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642 CHAPTER 12 THE CHI-SQUARE DISTRIBUTION AND THE ANALYSIS OF FREQUENCIES
aware of another variable that may be associated with the disease, with the risk factor,
or with both in such a way that the true relationship between the disease status and the
risk factor is masked. Such a variable is called a confounding variable. For example,
experience might indicate the possibility that the relationship between some disease and
a suspected risk factor differs among different ethnic groups. We would then treat ethnic
membership as a confounding variable. When they can be identified, it is desirable
to control for confounding variables so that an unambiguous measure of the relationship
between disease status and risk factor may be calculated. A technique for accomplishing
this objective is the Mantel–Haenszel (22) procedure, so called in recognition
of the two men who developed it. The procedure allows us to test the null hypothesis
that there is no association between status with respect to disease and risk factor status.
Initially used only with data from retrospective studies, the Mantel–Haenszel procedure
is also appropriate for use with data from prospective studies, as discussed by
Mantel (23).
In the application of the Mantel–Haenszel procedure, case and control subjects are
assigned to strata corresponding to different values of the confounding variable. The data
are then analyzed within individual strata as well as across all strata. The discussion that
follows assumes that the data under analysis are from a retrospective or a prospective
study with case and noncase subjects classified according to whether they have or do not
have the suspected risk factor. The confounding variable is categorical, with the different
categories defining the strata. If the confounding variable is continuous it must be categorized.
For example, if the suspected confounding variable is age, we might group
subjects into mutually exclusive age categories. The data before stratification may be
displayed as shown in Table 12.7.3.
Application of the Mantel–Haenszel procedure consists of the following steps.
1. Form k strata corresponding to the k categories of the confounding variable. Table
12.7.5 shows the data display for the ith stratum.
2. For each stratum compute the expected frequency e i of the upper left-hand cell of
Table 12.7.5 as follows:
e i = 1a i + b i 21a i + c i 2
n i
(12.7.5)
TABLE 12.7.5 Subjects in the ith Stratum of a Confounding
Variable Classified According to Status Relative to a Risk
Factor and Whether They Are Cases or Controls
Sample
Risk Factor Cases Controls Total
Present
Absent
a i
c i
b i
d i
a i + b i
c i + d i
Total
a i + c i
b i + d i
n i