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

tel-00703797, version 2 - 7 Jun 2012
Chapitre 1. Sur la nécessité d’un modèle de marché
Let us first regress the bronchitis indicator on all variables
⎛ ⎞
⎛ ⎞
Y =
⎜
⎝
B1
.
Bn
⎟
⎠ and X =
⎜
⎝.
1 P1 C1
.
.
1 Pn Cn
with a logit link function. The regression summary is given below
⎟
⎠ ,
Call: glm(formula = bron ~ 1 + cigs + poll, family = binomial)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.4023 -0.5606 -0.4260 -0.3155 2.3594
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -10.08491 2.95100 -3.417 0.000632 ***
cigs 0.21169 0.03813 5.552 2.83e-08 ***
poll 0.13176 0.04895 2.692 0.007113 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Null deviance: 221.78 on 211 degrees of freedom
Residual deviance: 174.21 on 209 degrees of freedom - AIC: 180.21
So the GLM fit seems good because all variables (including intercept) are significant with
a very low p-value. However the plot of residuals ∗ (see Figure 1.8a) against fitted values † is
quite puzzling. Two distinct curves are shown: one for ill patients and the other for healthy
ones.
When categorizing the P variable, we lose information but we transform binary data into
binomial data. This makes the fit better on this aspect, see Figure 1.8b. So for the same data,
with the same (significant) variables, the two analyses of residuals lead to different conclusions.
Hence, conclusions of residual analysis must be taken with great care.
GLM outputs of Section 1.3.1
See below the summary table with coefficients values, standard errors, z-statistics and pvalue.
For confidentiality reason, all the deviance and AIC statistics shown in this paper have
been scaled by the same positive coefficient.
Here follows the regression summary when variables are categorical.
Call: glm(formula = did_lapse ~ agepolgroup2 + priceratio:agegroup4 +
priceratio * (gender + agevehgroup2 + prembeforegroup2),
family = binomial(), data = workdata)
Deviance Residuals:
Min 1Q Median 3Q Max
-3.1587 -0.6633 -0.6060 -0.5193 2.8747
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.522477 0.120852 -20.873 < 2e-16 ***
∗. Working residuals are ˆɛi = Yi − ˆπi. Note that using other residual types, Pearson, Studentized, do not
change this behavior.
†. Fitted values are ˆπi.
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