Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

Corresponding 2 2 correlation
matrix

C¼

1 corrðY 1 ;Y 2 Þ

corrðY 1 ;Y 2 Þ 1

"

Diagonal matrix: has 0 in all non-
diagonal entries.

Diagonal 22 matrix with var-
iances on diagonal

D¼

varðY 1 Þ 0

0 varðY 2 Þ

"

Can extend toNNmatrices

Matrices symmetric: (i, j)¼(j, i)
element

covðY 1 ;Y 2 Þ¼covðY 2 ;Y 1 Þ

corrðY 1 ;Y 2 Þ¼corrðY 2 ;Y 1 Þ

Relationship between covariance
and correlation expressed as

covðY 1 ;Y 2 Þ

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

varðY 1 Þ

p

½corrðY 1 ;Y 2 ފ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

varðY 2 Þ

p

Matrix version:V¼D

(^12)
CD
(^12)
;
whereD
(^12)
D
(^12)
¼D
Logistic regression

D¼

m 1 ð 1 m 1 Þ 0

0 m 2 ð 1 m 2 Þ

"

;

where

varðYiÞ¼mið 1 miÞ

mi¼g^1 ðX;bÞ

The corresponding 22 correlation matrix (C)

is also shown at left. Note that the covariance

between a variable and itself is the variance

of that variable [e.g., cov(Y 1 ,Y 1 )¼var(Y 1 )],

so that the correlation between a variable and

itself is 1.

Adiagonal matrixhas a 0 in all nondiagonal

entries.

A22 diagonal matrix (D) with the variances

along the diagonal is of the form shown at left.

The definitions ofV,C, andDcan be extended

from 2 2 matrices toNNmatrices. A

symmetric matrixis a square matrix in which

the (i, j) element of the matrix is the same value

as the (j, i) element. The covariance of (Yi,Yj)is

the same as the covariance of (Yj,Yi); thus the

covariance and correlation matrices are sym-

metric matrices.

The covariance betweenY 1 andY 2 equals the

standard deviation ofY 1 , times the correlation

betweenY 1 andY 2 , times the standard devia-

tion ofY 2.

The relationship between covariance and cor-

relation can be similarly expressed in terms of

the matricesV,C, andDas shown on the left.

For logistic regression, the variance of the

response Yiequalsmitimes (1mi). The cor-

responding diagonal matrix (D) hasmi(1mi)

for the diagonal elements and 0 for the off-

diagonal elements. As noted earlier, the mean

(mi) is expressed as a function of the covariates

and the regression parameters [g^1 (X,b)].

508 14. Logistic Regression for Correlated Data: GEE