[R] logistic regression - glm() - example in Dalgaard's book ISwR

[Paulo Barata]

Dear R-list members,

I would like to pose a question about the use and results
of the glm() function for logistic regression calculations.

The question is based on an example provided on p. 229
in P. Dalgaard, Introductory Statistics with R, 2nd. edition,
Springer, 2008. By means of this example, I was trying to
practice the different ways of entering data in glm().

In his book, Dalgaard provides data from a case-based study
about hypertension summarized in the form of a table. He shows
two ways of entering the response (dependent) variable data
in glm(): (1) as a matrix of successes/failures (diseased/
healthy); (2) as the proportion of people diseased for each
combination of independent variable's categories.

I tried to enter the response variable in glm() in a third
way: by reconstructing, from the table, the original data
in a case-based format, that is, a data frame in which
each row shows the data for one person. In this situation,
the response variable would be coded as a numeric 0/1 vector,
0=failure, 1=success, and so it would be entered in glm() as
a numeric 0/1 vector.

The program below presents the calculations for each of the
three ways of entering data - the first and second methods
were just copied from Dalgaard's book.

The three methods produce the same results with regard
to the estimated coefficients, when the output is seen
with five decimals (although regression 3 seems to have
produced slightly different std.errors).

My main question is: Why are the residual deviance, its
degrees of freedom and the AIC produced by regression 3
completely different when compared to those produced by
regressions 1 and 2 (which seem to produce equal results)?
It seems that the degrees of freedom in regressions 1
and 2 are based on the size (number of rows) of table d
(see the output of the program below), but this table is
just a way of summarizing the data. The degrees of
freedom in regressions 1 and 2 are not based on the
actual number of cases (people) examined, which is n=433.

I understand that no matter the way of entering the data
in glm(), we are always analyzing the same data, which
are those presented in table format on Dalgaard's page
229 (these data are in data.frame d in the program below).
So I understand that the three ways of entering data
in glm() should produce the same results.

Secondarily, why are the std.errors in regression 3 slightly
different from those calculated in regressions 1 and 2?

I am using R version 2.11.1 on Windows XP.

Thank you very much.

Paulo Barata

##== begin =================================================

## data in: P. Dalgaard, Introductory Statistics with R,
## 2nd. edition, Springer, 2008
## logistic regression - example in Dalgaard's Section 13.2,
## page 229

rm(list=ls())

## data provided on Dalgaard's page 229:
no.yes <- c("No","Yes")
smoking <- gl(2,1,8,no.yes)
obesity <- gl(2,2,8,no.yes)
snoring <- gl(2,4,8,no.yes)
n.tot <- c(60,17,8,2,187,85,51,23)
n.hyp <- c(5,2,1,0,35,13,15,8)

d <- data.frame(smoking,obesity,snoring,n.tot,n.hyp)
## d is the data to be analyzed, in table format
## d is the first table on Dalgaard's page 229
## n.tot = total number of cases
## n.hyp = people with hypertension
d

## regression 1: Dalgaard's page 230
## response variable entered in glm() as a matrix of
## successes/failures
hyp.tbl <- cbind(n.hyp,n.tot-n.hyp)
regression1 <- glm(hyp.tbl~smoking+obesity+snoring,
                    family=binomial("logit"))

## regression 2: Dalgaard's page 230
## response variable entered in glm() as proportions
prop.hyp <- n.hyp/n.tot
regression2 <- glm(prop.hyp~smoking+obesity+snoring,
                    weights=n.tot,family=binomial("logit"))

## regression 3 (well below): data entered in glm()
## by means of 'reconstructed' variables
## variables with names beginning with 'r' are
## 'reconstructed' from data in data.frame d.
## The objective is to reconstruct the original
## data from which the table on Dalgaard's page 229
## has been produced

rsmoking <- c(rep('No',d[1,4]),rep('Yes',d[2,4]),
               rep('No',d[3,4]),rep('Yes',d[4,4]),
               rep('No',d[5,4]),rep('Yes',d[6,4]),
               rep('No',d[7,4]),rep('Yes',d[8,4]))
rsmoking <- factor(rsmoking)
length(rsmoking)  # just a check

robesity <- c(rep('No', d[1,4]),rep('No', d[2,4]),
               rep('Yes',d[3,4]),rep('Yes',d[4,4]),
               rep('No', d[5,4]),rep('No', d[6,4]),
               rep('Yes',d[7,4]),rep('Yes',d[8,4]))
robesity <- factor(robesity)
length(robesity)  # just a check

rsnoring <- c(rep('No', d[1,4]),rep('No', d[2,4]),
               rep('No', d[3,4]),rep('No', d[4,4]),
               rep('Yes',d[5,4]),rep('Yes',d[6,4]),
               rep('Yes',d[7,4]),rep('Yes',d[8,4]))
rsnoring <- factor(rsnoring)
length(rsnoring)  # just a check

rhyp <- c(rep(1,d[1,5]),rep(0,d[1,4]-d[1,5]),
           rep(1,d[2,5]),rep(0,d[2,4]-d[2,5]),
           rep(1,d[3,5]),rep(0,d[3,4]-d[3,5]),
           rep(1,d[4,5]),rep(0,d[4,4]-d[4,5]),
           rep(1,d[5,5]),rep(0,d[5,4]-d[5,5]),
           rep(1,d[6,5]),rep(0,d[6,4]-d[6,5]),
           rep(1,d[7,5]),rep(0,d[7,4]-d[7,5]),
           rep(1,d[8,5]),rep(0,d[8,4]-d[8,5]))
length(rhyp)      # just a check

sum(n.tot)  # just a check

## data.frame(rsmoking,robesity,rsnoring,rhyp) would provide
## the data in a case-based format - each row would present
## data for one case (one person), with response variable
## coded 0/1 for No/Yes
## The five first cases (people in the study):
data.frame(rsmoking,robesity,rsnoring,rhyp)[1:5,]

## regression 3: response variable entered in glm()
## as a numeric 0/1 vector
regression3 <- glm(rhyp~rsmoking+robesity+rsnoring,
                    family=binomial("logit"))

summary(regression1)
summary(regression2)
summary(regression3)

## note different residual deviance, degrees of freedom
## and AIC between regression 3 and regressions 1 and 2.

##== end =================================================

----------------------------------------------------------
Paulo Barata
Fundacao Oswaldo Cruz - Oswaldo Cruz Foundation
Rua Leopoldo Bulhoes 1480 - 8A
21041-210  Rio de Janeiro - RJ
Brazil

E-mail: pbarata at infolink.com.br
Alternative e-mail: paulo.barata at ensp.fiocruz.br