Hurdle model

A hurdle model is a class of statistical models where a random variable is modelled using two parts, the first of which is the probability of attaining the value 0, and the second part models the probability of the non-zero values. The use of hurdle models is 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 binomial³ models for the non-zero counts.

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