IRT-ZIP Modeling for Multivariate Zero-Inflated Count ...

IRT-ZIP Modeling for Multivariate Zero-Inflated Count Data Author(s): Lijuan Wang Source: Journal of Educational and Behavioral Statistics, Vol. 35, No. 6 (December 2010), pp. 671-692 Published by: American Educational Research Association and American Statistical Association Stable URL: http://www.jstor.org/stable/40959474 Accessed: 11-04-2016 18:41 UTC REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/40959474?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms

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Journal of Educational and Behavioral Statistics

December 2010, Vol. 35, No. 6, pp. 671-692 DOI: 10.3102/1076998610375838

© 2010 AERA, http://jebs.aera.net

IRT-ZIP Modeling for Multivariate Zero-Inflated Count Data Lijuan Wang University of Notre Dame

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.

Keywords: multivariate zero-inflated count data; IRT-ZIP; item properties

In psychological and educational research, counts of a behavior or an event in a time interval of specified length often are collected through observations, surveys, or experiments. Cameron and Trivedi (1998) defined an event count as the number of times an event occurs, which is a realization of a nonnegative integer-

valued random variable. For example, in a classroom observational study, frequencies of specific behaviors such as effective interactions between students and their teacher are observed and recorded over a period of 10 minutes. Or, in a substance use study, the number of cigarettes smoked by teenagers in a day during the last 30 days can be collected via a survey questionnaire. The Poisson regression model (e.g., Cox & Lewis, 1966; Frome, Kutner, & Beauchamp, 1973; Gart, 1964; McCullagh & Neider, 1989) is a typical method for analyzing cross-

sectional count data. The mixed-effects Poisson factor model (Böckenholt, Kamakura, & Wedel, 2003) can be applied to analyze multivariate count data. Count data that contain excessive numbers of zeros are called zero-inflated

count data. Zero-inflated count data usually come from two distinct processes. One process is related to whether the event has a chance to happen, and the other process is related to the count of the event, given that the event does have a chance to happen. For example, in investigating the number of cigarettes smoked by a participant in a day during the last 30 days, some participants report zeros 671

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Wang

because they are nonsmokers and some participants report zeros because they happen not to smoke during that period although they are smokers. We define the

former zeros (qualitative zeros) as perfect zero states and the latter zeros (quantitative zeros) as zero counts following a Poisson process. Zero-inflated count data can be analyzed using the zero-inflated Poisson model (ZIP model; Lambert, 1992). The ZIP model with random effects (ZIPR model; e.g., Hall, 2000; Lee,

Wang, Scott, Yau, & McLachlan, 2006; Liu & Powers, 2007; Min & Agresti, 2005; Rabe-Hesketh & Skrondal, 2007) can be used to analyze multilevel or longitudinal zero-inflated count data. What to do with measurement errors is always an issue for social and behavioral research. A typical method to reduce the effect of measurement errors is to measure a latent construct by collecting multivariate data using multiple man-

ifest variables. For continuous multivariate data, factor analyses are the typical

methods for examining the relationship between manifest variables and latent factors (e.g., Cattell, 1978; Fuller, 1987; McDonald, 1985). For categorical multivariate data, item response theory models (IRT models) are the typical methods to analyze the psychometric properties of items and to obtain latent trait scores for the measured constructs by modeling the relationship between items and trait scores (e.g., Embretson & Reise, 2000; Lord, 1980; Lord & Novick, 1968; Rasch,

1960). However, how to model multivariate zero-inflated count data to analyze psychometric properties of zero-inflated count items and how to obtain person trait scores from multiple zero-inflated count items were not discussed in previous lit-

erature. When analyzing multivariate zero-inflated count data, although one can form and model composite scores, a disadvantage of this approach is that the distribution of the composite scores is extremely skewed right due to excess zeros and the weights of items forming the composite scores are usually arbitrarily set to be 1. Therefore, the purposes of this study are to (a) introduce a model for

multivariate zero-inflated count data to analyze psychometric properties of zero-inflated count items and to predict person trait scores from multiple items; (b) explore an empirical data set to illustrate the application of the model; and (c) examine the performance of the proposed model via a simulation study with different sample sizes.

Review of Precursors of the Models ZIP Model

For handling univariate zero-inflated count data, Lambert (1992) introduced the ZIP model. The ZIP model uses two link functions, a logit link function and a log link function, to capture statistical features of the two processes: the perfect

zero state and the Poisson process, for zero-inflated count data. The model is expressed as follows: 672

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IRT-ZIP Modeling

JO, with probability 1 - p¡

1 ~ 'Poisson(''), with probability p¡ log(A,) = Xfi iogit(pi)=z/ir,

where Yt is the observed datum for person i (i = 1,2, . . . , n). Therefore, we have

?v(Yi = 0) = (1 -Pi) +Pie-X> andPr (ÏJ = yt) = Pi6 ^ ¿ , yt = 1,2,... In this model, the logit link is used to predict the probability of subject i (pt) to be in the Poisson process. In addition, the log link is used to predict the expected mean (A,-) of event counts for subject /, given that the subject is not in the perfect

zero state, but rather in the Poisson process, x and z are the vectors of covariates

hypothesized to explain interindividual differences in the expected counts and interindividual differences in the types of states, x and z may or may not be the same sets of variables, ß and y are the associated regression parameters.

ZIP Model With Random Effects For handling nested zero-inflated count data, the ZIP models with random effects (the ZIPR models or multilevel ZIP models) were developed and applied

to analyze empirical data (e.g., Hall, 2000; Lee et al., 2006). For example, a modified version of the ZIPR model introduced by Lee, Wang, Scott, Yau, and

McLachlan (2006) can be expressed as follows:

Í 0, with probability l-py y ~ | Poisson (Xij) , with probability py l0g(A,y) = X^.ß + W;

logit(pij)=z^y + Ui, where Xy and ztJ represents theyth observations of the /th individual (i = 1, 2, . . . , n;

y=l,2, . . . , m). ß and y are the corresponding fixed-effects regression parameters. The vectors w,- and u,- are the subject random effects or random intercepts in two

equations that are assumed to be independently and normally distributed. Hall's (2000) model is a reduced model of the above model, which contains a random intercept parameter only in the log equation. By including random effects into the ZIP model, we can account for dependence of observations within clusters. How-

ever, a limitation of the ZIPR model is that it only analyzes univariate zeroinflated count variable and item information cannot be obtained because no item

parameters are included in the model. 673

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Wang

Item Response Models For analyzing multivariate categorical data, item response modeling (e.g., Embretson & Reise, 2000; Lord, 1980) probably is the most widely used method, especially when individual trait scores and item psychometric features are of interest. There is a series of item response models for different kinds of catego-

rical data. Here, we focus only on the 2-parameter (2PL) item response model for binary data, because it is most pertinent to this study. The 2PL model is used to predict the probability (py) of a person endorsing an item while considering the item- varying parameters, ßj for item location (or difficulty) parameters and olj for item slope (or discrimination) parameters, which allow for different weights for different items, and the person- varying latent trait variable 0,. The 2PL model is

expressed as

lo&Hpv) = aJ(Oi-ßj). As ay increases, the item discriminates more precisely among trait levels. When OLj is fixed to be 1, the 2PL model is reduced to be a Rasch model (Rasch, 1960) or

a 1PL model. As ßj increases, the item is more and more difficult to endorse. This 2PL model has been shown to be mathematically equivalent to the confirmatory

factor analysis model for binary data (Takane & de Leeuw, 1987). The IRT models can be expressed as generalized mixed/multilevel models (e.g., Adams,

Wilson, & Wu, 1997; Rijmen, Tuerlinckx, De Boeck, & Kuppens, 2003). Raudenbush, Johnson, and Sampson (2003) also reexpressed the Rasch model and the 2PL model as a two-level logistic model by including dummy variables indicating item numbers.

Proposed Model In this section, we propose an IRT zero-inflated Poisson model (IRT-ZIP model) to handle multivariate zero-inflated count data. In the model, two sets of item parameters (item slope parameters and item location parameters) are used to explain item characteristics, and a common latent trait variable is included to obtain information on person trait scores. The model is described as follows. The observed zero-inflated count data indicate

f0, with probability 1 - ptj ij ~ 'Poisson('ij), with probability p¡j Therefore,

Pr(Yij = 0) = ('-pij)+pije-x«

Pt/e^M y (la) Pr{Yy=yv)=P9 Pt/e^M yvl y , ^ = 1,2,.... The parameters ptj and Xy can be modeled by 674

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IRT-ZIP Modeling increased over time for this NLSY97 sample. In other words, more and more participants became involved with substance use and the level of substance use was generally increasing with age. Results From the IRT-ZIP Models

To investigate psychometric properties of the 3 items and analyze how the item parameters change over time, the IRT-ZIP models were fitted to data of

each wave. Table 2 presents the item parameter estimates with fixed slope parameters (=1) and Table 3 displays the item parameter estimates without any parameter constraints. Key findings from Tables 2 and 3 are as follows. First, based on the likelihood ratio test, the models without any parameter constraints significantly fitted the data better than the models with parameter constraints for all eight waves. This indicates that the discrimination ability of different items in estimating trait levels differs by items. However, this conclusion is tempered by the fact that the differences in the fit between the two models

are not very large compared to the large sample size in the study. Second, the order of magnitude of the location parameter estimates was exam-

ined within each wave. The results showed that the orders of magnitude of the

location parameter estimates were similar across eight waves and between the two nested models. For example, for the location parameter estimates of the log

equations, the estimates of the alcohol use items were larger than those of the cigarette use items and the marijuana use items. This indicates that it is less likely

for a person with a fixed substance use level to reach the same amount of expected frequency in alcohol use as in cigarette use and marijuana use. This result was consistent with our descriptive statistics in the rows of "use frequencies" in Table 1 ? For the location parameters of the logit equations, the estimates

of the marijuana use items are all larger than those of the cigarette use items, while the estimates of the cigarette use items are larger than those of the alcohol

use items. This indicates that it is less likely for a person with a fixed substance use level to be in the Poisson process on marijuana use than on cigarette use and

on alcohol use. In other words, it is less likely on average for a person to be involved in marijuana use than cigarette use or alcohol use. And this result is also consistent with the descriptive statistics in the first six rows in Table 1 .

Third, after comparing the item location parameter estimates across eight waves, we find that the three estimates of the log equations are generally getting smaller and smaller, which means that overall the sample had increasing frequencies in substance use. This result is also consistent with the descriptive statistics

on "use frequencies" in Table 1. The location parameter estimates of the logit equations for the cigarette use item and alcohol use item are generally decreasing over time, which also means that the participants had increasing likelihoods of becoming involved with substance use or being substance users. The location parameter estimates of the logit equation for marijuana use fluctuated over time 681

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Pearson correlations were calculated between the person trait estimates and the true values when the sample size is 2,000. The average correlation was .81 (SD = 0.01). The average correlation between the composite scores and the true

values also was calculated and was .54 (SD = 0.05). The correlations indicated that the proposed method should be more appropriate than the composite score method for estimating the substance use of youths.

In terms of accuracies of predictions on the probabilities of being in the Poisson state (pi) and the expected counts (A,-), Pearson correlations between the

predictions from the model and the true values were calculated and averaged across different replications with the sample size of 2,000. The correlations for the probabilities were .85, .79, and .89, respectively, and the correlations for the expected counts are .94, .92, and .93 for the 3 items, respectively. The correlations from the simulation studies are higher than the corresponding correlations from the empirical data mainly because that the simulated data were simulated from the proposed model and thus followed the model assumptions more strictly. Discussion and Conclusion

This study introduced an IRT-ZIP model to analyze multivariate zero-inflated count data using two link functions: a log link function and a logit link function. By including two link functions, the probability of being in the Poisson process and the

expected frequency of an event given that participant is in the Poisson process can

be modeled simultaneously. A latent person trait variable is included in the model to measure latent trait scores. Both item slope and location parameters are included

in the model to investigate psychometric properties of the multiple items.

The model was applied to analyze the substance use data to illustrate the applications. The empirical results provided rich information regarding both the sub-

stance use of adolescence and the model performance. The order of magnitude of the item location parameter estimates in the logit equations indicates that it is less

likely for a person with a fixed substance use level to be involved with marijuana

use than cigarette use and alcohol use. This result is partly consistent with the "gateway hypothesis" in substance use research (Kandel, 2003; Morral, McCaffrey, & Paddock, 2002). Kandel (2003) described the three propositions in the "gateway hypothesis," which are sequencing (fixed relationship between two substances such that one substance is regularly initiated before the other), association (initiation of one substance increases the likelihood of initiation of

the second substance), and causation (a controversial proposition, use of the first substance causes the use of the second substance). The application of the IRT-ZIP model to the substance use data in this study is based on the association proposition and the results were consistent with the sequencing proposition from this naturalistic population example with a novel method. Overall, the associations and differences among the 3 items measuring latent substance use levels

were investigated successfully using the IRT-ZIP model. The consistency 688

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IRT-ZIP Modeling between some of the current findings and previous findings supports the reason-

ableness and reliability of the results from the IRT-ZIP model. Simulation studies also were conducted to examine the performance of the model under the empirical scenario. The simulation results showed that, similar

to regular IRT models (Thissen & Wainer, 1982, 2001), fitting the IRT-ZIP model requires large sample sizes to obtain stable item parameter estimates. For a 2PL IRT model, rules of thumb for the minimum number of examinees required

for accurate parameter estimation range from 500 (Hulin, Lissak, & Drasgow, 1982) to 1,000 (Ree & Jensen, 1980). For the proposed IRT-ZIP model, in light of the tested simulation conditions, a sample size of 1,000 seems large enough to recover the parameter estimates. For large-scale psychological and educational observational or survey research, this might not be a big issue. However, in many practical psychological and educational experiments, only a small number of participants' data are collected and thus possibly preclude or at least limit the use of the proposed method. In terms of recovering the person trait scores, the performance of the IRT-ZIP model was better than the composite score method. This is because the IRT-ZIP model considers the natural features of the multivariate

zero-inflated count data. By including two link functions, we can model two pro-

cesses more accurately and validly. The model thus can be useful in large-scale psychological or educational settings where count data on the number of behaviors are available.

Limitations and Future Directions

There are several issues that should be investigated in the future. First, the

empirical data characteristics include a large sample size (N > 7,000) and repeated measures making them particularly apt for illustrating the application of the proposed model. However, the data were limited to only 3 items. With 3 items, the correlation between the person trait estimates and the true values was

only about .81 under the empirical data scenario, in accord with the simulation results. Further simulations should examine the performance of the model on estimating both item parameters and trait scores with larger numbers of items and

different proportions of zeros. Examples of applying the proposed models to problem behaviors or other psychological research with a larger number of items can also be included in future studies. Second, because the empirical data contain repeated measurements, change patterns of substance use levels from adoles-

cence to young adulthood could be investigated here. The proposed IRT-ZIP model can be extended to a three-level mixed-effects model, which involves estimating item parameters, latent person trait scores, and longitudinal change patterns simultaneously in future studies. Third, in the current proposed model, only one common latent variable is included. However, it is possible to include

two latent variables to investigate the propensity and level scores individually. Future studies can also be done to investigate the multidimensionality. 689

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Wang In summary, both the simulation results and the empirical results supported

the conclusion that the proposed IRT-ZIP model is a valid and feasible method for analyzing multivariate zero-inflated count data more precisely and reasonably. The work so far should encourage researchers to apply the IRT-ZIP model to more substantive research areas and thereby answer more interesting empirical research questions.

Appendix

SAS Scripts for Fitting an IRT-ZIP Model proc nlmixed data=realdata qpoints=5; bounds al-a6>0;

parms al = l a2=l a3=l a4=l a5=l a6=l betall=0 betal2=0 betal3=0 beta21=0 beta22=0 beta23=0;

slopel=al*il+a2*i2+a3*i3; Slope2=a4*il+a5*i2+a6*i3; etal=slopel*(theta-betall*il-betal2*i2-betal3*i3); eta2=slope2*(theta-beta2 1 *i 1 -beta22*i2-beta23 *i3);

lambda=exp(eta 1 );

p=exp(eta2)/(l+exp(eta2)); IF (resp=0) then pai=(l-p)-h p*exp(-l*lambda);

ELSEpai=p*exp(-l*lambda)*(lambda**resp)/gamma(resp+l); II=LOG(PAI);

model resp-GENERAL(II); random theta^normal(0,l) subjected; predict theta out=personparm;

predict p out=pout; predict lambda out=lambdaout; run;

Acknowledgment The author is grateful for the helpful comments on this study from John R.

Nesselroade, John J. McArdle, Steven M. Boker, Karen M. Schmidt, Xiaohui Wang, and the reviewers. Notes

1 . If the answer from Question 1 is 0, then the value of this variable is set to be

0; if the answer from question 1 is not 0 but the answer from Question 2 is 0, then

the value is also set to be 0; otherwise, the value is set to be the answer from Question 3. 690

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IRT-ZIP Modeling 2. It should be mentioned that for the marijuana use data, the observed frequency represents 30 days, and for the alcohol use and cigarette use, the frequency represents 1 day.

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SUGI 24 Conference Proceedings, Paper 287. Cary, NC: SAS Institute. Author

LIJUAN WANG, PhD, is an assistant professor at the University of Notre Dame, Notre Dame, IN, 46556; Email: [email protected]. Her research interests are longitudinal data analysis and analysis of nonnormal data.

Manuscript received May 12, 2009 Revision received January 17, 2010 Accepted February 28, 2010

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