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

Answers to
Practice
Exercises

1. logit P(X)¼aþb 1 HTþb 2 HSþb 3 CTþb 4 AGE
   þb 5 SEX.


2. logit P(X)¼aþb 1 HTþb 2 HSþb 3 CTþb 4 AGE
   þb 5 SEXþb 6 HTHSþb 7 HTCT
   þb 8 HTAGEþb 9 HTSEX
   þb 10 HSCTþb 11 HSAGE
   þb 12 HSSEXþb 13 CTAGE
   þb 14 CTSEXþb 15 AGESEX.


3. H 0 :b 6 ¼b 7 ¼...¼b 15 ¼0, i.e., the coefficients of all
   product terms are zero.
   Likelihood ratio statistic: LR¼2lnL^ 1 (2lnL^ 2 ),
   whereL^ 1 is the maximized likelihood for the reduced
   model (i.e., Exercise 1 model) andL^ 2 is the maximized
   likelihood for the full model (i.e., Exercise 2 model).
   Distribution of LR statistic: chi square with 10 degrees
   of freedom.


4. H 0 :b 1 ¼0, whereb 1 is the coefficient of HT in the no
   interaction model; alternatively, this null hypothesis
   can be stated asH 0 :OR¼1, where OR denotes the
   odds ratio for the effect of HT adjusted for the other
   four variables in the no interaction model.
   Likelihood ratio statistic: LR¼2lnL^ 0 (2lnL^ 1 ),
   whereL^ 0 is the maximized likelihood for the reduced
   model (i.e., Exercise 1 model less the HT term and its
   corresponding coefficient) andL^ 1 is the maximized
   likelihood for the full model (i.e., Exercise 1 model).
   Distribution of LR statistic: approximately chi square
   with one degree of freedom.


5. The null hypothesis for the Wald test is the same as
   that given for the likelihood ratio test in Exercise 4.H 0 :
   b 1 ¼0 or, equivalently,H 0 :OR¼1, where OR denotes
   the odds ratio for the effect of HT adjusted for the
   other four variables in the no interaction model.
   Wald test statistic:Z=^b 1 =s^b 1 , whereb 1 is the coefficient
   of HT in the no interaction model.
   Distribution of Wald statistic: approximately normal
   (0, 1) underH 0 ; alternatively, the square of the Wald
   statistic, i.e.,Z^2 , is approximately chi square with one
   degree of freedom.


6. The sample size for this study is 12,742, which is very
   large; consequently, the Wald and LR test statistics
   should be approximately the same.


7. The odds ratio of interest is given by eb^1 , whereb 1 is
   the coefficient of HT in the no interaction model; a
   95% confidence interval for this odds ratio is given by
   the following formula:

expb^ 1  1 : 96

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dvar^b 1

r 

;

162 5. Statistical Inferences Using Maximum Likelihood Techniques