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An Inequality for Correlations in Unidimensional Monotone Latent Variable Models for Binary Variables

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Abstract

It is shown that a unidimensional monotone latent variable model for binary items implies a restriction on the relative sizes of item correlations: The negative logarithm of the correlations satisfies the triangle inequality. This inequality is not implied by the condition that the correlations are nonnegative, the criterion that coefficient H exceeds 0.30, or manifest monotonicity. The inequality implies both a lower bound and an upper bound for each correlation between two items, based on the correlations of those two items with every possible third item. It is discussed how this can be used in Mokken’s (A theory and procedure of scale-analysis, Mouton, The Hague, 1971) scale analysis.

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Acknowledgements

This research was supported by Ellis Statistical Consultations. The author wishes to thank the Associate Editor for suggesting a more general formulation of the theorems, and to the Editor for his suggestions about the organization of the manuscript.

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  1. School of Psychology and Artificial Intelligence, Radboud University Nijmegen, P.O. Box 9104, 6500 HE, Nijmegen, The Netherlands

    Jules L. Ellis

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  1. Jules L. Ellis
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Correspondence to Jules L. Ellis.

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Ellis, J.L. An Inequality for Correlations in Unidimensional Monotone Latent Variable Models for Binary Variables. Psychometrika 79, 303–316 (2014). https://doi.org/10.1007/s11336-013-9341-5

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  • DOI: https://doi.org/10.1007/s11336-013-9341-5

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