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

$\Pr(x=0)=\theta$

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

where $p_{x\neq 0}(x)$ is a truncated probability distribution function, truncated at 0.

Hurdle models were introduced by John G. Cragg in 1971, 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, and negative binomial 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 $p(x=0)$ and the mixture weight $\pi$. Specifically, a zero-inflated model for a random variable x is

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

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

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

See also

• Zero-inflated model
• Truncated normal hurdle model