Hurdle model

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

${\displaystyle \Pr(x=0)=\theta }$

${\displaystyle \Pr(x\neq 0)=p_{x\neq 0}(x)}$

where ${\displaystyle p_{x\neq 0}(x)}$ 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 ${\displaystyle p(x=0)}$ and the mixture weight ${\displaystyle \pi }$. Specifically, a zero-inflated model for a random variable x is

${\displaystyle \Pr(x=0)=\pi +(1-\pi )\times p(x=0)}$

${\displaystyle \Pr(x=h_{i})=(1-\pi )\times p(x=h_{i})}$

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

See also

• Zero-inflated model
• Truncated normal hurdle model

References

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. JSTOR   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. CiteSeerX   10.1.1.296.3503 . doi:10.1191/1471082X05st084oa. S2CID   2400918.