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

Answers to
Practice
Exercises

1. The odds ratio for the data in Table 1 and Table 2 is 9.
   The correlation for the data in Table 1 is
   ½ð 3 Þðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 Þð 4 Þð 4 ފ
   ð 4 Þð 4 Þð 4 Þð 4 Þ
   p ¼ 0 : 5 , and for the data in Table 2 ,itis
   ½ð 9 Þðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 Þð 10 Þð 10 ފ
   ð 10 Þð 2 Þð 10 Þð 2 Þ

p ¼ 0 : 4 So, the odds ratios are the same

but the correlations are different.

2. IfB¼C¼0, thenM 1 ¼N 1 ¼AandM 0 ¼N 0 ¼DandT
   ¼AþD.

corr¼

ATM 1 N 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M 1 M 0 N 1 N 0

p ¼

AðAþDÞAD

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðADÞðADÞ

p ¼ 1 :

The corresponding odds ratio is infinity. Even if just

one of the cells (BorC) were zero, the odds ratio

would still be infinity, but the corresponding correla-

tion would be less than 1.

IfA¼D¼0, thenM 1 ¼N 0 ¼BandM 0 ¼N 1 ¼Cand

T¼BþC.

corr¼

ATM 1 N 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M 1 M 0 N 1 N 0

p ¼

0 ðBþCÞBC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðBCÞðBCÞ

p ¼ 1 :

The corresponding odds ratio is zero.

3. IfAD¼BC,thenD¼(BC)/AandT¼AþBþCþ(BC)/A.

corr¼

ATM 1 N 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M 1 M 0 N 1 N 0

p

¼

A½AþBþCþðBC=Aފ½ðAþBÞðAþCފ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M 1 M 0 N 1 N 0

p ¼ 0

OR¼ADBC¼ADAD¼ 1 (indicating no association between

YjandYk).

4. exp(b 1 þb 1 i)


5. 
   

1

1 þexp½ðb 0 þb 0 iފ

6. exp(b 1 ) sincebliis a random variable with a mean of 0
   (compare to Exercise 4).


7. The variance components and unstructured covari-
   ance structures allow for variance heterogeneity.


8. The unstructured covariance structure.


9. The dimensions of theGmatrix are 22 and the
   dimensions of theRmatrix are 55 for subjecti.


10. The odds ratio is 3.0 for both Subject 1 and Subject
    2 separately, whereas the odds ratio is 2.83 after pool-
    ing the data. The pooled odds ratio is smaller.


11. The baseline risk for Subject 1 is 0.25, whereas the
    baseline risk for Subject 2 is 0.10. There is a difference
    in the baseline risk (although there are only two sub-
    jects).Note. In general, assuming there is heterogeneity

Answers to Practice Exercises 597