# aliquote

## < a quantity that can be divided into another a whole number of time />

While browsing questions related to psychometrics posted late in 2012 on Cross Validated, I noticed two questions dealing with hierarchical ωh.

These questions come from the use of William Revelle‘s psych package, which offers a very nice toolkit for serious psychometrics, especially work related to factor analysis. Just take a look at some of his Psychology 454 syllabus (PDF) to get an idea of what’s available in psych.

The ωh measure was popularized by Zinbarg, Revelle and coll few years ago. In my answer to this question on Cross Validated, Omega vs. alpha reliability, I neglected to mention that Cronbach’s alpha originated in the work of Guttman (as λ3), as pointed out by McDonald (p. 92), although this is something I mentioned in an earlier answer of mine. Again, William Revelle offer some nice readings on Psychometric Theory, including Reliability Theory (starting p. 44 ff). I haven’t use ωh myself but it is clear that classical measures of internal consistency suffer from many drawbacks (especially in relation to unidimensionality and equal loadings assumption, as found in my other posts or in several articles(4,5)) which are largely ignored in several papers dealing with the validation of health questionnaires that I happen to read.

A quick literature review suggests that some applied papers(6) relied on ωh to assess the amount of general factor saturation. On a related point, Graham(7) discussed the use of SEM and congeneric estimate of reliability when tau-equivalence assumption of Cronbach’s alpha fails. Cronbach’s alpha and coefficient ω use to be compared as discrepancy between their values reflect the extent to which the reliability estimate is influenced by allowing group factors to figure into true score variation.(8)

### References

1. Zinbarg, R.E., Revelle, W., and Yovel, I. (2007). Estimating ωh for structures containing two group factors: Perils and prospects. Applied Psychological Measurement, 31(2), 135–157.
2. 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 ωh. Applied Psychological Measurement, 30(2), 121–144.
3. McDonald, R.P. (1999). Test theory: A unified treatment. Mahwah, NJ: Lawrence Erlbaum.
4. Falissard, B. (1999). The unidimensionality of a psychiatric scale: a statistical point of view. International Journal of Methods in Psychiatric Research, 8(3), 162-167.
5. Tavakol, M. and Dennick, R. (2011). Making sense of Cronbach’s alpha. International Journal of Medical Education, 2, 53-55.
6. Steer, R.A. (2009). Amount of General Factor Saturation in the Beck Anxiety Inventory Responses of Outpatients with Anxiety Disorders. Journal of Psychopathology and Behavioral Assessment, 31, 112–118.
7. Graham, J.M. (2006). Congeneric and (Essentially) Tau-Equivalent Estimates of Score Reliability. Educational and Psychological Measurement, 66(6), 930-944.
8. Reise, S.P., Moore, T.M., and Haviland, M.G. (2010). Bifactor Models and Rotations: Exploring the Extent to which Multidimensional Data Yield Univocal Scale Scores. Journal of Personality Assessment, 92(6), 544-559.