IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v79y2014i2p303-316.html
   My bibliography  Save this article

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

Author

Listed:
  • Jules Ellis

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. Copyright The Psychometric Society 2014

Suggested Citation

  • Jules Ellis, 2014. "An Inequality for Correlations in Unidimensional Monotone Latent Variable Models for Binary Variables," Psychometrika, Springer;The Psychometric Society, vol. 79(2), pages 303-316, April.
    • Handle: RePEc:spr:psycho:v:79:y:2014:i:2:p:303-316
    DOI: 10.1007/s11336-013-9341-5
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s11336-013-9341-5
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s11336-013-9341-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Yoav Benjamini & Abba M. Krieger & Daniel Yekutieli, 2006. "Adaptive linear step-up procedures that control the false discovery rate," Biometrika, Biometrika Trust, vol. 93(3), pages 491-507, September.
    2. van der Ark, L. Andries, 2007. "Mokken Scale Analysis in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 20(i11).
    3. Jinming Zhang & William Stout, 1999. "The theoretical detect index of dimensionality and its application to approximate simple structure," Psychometrika, Springer;The Psychometric Society, vol. 64(2), pages 213-249, June.
    4. De Gooijer, Jan G. & Yuan, Ao, 2011. "Some exact tests for manifest properties of latent trait models," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 34-44, January.
    5. Kun Liang & Dan Nettleton, 2012. "Adaptive and dynamic adaptive procedures for false discovery rate control and estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(1), pages 163-182, January.
    6. Maydeu-Olivares, Albert & Joe, Harry, 2005. "Limited- and Full-Information Estimation and Goodness-of-Fit Testing in 2n Contingency Tables: A Unified Framework," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1009-1020, September.
    7. William Meredith, 1993. "Measurement invariance, factor analysis and factorial invariance," Psychometrika, Springer;The Psychometric Society, vol. 58(4), pages 525-543, December.
    8. Marco Scarsini & Fabio Spizzichino, 1999. "Simpson-type paradoxes, dependence, and ageing," Post-Print hal-00540264, HAL.
    9. Edward Ip, 2001. "Testing for local dependency in dichotomous and polytomous item response models," Psychometrika, Springer;The Psychometric Society, vol. 66(1), pages 109-132, March.
    10. Paul Rosenbaum, 1984. "Testing the conditional independence and monotonicity assumptions of item response theory," Psychometrika, Springer;The Psychometric Society, vol. 49(3), pages 425-435, September.
    11. Michele Guindani & Peter Müller & Song Zhang, 2009. "A Bayesian discovery procedure," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 905-925, November.
    12. John D. Storey, 2007. "The optimal discovery procedure: a new approach to simultaneous significance testing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(3), pages 347-368, June.
    13. Alberto Maydeu-Olivares & Rosa Montaño, 2013. "How Should We Assess the Fit of Rasch-Type Models? Approximating the Power of Goodness-of-Fit Statistics in Categorical Data Analysis," Psychometrika, Springer;The Psychometric Society, vol. 78(1), pages 116-133, January.
    14. Jinming Zhang, 2007. "Conditional Covariance Theory and Detect for Polytomous Items," Psychometrika, Springer;The Psychometric Society, vol. 72(1), pages 69-91, March.
    15. Helmut Finner & Veronika Gontscharuk, 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, Royal Statistical Society, vol. 71(5), pages 1031-1048, November.
    16. L. Ark & Marcel Croon & Klaas Sijtsma, 2008. "Mokken Scale Analysis for Dichotomous Items Using Marginal Models," Psychometrika, Springer;The Psychometric Society, vol. 73(2), pages 183-208, June.
    17. van der Ark, L. Andries, 2012. "New Developments in Mokken Scale Analysis in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i05).
    18. Cees Glas, 1988. "The derivation of some tests for the rasch model from the multinomial distribution," Psychometrika, Springer;The Psychometric Society, vol. 53(4), pages 525-546, December.
    19. Francesco Bartolucci & Antonio Forcina, 2005. "Likelihood inference on the underlying structure of IRT models," Psychometrika, Springer;The Psychometric Society, vol. 70(1), pages 31-43, March.
    20. Jules Ellis & Brian Junker, 1997. "Tail-measurability in monotone latent variable models," Psychometrika, Springer;The Psychometric Society, vol. 62(4), pages 495-523, December.
    21. William Meredith & Roger Millsap, 1992. "On the misuse of manifest variables in the detection of measurement bias," Psychometrika, Springer;The Psychometric Society, vol. 57(2), pages 289-311, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Michael Brusco & Hans-Friedrich Köhn & Douglas Steinley, 2015. "An Exact Method for Partitioning Dichotomous Items Within the Framework of the Monotone Homogeneity Model," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 949-967, December.
    2. Rudy Ligtvoet, 2022. "Incomplete Tests of Conditional Association for the Assessment of Model Assumptions," Psychometrika, Springer;The Psychometric Society, vol. 87(4), pages 1214-1237, December.
    3. Jules L. Ellis & Klaas Sijtsma, 2023. "A Test to Distinguish Monotone Homogeneity from Monotone Multifactor Models," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 387-412, June.
    4. Riet Bork & Raoul P. P. P. Grasman & Lourens J. Waldorp, 2018. "Unidimensional factor models imply weaker partial correlations than zero-order correlations," Psychometrika, Springer;The Psychometric Society, vol. 83(2), pages 443-452, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jules L. Ellis & Klaas Sijtsma, 2023. "A Test to Distinguish Monotone Homogeneity from Monotone Multifactor Models," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 387-412, June.
    2. Rudy Ligtvoet, 2022. "Incomplete Tests of Conditional Association for the Assessment of Model Assumptions," Psychometrika, Springer;The Psychometric Society, vol. 87(4), pages 1214-1237, December.
    3. J. Straat & L. Ark & Klaas Sijtsma, 2013. "Comparing Optimization Algorithms for Item Selection in Mokken Scale Analysis," Journal of Classification, Springer;The Classification Society, vol. 30(1), pages 75-99, April.
    4. Michael Brusco & Hans-Friedrich Köhn & Douglas Steinley, 2015. "An Exact Method for Partitioning Dichotomous Items Within the Framework of the Monotone Homogeneity Model," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 949-967, December.
    5. Jesper Tijmstra & Maria Bolsinova, 2019. "Bayes Factors for Evaluating Latent Monotonicity in Polytomous Item Response Theory Models," Psychometrika, Springer;The Psychometric Society, vol. 84(3), pages 846-869, September.
    6. César Merino-Soto & Arturo Juárez-García & Guillermo Salinas Escudero & Filiberto Toledano-Toledano, 2022. "Parametric and Nonparametric Analysis of the Internal Structure of the Psychosocial Work Processes Questionnaire (PROPSIT) as Applied to Workers," IJERPH, MDPI, vol. 19(13), pages 1-23, June.
    7. Mark Reiser & Silvia Cagnone & Junfei Zhu, 2023. "An Extended GFfit Statistic Defined on Orthogonal Components of Pearson’s Chi-Square," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 208-240, March.
    8. Yang Liu & Ji Seung Yang & Alberto Maydeu-Olivares, 2019. "Restricted Recalibration of Item Response Theory Models," Psychometrika, Springer;The Psychometric Society, vol. 84(2), pages 529-553, June.
    9. Alberto Maydeu-Olivares & Rosa Montaño, 2013. "How Should We Assess the Fit of Rasch-Type Models? Approximating the Power of Goodness-of-Fit Statistics in Categorical Data Analysis," Psychometrika, Springer;The Psychometric Society, vol. 78(1), pages 116-133, January.
    10. Merkle, Edgar C. & Steyvers, Mark & Mellers, Barbara & Tetlock, Philip E., 2017. "A neglected dimension of good forecasting judgment: The questions we choose also matter," International Journal of Forecasting, Elsevier, vol. 33(4), pages 817-832.
    11. A. Béguin & C. Glas, 2001. "MCMC estimation and some model-fit analysis of multidimensional IRT models," Psychometrika, Springer;The Psychometric Society, vol. 66(4), pages 541-561, December.
    12. Edward Ip & Yuchung Wang & Paul Boeck & Michel Meulders, 2004. "Locally dependent latent trait model for polytomous responses with application to inventory of hostility," Psychometrika, Springer;The Psychometric Society, vol. 69(2), pages 191-216, June.
    13. Forcina, Antonio, 2008. "Identifiability of extended latent class models with individual covariates," Computational Statistics & Data Analysis, Elsevier, vol. 52(12), pages 5263-5268, August.
    14. Francisco J. Conejo & Lawrence F. Cunningham & Clifford E. Young, 2020. "Revisiting the Brand Luxury Index: new empirical evidence and future directions," Journal of Brand Management, Palgrave Macmillan, vol. 27(1), pages 108-122, January.
    15. Edsel Peña & Joshua Habiger & Wensong Wu, 2015. "Classes of multiple decision functions strongly controlling FWER and FDR," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(5), pages 563-595, July.
    16. van der Ark, L. Andries, 2012. "New Developments in Mokken Scale Analysis in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i05).
    17. Albert Maydeu-Olivares & Harry Joe, 2006. "Limited Information Goodness-of-fit Testing in Multidimensional Contingency Tables," Psychometrika, Springer;The Psychometric Society, vol. 71(4), pages 713-732, December.
    18. Francesco Bartolucci & Antonio Forcina, 2005. "Likelihood inference on the underlying structure of IRT models," Psychometrika, Springer;The Psychometric Society, vol. 70(1), pages 31-43, March.
    19. Bastiaan Bruinsma, 2020. "A comparison of measures to validate scales in voting advice applications," Quality & Quantity: International Journal of Methodology, Springer, vol. 54(4), pages 1299-1316, August.
    20. Jesper Tijmstra & David Hessen & Peter Heijden & Klaas Sijtsma, 2013. "Testing Manifest Monotonicity Using Order-Constrained Statistical Inference," Psychometrika, Springer;The Psychometric Society, vol. 78(1), pages 83-97, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:psycho:v:79:y:2014:i:2:p:303-316. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.