Hurdle model

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Short description: Class of statistical models

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


  1. 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. 
  2. 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. 
  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. 
  4. 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.