Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Risk Classification 75
used). The ratio of the deviance to the number of degrees of freedom should be close to
1 to indicate goodness-of-fit. Here, we obtain 05357 (the deviance is equal to 7764.83).
Note however that the data should be grouped to make the Chi-square approximation more
reliable, so that we cannot use this statistic at the present stage.
An important aspect of insurance ratemaking with generalized regression models is the
selection of explanatory variables in the model. Changes in goodness-of-fit statistics are
often used to evaluate the contribution of subsets of explanatory variables to a particular
model. One strategy for variable selection is to fit a sequence of models, beginning with
a simple model with only an intercept term, and then include one additional explanatory
variable in each successive model. The importance of the additional explanatory variable
can be measured by the difference in fitted log-likelihoods between successive models.
Asymptotic tests computed by the GENMOD procedure enable the actuary to assess the
statistical significance of the additional term (this is called Type I analysis in SAS R ).
Another strategy (adopted here) consists of starting from a model incorporating all the
available information, and then excluding the irrelevant explanatory variables. To this end,
the GENMOD procedure generates a Type 3 analysis (analogous to Type III sums of squares
in the GLM procedure). A Type 3 analysis does not depend on the order in which the terms
for the model are specified (in contrast to the Type 1 analysis).
Type 3 analysis compares the complete model (that is the model which includes all the
specified variables) with the different submodels obtained by deleting one of the explanatory
variables. It enables the actuary to test the relevance of one variable taking all the others
into account. It roughly corresponds to the backward approach: at each step, we exclude the
variable with the largest p-value until no more can be excluded (i.e. until all the p-values
are smaller than a fixed threshold, generally 5 %). Note that the Type 3 analysis works with
the variables and not with the levels of these variables. Indeed, it is possible to obtain a
relevant variable for which some levels are not relevant. The results of the Type 3 analysis
are as follows:
Source DF Chi-square Pr > Chi-sq
Gender ∗ Age 7 7516