showcase the utility of models designed to incorporate zeros from multiple generating processes. We will examine predictors of absences as a vehicle

Transcription

1 Lauren Porter, Gloria Yeomans-Maldonado, Ann A. O Connell Not All Zero s Are Created Equal: Zero-Inflated and Hurdle Models for Counts with Excess Zeros Background: School absenteeism has been shown to be associated with a variety of individual, family and schoolbased factors (Kearney, 2008). In the general student population, the distribution of counts of student absences is usually characterized by a marked positive skew due to a high frequency of zero absences. Studies examining predictors of absenteeism have used varied approaches to address the distribution of student absences and the preponderance of excess zeros including dichotomization of the outcome (e.g., normal- vs high-absence) ordinal grouping (e.g., no/low/high absences) or analyses of subgroups of students that ignore and omit non- or low-absentee individuals. Our aim in this paper is to contribute to the methodological literature for examining absenteeism and clarify the benefits and challenges of using count models for data with excess zeros. In particular, we distinguish between zero-inflated and hurdle models, for which the generation of zeros are differentially conceived, and demonstrate their application using data from the National Longitudinal Survey of Youth. Traditional regression-based methods for working with count data, such as ordinary least squares models, assume a normal distribution of the residuals and present the risk of under-inflating the influence of excess zeros (Hu, Pavlicova, & Nunes, 2011). Poisson regression is a model specifically for counts that is free of the assumption of normal residuals, but carries its own assumption about the shape of the distribution; namely, that the mean and variance of the count distribution are equal. This property is rarely met in practice, particularly when excess zeros are present. The Poisson distribution is a discrete probability distribution expressing the chances of a given number of events occurring in a fixed interval of time (Haight, 1967) and can be considered as a special case of the Negative Binomial (NB) distribution. The NB distribution is also a discrete probability distribution for the number of cases falling into one of two outcome categories, without the Poisson constraint of the mean and variance being identical. A limitation of NB regression is the potential confound between whether the appearance of overdispersion is a result of actual overdispersion or an error in the systematic part of the model (Berk & MacDonald, 2008). An alternative to Poisson models for count data with or without adjustment for overdispersion, zeroinflated models provide a way of treating data with a preponderance of zeroes. These models consist of two distributions: one with only zeroes, and another with both zeroes and non-zeroes (Lambert, 1992). This model is essentially a mixture of a Bernoulli distribution and a Poisson distribution for which the count model has a nonzero probability of generating zeroes. Limitations of this approach include the assumption that the zeroes all came from the same place (SAS, 2015). Hurdle regression, originating with Cragg (1971) and later popularized by Mullahy (1986) comprises two separate processes in one analysis: both binary and count processes. Bernoulli probability governs the binary outcome and, for non-zero cases, the conditional distribution of the positives is governed by a truncated-at-zero count data model. One instance of hurdle models in educational literature assesses misreported binary outcomes in randomized control trials (Schochet, 2013). An addition example uses zero-inflated and overdispersed count models to explore school suspensions (Desjardins, 2016). We will demonstrate the utility of these approaches for absenteeism data. Purpose: We focus on the defining features of the Zero-Inflated and Hurdle models as applied to educational data, with both Poisson and Negative Binomial assumptions. Our empirical illustration will

2 showcase the utility of models designed to incorporate zeros from multiple generating processes. We will examine predictors of absences as a vehicle to highlight the usefulness and applicability of these approaches. Sample and Data Collection: Pre-existing public-use data were used for this research. Administrative records documenting 9th grade absences from the National Longitudinal Survey of Youth (NLSY79) were analyzed, consisting of a nationally representative sample of 12,686 individuals years old initially interviewed in Cases with missing data for 9 th grade absences were excluded from analysis. Research Design and Analysis: Data were analyzed through the following regression methods: Poisson, Negative Binomial, Zero- Inflated Poisson, Zero-Inflated Negative Binomial, and Hurdle Negative Binomial. The results of these analyses will serve as a demonstration of how these models can be best applied in education research for absenteeism. Finite Mixture Modeling (FMM) in SAS 9.4 was used to analyze the data. Variables include the dependent variable (number of student absences), and four predictors: measure of general health, enrollment in remedial English, household income, and how many time the student was charged for an illegal act (Charlton et al., 1991; Havik, Bru, & Ertesvåg, 2015).For space considerations, individual and family-context are demonstrated here; our presentation will also examine school-context predictors. Results: Figure 1 displays the dependent variable, demonstrating substantial amounts of zeroes. Table 1 presents parameter estimates for the models. The Poisson Hurdle model experienced convergence errors; these errors will be addressed leading up to the presentation. One variable, the number of times a student is charged with illegal activity, is a significant predictor across all models. Controlling for all other variables in the model, every charge of illegal activity resulted in an expected increase of absences. More telling is the difference in the mixing probabilities between Zero-inflated and Hurdle models. The Negative Binomial Hurdle model indicates that 25.72% of students individual position on this distribution is accounted for by either the binary or truncated negative binomial processes. The Zero-Inflated Negative Binomial model indicates that 98.25% of students individual position on this distribution is accounted for by either the binary or truncated negative binomial processes. Conclusions: The Zero-Inflated and Hurdle model s utility lies in their ability to handle zeroes in ways that are useful to educational researchers. The Hurdle model assumes multiple data-generating processes are at play, which is not an assumption that can typically be made without first-hand knowledge of the data collection process and outcome measure. Additional features of both approaches will be highlighted during the presentation, including the importance of data collection knowledge, a potential source of bias in handling cases with excess zeroes.

3 References Berk, R., & MacDonald, J. M. (2008). Overdispersion and poisson regression. Journal of Quantitative Criminology, 24(3), Charlton, A., Larcombe, I. J., Meller, S. T., Jones, P. H. M., Mott, M. G., Potton, M. W., Walker, J. J. P. (1991). Absence from school related to cancer and other chronic conditions, (36), Cragg, J. G. (1971). Some Statistical Models for Limited Dependent Variables with Application to the Demand for Durable Goods Author ( s ): John G. Cragg. Econometrica, 39(5), Desjardins, C. D., & Desjardins, C. D. (2016). Modeling Zero-Inflated and Overdispersed Count Data : An Empirical Study of School Suspensions Modeling Zero-Inflated and Overdispersed Count Data : An Empirical Study of School Suspensions, 973(July). Haight, F. A. (1967). Handbook of the Poisson distribution. New York: Wiley. Havik, T., Bru, E., & Ertesvåg, S. K. (2015). truancy-related reasons for school non-attendance, (123), Hu, M.-C., Pavlicova, M., & Nunes, E. V. (2011). Zero-Inflated and Hurdle Models of Count Data with Extra Zeros: Examples from an HIV-Risk Reduction Intervention Trial. The American Journal of Drug and Alcohol Abuse, 37(5), Lambert, D. (1992). Zero-Inflated Poisson With an Regression, in Manufacturing to Defects Application. Technometrics, 34(1), Mullahy, J. (1986). Specification and testing of some modified count data models. Journal of Econometrics, 33, National Longitudinal Survey of Youth. The NLSY79 Sample: An Introduction. Obtained August 31, from sample-introduction. SAS. Usage Note 48506: Fitting Hurdle Models. Obtained Jan, 15, 2015 from Schochet, P. Z. (2013). A Statistical Model for Misreported Binary Outcomes in Clustered RCTs of Education Interventions. Journal of Educational and Behavioral Statistics, 38(5),

4 Figure 1: NLSY Data Histogram Table 1 Model Parameter Estimates Model Estimate SE p-value Absences Mixing Probabilities Poisson Intercept < N/A General Health < Remedial English Activities < K) < Zero-Inflated Poisson Intercept < Component 1 =.9325 General Health < Component 2 = Remedial English < Activities < K) < Negative Binomial Intercept < N/A

5 General Health Remedial English Activities K) Zero-Inflated Negative Binomial Intercept < Component 1 =.9825 General Health Component 2 = Remedial English Activities K) Hurdle Negative Binomial Intercept < Component 1 =.2572 General Health Component 2 = Remedial English Activities K)