A New Statistic for Evaluating Item Response Theory Models for Ordinal Data. CRESST Report 839.

.....................................................................................................................................4 Introduction ................................................................................................................................4 Some Notation ...........................................................................................................................7 Maximum Likelihood Estimation of IRT Models .....................................................................8 Limited-information Goodness-of-fit Testing ...........................................................................9 Distribution of Multinomial Residuals under Maximum Likelihood Estimation..................9 First Order Margins .............................................................................................................10 Second Order Margins .........................................................................................................12 Existing Test Statistics: and .................................................................................14 The Proposed Test Statistic ..................................................................................................16 A Measure of Model Error ...................................................................................................17 Simulations ..............................................................................................................................18 Type I Error Rate .................................................................................................................19 Power ...................................................................................................................................20 Analysis of Empirical Data ......................................................................................................22 Discussion ................................................................................................................................24 References ................................................................................................................................26

[1]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[2]  M. Browne.  Asymptotically distribution-free methods for the analysis of covariance structures. , 1984, The British journal of mathematical and statistical psychology.

[3]  M. Edelen,et al.  Methodology for developing and evaluating the PROMIS smoking item banks. , 2014, Nicotine & tobacco research : official journal of the Society for Research on Nicotine and Tobacco.

[4]  Albert Maydeu-Olivares.  Goodness-of-Fit Assessment of Item Response Theory Models , 2013 .

[5]  D. Thissen,et al.  Limited-information goodness-of-fit testing of item response theory models for sparse 2 tables. , 2006, The British journal of mathematical and statistical psychology.

[6]  D. Bartholomew,et al.  A goodness of fit test for sparse 2p contingency tables. , 2002, British Journal of Mathematical & Statistical Psychology.

[7]  Harry Joe,et al.  Limited Information Goodness-of-Fit Testing in Multidimensional Contingency Tables , 2005 .

[8]  David J. Bartholomew,et al.  The Goodness of Fit of Latent Trait Models in Attitude Measurement , 1999 .

[9]  F. Samejima.  Estimation of latent ability using a response pattern of graded scores , 1969 .

[10]  Albert Maydeu-Olivares,et al.  Limited Information Goodness-of-fit Testing in Multidimensional Contingency Tables , 2005 .

[11]  D. Darling,et al.  A Test of Goodness of Fit , 1954 .

[12]  Li Cai,et al.  Limited-information goodness-of-fit testing of hierarchical item factor models. , 2013, The British journal of mathematical and statistical psychology.

[13]  M. Browne,et al.  Alternative Ways of Assessing Model Fit , 1992 .

[14]  David Thissen,et al.  A response model for multiple choice items , 1984 .

[15]  R. D. Bock,et al.  Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm , 1981 .

[16]  M. Edelen,et al.  Toward a more systematic assessment of smoking: development of a smoking module for PROMIS®. , 2012, Addictive behaviors.

[17]  M. R. Novick,et al.  Statistical Theories of Mental Test Scores. , 1971 .

[18]  Mark Reiser,et al.  Analysis of residuals for the multionmial item response model , 1996 .

[19]  Harry Joe,et al.  A General Family of Limited Information Goodness-of-Fit Statistics for Multinomial Data , 2010 .

[20]  F. Samejima.  Estimation of latent ability using a response pattern of graded scores , 1968 .