Calculating/understanding variance-covariance matrix of logistic regression (lrm $var)

On Thu, 29 Jan 2004 02:34:27 +0100 (CET)

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

"Karl" == Karl Knoblick <karlknoblich at yahoo.de>
    on Thu, 29 Jan 2004 02:34:27 +0100 (CET) writes:

Karl> Hallo!
    Karl> I want to understand / recalculate what is done to get
    Karl> the CI of the logistic regression evaluated with lrm.
    Karl> As far as I came back, my problem is the
    Karl> variance-covariance matrix fit$var of the fit
    Karl> (fit<-lrm(...), fit$var). Here what I found and where
    Karl> I stucked:

    Karl> -----------------
    Karl> library(Design)

    .....

The usual ("official") R (and S) way for this is using  
    r <- glm(..., family = binomial)
with  predict(r, .., se.fit=TRUE) 
and   vcov(r)
giving the variance-covariance matrix,
calling the vcov.glm(.) method in this case, which it self
mainly relies on summary.glm(.).

---

As you see yourself,  lrm() is from a particular CRAN package by
Prof Frank Harrell and if you really want that, you should ask
the package author -- as you are told in the posting guide
(you should read! -- see the last line of every R-help message).

Regards,
Martin Maechler <maechler at stat.math.ethz.ch>	http://stat.ethz.ch/~maechler/
Seminar fuer Statistik, ETH-Zentrum  LEO C16	Leonhardstr. 27
ETH (Federal Inst. Technology)	8092 Zurich	SWITZERLAND
phone: x-41-1-632-3408		fax: ...-1228			<><