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

A new estimator of Zumbo’s Ordinal Alpha: a copula approach 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quality & Quantity Springer Journals

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

, Volume 49 (3) – Oct 11, 2014
13 pages

/lp/springer_journal/a-new-estimator-of-zumbo-s-ordinal-alpha-a-copula-approach-ti0yQr4P12
Publisher
Springer Journals
Subject
Social Sciences, general; Methodology of the Social Sciences; Social Sciences, general
ISSN
0033-5177
eISSN
1573-7845
D.O.I.
10.1007/s11135-014-0114-8
Publisher site
See Article on Publisher Site

### 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.

### Journal

Quality & QuantitySpringer Journals

Published: Oct 11, 2014

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