The link function
g
for a generalized linear model relates the
linear predictor
to the mean response
by
Links are implemented as objects. Table 1 lists the pre-defined link functions, along with the expressions used to return link objects.
Table 1:
Link Functions and Expression for Obtaining Link Objects
With one exception, the pre-defined links require no parameters.
These link objects can therefore be shared among models. The exception
is the power link. Links for binomial models are defined for
n = 1
trials and assume
.
Link objects inherit from the ="2D glim-link-proto=-1 prototype. The ="2D log-link=-1 object, for example, is constructed by
(defproto log-link () () glim-link-proto)Since this prototype can be used directly in model objects, the convention of having prototype names end in ="2D -proto=-1 is not used. The ="2D glim-link-proto=-1 prototype provides a ="2D :print=-1 method that should work for most link functions. The ="2D log-link=-1 object prints as
> log-link #<Glim Link Object: LOG-LINK>
The
="2D glim-proto=-1
computing methods assume that a link object
responds to three messages:
(send
link
:eta
mu
)
(send
link
:means
eta
)
(send
link
:derivs
mu
)
The ="2D :eta=-1 method returns a sequence of linear predictor values for a particular mean sequence. The ="2D :means=-1 method is the inverse of ="2D :eta=-1 : it returns mean values for specified values of the linear predictor. The ="2D :derivs=-1 method returns the values of
at the specified mean values. As an example, for the ="2D log-link=-1 object these three methods are defined as
(defmeth log-link :eta (mu) (log mu)) (defmeth log-link :means (eta) (exp eta)) (defmeth log-link :derivs (mu) (/ mu))
Alternative link structures can be constructed by setting up a new prototype and defining appropriate ="2D :eta=-1 , ="2D :means=-1 , and ="2D :derivs=-1 methods. Parametric link families can be implemented by providing one or more slots for holding the parameters. The power link is an example of a parametric link family. The power link prototype is defined as
(defproto power-link-proto '(power) () glim-link-proto)The slot ="2D power=-1 holds the power exponent. An accessor method is defined by
(defmeth power-link-proto :power () (slot-value 'power))and the ="2D :isnew=-1 initialization method is defined to require a power argument:
(defmeth power-link-proto :isnew (power) (setf (slot-value 'power) power))Thus a power link for a particular exponent, say the exponent 2, can be constructed using the expression
(send power-link-proto :new 2)
To complete the power link prototype, we need to define the three required methods. They are defined as
(defmeth power-link-proto :eta (mu) (^ mu (send self :power)))
(defmeth power-link-proto :means (eta) (^ eta (/ (slot-value 'power))))and
(defmeth power-link-proto :derivs (mu)
(let ((p (slot-value 'power)))
(* p (^ mu (- p 1)))))
The definition of the
="2D :means=-1
method could be improved to allow
negative arguments when the power is an odd integer. Finally, the
="2D :print=-1
method is redefined to reflect the value of the
exponent:
(defmeth power-link-proto :print (&optional (stream t)) (format stream ``#<Glim Link Object: Power Link (~s)>'' (send self :power)))Thus a square link prints as
> (send power-link-proto :new 2) #<Glim Link Object: Power Link (2)>
Anthony Rossini