> r=glm(dist~speed,data=cars,family=poisson) > P=predict(r,type="response", + newdata=data.frame(speed=seq(-1,35,by=.2))) > plot(cars,xlim=c(0,31),ylim=c(0,170)) > abline(v=30,lty=2) > lines(seq(-1,35,by=.2),P,lwd=2,col="red") > P0=predict(r,type="response",se.fit=TRUE, + newdata=data.frame(speed=30)) > points(30,P1$fit,pch=4,lwd=3)
Let denote the maximum likelihood estimator of . Then
where is Fisher information of (from standard maximum likelihood theory). Recall that
where computation of those values is based on the following calculations
In the case of the log-Poisson regression
Let us get back to our initial problem.
- confidence interval for the linear combination
Then, since as an asymptotic multivariate distribution, any linear combination of the parameters will also be normal, i.e.
has a normal distribution, centered on , with variance where is the variance of . All those quantities can be easily computed. First, we can get the variance of the estimators
Hence, if we compare with the output of the regression,
> summary(reg)$cov.unscaled (Intercept) speed (Intercept) 0.0066870446 -3.474479e-04 speed -0.0003474479 1.940302e-05 > V [,1] [,2] [1,] 0.0066871228 -3.474515e-04 [2,] -0.0003474515 1.940318e-05Based on those values, it is easy to derive the standard deviation for the linear combination,
And once we have the standard deviation, and normality (at least asymptotically), confidence intervals are derived, and then, taking the exponential of the bounds, we get confidence interval
- delta method
The delta method gives us (asymptotic) normality, so once we have a standard deviation, we get the confidence interval.
Note that those quantities - obtained with two different approaches - are rather close here
> exp(P2$fit-1.96*P2$se.fit) 1 138.8495 > P1$fit-1.96*P1$se.fit 1 137.8996 > exp(P2$fit+1.96*P2$se.fit) 1 173.9341 > P1$fit+1.96*P1$se.fit 1 172.9101
- bootstrap techniques