Omega vs. alpha reliability

Question

I wonder if someone can explain what is the main difference between omega and alpha reliabilities?

I understand an omega reliability is based on hierarchical factor model as shown in the following picture, and alpha uses average inter-item correlations.

What I don't understand is, in what condition, omega reliability coefficient would be higher than alpha coefficient, and vice versa?

Can I assume if the correlations between the subfactors and the variables are higher, the omega coefficient would also be higher (as shown in the above picture)?

Any advice is appreciated!

Answer 1

The $\omega_h$ (hierarchical) coefficient gives the proportion of variance in scale scores accounted for by a general factor (1,2), usually from a second-order factor analysis. However, if any zero-order dimensions are reflected in such scales, $\omega_h$ will be less than Cronbach's $\alpha$ (which should only be used with unidimensional scales in any case). It is only when the measurement instrument is so-called tau-equivalent (equal factor loadings but possibly unequal but uncorrelated errors) that $\alpha=\omega_h$. This was early demonstrated by McDonald. Regardless of the indicator used, low values indicate that it makes no sense to compute a sum score (i.e., to add contribution of each item score together to derive a composite score).

To sum up, correlated measurement errors, multidimensionality or unequal factor loadings make both indicators likely to diverge, with hierarchical $\omega_h$ being the reliability measure to use, following Revelle and coworkers' past work (see (1) for more discussion about that).

References

1. Zinbarg, R.E., Revelle, W., and Yovel, I. (2007). Estimating $\omega_h$ for structures containing two group factors: Perils and prospects. Applied Psychological Measurement, 31(2), 135–157.
2. McDonald, R.P. (1999). Test theory: A unified treatment. Mahwah, NJ: Lawrence Erlbaum.
3. Zinbarg, R.E., Yovel, I., Revelle, W., and McDonald, R.P (2006). Estimating Generalizability to a Latent Variable Common to All of a Scale’s Indicators: A Comparison of Estimators for $\omega_h$. Applied Psychological Measurement, 30(2), 121–144.

Answer 2

Cronbach’s alpha depends on the assumption that each indicator variable contributes equally to the factor, i.e., all (unstandardized) loadings must be the same (tau-equivalence). If this assumption is violated, true reliability will be underestimated.

The second assumption for alpha is that the error variances of the indicators must be uncorrelated. In other words, a single factor must account for all the common variance of the indicators. If this is not the case, alpha will overestimate reliability.

Omega does not require tau-equivalence or uncorrelated error variances. There are two versions of omega. The first is used when error variances are uncorrelated, the second if they are correlated. Omega and alpha will yield the same result if the assumptions of alpha are not violated by the data.