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A new estimator of Zumbo’s Ordinal Alpha: a copula approach

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Abstract

We aim to propose a new estimator of the reliability index for polytomous ordinal items, introduced by Zumbo et al. (J Mod Appl Stat Methods 6:21–29, 2007), who suggested a modification of the classical Cronbach’s \(\alpha \) indicator to be used in presence of ordinal variables. Zumbo et al. introduced an underlying variable conceptualization of the Cronbach’s reliability index and suggested that one can estimate this underlying variable index, Zumbo’s Ordinal Alpha, using ML estimation of the polychoric correlations. Our proposal relaxes the assumption made by Zumbo et al., that the ordinal variables have an underlying multinormal distribution, by using a copula framework. The proposed estimator builds upon the Spearman grade correlation coefficient on a transformation of the ordinal variables, calculated by the copula function. An empirical version of our proposal, defined by means of the empirical copula, is also presented. Some examples and simulation studies are presented.

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Notes

  1. See (Nelsen (2006), pp. 169–170) for the definition of the grade correlation coefficient for continuous random variables.

References

  • Bartolomew, D.J.: The Statistical Approach to Social Measurement. Academic Press, San Diego (1996)

    Google Scholar 

  • Bollen, K.A.: Structural Equations with Latent Variables. Wiley, New York (1989)

    Book  Google Scholar 

  • Cortina, J.M.: What is coefficient alpha? An examination of theory and applications. J. Appl. Psychol. 78, 98–104 (1993)

    Article  Google Scholar 

  • Cronbach, L.J.: Coefficient alpha and the internal structure of tests. Psychometrika 16, 297–334 (1951)

    Article  Google Scholar 

  • Drasgow, F.: Polychoric and polyserial correlations, The Encyclopedia of Statistics, 7:68–74, John Wiley, (1986)

  • Ekström, J.: Contributions to the Theory of Measures of Association for Ordinal Variables, Digital Comprehensive Summaries of Uppsala Dissertation from the Faculty of Social Sciences, n. 50. ACTA Universitatis Upsaliensis, Uppsala (2009)

    Google Scholar 

  • Gadermann, A.M., Zumbo, B.: Estimating ordinal reliability for Likert-type and ordinal item response data: A conceptual, empirical, and practical guide, Practical Assessment, Research & Evaluation, 17(3) (2012)

  • Gelin, M.N., Beasley, T.M., Zumbo, B.D.: What is the impact on scale reliability and exploratory factor analysis of a Pearson correlation matrix when some respondents are not able to follow the rating scale?, Paper presented at the Annual Meeting of the American Educational Research Association. (AERA), Chicago (2003)

  • Liu, Y., Wu, A., Zumbo, B.: The impact of outliers on Cronbach’s coefficient alpha estimate of reliability: ordinal/rating scale item responses. Educ. Psychol. Meas. 70(1), 5–21 (2010)

    Article  Google Scholar 

  • Martinson, E.O., Hamdan, M.A.: Maximum likelihood and some other asymptotically efficient estimators of correlation in two way contingency tables. J. Stat. Comput. Simul. 1, 45–54 (1971)

    Article  Google Scholar 

  • Maydeu Olivares, A., Coffman, D.L., Hartmann, W.M.: Asymptotically distribution free (ADF) interval estimation of coefficient alpha. Psychol. Methods 12, 157–176 (2007)

    Article  Google Scholar 

  • Nelsen, R.B.: An Introduction to Copulas. Springer, New York (2006)

    Google Scholar 

  • Olsson, U.: Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika 44, 443–460 (1979)

    Article  Google Scholar 

  • Pearson, K.: Mathematical contribution to the theory of evolution, Philosophical Transactions of the Royal Society of London. Ser. A 195, 1–47 (1900)

    Google Scholar 

  • Schmitt, N.: Uses and abuses of coefficient alpha. Psychol. Assess. 8, 350–353 (1996)

    Article  Google Scholar 

  • Zumbo, B.D., Gadermann, A.M., Zeisser, C.: Ordinal versions of coefficient Alpha and Theta for Likert rating scales. J. Mod. Appl. Stat. Methods 6, 21–29 (2007)

    Google Scholar 

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Correspondence to Andrea Bonanomi.

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Bonanomi, A., Cantaluppi, G., Nai Ruscone, M. et al. A new estimator of Zumbo’s Ordinal Alpha: a copula approach. Qual Quant 49, 941–953 (2015). https://doi.org/10.1007/s11135-014-0114-8

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