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Mean Comparison: Manifest Variable Versus Latent Variable

Published online by Cambridge University Press:  01 January 2025

Ke-Hai Yuan  and
Peter M. Bentler
Ke-Hai Yuan*
Affiliation:
University of Notre Dame
Peter M. Bentler
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to Ke-Hai Yuan (kyuan@nd.edu), Department of Psychology, University of Notre Dame, Notre Dame, IN 46556, USA.
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Abstract

Mean comparisons are of great importance in the application of statistics. Procedures for mean comparison with manifest variables have been well studied. However, few rigorous studies have been conducted on mean comparisons with latent variables, although the methodology has been widely used and documented. This paper studies the commonly used statistics in latent variable mean modeling and compares them with parallel manifest variable statistics. Our results indicate that, under certain conditions, the likelihood ratio and Wald statistics used for latent mean comparisons do not always have greater power than the Hotelling T2 statistics used for manifest mean comparisons. The noncentrality parameter corresponding to the T2 statistic can be much greater than those corresponding to the likelihood ratio and Wald statistics, which we find to be different from those provided in the literature. Under a fixed alternative hypothesis, our results also indicate that the likelihood ratio statistic can be stochastically much greater than the corresponding Wald statistic. The robustness property of each statistic is also explored when the model is misspecified or when data are nonnormally distributed. Recommendations and advice are provided for the use of each statistic.

Keywords

noncentrality parameterlikelihood ratio statisticWald statisticempirical powerparameter biasasymptotic robustness
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 1 , March 2006 , pp. 139 - 159
Copyright
Copyright © 2006 The Psychometric Society

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Footnotes

The research was supported by NSF grant DMS-0437167 and Grant DA01070 from the National Institute on Drug Abuse.

We would like to thank three referees for suggestions that helped in improving the paper.

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