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.