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Resampling-Based Inference Methods for Comparing Two Coefficients Alpha

Published online by Cambridge University Press:  01 January 2025

Markus Pauly ,
Maria Umlauft  and
Ali Ünlü
Markus Pauly*
Affiliation:
Ulm University
Maria Umlauft
Affiliation:
Ulm University
Ali Ünlü
Affiliation:
Technical University of Munich
*
Correspondence should be made to Markus Pauly, Institute of Statistics, Ulm University, Ulm, Germany. Email: markus.pauly@uni-ulm.de; URL: http://www.uni-ulm.de/mawi/statistics/team/professors/prof-dr-markus-pauly.html
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Abstract

The two-sample problem for Cronbach’s coefficient αC\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha _C$$\end{document}, as an estimate of test or composite score reliability, has attracted little attention compared to the extensive treatment of the one-sample case. It is necessary to compare the reliability of a test for different subgroups, for different tests or the short and long forms of a test. In this paper, we study statistical procedures of comparing two coefficients αC,1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha _{C,1}$$\end{document} and αC,2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha _{C,2}$$\end{document}. The null hypothesis of interest is H0:αC,1=αC,2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_0 : \alpha _{C,1} = \alpha _{C,2}$$\end{document}, which we test against one-or two-sided alternatives. For this purpose, resampling-based permutation and bootstrap tests are proposed for two-group multivariate non-normal models under the general asymptotically distribution-free (ADF) setting. These statistical tests ensure a better control of the type-I error, in finite or very small sample sizes, when the state-of-affairs ADF large-sample test may fail to properly attain the nominal significance level. By proper choice of a studentized test statistic, the resampling tests are modified in order to be valid asymptotically even in non-exchangeable data frameworks. Moreover, extensions of this approach to other designs and reliability measures are discussed as well. Finally, the usefulness of the proposed resampling-based testing strategies is demonstrated in an extensive simulation study and illustrated by real data applications.

Keywords

bootstrapcoefficient alphaCronbach’s alphanon-normalitypermutationreliabilityresampling-based inference
Type
Original Paper
Information
Psychometrika , Volume 83 , Issue 1 , March 2018 , pp. 203 - 222
Copyright
Copyright © 2017 The Psychometric Society

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Footnotes

The work of Markus Pauly and Maria Umlauft was supported by the German Research Foundation project DFG-PA 2409/3-1.

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11336-017-9601-x) contains supplementary material, which is available to authorized users.

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