Calculating/understanding variance-covariance matrix of logistic regression (lrm $var)
On Thu, 29 Jan 2004 02:34:27 +0100 (CET)
Karl Knoblick <karlknoblich at yahoo.de> wrote:
Hallo!
I want to understand / recalculate what is done to get
the CI of the logistic regression evaluated with lrm.
As far as I came back, my problem is the
variance-covariance matrix fit$var of the fit
(fit<-lrm(...), fit$var). Here what I found and where
I stucked:
-----------------
library(Design)
# data
D<-c(rep("a", 20), rep("b", 20))
V<-0.25*(1:40)
V[1]<-25
V[40]<-15
data<-data.frame(D, V)
d<-datadist(data)
options(datadist="d")
# Fit
fit<-lrm(D ~ V, data=data, x=TRUE, se.fit=TRUE)
plot(fit, conf.int=0.95) # same as plot(fit)
# calculation of upper and lower CI (pred$lower,
pred$upper)
pred<-predict(fit, data.frame(V=V), conf.int=0.95,
se.fit=TRUE)
points(V, pred$upper, col=2, pch=3) # to check
# looking in function predict, the CI are calculated
with the se
# using fit$var:
X<-cbind(rep(1, length(fit$x)), fit$x) # fit$x are the
V
cov<-fit$var # <- THIS I DO NOT UNDERSTAND (***) s.
below
se <- drop(sqrt(((X %*% cov) * X) %*% rep(1,
ncol(X))))
# check if it is the same
min(se - pred$se.fit) # result: 0
max(se - pred$se.fit) # result: 0
# looking at the problem:
cov
-----------------
Result:
Intercept V
Intercept 0.7759040 -0.12038969
V -0.1203897 0.02274177
(***)
fit$var is the estimated variance-covariance matrix.
How is it calculated? (Meaning of intercept and x?)
Does anybody know how calculationg this "by hand" or
can give me a reference (preferable in the internet)?
Thanks!
Karl.
Karl:
I'm not clear why you quoted the other code as your entire question is
about the basic quantity fit$var. fit$var is the inverse of the observed
information matrix at the final regression coefficient estimates. This is
a very standard approach and is detailed in most books on logistic
regression or glms. It is related to the Newton-Raphson iterative
algorithm for maximizing the likelihood. The information matrix is like
the sums of squares and cross-product matrix in ordinary regression except
for a weigh of the form P*(1-P) where P is a row's estimated probability
of even from the final iteration.
---
Frank E Harrell Jr Professor and Chair School of Medicine
Department of Biostatistics Vanderbilt University