Let λxtc denote the systematic component and εxtc the random component. The model is:
nxtc = λxtc + εxtc (1)
With E(nxtc) = λxtc
E(εxtc) = 0.
The parameter λxtc of the Poisson distribution and λxtc are assumed to satisfy a model
that is loglinear in a set Θ of unknown parameters. One parameter is associated with
each of the ages, cohorts and periods. The systematic component is
λxtc=Lxtcκαxβtτc exp γZxtc (2)
where Θ = {κ, αx, βt, τc, γ}, γ being a k-length vector, Lxtc is the duration of exposure
assumed to be given, and Zxtc is a vector of covariates Z(k)xtc, k=1,..,K. Model (2) is the
multiplicative formulation of the loglinear model. The additive formulation is obtained
by taking the natural logarithm of both sides. In that case, the ln of the dependent
variable is linear in the parameters.
The unknown parameters must be determined from the data. This may be done using the
method of maximum likelihood. To evaluate the goodness of fit of the model, we
compare the likelihood achieved by the current model to the maximum of the likelihood
achievable (i.e. the likelihood achieved by the full model). The logarithm of the ratio is
known as the scaled deviance. The deviance is proportional to twice the difference
between the loglikelihoods:
S(n, λ) = -2 ln [L(λ,n)∕L(n,n)] = 2[ln L(n.n) - ln L(λ,n)] (3)
Large values of S indicate low values of L(λ,n) relative to the full model, increasing
lack of fit. For the Poisson distribution, the deviance is
S(n,λ)=2∑xtc[nxtcln(nxtc∕λxtc)-(nxtc - λxtc)] (4)
If a constant term 0, which is known as the nuisance parameter, is included in the
model it is generally the case that Σ(nxtc-λxtc) = 0 so that
D(n,λ)=S(n, λ) 0 (5)
may be written in the more usual form of the loglikelihood ratio which is often used as a
test in the analysis of contingency tables
D(n,λ)=2∑xtcnxtcln(nxtc∕λxtc) (6)
In order to determine the unknown Θ parameters with maximum likelihood, we need to
maximize the loglikelihood function with respect to the parameters. This results in a set
of normal equations which need to be solved for the unknown parameters. The GLIM
package, which uses generalized weighted least squares, was applied. The weights are