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

Block diagonal matrix: subject-
specific correlation matrices form
blocks (Bi)

B 1 0

B 2

0B 3

2

6

4

3

7

5 where Bi¼ith block

We illustrate the form of the correlation matrix

in which responses are correlated within sub-

jects and independent between subjects. For

simplicity, consider a dataset with information

on only three subjects in which there are four

responses recorded for each subject. There are

12 observations (3 times 4) in all. The correla-

tion between responses from two different sub-

jects is 0, whereas the correlation between

responses from the same subject (i.e., thejth

andkth response from subjecti)isrijk.

This correlation matrix is called ablock diago-

nal matrix, where subject-specific correlation

matrices are the blocks along the diagonal of

the matrix.

The correlation matrix in the preceding exam-

ple contains 18 correlation parameters (6 per

cluster) based on only 12 observations. In this

setting, each subject has his or her own distinct

set of correlation parameters.

EXAMPLE

18 rs (6 per cluster/subject) but 12

observations

Subject i: {ri 12 , ri 13 , ri 14 , ri 23 , ri 24 , ri 34 }

EXAMPLE

Three subjects; four observations each

Within-cluster correlation betweenjth

andkth response from subjecti¼rijk

Between-subject correlations¼ 0

1 r 112 r 113 r 114 00 00 00 00

00 00 00 00

00 00 00 00

00 00 00 00

00 0 0 00 00

00 0 0 00 00

00 0 0 00 00

00 0 0 00 00

00 00

00 00

00 00

00 00

00 0 0

00 0 0

00 0 0

00 0 0

r 112 1 r 123 r 124

r 113 r 123 1 r 134

r 114 r 124 r 134 1

1 r 212 r 213 r 214

r 212 1 r 223 r 224

r 213 r 223 1 r 234

r 214 r 224 r 234 1

r 334 1

1 r 312 r 313 r 314

r 312 1 r 323 r 324

r 313 r 323 1 r 334

r 314 r 324

blocks

Presentation: VII. Correlation Structure 509