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

wheredvarð^b 1 Þis obtained from the variance–covar-

iance matrix or, alternatively, by squaring the value

of the standard error for^b 1 provided by the computer

in the listing of variables and their estimated coeffi-

cients and standard errors.

8. H 0 :bCAT¼0 in the no interaction model (Model I), or
   alternatively,H 0 :OR¼1, where OR denotes the odds
   ratio for the effect of CAT on CHD status, adjusted for
   the five other variables in Model I.
   Test statistic: Wald statisticZ¼^bCAT=s^bCAT, which is
   approximately normal (0, 1) under H 0 , or alterna-
   tively,Z^2 is approximately chi square with one degree
   of freedom underH 0.
   Test computation: Z¼ 0 : 5978 = 0 : 3520 ¼ 1 : 70 ; alterna-
   tively, Z^2 ¼ 2.88; the one-tailed P-value is 0.0894/
   2 ¼0.0447, which is significant at the 5% level.


9. The point estimate of the odds ratio for the effect
   of CAT on CHD adjusted for the other variables in
   Model I is given by e0.5978¼1.82. The 95% interval
   estimate for the above odds ratio is given by

expb^CAT 1 : 96

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dvar ^bCAT

r 

¼ðÞ 0 : 5978  1 : 96  0 : 3520

¼expðÞ 0 : 5978  0 : 6899

¼ e^0 :^0921 ;e^1 :^2876



¼ðÞ 0 : 91 ; 3 : 62 :

10. The null hypothesis for the likelihood ratio test for the
    effect of CC:H 0 :bCC¼0 wherebCCis the coefficient of
    CC in model II.
    Likelihood ratio statistic: LR =2lnL^I(2lnL^II)
    whereL^IandL^IIare the maximized likelihood func-
    tions for Models I and II, respectively. This statistic
    has approximately a chi-square distribution with one
    degree of freedom under the null hypothesis.
    Test computation: LR¼400.4357.0¼43.4. The
    P-value is 0.0000 to four decimal places. BecausePis
    very small, the null hypothesis is rejected and it is con-
    cluded that there is a significant effect of the CC variable,
    i.e., there is significant interaction of CHL with CAT.


11. The null hypothesis for the Wald test for the effect of
    CC is the same as that for the likelihood ratio test:H 0 :
    bCC¼0, wherebCCis the coefficient of CC in model II.
    Wald statistic:Z¼^bCC=s^bCC, which is approximately
    normal (0, 1) underH 0 , or alternatively,Z^2 is approxi-
    mately chi square with one degree of freedom underH 0.

Answers to Practice Exercises 163