Hostname: page-component-6bf8c574d5-qdpjg Total loading time: 0 Render date: 2025-02-27T02:06:34.310Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  01 January 2025

Jules L. Ellis
Jules L. Ellis*
Affiliation:
Radboud University Nijmegen
*
Requests for reprints should be sent to Jules L. Ellis, School of Psychology and Artificial Intelligence, Radboud University Nijmegen, P.O. Box 9104, 6500 HE Nijmegen, The Netherlands. E-mail: j.ellis@psych.ru.nl
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

nonparametric IRTconditional associationpartial correlationmonotonicityMokken scale analysis
Type
Original Paper
Information
Psychometrika , Volume 79 , Issue 2 , April 2014 , pp. 303 - 316
Copyright
Copyright © 2013 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baba, K., Shibata, R., Sibuya, M. (2004). Partial correlation and conditional correlation as measures of conditional independence. Australian & New Zealand Journal of Statistics, 46, 657–664CrossRefGoogle Scholar
Bartolucci, F., Forcina, A. (2000). A likelihood ratio test for MTP2 within binary variables. The Annals of Statistics, 28, 1206–1218CrossRefGoogle Scholar
Bartolucci, F., Forcina, A. (2005). Likelihood inference on the underlying structure of IRT models. Psychometrika, 70, 31–43CrossRefGoogle Scholar
Benjamini, Y., Krieger, A.M., Yekutieli, D. (2006). Adaptive linear step-up procedures that control the false discovery rate. Biometrika, 93, 491–507CrossRefGoogle Scholar
Benjamini, Y., Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. The Annals of Statistics, 29, 1165–1188CrossRefGoogle Scholar
Davenport, E.C. Jr., El-Sanhurry, N.A. (1991). Phi/Phimax: review and synthesis. Educational and Psychological Measurement, 51, 821–828CrossRefGoogle Scholar
De Gooijer, J.G., Yuan, A. (2011). Some exact tests for manifest properties of latent trait models. Computational Statistics & Data Analysis, 55, 34–44CrossRefGoogle ScholarPubMed
Ellis, J.L. (1993). Subpopulation invariance of patterns in covariance matrices. British Journal of Mathematical & Statistical Psychology, 46, 231–254CrossRefGoogle Scholar
Ellis, J.L., Junker, B.W. (1997). Tail-measurability in monotone latent variable models. Psychometrika, 62, 495–523CrossRefGoogle Scholar
Finner, H., Gontscharuk, V. (2009). Controlling the familywise error rate with plug-in estimator for the proportion of true null hypotheses. Journal of the Royal Statistical Society. Series B, 71, 1031–1048CrossRefGoogle Scholar
Glas, C.A.W. (1988). The derivation of some tests for the Rasch model from the multinomial distribution. Psychometrika, 53, 525–546CrossRefGoogle Scholar
Guindani, M., Müller, P., Zhang, S. (2009). A Bayesian discovery procedure. Journal of the Royal Statistical Society. Series B, 71, 905–925CrossRefGoogle ScholarPubMed
Holland, P.W., Rosenbaum, P.R. (1986). Conditional association and unidimensionality in monotone latent variable models. The Annals of Statistics, 14, 1523–1543CrossRefGoogle Scholar
Ip, E.H. (2001). Testing for local dependency in dichotomous and polytomous item response models. Psychometrika, 66, 109–132CrossRefGoogle Scholar
Junker, B.W. (1993). Progress in characterizing strictly unidimensional IRT representations. The Annals of Statistics, 21, 1359–1378Google Scholar
Junker, B.W., Ellis, J.L. (1997). A characterization of monotone unidimensional latent variable models. The Annals of Statistics, 25, 1327–1343CrossRefGoogle Scholar
Junker, B.W., Sijtsma, K. (2000). Latent and manifest monotonicity in item response models. Applied Psychological Measurement, 24, 65–81CrossRefGoogle Scholar
Junker, B.W., Sijtsma, K. (2001). Nonparametric item response theory in action: an overview of the special issue. Applied Psychological Measurement, 22, 211–220CrossRefGoogle Scholar
Lawrance, A.J. (1976). On conditional and partial correlation. American Statistician, 30, 146–149CrossRefGoogle Scholar
Liang, K., Nettleton, D. (2012). Adaptive and dynamic adaptive procedures for false discovery rate control and estimation. Journal of the Royal Statistical Society. Series B, 74, 163–182CrossRefGoogle Scholar
Loevinger, J. (1948). The technique of homogeneous tests compared with some aspects of “scale analysis” and factor analysis. Psychological Bulletin, 45, 507–530CrossRefGoogle Scholar
Maydeu-Olivares, A., Joe, H. (2005). Limited and full information estimation and goodness-of-fit testing in 2n tables: a unified approach. Journal of the American Statistical Association, 100, 1009–1020CrossRefGoogle Scholar
Maydeu-Olivares, A., Montaño, R. (2013). How should we assess the fit of Rasch-type models? Approximating the power of goodness-of-fit statistics in categorical data analysis. Psychometrika, 78(1), 116133CrossRefGoogle ScholarPubMed
Meinshausen, N., Rice, J. (2006). Estimating the proportion of false null hypotheses among a large number of independently tested hypotheses. The Annals of Statistics, 34, 373–393CrossRefGoogle Scholar
Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58, 525–543CrossRefGoogle Scholar
Meredith, W., Millsap, R.E. (1992). On the misuse of manifest variables in the detection of measurement bias. Psychometrika, 57, 289–311CrossRefGoogle Scholar
Mittal, Y. (1991). Homogeneity of subpopulations and Simpson’s paradox. Journal of the American Statistical Association, 86, 167–172CrossRefGoogle Scholar
Mokken, R.J. (1971). A theory and procedure of scale-analysis, The Hague: MoutonCrossRefGoogle Scholar
Mokken, R.J., Lewis, C. (1982). A nonparametric approach to the analysis of dichotomous item responses. Applied Psychological Measurement, 6, 417–430CrossRefGoogle Scholar
Molenaar, I.W., Sijtsma, K. (2000). User’s manual MSP5 for Windows [software manual], Groningen: iec ProGAMMAGoogle Scholar
Rosenbaum, P.R. (1984). Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika, 49, 425–435CrossRefGoogle Scholar
Scarsini, M., Spizzichino, F. (1999). Simpson-type paradoxes, dependence, and ageing. Journal of Applied Probability, 36, 119–131CrossRefGoogle Scholar
Sijtsma, K., Emons, W.H.M., Bouwmeester, S., Nyklicek, I., Roorda, L.D. (2008). Nonparametric IRT analysis of quality of life scales and its application to the world health organization quality of life scale (WHOQOL-Bref). Quality of Life Research, 17, 275–290CrossRefGoogle Scholar
Sijtsma, K., Meijer, R.R. (2007). Nonparametric item response theory and special topics. In Rao, C.R., Sinharay, S. (Eds.), Psychometrics, Amsterdam: Elsevier 719746Google Scholar
Storey, J.D. (2007). The optimal discovery procedure: a new approach to simultaneous significance testing. Journal of the Royal Statistical Society. Series B, 69, 347–368CrossRefGoogle Scholar
Stout, W.F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293–325CrossRefGoogle Scholar
Stout, W., Habing, B., Douglas, J., Kim, H., Roussos, L., Zhang, J. (1996). Conditional covariance-based nonparametric multidimensional assessment. Applied Psychological Measurement, 20, 331–354CrossRefGoogle Scholar
Ten Holt, J.C., Van Duijn, M.A.J., Boomsma, A. (2010). Scale construction and evaluation in practice: a review of factor analysis versus item response theory applications. Psychological Test and Assessment Modeling, 52, 272–297Google Scholar
Van der Ark, L.A. (2007). Mokken scale analysis in R. Journal of Statistical Software, 20(11), 119 Retrieved from http://www.jstatsoft.orgGoogle Scholar
Van der Ark, L.A. (2012). New developments in Mokken scale analysis in R. Journal of Statistical Software, 48(5), 127 Retrieved from http://www.jstatsoft.orgGoogle Scholar
Van der Ark, L.A., Croon, M.A., Sijtsma, K. (2008). Mokken scale analysis for dichotomous items using marginal models. Psychometrika, 73, 183–208CrossRefGoogle ScholarPubMed
Van der Ven, A.H.G.S., Ellis, J.L. (2000). A Rasch analysis of Raven’s standard progressive matrices. Personality and Individual Differences, 29, 45–64CrossRefGoogle Scholar
Yuan, A., Clarke, B. (2001). Manifest characterization and testing for certain latent properties. The Annals of Statistics, 29, 876–898Google Scholar
Yule, G.U. (1903). Notes on the theory of association of attributes in statistics. Biometrika, 2, 121–134CrossRefGoogle Scholar
Zhang, J. (2007). Conditional covariance theory and DETECT for polytomous items. Psychometrika, 72, 69–91CrossRefGoogle Scholar
Zhang, J., Stout, W.E. (1999). The theoretical DETECT index of dimensionality and its application to approximate simple structure. Psychometrika, 64, 213–249CrossRefGoogle Scholar