6.10 The Hurdle model

Proposed first by (Crawley 2013), this model can take into account the fact that the data has more zeros and also can handle the overdispersion problem. It has two components (or steps), truncated count component defined by the chosen discrete distribution such as poisson or negative binomial, and a hurdle components models zero vs larger counts (that uses censored count distribution or binomial model). In other words, this models assumes that two population distributions underlying the data, one distribution for zero values, and another different distribution for the positive values, such that we will end up with two models.

The general form of the hurdle model assumes that each of those models has different (or the same) of the regressors set. Thus, if the regressors set of the zero model is \(Z\), and for the second model is \(X\), then the model is given by:

\[\begin{equation} f_h(y|X,Z,\beta, \gamma)=\begin{cases}f_{zero}(y=0|Z,\gamma) \qquad y=0 \\ \frac{\bigg[1-f_{zero}(y=0|z,\gamma)\bigg]f_{count}(y|X, \beta)}{\bigg[1-f_{count}(y=0|X,\beta)\bigg]} \quad y>0 \end{cases} \tag{6.78} \end{equation}\]

Where the \(f_{count}\) model could be a Poisson or Negative binomial model. For instance if we use the Poisson model, the above formulation will be rewritten as follows:

\[\begin{equation} f_h(y|X,Z,\beta, \gamma)= \begin{cases} exp(-z_i\gamma) \qquad y=0 \\ \frac{(1-exp(-z_i\gamma))\lambda_i^{y_i} exp(-\lambda_i)}{y_i!(1-exp(-\lambda_i))} \quad y>0 \end{cases} \tag{6.79} \end{equation}\]

Using again the log link function (6.69), the log-likelihood function will be:

\[\begin{equation} l(\beta, \gamma)=\sum(log(1-exp(-z_i\gamma))-y_ix_i\beta-exp(x_i\beta)-log(y_i!)-log(1-exp(exp(x_i\beta)))) \tag{6.80} \end{equation}\]

Then we can solve this equation using the numerical methods.