calc_omega() computes McDonald's \(\omega_\mathrm{total}\) for a
unidimensional scale using a one-factor confirmatory factor analysis
(CFA) fit with lavaan using MLR estimation. Missing data can be
handled via full information maximum likelihood (FIML).
Value
A numeric scalar giving McDonald;s \(\omega_\mathrm{total}\) (typically in [0, 1]; small estimation anomalies can yield values slightly outside this range).
Details
The model is f =~ item1 + item2 + ... + itemK with std.lv = TRUE
(i.e., \(\mathrm{Var}(f)=1\)).
\(\omega = \frac{(\sum_i \lambda_i)^2} {(\sum_i \lambda_i)^2 + \sum_i \theta_i}\), where \(\lambda_i\) are unstandardized loadings and \(\theta_i\) are unstandardized residual variances of the indicators.
The function validates inputs and fails with informative messages if the CFA does not converge or if implied residual variances are invalid. Mixed-sign loadings trigger a warning (often indicates reverse-keying is needed).
References
McDonald, R. P. (1999). Test Theory: A Unified Treatment. Lawrence Erlbaum.
Green, S. B., & Yang, Y. (2009). Reliability of summed item scores using structural equation modeling: An alternative to coefficient alpha. Psychometrika, 74(1), 155–167.
Zinbarg, R. E., Revelle, W., Yovel, I., & Li, W. (2005). Cronbach's α, Revelle's β, and McDonald's ωH. Psychometrika, 70(1), 123–133.