Distribution Family and Canonical Link

The GAM Procedure

Distribution Family and Canonical Link

For each distribution, more than one link can exist. Different link functions may result in slight differences in estimates for parametric models. However, the difference will be less pronounced for nonparametric models because of the flexibility of nonparametric model forms. To simplify the calculation, the GAM procedure uses the canonical link.

The GAM procedure can fit the data from the Gaussian and binomial distributions.

The Gaussian Model

With this model, the link function is the identity function, and the generalized additive model is the additive model.

The Binomial and Logistic Models

A binomial response model assumes that the proportion of successes Y is such that Y has a Bin(n, p(x)) distribution. The Bin(n, p(x)) refers to the binomial distribution with parameters n and p(x). Often the data are binary, in which case n=1. The canonical link is

$g(p)=log \frac{p}{n-p}=\eta$

The Poisson Model

The link function for the Poisson model is the log function. Assume the mean of the Poisson distribution is $\mu(x)$ , the dependence of $\mu(x)$ and independent variable x₁, ... ,x_k is

$g(\mu)=log(\mu)=\eta$

The Gamma Model

Let the mean of the Gamma distribution be $\mu(x)$ . The canonical link function for the Gamma distribution is -1/ $\mu(x)$ . Therefore, the relationship between $\mu(x)$ and the independent variable x₁, ... ,x_k is

$g(\mu)=-\frac{1}{\mu}=\eta$

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