is consistent with the alpha function in
the psych package. When indicators are ordinal, reliability
additionally applies the standard alpha calculation to the polychoric
correlation matrix to return Zumbo et al.'s (2007) "ordinal alpha".
Coefficient omega for categorical items is calculated using Green and Yang's
(2009, formula 21) approach. Three types of coefficient omega indicate
different methods to calculate item total variances. The original formula
from Green and Yang is equivalent to \(\omega_3\) in this function.
Green and Yang did not propose a method for
calculating reliability with a mixture of categorical and continuous
indicators, and we are currently unaware of an appropriate method.
Therefore, when reliability detects both categorical and continuous
indicators of a factor, an error is returned. If the categorical indicators
load on a different factor(s) than continuous indicators, then reliability
will still be calculated separately for those factors, but
return.total must be FALSE (unless omit.factors is used
to isolate factors with indicators of the same type).