IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v78y2013i1p116-133.html
   My bibliography  Save this article

How Should We Assess the Fit of Rasch-Type Models? Approximating the Power of Goodness-of-Fit Statistics in Categorical Data Analysis

Author

Listed:
  • Alberto Maydeu-Olivares
  • Rosa Montaño

Abstract

We investigate the performance of three statistics, R 1 , R 2 (Glas in Psychometrika 53:525–546, 1988 ), and M 2 (Maydeu-Olivares & Joe in J. Am. Stat. Assoc. 100:1009–1020, 2005 , Psychometrika 71:713–732, 2006 ) to assess the overall fit of a one-parameter logistic model (1PL) estimated by (marginal) maximum likelihood (ML). R 1 and R 2 were specifically designed to target specific assumptions of Rasch models, whereas M 2 is a general purpose test statistic. We report asymptotic power rates under some interesting violations of model assumptions (different item discrimination, presence of guessing, and multidimensionality) as well as empirical rejection rates for correctly specified models and some misspecified models. All three statistics were found to be more powerful than Pearson’s X 2 against two- and three-parameter logistic alternatives (2PL and 3PL), and against multidimensional 1PL models. The results suggest that there is no clear advantage in using goodness-of-fit statistics specifically designed for Rasch-type models to test these models when marginal ML estimation is used. Copyright The Psychometric Society 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:psycho:v:78:y:2013:i:1:p:116-133
    DOI: 10.1007/s11336-012-9293-1
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s11336-012-9293-1
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s11336-012-9293-1?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. David Thissen, 1982. "Marginal maximum likelihood estimation for the one-parameter logistic model," Psychometrika, Springer;The Psychometric Society, vol. 47(2), pages 175-186, June.
    2. R. Bock & Murray Aitkin, 1981. "Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm," Psychometrika, Springer;The Psychometric Society, vol. 46(4), pages 443-459, December.
    3. C. Glas & N. Verhelst, 1989. "Extensions of the partial credit model," Psychometrika, Springer;The Psychometric Society, vol. 54(4), pages 635-659, September.
    4. Teugels, Jozef L, 1990. "Some representations of the multivariate Bernoulli and binomial distributions," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 256-268, February.
    5. 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.
    6. 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.
    7. Albert Satorra & Willem Saris, 1985. "Power of the likelihood ratio test in covariance structure analysis," Psychometrika, Springer;The Psychometric Society, vol. 50(1), pages 83-90, March.
    8. Mark Reiser, 1996. "Analysis of residuals for the multionmial item response model," Psychometrika, Springer;The Psychometric Society, vol. 61(3), pages 509-528, September.
    9. David J. Bartholomew & Panagiota Tzamourani, 1999. "The Goodness of Fit of Latent Trait Models in Attitude Measurement," Sociological Methods & Research, , vol. 27(4), pages 525-546, May.
    10. Hans Irtel, 1995. "An extension of the concept of specific objectivity," Psychometrika, Springer;The Psychometric Society, vol. 60(1), pages 115-118, March.
    11. Arnold Wollenberg, 1982. "Two new test statistics for the rasch model," Psychometrika, Springer;The Psychometric Society, vol. 47(2), pages 123-140, June.
    12. Agresti, Alan & Yang, Ming-Chung, 1987. "An empirical investigation of some effects of sparseness in contingency tables," Computational Statistics & Data Analysis, Elsevier, vol. 5(1), pages 9-21.
    13. Harry Joe & Alberto Maydeu-Olivares, 2010. "A General Family of Limited Information Goodness-of-Fit Statistics for Multinomial Data," Psychometrika, Springer;The Psychometric Society, vol. 75(3), pages 393-419, September.
    14. 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.
    15. Erling Andersen, 1973. "A goodness of fit test for the rasch model," Psychometrika, Springer;The Psychometric Society, vol. 38(1), pages 123-140, March.
    16. Anders Christoffersson, 1975. "Factor analysis of dichotomized variables," Psychometrika, Springer;The Psychometric Society, vol. 40(1), pages 5-32, March.
    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. Felix Zimmer & Clemens Draxler & Rudolf Debelak, 2023. "Power Analysis for the Wald, LR, Score, and Gradient Tests in a Marginal Maximum Likelihood Framework: Applications in IRT," Psychometrika, Springer;The Psychometric Society, vol. 88(4), pages 1249-1298, December.
    2. 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.
    3. Clemens Draxler & Rainer Alexandrowicz, 2015. "Sample Size Determination Within the Scope of Conditional Maximum Likelihood Estimation with Special Focus on Testing the Rasch Model," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 897-919, December.
    4. 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.

    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. 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.
    2. Wu, Jianmin & Bentler, Peter M., 2013. "Limited information estimation in binary factor analysis: A review and extension," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 392-403.
    3. 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.
    4. 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.
    5. Li Cai, 2010. "A Two-Tier Full-Information Item Factor Analysis Model with Applications," Psychometrika, Springer;The Psychometric Society, vol. 75(4), pages 581-612, December.
    6. David J. Hessen, 2010. "Likelihood Ratio Tests for Special Rasch Models," Journal of Educational and Behavioral Statistics, , vol. 35(6), pages 611-628, December.
    7. Clemens Draxler & Rainer Alexandrowicz, 2015. "Sample Size Determination Within the Scope of Conditional Maximum Likelihood Estimation with Special Focus on Testing the Rasch Model," Psychometrika, Springer;The Psychometric Society, vol. 80(4), pages 897-919, December.
    8. Vassilis Vasdekis & Silvia Cagnone & Irini Moustaki, 2012. "A Composite Likelihood Inference in Latent Variable Models for Ordinal Longitudinal Responses," Psychometrika, Springer;The Psychometric Society, vol. 77(3), pages 425-441, July.
    9. Clemens Draxler, 2018. "Bayesian conditional inference for Rasch models," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 102(2), pages 245-262, April.
    10. Clemens Draxler, 2010. "Sample Size Determination for Rasch Model Tests," Psychometrika, Springer;The Psychometric Society, vol. 75(4), pages 708-724, December.
    11. Nuo Xi & Michael W. Browne, 2014. "Contributions to the Underlying Bivariate Normal Method for Factor Analyzing Ordinal Data," Journal of Educational and Behavioral Statistics, , vol. 39(6), pages 583-611, December.
    12. Silvia cagnone & Stefania Mignani, 2007. "Assessing the goodness of fit of a latent variable model for ordinal data," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 337-361.
    13. Cees Glas, 1999. "Modification indices for the 2-PL and the nominal response model," Psychometrika, Springer;The Psychometric Society, vol. 64(3), pages 273-294, September.
    14. C. Glas & Anna Dagohoy, 2007. "A Person Fit Test For Irt Models For Polytomous Items," Psychometrika, Springer;The Psychometric Society, vol. 72(2), pages 159-180, June.
    15. Haruhiko Ogasawara, 2013. "Asymptotic properties of the Bayes modal estimators of item parameters in item response theory," Computational Statistics, Springer, vol. 28(6), pages 2559-2583, December.
    16. Salim Moussa, 2016. "A two-step item response theory procedure for a better measurement of marketing constructs," Journal of Marketing Analytics, Palgrave Macmillan, vol. 4(1), pages 28-50, March.
    17. Yoshio Takane & Jan Leeuw, 1987. "On the relationship between item response theory and factor analysis of discretized variables," Psychometrika, Springer;The Psychometric Society, vol. 52(3), pages 393-408, September.
    18. Dylan Molenaar, 2015. "Heteroscedastic Latent Trait Models for Dichotomous Data," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 625-644, September.
    19. Harry Joe & Alberto Maydeu-Olivares, 2010. "A General Family of Limited Information Goodness-of-Fit Statistics for Multinomial Data," Psychometrika, Springer;The Psychometric Society, vol. 75(3), pages 393-419, September.
    20. Erling Andersen, 1995. "Residualanalysis in the polytomous rasch model," Psychometrika, Springer;The Psychometric Society, vol. 60(3), pages 375-393, September.

    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:78:y:2013:i:1:p:116-133. 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.