7.6 Logistic Regression 121esized that the nonmenopausal women would be at greater risk for
anemia than the postmenopausal women. By risk we mean the probabil-
ity of being anemic given whether or not you are postmenopausal. So
we are saying that we expect that just conditioning on nonmenopausal
or postmenopausal, we would expect the conditional probability to be
higher for nonmenopausal women.
Another common way to look at the difference in risks such as
anemia when comparing two groups like this is the odds ratio, say
O 1 / O 2 , where O 1 = π 1 /(1 − π 1 ) and O 2 = π 2 /(1 − π 2 ). O 1 and O 2 are
called the odds — say 1 denotes nonmenopausal women and 2 denotes
postmenopausal women. Relative risk is π 1 / π 2. So when π 1 and π 2 are
small, 1 − π 1 and 1 − π 2 are close to 1, and the odds ratio and relative
risk are nearly the same. But when they are not small, the two measures
can differ. Lachin ( 2000 ) is an excellent text on biostatistics that empha-
sizes relative risks and odds ratios. So it is a great source to use to clear
up any confusion you might have.
Campbell and Machin only used the age dichotomized at 30, and
estimated that the regression parameter for age group was 1.4663, with
a standard error of 1.1875. The Wald test is the analog in logistic regres-
sion to the t - test for signifi cance of the parameter. The value of the
Wald statistic was 1.5246, which translates to a p - value of 0.2169. So
at least for the two age groups, there was not a statistically signifi cant
difference. However, age could still be an important factor if the cut
point should be different or if age is left on a continuous scale. Also,
it may be that there is too much patient to patient variability for 20
women to be an adequate sample size. Also, age is correlated with
menopause. So it may be that age would be far more important if the
dichotomous menopause variable were not included in the model.
The result of performing the logistic regression using the actual
ages, as was done by Chernick and Friis ( 2003 ), gives a coeffi cient of
− 0.2077, with a standard deviation of 0.1223, indicating a possible
decrease in the risk of anemia with increasing age. The Wald statistic
is 2.8837, corresponding to a p - value of 0.0895. This is signifi cant at
the 10% level, but not at the 5% level. It could well be that we would
fi nd greater signifi cance with a larger sample of women. The choice of
30 to dichotomize was probably a bad choice. We note that 6 of the 15
women over 30 were not menopausal, and the coeffi cient of 1.4663
was in the opposite direction of what the alternative hypothesis would
suggest.