Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
Model adequacy
1.2. GLMs, a brief introduction
The deviance, which is one way to measure the model adequacy with the data and generalizes
the R 2 measure of linear models, is defined by
D(y, ˆπ) = 2(ln(L(y1, . . . , yn, y1, . . . , yn)) − ln(L(ˆπ1, . . . , ˆπn, y1, . . . , yn))),
where ˆπ is the estimate of the beta vector. The “best” model is the one having the lowest
deviance. However, if all responses are binary data, the first term can be infinite. So in
practice, we consider the deviance simply as
D(y, ˆπ) = −2 ln(L(ˆπ1, . . . , ˆπn, y1, . . . , yn)).
Furthermore, the deviance is used as a relative measure to compare two models. In most softwares,
in particular in R, the GLM fitting function provides two deviances: the null deviance
and the deviance. The null deviance is the deviance for the model with only an intercept or if
not offset only, i.e. when p = 1 and X is only an intercept full of 1 ∗ . The (second) deviance
is the deviance for the model D(y, ˆπ) with the p explanatory variables. Note that if there are
as many parameters as there are observations, then the deviance will be the best possible, but
the model does not explain anything.
Another criterion introduced by Akaike in the 70’s is the Akaike Information Criterion
(AIC), which is also an adequacy measure of statistical models. Unlike the deviance, AIC
aims to penalized overfitted models, i.e. models with too many parameters (compared to the
length of the dataset). AIC is defined by
AIC(y, ˆπ) = 2k − ln(L(ˆπ1, . . . , ˆπn, y1, . . . , yn)),
where k the number of parameters, i.e. the length of β. This criterion is a trade-off between
further improvement in terms of log-likelihood with additional variables and the additional
model cost of including new variables. To compare two models with different parameter
numbers, we look for the one having the lowest AIC.
In a linear model, the analysis of residuals (which are assumed to be identical and independent
Gaussian variables) may reveal that the model is unappropriate. Typically we can
plot the fitted values against the fitted residuals. For GLMs, the analysis of residuals is much
more complex, because we loose the normality assumption. Furthermore, for binary data, i.e.
not binomial data, the plot of residuals exhibits straight lines, which are hard to interpret, see
Appendix 1.8.2. We believe that the residual analysis is not appropriate for binary regressions.
Variable selection
From the normal asymptotic distribution of the maximum likelihood estimator, we can
derive confidence intervals as well as hypothesis tests for coefficents. Therefore, a p-value is
available for each coefficient of the regression, which help us to keep only the most significant
variable. However, as removing one variable impacts the significance of other variables, it can
be hard to find the optimal set of explanatory variables.
There are two approaches: either a forward selection, i.e. starting from the null model, we
add the most significant variable at each step, or a backward elimination, i.e. starting from
the full model, we remove the least significant variable at each step.
∗. It means all the heterogeneity of data comes from the random component.
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