John Schmid and John M. Leiman (1957) discuss how to transform a hierarchical factor structure to a bifactor structure. Schmid contains the example 12 x 12 correlation matrix. schmid.leiman is a 12 x 12 correlation matrix with communalities on the diagonal. This can be used to show the effect of correcting for attenuation. Two additional data sets are taken from Chen et al. (2006).

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Two artificial correlation matrices from Schmid and Leiman (1957). One real and one artificial covariance matrices from Chen et al. (2006).

Schmid: a 12 x 12 artificial correlation matrix created to show the Schmid-Leiman transformation.

schmid.leiman: A 12 x 12 matrix with communalities on the diagonal. Treating this as a covariance matrix shows the 6 x 6 factor solution

Chen: An 18 x 18 covariance matrix of health related quality of life items from Chen et al. (2006). Number of observations = 403. The first item is a measure of the quality of life. The remaining 17 items form four subfactors: The items are (a) Cognition subscale: “Have difficulty reasoning and solving problems?" “React slowly to things that were said or done?"; “Become confused and start several actions at a time?" “Forget where you put things or appointments?"; “Have difficulty concentrating?" (b) Vitality subscale: “Feel tired?" “Have enough energy to do the things you want?" (R) “Feel worn out?" ; “Feel full of pep?" (R). (c) Mental health subscale: “Feel calm and peaceful?"(R) “Feel downhearted and blue?"; “Feel very happy"(R) ; “Feel very nervous?" ; “Feel so down in the dumps nothing could cheer you up? (d) Disease worry subscale: “Were you afraid because of your health?"; “Were you frustrated about your health?"; “Was your health a worry in your life?" .

West: A 16 x 16 artificial covariance matrix from Chen et al. (2006).

John Schmid Jr. and John. M. Leiman (1957), The development of hierarchical factor solutions.Psychometrika, 22, 83-90.

F.F. Chen, S.G. West, and K.H. Sousa.(2006) A comparison of bifactor and second-order models of quality of life. Multivariate Behavioral Research, 41(2):189-225, 2006.

Y.-F. Yung, D.Thissen, and L.D. McLeod. (1999) On the relationship between the higher-order factor model and the hierarchical factor model. Psychometrika, 64(2):113-128, 1999.

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