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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.

enter image description here

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!

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  • $\begingroup$ I provided some discussion on the use of Cronbach's alpha vs. other indexes of reliability on this related thread: Assessing reliability of a questionnaire: dimensionality, problematic items, and whether to use alpha, lambda6 or some other index?. The response to your first question can be found in Revelle's articles published in Psychometrika. $\endgroup$
    – chl
    Oct 11, 2012 at 10:23
  • $\begingroup$ Hi I have read Revelle's paper, but I don't think I have fully understood it. That was the reason I posted it here and hoping that someone can point to the right direction. I have computed both omega and alpha reliability analysis for a set of data, sometimes, the omega coefficient is higher, sometimes, the alpha is higher - and I don't really understand why is the case. $\endgroup$
    – user11820
    Oct 12, 2012 at 8:29

2 Answers 2

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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.
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  • $\begingroup$ Omega is typically higher than Alpha, not lower, however. $\endgroup$ Jun 6, 2022 at 14:04
  • $\begingroup$ Why "usually from a second-order factor analysis"? Wouldn't a single factor often show up in first-order analysis? $\endgroup$
    – rolando2
    Mar 29 at 14:26
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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.

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  • $\begingroup$ Cronbach's alpha does not involve assumptions like unidimensionality. Its definition assumes no statistical model or distribution, merely the existence of at least two item scores, which can be summed to create a total score. $\endgroup$ Dec 19, 2019 at 20:14
  • $\begingroup$ What Cronbach's alpha measures is reliability when a scale is considered unidimensional. That sounds the same to me as the assumption of unidimensionality. $\endgroup$
    – rolando2
    Mar 29 at 14:24

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