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

terms of theith subject’s mean response:

logitmi¼PðD¼ 1 jRXÞ¼b 0 þb 1 RXiþb 0 iþb 1 iRXi;

whereb 0 ifollows a normal distribution with mean 0 and

variancesb 02 ,b 1 ifollows a normal distribution with mean

0 and variancesb 12 and where the covariance matrix ofb 0 i

andb 1 iis a 22 matrix,G.

It may be helpful to restate the model by rearranging the

parameters such that the intercept parameters (fixed and

random effects) and the slope parameters (fixed and ran-

dom effects) are grouped together:

logitmi¼PðD¼ 1 jRXÞ¼ðb 0 þb 0 iÞþðb 1 þb 1 iÞRXi

4. Use the model to obtain the odds ratio for RX¼1 vs.
   RX¼0 for subjecti.


5. Use the model to obtain the baselineriskfor subjecti
   (i.e., risk when RX¼0).


6. Use the model to obtain the odds ratio (RX¼1 vs. RX¼

0. averaged over all subjects.
   Below are three examples of commonly used covariance
   structures represented by 33 matrices. The elements are
   written in terms of the variance (s^2 ), standard deviation
   (s), and correlation (r). The covariance structures are pre-
   sented in this form in order to contrast their structures
   with the correlation structures presented in Chap. 14. A
   covariance structure not only contains correlation para-
   meters but variance parameters as well.

Variance Compound Unstructured

components symmetric

s^2100

s^220

00 s^23

2

4

3

5

s^2 s^2 rs^2 r

s^2 rs^2 s^2 r

s^2 rs^2 rs^2

2

4

3

5

s^21 s 1 s 2 r 12 s 1 s 3 r 13

s 1 s 2 r 12 s^22 s 2 s 3 r 23

s 1 s 3 r 13 s 2 s 3 r 23 s^23

2

4

3

5

The compound symmetric covariance structure has the

additional constraint thatr0,

7. Which of the above covariance structures allow for
   variance heterogeneity within a cluster?


8. Which of the presented covariance structures allow for
   both variance heterogeneity and correlation within a
   cluster.


9. Consider a study in which there are five responses per
   subject. If a model contains two subject-specific ran-
   dom effects (for the intercept and slope), then for sub-
   jecti, what are the dimensions of theGmatrix and of
   theRmatrix?

592 16. Other Approaches for Analysis of Correlated Data