Sage Journals: Discover world-class research

Abstract

Unidimensional, item response theory (IRT) models assume a single homogeneous population. Mixture IRT (MixIRT) models can be useful when subpopulations are suspected. The usual MixIRT model is typically estimated assuming a normally distributed latent ability. Research on normal finite mixture models suggests that latent classes potentially can be extracted, even in the absence of population heterogeneity, if the distribution of the data is non-normal. In this study, the authors examined the sensitivity of MixIRT models to latent non-normality. Single-class IRT data sets were generated using different ability distributions and then analyzed with MixIRT models to determine the impact of these distributions on the extraction of latent classes. Results suggest that estimation of mixed Rasch models resulted in spurious latent class problems in the data when distributions were bimodal and uniform. Mixture two-parameter logistic (2PL) and mixture three-parameter logistic (3PL) IRT models were found to be more robust to latent non-normality.

Keywords

mixture IRT model latent non-normality over-extraction spurious latent class MCMC estimation

Get full access to this article

View all access options for this article.

References

Akaike

(1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716-723.

Alexeev

Templin

Cohen

A. S.

(2011). Spurious latent classes in the mixture Rasch model. Journal of Educational Measurement, 48, 313-332.

Andersen

E. B.

Madsen

(1977). Estimating the parameters of a latent population distribution. Psychometrika, 42, 357-374.

Bauer

D. J.

Curran

P. J.

(2003). Distributional assumptions of growth mixture models: Implications for over-extraction of latent trajectory classes. Psychological Methods, 8, 338-363.

Bock

R. D.

Aitkin

(1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443-459.

Bolt

D. M.

Cohen

A. S.

Wollack

J. A.

(2002). Item parameter estimation under conditions of test speededness: Application of a mixture Rasch model with ordinal constraints. Journal of Educational Measurement, 39, 331-348.

Cho

S.-J.

Cohen

A. S.

Kim

S.-H.

(2013). Markov chain Monte Carlo estimation of a mixture item response model. Journal of Statistical Computation and Simulation, 83, 278-306.

Congdon

(2003). Applied Bayesian modelling. New York, NY: Wiley.

Fleishman

A. I.

(1978). A method for simulating non-normal distributions. Psychometrika, 43, 521-532.

10.

Foy

Arora

Stanco

G. M.

(2013). TIMSS 2011 user guide for the international database. Chestnut Hill, MA: Boston College.

11.

Gelman

Rubin

D. B.

(1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457-472.

12.

Gregoire

T. G.

Driver

B. L.

(1987). Analysis of ordinal data to detect population differences. Psychological Bulletin, 101, 159-165.

13.

Kang

Cohen

A. S.

(2007). IRT model selection methods for dichotomous items. Applied Psychological Measurement, 31, 331-358.

14.

Cohen

A. S.

Kim

S.-H.

Cho

S.-J.

(2009). Model selection methods for mixture dichotomous IRT models. Applied Psychological Measurement, 33, 353-373.

15.

Lord

F. M.

Novick

M. R.

(1968). Statistical theories of mental test scores. Reading, MA: Addison-Wesley.

16.

McLachlan

Peel

(2000). Finite mixture models. New York, NY: Wiley.

17.

Micceri

(1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105, 156-166.

18.

Mislevy

R. J.

Verhelst

(1990). Modeling item responses when different subjects employ different solution strategies. Psychometrika, 55, 195-215.

19.

Muthén

(2008). Latent variable hybrids: Overview of old and new models. In Hancock

G. R.

Samuelsen

K. M.

(Eds.), Advances in latent variable mixture models (pp. 1-24). Charlotte, NC: Information Age.

20.

Nylund

K. L.

Asparouhov

Muthén

B. O.

(2007). Deciding on the number of classes in latent class analysis and growth mixture modeling: A Monte Carlo simulation study. Structural Equation Modeling, 14, 535-569.

21.

Patz

R. J.

Junker

B. W.

(1999). A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, 24, 146-178.

22.

Pearson

E. S.

Please

N. W.

(1975). Relation between the shape of population distribution and the robustness of four simple test statistics. Biometrika, 62, 223-241.

23.

Preinerstorfer

Formann

A. K.

(2011). Parameter recovery and model selection in mixed Rasch models. British Journal of Mathematical and Statistical Psychology, 65, 251-262.

24.

Rasch

(1960). Probabilistic models for some intelligence and attainment tests. Copenhagen, Denmark: Nielsen & Lydiche.

25.

Reckase

M. D.

(1979). Unifactor latent trait models applied to multifactor tests: Results and implications. Journal of Educational Statistics, 4, 207-230.

26.

Reckase

M. D.

(2009). Multidimensional item response theory. New York, NY: Springer.

27.

Rost

(1990). Rasch models in latent classes: An integration of two approaches to item analysis. Applied Psychological Measurement, 14, 271-282.

28.

Samuelsen

K. M.

(2005). Examining differential item functioning from a latent class perspective (Doctoral dissertation). University of Maryland, College Park.

29.

Schwarz

(1978). Estimating the dimension of a model. Annals of Statistics, 6, 461-464.

30.

Spiegelhalter

Thomas

Best

(2003). WinBUGS (Version 1.4) [Computer software]. Cambridge, UK: Biostatistics Unit, Institute of Public Health.

31.

Spiegelhalter

D. J.

Best

N. G.

Carlin

B. P.

van der Linde

(2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64, 583-639.

32.

Thissen

(1991). MULTILOG user’s guide: Multiple categorical item analysis and test scoring using item response theory. Chicago, IL: Scientific Software International.

33.

Tofighi

Enders

C. K.

(2007). Identifying the correct number of classes in a growth mixture model. In Hancock

G. R.

Samuelsen

K. M.

(Eds.), Mixture models in latent variable research (pp. 317-341). Greenwich, CT: Information Age.

34.

Woods

C. M.

(2004). Item response theory with estimation of the latent population distribution using spline-based densities (Unpublished doctoral dissertation). University of North Carolina at Chapel Hill.

35.

Woods

C. M.

(2006a). Ramsay-curve item response theory (RC-IRT) to detect and correct for nonnormal latent variables. Psychological Methods, 11, 253-270.

36.

Woods

C. M.

(2006b). RCLOG v.2: Software for item response theory parameter estimation with the latent population distribution represented using spline-based densities (Technical report). St. Louis, MO: Washington University.

37.

Jia

(2011). The sensitivity of parameter estimates to the latent ability distribution (ETS RR-11-40). Retrieved from http://www.ets.org/Media/Research/pdf/RR-11-40.pdf

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.07 MB

The Impact of Non-Normality on Extraction of Spurious Latent Classes in Mixture IRT Models

Abstract

Keywords

Get full access to this article

References

Supplementary Material