# Hurdle model

__: Class of statistical models__

**Short description**A **hurdle model** is a class of statistical models where a random variable is modelled using two parts, the first which is the probability of attaining value 0, and the second part models the probability of the non-zero values. The use of hurdle models are often motivated by an excess of zeroes in the data, that is not sufficiently accounted for in more standard statistical models.

In a hurdle model, a random variable *x* is modelled as

- [math]\displaystyle{ \Pr (x = 0) = \theta }[/math]
- [math]\displaystyle{ \Pr (x \ne 0) = p_{x \ne 0}(x) }[/math]

where [math]\displaystyle{ p_{x \ne 0}(x) }[/math] is a truncated probability distribution function, truncated at 0.

Hurdle models were introduced by John G. Cragg in 1971,^{[1]} where the non-zero values of *x* were modelled using a normal model, and a probit model was used to model the zeros. The probit part of the model was said to model the presence of "hurdles" that must be overcome for the values of x to attain non-zero values, hence the designation *hurdle model*. Hurdle models were later developed for count data, with Poisson, geometric,^{[2]} and negative binomial^{[3]} models for the non-zero counts .

## Relationship with zero-inflated models

Hurdle models differ from zero-inflated models in that zero-inflated models model the zeros using a two-component mixture model. With a mixture model, the probability of the variable being zero is determined by both the main distribution function [math]\displaystyle{ p(x = 0) }[/math] and the mixture weight [math]\displaystyle{ \pi }[/math]. Specifically, a zero-inflated model for a random variable *x* is

- [math]\displaystyle{ \Pr (x = 0) = \pi + (1 - \pi) \times p(x = 0) }[/math]

- [math]\displaystyle{ \Pr (x = h_i) = (1 - \pi) \times p(x = h_i) }[/math]

where [math]\displaystyle{ \pi }[/math] is the mixture weight that determines the amount of zero-inflation. A zero-inflated model can only increase the probability of [math]\displaystyle{ \Pr (x = 0) }[/math], but this is not a restriction in hurdle models.^{[4]}

## See also

## References

- ↑ Cragg, John G. (1971). "Some Statistical Models for Limited Dependent Variables with Application to the Demand for Durable Goods".
*Econometrica***39**(5): 829–844. doi:10.2307/1909582. - ↑ Mullahy, John (1986). "Specification and testing of some modified count data models".
*Journal of Econometrics***33**(3): 341–365. doi:10.1016/0304-4076(86)90002-3. - ↑ Welsh, A. H.; Cunningham, R. B.; Donnelly, C. F.; Lindenmayer, D. B. (1996). "Modelling the abundance of rare species: statistical models for counts with extra zeros".
*Ecological Modelling***88**(1–3): 297–308. doi:10.1016/0304-3800(95)00113-1. - ↑ Min, Yongyi; Agresti, Alan (2005). "Random effect models for repeated measures of zero-inflated count data".
*Statistical Modelling***5**(1): 1–19. doi:10.1191/1471082X05st084oa.

Original source: https://en.wikipedia.org/wiki/Hurdle model.
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