Hi Linda and/or Bengt, I have conducted a confirmatory factor analysis the neuroticism scale from the Eysenck Personality Questionnaire which is made up of binary items. One of our aims is to estimate coefficient omega_hieararchical - an index first identified by Rod McDonald and that I have published some work on recently (Zinbarg et al., 2005, Psychometrika, 70, 123-133; Zinbarg et al., 2006, Applied Psychological Measurement, 30, 121 – 144). To estimate omega_h, one sums the unstandardized factor loadings on a general factor, squares this sum (to sum the reproduced covariance matrix that is reproduced on the basis of the general factor) and then divides the sum by the observed variance in the total scores obtained by summing all the items on the scale. This yields an estimate of the proportion of variance in total scores that is attributable to the general factor running through the item set. When we do this with the factor loadings from the CFA we ran in Mplus specifying all the items as being categorical, we get a nonsensical value for omega_h (> 1). I suspect that the reason is that the loadings in the analysis of categorical items have at least a somewhat different meaning than the loadings in the analysis of continuous items. Any thoughts you might have on the appropriate way to estimate omega_h for categorical items would be greatly appreciated. Thanks very much! Rick Zinbarg
as a follow-up to my question, I had an idea that I was hoping to check out with you. In principle, an alternative way to estimate omega_h would be to predict each participant's total score on the basis of his/her standing on the general factor, obtain the variance in these predicted scores and divide it by the observed variance in the actual total scores. Using CFA with categorical items, is the way that I would estimate each participant's response to each item reproduced on the basis of his/her standing on the general factor be to save factor scores for the participant's, multiply each person's factor score on the general factor by the item's loading on the general factor and if this product exceeds the item's threshold then set the person's response on that item equal to 1 and otherwise to 0? If so, I could estimate each person's item level responses and then simply sum them to predict his/her total score on the basis of the general factor. Thanks!
If you want the proportion of the variance in the sum of say binary items that is accounted for by a general factor, it seems that you would have to take into account that the general factor has a non-linear influence on this sum via the probit regression. It would seem feasible to generalize to this binary (or polytomous) case, and preferable to do so without going via estimated factor scores.
thanks - I am not sure how to take the non-linear influence into account. Any insights you could share would be most appreciated. / P.S. As others have expressed before, thanks for thid awesome discussion board!
Hi Bengt and Linda, I had another idea. To estimate omega_h with categorical items, what if one added a variable to the model corresponding to the total score of the scale? I know that Raykov does this - adding a phantom variable corresponding to the total score - into CFA models to estimate factor analytically based reliability estimates. One could then estimate the correlation between the total score and the general factor in a hierarchical or higher-order CFA. In principle this seems to me like it should work (and then one could simply square the correlation to get omega_h) but when I try doing this with some simulated continuous data sets I have created (either by creating a total score in SPSS and adding this variable as another observed in my data file or by creating a "factor" that is loaded on by all items with all the loadings constrained to equal 1) I run into problems. I get warnings such as the PSI matrix is not positive definite (when I create a "factor" coresponding to the total score) or the sample covariance of the independent variables is singular (when the total score is an observed variable in my data file and I try to estimate its regression with the general factor)