This study introduces an item response theory–zero-inflated Poisson (IRT–ZIP) model to investigate psychometric properties of multiple items and predict individuals' latent trait scores for multivariate zero-inflated count data. In the model, two link functions are used to capture two processes of the zero-inflated count data. Item parameters are included to investigate item performance from both propensity and level perspectives. The application of the model was illustrated by analyzing the substance use data from the National Longitudinal Study of Youth. A simulation study based on the empirical data analysis scenario showed that the item parameters can be recovered accurately and precisely with adequate sample sizes. Limitations and future directions are discussed.
AdamsR. J.WilsonM.WuM. L. (1997). Multilevel item response models: An approach to errors in variables regression. Journal of Educational and Behavioral Statistics, 22, 47–76.
2.
BöckenholtU.KamakuraW. A.WedelM. (2003). The structure of self-reported emotional experiences: A mixed-effects Poisson factor model. British Journal of Mathematical & Statistical Psychology, 56, 215–229.
3.
CameronC.TrivediP. K. (1998). Regression analysis of count data, Econometric Society Monograph No. 30. Cambridge, England: Cambridge University Press.
4.
CattellR. B. (1978). The scientific use of factor analysis. New York, NY: Plenum.
5.
CoxD. R.LewisP. A. W. (1966). The statistical analysis of series of events. (Methuen Monograph). London, England: Methuen.
6.
EmbretsonS. E.ReiseS. P. (2000). Item response theory for psychologists. Mahwah, NJ: Erlbaum.
7.
FromeE. L.KutnerM. H.BeauchampJ. J. (1973). Regression analysis of Poisson-distributed data. Journal of the American Statistical Association, 68, 935–940.
8.
FullerW. A. (1987). Measurement error models. New York, NY: Wiley.
9.
GartJ. J. (1964). The analysis of Poisson regression with an application in virology. Biometrika, 51, 517–521.
10.
HallD. B. (2000). Zero-inflated Poisson and binomial regression with random effects: A case study. Biometrics, 56, 1030–1039.
11.
HulinC. L.LissakR. I.DrasgowF. (1982). Recovery of two- and three- parameter logistic item characteristic curves: A Monte Carlo study. Applied Psychological Measurement, 6, 249–260.
12.
KandelD. B. (2003). Does marijuana use cause the use of other drugs?. Journal of the American Medical Association, 289, 482–483.
13.
LambertD. (1992). Zero-inflated Poisson regression, with an application to the defects in manufacturing. Technometrics, 34, 1–14.
14.
LeeA. H.WangK.ScottJ. AYauK. K. W.McLachlanG. J. (2006). Multi-level zero-inflated Poisson regression modelling of correlated count data with excess zeros. Statistical Methods in Medical Research, 15, 47–61.
15.
LiuH.PowersD. A. (2007). Growth curve models for zero-inflated count data: An application to smoking behavior. Structural Equation Modeling, 14, 247–279.
16.
LordF. M. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Erlbaum.
17.
LordF. M.NovickM. R. (1968). Statistical theories of mental test scores. Reading, MA: Addison-Wesley.
18.
McCullaghP.NelderJ. A. (1989). Generalized linear models. 2nd ed.London, England: Chapman and Hall.
19.
McDonaldR. P. (1985). Factor analysis and related methods. Hillsdale, NJ: Erlbaum.
20.
MinY. Y.AgrestiA. (2005). Random effect models for repeated measures of zero-inflated count data. Statistical Modeling, 5, 1–19.
21.
MorralA. R.McCafferyD. F.PaddockS. M. (2002). Reassessing the marijuana gateway effect. Addiction, 97, 1493–1504.
22.
PinheiroJ. C.BatesD. M. (1995). Approximation to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics, 4, 12–35.
23.
Rabe-HeskethS.SkrondalA. (2007). Multilevel and latent variable modeling with composite links and exploded likelihoods. Psychometrika, 72, 123–140.
24.
RaschG. (1960). Probabilistic models for some intelligence and attainment tests. Chicago, IL: University of Chicago Press.
25.
RaudenbushS. W.JohnsonC.SampsonR. (2003). A multivariate, multilevel Rasch model with applications to self-reported criminal behavior. Sociological Methodology, 33, 169–211.
26.
ReeM. J.JensenH. E. (1980). Effects of sample size on linear equating of item characteristic curve parameters. In WeissD. J. (Ed.), Proceedings of the 1979 computerized adaptive testing conference. Minneapolis: University of Minnesota.
27.
RijmenF.TuerlinckxF.De BoeckP.KuppensP. (2003). A nonlinear mixed model framework for item response theory. Psychological Methods, 8, 185–205.
28.
TakaneY.de LeeuwJ. (1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrica, 52, 393–408.
29.
ThissenD.WainerH. (1982). Some standard errors in item response theory. Psychometrica, 47, 397–412.
30.
ThissenD.WainerH. (2001). Test scoring. Hillsdale, NJ: Erlbaum.
31.
WolfingerR. (1999). Fitting nonlinear mixed models with the new NLMIXED procedure. In SUGI 24 Conference Proceedings, Paper 287. Cary, NC: SAS Institute.
