Canonical link functions

A type of link function known as the canonical link function is given by g = b'-1, where b is obtained from the general form for the density function for exponential distributions given above. For the distributions we have considered, the canonical link functions are given by

Distribution

g(fi) = b'~

-V)

Name

Bernoulli

log(ß/(1

- ß))

Logit

Binomial

log(ß/(1

- ß))

Logit

Poisson

log(ß)

Log

Poisson with offset

log(ß)

Log

In most situations, use of the canonical link function will lead to a satisfactory analysis model. However, we should also mention that there is not a strict requirement for canonical link functions to be used in the GLM and non-canonical link functions are also available. These are not derived from the density function but still map the data from their original scale onto the real scale. For example, a link function known as the probit function, g(ß) = $—1(ß) (where $ is the cumulative normal density function), is sometimes used for binary data recorded in toxicology experiments, since values of ß corresponding to specific probabilities can easily be obtained using the normal density function. Despite not being canonical, this link function does still map the original range of the data (0 to 1) to —to to to as required for the GLM. In this book we will not be considering non-canonical link functions. However, further information can be found in McCullagh andNelder (1989).

We earlier specified a general formula for the variance in the GLM as var(y) = ab"($). Using the relationship g = b'—1, for canonical link functions we can now equivalently write the variance in terms of ß and the function g as

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