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568 CHAPTER 11 REGRESSION ANALYSIS: SOME ADDITIONAL TECHNIQUES
odds, which is frequently used by those who place bets on the outcomes of sporting events
or participate in other types of gambling activities. Using probability terminology, we may
define odds as follows.
DEFINITION
The odds for success are the ratio of the probability of success to the
probability of failure.
The odds ratio is a measure of how much greater (or less) the odds are for subjects
possessing the risk factor to experience a particular outcome. This conclusion
assumes that the outcome is a rare event. For example, when the outcome is the contracting
of a disease, the interpretation of the odds ratio assumes that the disease is rare.
Suppose, for example, that the outcome variable is the acquisition or nonacquisition
of skin cancer and the independent variable (or risk factor) is high levels of exposure to
the sun. Analysis of such data collected on a sample of subjects might yield an odds ratio
of 2, indicating that the odds of skin cancer are two times higher among subjects with high
levels of exposure to the sun than among subjects without high levels of exposure.
Computer software packages that perform logistic regression frequently provide as
part of their output estimates of b 0 and b 1 and the numerical value of the odds ratio.
As it turns out the odds ratio is equal to exp 1b 1 2.
EXAMPLE 11.4.1
LaMont et al. (A-9) tested for obstructive coronary artery disease (OCAD) among 113 men
and 35 women who complained of chest pain or possible equivalent to their primary care
physician. Table 11.4.2 shows the cross-classification of OCAD with gender. We wish
to use logistic regression analysis to determine how much greater the odds are of finding
OCAD among men than among women.
Solution:
We may use the SAS ® software package to analyze these data. The independent
variable is gender and the dependent variable is status with respect
to having obstructive coronary artery disease (OCAD). Use of the SAS ®
command PROC LOGIST yields, as part of the resulting output, the statistics
shown in Figure 11.4.1.
TABLE 11.4.2 Cases of Obstructive Coronary
Artery Disease (OCAD) Classified by Sex
Disease Males Females Total
OCAD present 92 15 107
OCAD not present 21 20 41
Total 113 35 148
Source: Matthew J. Budoff, M.D. Used with permission.