RE: st: calculate alpha after polychoric factor analysis

[Cameron McIntosh <cnm100@hotmail.com>]

However, I should note that model-based reliability is a much better alternative to alpha (which could now be consigned to the museum of reliability estimation):
Sijtsma, K. (2009a). On the use, the misuse, and the very limited usefulness of Cronbach's alpha. Psychometrika, 74(1), 107-120.

Sijtsma, K. (2009b). Reliability beyond theory and Into practice. Psychometrika, 74(1), 169–173.

Bentler, P.M. (2009). Alpha, dimension-free, and model-based internal consistency reliability. Psychometrika, 74(1), 137-143. http://aqm.gseis.ucla.edu/Papers%20pdf%20format/Bentler%20Alpha%20Dimension-free%20Psychometrika%2008-1.pdf

Revelle, W., & Zinbarg, R.E. (2009). Coefficients alpha, beta, omega and the glb: comments on Sijtsma. Psychometrika, 74(1), 145-154.

Green, S.B., & Yang, Y. (2009a). Commentary on coefficient alpha: a cautionary tale. Psychometrika, 74(1), 121-135.

Green, S.B., & Yang, Y. (2009b). Reliability of summed item scores using structural equation modeling: An alternative to coefficient alpha. Psychometrika, 74(1), 155-167.

Yang, Y., & Green, S.B. (2010). A note on structural equation modeling estimates of reliability. Structural Equation Modeling, 17(1), 66-81. 

Ogasawara, H. (2009). On the estimators of model-based and maximal reliability. Journal of Multivariate Analysis, 100(6), 1232-1244.

Raykov, T. (1997). Estimation of composite reliability for congeneric measures. Applied Psychological Measurement, 21, 173-184. 

I would suggest that Seyi estimate reliability directly from the factor model parameters rather than the inter-item polychoric correlation matrix.
My two cents,
Cam
----------------------------------------
> From: cnm100@hotmail.com
> To: statalist@hsphsun2.harvard.edu
> Subject: RE: st: calculate alpha after polychoric factor analysis
> Date: Tue, 14 Feb 2012 21:58:08 -0500
>
> You may also want to see a more recent paper:
> Gadermann, A.M., Guhn, M., & Zumbo, B.D. (2012). Estimating ordinal reliability for Likert-type and ordinal item response data: A conceptual, empirical, and practical guide. Practical Assessment, Research & Evaluation, 17(3).    http://pareonline.net/pdf/v17n3.pdf
>
> Cam
> ----------------------------------------
> > From: jcoveney@bigplanet.com
> > To: statalist@hsphsun2.harvard.edu
> > Subject: Re: st: calculate alpha after polychoric factor analysis
> > Date: Wed, 15 Feb 2012 11:54:31 +0900
> >
> > Seyi Soremekun wrote:
> >
> > Thanks (and for the polychoric command itself). The sample size is not an issue
> > - all respondents answered all the questions, displaying -n- was just a check
> > for me (if that is what you meant in your note?)
> >
> > I'm not sure about the appropriateness of using alpha either, however, I know
> > that ordinal alphas are available in other types of software (Basto and Pereira
> > 2012 J. Statistical Software 46(4)), and wondered if there was something I could
> > use in Stata.
> > I was hoping that I could ask stata to calculate the alpha on the saved matrix
> > rather than a list of variables.
> >
> > --------------------------------------------------------------------------------
> >
> > The Basto and Pereira article references Bruno D. Zumbo, Anne M. Gadermann and
> > Cornelia Zeisser, Ordinal Versions of Coefficients Alpha and Theta for Likert
> > Rating Scales. _Journal of Modern Applied Statistical Methods_ 6(1): 21--9,
> > 2007), which in turn references earlier work for the formula it displays for
> > ordinal coefficient alpha. If I've got the formula correctly implemented below,
> > then you can try something like what is illustrated. It shouldn't be too
> > difficult to morph it into an ado-file, but you'd probably want to verify its
> > correctness with a couple of worked examples.
> >
> > Joseph Coveney
> >
> > version 11.2
> >
> > set more off
> > set seed `=date("2012-02-14", "YMD")'
> > tempname C
> > matrix define `C' = I(4) * 0.45 + J(4, 4, 0.55)
> > drawnorm latent1 latent2 latent3 latent4, corr(`C') double n(100) clear
> > forvalues i = 1/4 {
> > generate double u`i' = normal(latent`i')
> > generate byte manifest`i' = 0
> > quietly forvalues cut = 0.2(0.2)0.8 {
> > replace manifest`i' = manifest`i' + 1 if u`i' > `cut'
> > }
> > }
> > polychoric manifest*
> > matrix define `C' = r(R)
> > factormat `C', n(100) factors(1)
> >
> > tempname L Psi
> > matrix define `L' = e(L)
> > matrix define `Psi' = e(Psi)
> >
> > local p = rowsof(`L')
> >
> > tempname f f2 u2
> > scalar define `f' = 0
> > scalar define `f2' = 0
> > scalar define `u2' = 0
> > forvalues i = 1/`p' {
> > scalar define `f' = `f' + `L'[`i', 1]
> > scalar define `f2' = `f2' + `L'[`i', 1] * `L'[`i', 1]
> > scalar define `u2' = `u2' + `Psi'[1, `i']
> > }
> > scalar define `f' = `f' / `p'
> > scalar define `f2' = `f2' / `p'
> > scalar define `u2' = `u2' / `p'
> >
> > tempname pf2
> > scalar define `pf2' = `p' * `f' * `f'
> > scalar define alpha = `p' / (`p' - 1) * ///
> > (`pf2' - `f2') / (`pf2' + `u2')
> >
> > display in smcl as text "Ordinal alpha = " as result %06.4f alpha
> >
> > alpha latent*
> > alpha manifest*, std
> >
> > exit
> >
> >
> > *
> > * For searches and help try:
> > * http://www.stata.com/help.cgi?search
> > * http://www.stata.com/support/statalist/faq
> > * http://www.ats.ucla.edu/stat/stata/
>
> *
> * For searches and help try:
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> * http://www.ats.ucla.edu/stat/stata/
 		 	   		  
*
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