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

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  • Andrea Bonanomi
  • Gabriele Cantaluppi
  • Marta Nai Ruscone
  • Silvia Osmetti

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 $$\alpha $$ α 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. Copyright Springer Science+Business Media Dordrecht 2015

Suggested Citation

  • Andrea Bonanomi & Gabriele Cantaluppi & Marta Nai Ruscone & Silvia Osmetti, 2015. "A new estimator of Zumbo’s Ordinal Alpha: a copula approach," Quality & Quantity: International Journal of Methodology, Springer, vol. 49(3), pages 941-953, May.
  • Handle: RePEc:spr:qualqt:v:49:y:2015:i:3:p:941-953
    DOI: 10.1007/s11135-014-0114-8
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    References listed on IDEAS

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    1. Ulf Olsson, 1979. "Maximum likelihood estimation of the polychoric correlation coefficient," Psychometrika, Springer;The Psychometric Society, vol. 44(4), pages 443-460, December.
    2. Lee Cronbach, 1951. "Coefficient alpha and the internal structure of tests," Psychometrika, Springer;The Psychometric Society, vol. 16(3), pages 297-334, September.
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    1. Andrea Bonanomi & Marta Nai Ruscone & Silvia Angela Osmetti, 2017. "Defining subjects distance in hierarchical cluster analysis by copula approach," Quality & Quantity: International Journal of Methodology, Springer, vol. 51(2), pages 859-872, March.

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