Zero-inflated model

In statistics, a zero-inflated model is a statistical model based on a zero-inflated probability distribution, i.e. a distribution that allows for frequent zero-valued observations.

Zero-inflated Poisson

One well-known zero-inflated model is Diane Lambert's zero-inflated Poisson model, which concerns a random event containing excess zero-count data in unit time.[1] For example, the number of insurance claims within a population for a certain type of risk would be zero-inflated by those people who have not taken out insurance against the risk and thus are unable to claim. The zero-inflated Poisson (ZIP) model mixes two zero generating processes. The first process generates zeros. The second process is governed by a Poisson distribution that generates counts, some of which may be zero. The mixture is described as follows:

${\displaystyle \Pr(Y=0)=\pi +(1-\pi )e^{-\lambda }}$

${\displaystyle \Pr(Y=y_{i})=(1-\pi ){\frac {\lambda ^{y_{i}}e^{-\lambda }}{y_{i}!}},\qquad y_{i}=1,2,3,...}$

where the outcome variable ${\displaystyle y_{j}}$ has any non-negative integer value, ${\displaystyle \lambda }$ is the expected Poisson count for the ${\displaystyle i}$th individual; ${\displaystyle \pi }$ is the probability of extra zeros.

The mean is ${\displaystyle (1-\pi )\lambda }$ and the variance is ${\displaystyle \lambda (1-\pi )(1+\pi \lambda )}$.

Estimators of ZIP parameters

The method of moments estimators are given by[2]

${\displaystyle {\hat {\lambda }}_{mo}={\frac {s^{2}+m^{2}}{m}}-1,}$

${\displaystyle {\hat {\pi }}_{mo}={\frac {s^{2}-m}{s^{2}+m^{2}-m}},}$

where ${\displaystyle m}$ is the sample mean and ${\displaystyle s^{2}}$ is the sample variance.

The maximum likelihood estimator[3] can be found by solving the following equation

${\displaystyle m(1-e^{-{\hat {\lambda }}_{ml}})={\hat {\lambda }}_{ml}\left(1-{\frac {n_{0}}{n}}\right).}$

where ${\displaystyle {\frac {n_{0}}{n}}}$ is the observed proportion of zeros.

A closed form solution of this equation is given by[4]

${\displaystyle {\hat {\lambda }}_{ml}=W_{0}(-se^{-s})+s}$

with ${\displaystyle W_{0}}$ being the main branch of Lambert's W-function[5] and

${\displaystyle s={\frac {m}{1-{\frac {n_{0}}{n}}}}}$.

Alternatively, the equation can be solved by iteration.[6]

The maximum likelihood estimator for ${\displaystyle \pi }$ is given by

${\displaystyle {\hat {\pi }}_{ml}=1-{\frac {m}{{\hat {\lambda }}_{ml}}}.}$

Related models

In 1994, Greene considered the zero-inflated negative binomial (ZINB) model.[7] Daniel B. Hall adapted Lambert's methodology to an upper-bounded count situation, thereby obtaining a zero-inflated binomial (ZIB) model.[8]

Discrete pseudo compound Poisson model

If the count data ${\displaystyle Y}$ is such that the probability of zero is larger than the probability of nonzero, namely

${\displaystyle \Pr(Y=0)>0.5}$

then the discrete data ${\displaystyle Y}$ obey discrete pseudo compound Poisson distribution.[9]

In fact, let ${\displaystyle G(z)=\sum \limits _{n=0}^{\infty }P(Y=n)z^{n}}$ be the probability generating function of ${\displaystyle y_{i}}$. If ${\displaystyle p_{0}=\Pr(Y=0)>0.5}$, then ${\displaystyle |G(z)|\geqslant p_{0}-\sum \limits _{i=1}^{\infty }p_{i}=2p_{0}-1>0}$. Then from the Wiener–Lévy theorem,[10] ${\displaystyle G(z)}$ has the probability generating function of the discrete pseudo compound Poisson distribution.

We say that the discrete random variable ${\displaystyle Y}$ satisfying probability generating function characterization

${\displaystyle G_{Y}(z)=\sum \limits _{n=0}^{\infty }P(Y=n)z^{n}=\exp \left(\sum _{k=1}^{\infty }\alpha _{k}\lambda (z^{k}-1)\right),\quad (|z|\leq 1)}$

has a discrete pseudo compound Poisson distribution with parameters

${\displaystyle (\lambda _{1},\lambda _{2},\ldots )=(\alpha _{1}\lambda ,\alpha _{2}\lambda ,\ldots )\in \mathbb {R} ^{\infty }\left(\sum _{k=1}^{\infty }\alpha _{k}=1,\sum \limits _{k=1}^{\infty }|\alpha _{k}|<\infty ,\alpha _{k}\in \mathbb {R} ,\lambda >0\right).}$

When all the ${\displaystyle \alpha _{k}}$ are non-negative, it is the discrete compound Poisson distribution (non-Poisson case) with overdispersion property.

See also

• Poisson distribution
• Zero-truncated Poisson distribution
• Compound Poisson distribution
• Sparse approximation
• Hurdle model

Software

• pscl and brms R packages

References

1. Lambert, Diane (1992). "Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing". Technometrics. 34 (1): 1–14. doi:10.2307/1269547. JSTOR 1269547.
2. Beckett, Sadie; Jee, Joshua; Ncube, Thalepo; Washington, Quintel; Singh, Anshuman; Pal, Nabendu (2014). "Zero-inflated Poisson (ZIP) distribution: parameter estimation and applications to model data from natural calamities". Involve. 7 (6): 751–767. doi:10.2140/involve.2014.7.751.
3. Johnson, Norman L.; Kotz, Samuel; Kemp, Adrienne W. (1992). Univariate Discrete Distributions (2nd ed.). Wiley. pp. 312–314. ISBN 978-0-471-54897-3.
4. Dencks, Stefanie; Piepenbrock, Marion; Schmitz, Georg (2020). "Assessing Vessel Reconstruction in Ultrasound Localization Microscopy by Maximum-Likelihood Estimation of a Zero-Inflated Poisson Model". IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. doi:10.1109/TUFFC.2020.2980063.
5. Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996). "On the Lambert W Function". Advances in Computational Mathematics. 5 (1): 329–359. arXiv:1809.07369. doi:10.1007/BF02124750.
6. Böhning, Dankmar; Dietz, Ekkehart; Schlattmann, Peter; Mendonca, Lisette; Kirchner, Ursula (1999). "The zero-inflated Poisson model and the decayed, missing and filled teeth index in dental epidemiology". Journal of the Royal Statistical Society, Series A. 162 (2): 195–209. doi:10.1111/1467-985x.00130.
7. Greene, William H. (1994). "Some Accounting for Excess Zeros and Sample Selection in Poisson and Negative Binomial Regression Models". Working Paper EC-94-10: Department of Economics, New York University. SSRN 1293115.
8. Hall, Daniel B. (2000). "Zero-Inflated Poisson and Binomial Regression with Random Effects: A Case Study". Biometrics. 56 (4): 1030–1039. doi:10.1111/j.0006-341X.2000.01030.x.
9. Huiming, Zhang; Yunxiao Liu; Bo Li (2014). "Notes on discrete compound Poisson model with applications to risk theory". Insurance: Mathematics and Economics. 59: 325–336. doi:10.1016/j.insmatheco.2014.09.012.
10. Zygmund, A. (2002). Trigonometric Series. Cambridge: Cambridge University Press. p. 245.