Exposure with binomial responses

Last week, we’ve seen how to take into account the exposure to compute nonparametric estimators of several quantities (empirical means, and empirical variances) incorporating exposure. Let us see what can be done if we want to model a binomial response. The model here is the following: ,

• the number of claims  on the period  is unobserved
• the number of claims  on  is observed (as well as )

that can be visualize below

Consider the case where the variable of interest is not the number of claims, but simply the indicator of the occurrence of a claim. Then we wish to model the event versus , interpreted as non-occurrence and occurrence. Given the fact that we can only observe  versus . Having an inclusion is not enough to derive a model. Actually, with a Poisson process model, we can get easily that

With words, it means that the probability of not having a claim in the first six months of the year is the square root of not have a claim over a year. Which makes sense. Assume that the probability of not having a claim can be explained by some covariates, denoted , through some link function (using the GLM terminology),

Now, since we do observe  – and not  – we have

The dataset we will use is always the same

> sinistre=read.table("http://freakonometrics.free.fr/sinistreACT2040.txt",
+ header=TRUE,sep=";")
> sinistres=sinistre[sinistre$garantie=="1RC",]
> sinistres=sinistres[sinistres$cout>0,]
> contrat=read.table("http://freakonometrics.free.fr/contractACT2040.txt",
+ header=TRUE,sep=";")
> T=table(sinistres$nocontrat)
> T1=as.numeric(names(T))
> T2=as.numeric(T)
> nombre1 = data.frame(nocontrat=T1,nbre=T2)
> I = contrat$nocontrat%in%T1
> T1= contrat$nocontrat[I==FALSE]
> nombre2 = data.frame(nocontrat=T1,nbre=0)
> nombre=rbind(nombre1,nombre2)
> sinistres = merge(contrat,nombre)
> sinistres$nonsin = (sinistres$nbre==0)

The first model we can consider is based on the standard logistic approach, i.e.

That’s nice, but difficult to handle with standard functions. Nevertheless, it is always possible to compute numerically the maximum likelihood estimator of given .

> Y=sinistres$nonsin
> X=cbind(1,sinistres$ageconducteur)
> E=sinistres$exposition
> logL = function(beta){
+ 	pi=(exp(X%*%beta)/(1+exp(X%*%beta)))^E
+ 	-sum(log(dbinom(Y,size=1,prob=pi)))
+ }
> optim(fn=logL,par=c(-0.0001,-.001),
+ method="BFGS")
$par
[1] 2.14420560 0.01040707
$value
[1] 7604.073
$counts
function gradient 
      42       10 
$convergence
[1] 0
$message
NULL
> parametres=optim(fn=logL,par=c(-0.0001,-.001),
+ method="BFGS")$par

Now, let us look at alternatives, based on standard regression models. For instance a binomial-log model. Because the exposure appears as a power of the annual probability, everything would be fine if  was the exponential function (or  was the log link function), since

Now, if we try to code it, it starts quickly to be problematic,

> reg=glm(nonsin~ageconducteur+offset(exposition),
+ data=sinistresI,family=binomial(link="log")) 
Error: no valid set of coefficients has been found: please supply starting values

I tried (almost) everything I could, but I could not get rid of that error message,

> startglm=c(0,-.001)
> names(startglm)=c("(Intercept)","ageconducteur")
> etaglm=rep(-.01,nrow(sinistresI))
> etaglm[sinistresI$nonsin==0]=-10
> muglm=exp(etaglm)
> reg=glm(nonsin~ageconducteur+offset(exposition),
+ data=sinistresI,family=binomial(link="log"),
+ control = glm.control(epsilon=1e-5,trace=TRUE,maxit=50),
+ start=startglm,
+ etastart=etaglm,mustart=muglm)
Deviance = NaN Iterations - 1 
Error: no valid set of coefficients has been found: please supply starting values

So I decided to give up. Almost. Actually, the problem comes from the fact that is closed to 1. I guess everything would be nicer if we could work with probability close to 0. Which is possible, since

where is close to 0. So we can use Taylor’s expansion,

Here, the exposure does no longer appears as a power of the probability, but appears multiplicatively. Of course, there are higher order terms. But let us forget them (so far). If – one more time – we consider a log link function, then we can incorporate the exposure, or to be more specific, the logarithm of the exposure.

> regopp=glm((1-nonsin)~ageconducteur+offset(log(exposition)),
+ data=sinistresI,family=binomial(link="log"))

which now works perfectly.

Now, to see a final model, perhaps we should get back to our Poisson regression model since we do have a model for the probability that .

> regpois=glm(nbre~ageconducteur+offset(log(exposition)),
+ data=sinistres,family=poisson(link="log"))

We can now compare those three models. Perhaps, we should also include the prediction without any explanatory variable. For the second model (actually, it does run without any explanatory variable), we run

>  regreff=glm((1-nonsin)~1+offset(log(exposition)),
+ data=sinistres,family=binomial(link="log"))

so that the prediction is here

> exp(coefficients(regreff))
(Intercept) 
 0.06776376

This value is comparable with the logistic regression,

> logL2 = function(beta){
+ 	pi=(exp(beta)/(1+exp(beta)))^E
+ 	-sum(log(dbinom(Y,size=1,prob=pi)))}
> param=optim(fn=logL2,par=.01,method="BFGS")$par
> 1-exp(param)/(1+exp(param))
[1] 0.06747777

But is quite different from the Poisson model,

> exp(coefficients(glm(nbre~1+offset(log(exposition)),
+ data=sinistres,family=poisson(link="log"))))
(Intercept) 
 0.07279295

Let us produce a graph, to compare those models,

> age=18:100
> yml1=exp(parametres[1]+parametres[2]*age)/(1+exp(parametres[1]+parametres[2]*age))
> plot(age,1-yml1,type="l",col="purple")
> yp=predict(regpois,newdata=data.frame(ageconducteur=age,
+ exposition=1),type="response")
> yp1=1-exp(-yp)
> ydl=predict(regopp,newdata=data.frame(ageconducteur=age,
+ exposition=1),type="response")
> plot(age,ydl,type="l",col="red")
> lines(age,yp1,type="l",col="blue")
> lines(age,1-yml1,type="l",col="purple")
> abline(h=exp(coefficients(regreff)),lty=2)

Observe here that the three models are quite different. Actually, with two models, it is possible to run more complex regression, e.g. with splines, to visualize the impact of the age on the probability of having – or not – a car accident. If we compare the Poisson regression (still in red) and the log-binomial model, with Taylor’s expansion, we get

The next step is to see how to incorporate the exposure in a tree. But that’s another story…

Arthur Charpentier

Arthur Charpentier, professor in Montréal, in Actuarial Science. Former professor-assistant at ENSAE Paristech, associate professor at Ecole Polytechnique and assistant professor in Economics at Université de Rennes 1.  Graduated from ENSAE, Master in Mathematical Economics (Paris Dauphine), PhD in Mathematics (KU Leuven), and Fellow of the French Institute of Actuaries.

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