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Abstract

A new item response theory model for count data is introduced. In contrast to models in common use, it does not assume a fixed distribution for the responses as, for example, the Poisson count model and extensions do. The distribution of responses is determined by difficulty functions which reflect the characteristics of items in a flexible way. Sparse parameterizations are obtained by choosing fixed parametric difficulty functions, more general versions use an approximation by basis functions. The model can be seen as constructed from binary response models as the Rasch model or the normal-ogive model to which it reduces if responses are dichotomized. It is demonstrated that the model competes well with advanced count data models. Simulations demonstrate that parameters and response distributions are recovered well. An application shows the flexibility of the model to account for strongly varying distributions of responses.

Keywords

thresholds model latent trait models item response theory Rasch model normal-ogive model

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References

Anderson

D. A.

Aitkin

(1985). Variance component models with binary response: Interviewer variability. Journal of the Royal Statistical Society Series B, 47(2), 203–210. https://doi.org/10.1111/j.2517-6161.1985.tb01346.x

Doebler

Holling

(2016). A processing speed test based on rule-based item generation: An analysis with the rasch Poisson counts model. Learning and Individual Differences, 52, 121–128. https://doi.org/10.1016/j.lindif.2015.01.013

Eilers

P. H.

Marx

B. D.

(2021). Practical smoothing: The joys of P-splines. Cambridge University Press.

Eilers

P. H. C.

Marx

B. D.

(1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11(2), 89–121. https://doi.org/10.1214/ss/1038425655

Forthmann

Doebler

(2021). Reliability of researcher capacity estimates and count data dispersion: A comparison of Poisson, negative binomial, and Conway-Maxwell-Poisson models. Scientometrics, 126(4), 3337–3354. https://doi.org/10.1007/s11192-021-03864-8

Forthmann

Gühne

Doebler

(2020). Revisiting dispersion in count data item response theory models: The Conway–Maxwell–Poisson counts model. British Journal of Mathematical and Statistical Psychology, 73(1), 32–50. https://doi.org/10.1111/bmsp.12184

Forthmann

Holling

Çelik

Storme

Lubart

(2017). Typing speed as a confounding variable and the measurement of quality in divergent thinking. Creativity Research Journal, 29(3), 257–269. https://doi.org/10.1080/10400419.2017.1360059

Hinde

(1982). Compound Poisson regression models. In Gilchrist

(Ed.), GLIM 1982 International conference on generalized linear models (pp. 109–121). Springer-Verlag.

Huang

(2017). Mean-parametrized Conway–Maxwell–Poisson regression models for dispersed counts. Statistical Modelling, 17(6), 359–380. https://doi.org/10.1177/1471082x17697749

10.

Hung

L.-F.

(2012). A negative binomial regression model for accuracy tests. Applied Psychological Measurement, 36(2), 88–103. https://doi.org/10.1177/0146621611429548

11.

Jansen

M. G.

(1995). The Rasch Poisson counts model for incomplete data: An application of the EM algorithm. Applied Psychological Measurement, 19(3), 291–302. https://doi.org/10.1177/014662169501900307

12.

Jansen

M. G.

van Duijn

M. A.

(1992). Extensions of Rasch’s multiplicative Poisson model. Psychometrika, 57(3), 405–414. https://doi.org/10.1007/bf02295428

13.

Jansen

P. G.

Roskam

E. E.

(1986). Latent trait models and dichotomization of graded responses. Psychometrika, 51(1), 69–91. https://doi.org/10.1007/bf02294001

14.

Lord

F. M.

Novick

M. R.

(2008). Statistical theories of mental test scores. Information Age Publishing Inc.

15.

McCullagh

Nelder

J. A.

(1989). Generalized linear models (2nd ed.). Chapman & Hall.

16.

Rasch

(1960). Probabilistic models for some intelligence and attainment tests. Danish Institute for Educational Research.

17.

Rasch

(1961). On general laws and the meaning of measurement in psychology. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (4, pp. 321–333). Statistical Laboratory of the University of California.

18.

Roskam

E. E.

Jansen

P. G.

(1989). Conditions for Rasch-dichotomizability of the unidimensional polytomous rasch model. Psychometrika, 54(2), 317–332. https://doi.org/10.1007/bf02294523

19.

Ruppert

Wand

M. P.

Carroll

R. J.

(2009). Semiparametric regression during 2003 – 2007. Electronic Journal of Statistics, 3, 1193–1256. https://doi.org/10.1214/09-ejs525

20.

Samejima

(1973). Homogeneous case of the continuous response model. Psychometrika, 38(2), 203–219. https://doi.org/10.1007/bf02291114

21.

Samejima

(2016). Graded response model. In van der Linden

(Ed), Handbook of item response theory (pp. 95–108).

22.

Shmueli

Minka

T. P.

Kadane

J. B.

Borle

Boatwright

(2005). A useful distribution for fitting discrete data: Revival of the conway–maxwell–Poisson distribution. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(1), 127–142. https://doi.org/10.1111/j.1467-9876.2005.00474.x

23.

Silvia

P. J.

Beaty

R. E.

Nusbaum

E. C.

(2013). Verbal fluency and creativity: General and specific contributions of broad retrieval ability (gr) factors to divergent thinking. Intelligence, 41(5), 328–340. https://doi.org/10.1016/j.intell.2013.05.004

24.

Süß

H.-M.

Oberauer

Wittmann

W. W.

Wilhelm

Schulze

(2002). Working-memory capacity explains reasoning ability and a little bit more. Intelligence, 30(3), 261–288. https://doi.org/10.1016/s0160-2896(01)00100-3

25.

Tutz

(2021). Flexible predictive distributions from varying-thresholds modelling. Technical report. https://arxiv.org/abs/2103.13324

26.

Tutz

(2022). Item response thresholds models: A general class of models for varying types of items. Psychometrika. https://doi.org/10.1007/s11336-022-09865-7

27.

Vidakovic (1999). Statistical modelling by wavelets. Wiley Series in Probability and StatisticsWiley.

28.

Wand

M. P.

(2000). A comparison of regression spline smoothing procedures. Computational Statistics, 15(4), 443–462. https://doi.org/10.1007/s001800000047

29.

Wood

S. N.

(2006a). On confidence intervals for generalized additive models based on penalized regression splines. Australian & New Zealand Journal of Statistics, 48(4), 445–464. https://doi.org/10.1111/j.1467-842x.2006.00450.x

30.

Wood

S. N.

(2006b). Thin plate regression splines. Journal of the Royal Statistical Society, Series B, 65(1), 95–114. https://doi.org/10.1111/1467-9868.00374

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