A comparison of zero-inflated and hurdle models for modeling zero-inflated count data

Abstract

Counts data with excessive zeros are frequently encountered in practice. For example, the number of health services visits often includes many zeros representing the patients with no utilization during a follow-up time. A common feature of this type of data is that the count measure tends to have excessive zero beyond a common count distribution can accommodate, such as Poisson or negative binomial. Zero-inflated or hurdle models are often used to fit such data. Despite the increasing popularity of ZI and hurdle models, there is still a lack of investigation of the fundamental differences between these two types of models. In this article, we reviewed the zero-inflated and hurdle models and highlighted their differences in terms of their data generating processes. We also conducted simulation studies to evaluate the performances of both types of models. The final choice of regression model should be made after a careful assessment of goodness of fit and should be tailored to a particular data in question.

Keywords: Hurdle model; Model diagnosis; Zero deflation; Zero inflation.

Fig. 1

Percentage of zero deflation over all data points when the data are simulated from a HNB model of sample size n=300. In the left panel, the covariate x is a binary variable simulated from a Bernoulli distribution with probability parameter 0.5. In the right panel, the covariate is a continuou variable simulated from a standard normal distribution

Fig. 2

Probabilities of observing a zero (green), a sampling zero (blue) and their differences (black) against the covariate when the data are simulated from a HNB model with a binary covariate of sample size n=300. The intercepts for both the zero and truncated counts components are set as 1. The regression coefficients of the covariate for the zero (β₁) and the truncated counts component (α₁) are set as -2, -1.5, -1, -0.1, 0.1, 1, 1.5 and 2

Fig. 3

Probabilities of observing a zero (green), a sampling zero (blue) and their differences (black) against the covariate, when the data are simulated from a HNB model with a continuous covariate of sample size n=300. The intercepts for both the zero and truncated counts components are set as 1. The regression coefficients of the covariate for the zero (β₁) and the truncated counts component (α₁) are set as -2, -1.5, -1, -0.1, 0.1, 1, 1.5 and 2

Fig. 4

Mean of the standardized differences of the probability of being an excessive zero and the probability of being a sampling zero when data are simulated from a ZINB model of sample size n=300. In the left panel, the covariate is a binary variable from a Bernoulli random variable with probability parameter 0.5. In the right panel, the covariate is a continuous variable from a standard normal distribution

Fig. 5

Simulation results for the simulation setting #1 (true model: HNB model with a single binary covariate generated from a Bernoulli distribution with probability parameter 0.5). Evaluation criteria include $\overline{\Delta }$AIC (mean difference in AICs of the ZINB and HNB models); %ΔAIC>4 (percentage of the differences in AICs between the ZINB and HNB models that are above 4; percentage of Vuong’s test p-value <5% and percentage of the SW normality test of the RQRs for the ZINB model <5%

Fig. 6

Simulation results for the simulation setting #1 (true model: HNB model with a single continuous covariate generated from a standard normal distribution). Evaluation criteria include $\overline{\Delta }$AIC (mean difference in AICs of the ZINB and HNB models); %ΔAIC>4 (percentage of the differences in AICs between the ZINB and HNB models that are above 4; the percentage of Vuong’s test p-value <5% and the percentage of the SW normality test of the RQRs for the ZINB model <5%

Fig. 7

Simulation results for the simulation setting #2 (true model: ZINB model with a single binary covariate generated from a Bernoulli distribution with probability parameter 0.5). Evaluation criteria include $\overline{\Delta }$AIC (mean difference in AICs of the HNB and ZINB models); %ΔAIC>4 (percentage of the differences in AICs between the HNB and ZINB models that are above 4; the percentage of Vuong’s test p-value <5% and the percentage of the SW normality test of the RQRs for the HNB model <5%

Fig. 8

Simulation results for the simulation setting #2 (true model: ZINB model with a single continuous covariate generated from a standard normal distribution). Evaluation criteria include $\overline{\Delta }$AIC (mean difference in AICs of the HNB and ZINB models); %ΔAIC>4 (percentage of the differences in AICs between the HNB and ZINB models that are above 4; the percentage of Vuong’s test p-value <5% and the percentage of the SW normality test of the RQRs for the HNB model <5%