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

Practice
Exercises

The Practice Exercises presented here are primarily

designed to elaborate and expand on several concepts

that were briefly introduced in this chapter.

Exercises 1–3 relate to calculating odds ratios and their

corresponding correlations. Consider the following 2 2

table for two dichotomous responses (YjandYk). The cell

counts are represented byA,B,C, andD. The margins are

represented byM 1 ,M 0 ,N 1 , andN 0 and the total counts are

represented byT.

Yk¼ 1 Yk¼ 0 Total

Yj¼ 1 ABM 1 ¼AþB

Yj¼ 0 CDM 0 ¼CþD

Total N 1 ¼AþCN 0 ¼BþDT¼AþBþCþD

The formulas for calculating the correlation and odds ratio

betweenYjandYkin this setting are given as follows:

CorrðYj;YkÞ¼

ATM 1 N 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M 1 M 0 N 1 N 0

p ; OR¼

AD

BC

:

1. Calculate and compare the respective odds ratios and
   correlations betweenYjandYkfor the data summar-
   ized in Tables 1 and 2 to show that the same odds ratio
   can correspond to different correlations.

Table 1

Yk¼ 1 Yk¼ 0

Yj¼13 1

Yj¼01 3

Table 2

Yk¼ 1 Yk¼ 0

Yj¼19 1

Yj¼01 1

2. Show that ifboththeBandCcells are equal to 0, then
   the correlation betweenYjandYkis equal to 1 (assum-
   ingAandDare nonzero). What is the corresponding
   odds ratio if theBandCcells are equal to 0? Did both
   theB and C cells have to equal 0 to obtain this
   corresponding odds ratio? Also show that ifboththeA
   andDcells are equal to zero, then the correlation is
   equal to1. What is that corresponding odds ratio?


3. Show that ifAD¼BC, then the correlation betweenYj
   andYkis 0 and the odds ratio is 1 (assuming nonzero
   cell counts).
   Exercises 4–6 refer to a model constructed using the data
   from the Heartburn Relief Study. The dichotomous
   outcome is relief from heartburn (coded 1¼yes, 0¼no).
   The only predictor variable is RX (coded 1¼active treat-
   ment, 0¼standard treatment). This model containstwo
   subject-specific effects: one for the intercept (b 0 i) and the
   other (b 1 i) for the coefficient RX. The model is stated in

Practice Exercises 591