htl.html`The negative binomial (NB) distribution that predicts species abundance, ${tex`y`}, given mean, ${tex`\mu`}, and reciprocal dispersion, ${tex`\phi`}, has mass function:

${tex`\displaystyle \text{NB}(y_i | \mu_i,\phi) = \binom{y_i + \phi - 1}{y} \binom{\mu_i}{\mu_i + \phi}^y_i \binom{\phi}{\mu_i + \phi}^\phi, ~~\text{with}~~\text{E}[y_i] = \mu_i, \text{ and } \text{Var}[y_i] = \mu_i + \frac{\mu_i^2}{\phi}. `}

The zero-inflated NB with probability of excess zero linked to the mean (ZINBL) extends the NB distribution via:

${tex`\displaystyle \text{ZINBL}(y_i|\mu_i,\phi,\pi_i) = \begin{cases} \pi_i + (1 - \pi_i) \text{NB}(0 | \mu_i, \phi) & y_i = 0 \\[2mm] (1 - \pi_i) \text{NB}(y_i | \mu_i, \phi) & y_i > 0, \end{cases}~~~~\text{with}~\text{E}[y_i] = (1 - \pi_i)\mu_i, \text{ and } \text{Var}[y_i] = (1 - \pi_i)\left(\mu_i + \frac{\mu_i^2}{\phi}\right), `}

and where probability of excess zero, ${tex`\small \pi`}, is a decreasing logistic function of the NB mean:

${tex`\displaystyle \text{logit}~\pi_i = \gamma_0 - \gamma_1 \cdot \mu_i / H, `}

such that ${tex`\small \gamma_0`} and ${tex`\small \gamma_1`} control the log-odds intercept and slope of the probability of observing an excess zero given the mean of the distribution. Note that the mean ${tex`\small \mu`} is a modskurt function of an environmental variable ${tex`\small x`} with height parameter ${tex`\small H = \max \mu`} that can be adjusted below.

Finally, for ease of prior specification on the reciprocal dispersion of abundance ${tex`\small phi`}, we use a transformed variable,

${tex`\displaystyle \kappa = 1 / \sqrt{\phi}, `}

that models "overdispersion" directly, so that ${tex`\small \kappa \to 0`} leads to Poisson-like equidispersion (${tex`\small \text{E}[y] = \text{Var}[y]`}) and large ${tex`\small \kappa`} produces high levels of overdisperson. Refer to {{link to thesis?}} for the ecological motivations and mathematical details. `