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618 CHAPTER 12 THE CHI-SQUARE DISTRIBUTION AND THE ANALYSIS OF FREQUENCIES
Note that the SAS ® printout shows, in each cell, the percentage that cell frequency
is of its row total, its column total, and the grand total. Also shown, for each row and
column total, is the percentage that the total is of the grand total. In addition to the
statistic, SAS ® gives the value of several other statistics that may be computed from contingency
table data. One of these, the Mantel–Haenszel chi-square statistic, will be discussed
in a later section of this chapter.
Small Expected Frequencies The problem of small expected frequencies
discussed in the previous section may be encountered when analyzing the data of contingency
tables. Although there is a lack of consensus on how to handle this problem,
many authors currently follow the rule given by Cochran (5). He suggests that
for contingency tables with more than 1 degree of freedom a minimum expectation
of 1 is allowable if no more than 20 percent of the cells have expected frequencies
of less than 5. To meet this rule, adjacent rows and/or adjacent columns may be combined
when to do so is logical in light of other considerations. If is based on less
than 30 degrees of freedom, expected frequencies as small as 2 can be tolerated. We
did not experience the problem of small expected frequencies in Example 12.4.1, since
they were all greater than 5.
X 2 X 2
The 2 : 2 Contingency Table Sometimes each of two criteria of classification
may be broken down into only two categories, or levels. When data are crossclassified
in this manner, the result is a contingency table consisting of two rows and
two columns. Such a table is commonly referred to as a 2 * 2 table. The value of X 2
may be computed by first calculating the expected cell frequencies in the manner discussed
above. In the case of a 2 * 2 contingency table, however, X 2 may be calculated
by the following shortcut formula:
X 2 =
n1ad - bc2 2
1a + c21b + d21a + b21c + d2
(12.4.1)
where a, b, c, and d are the observed cell frequencies as shown in Table 12.4.5. When
we apply the 1r - 121c - 12 rule for finding degrees of freedom to a 2 * 2 table, the
result is 1 degree of freedom. Let us illustrate this with an example.
TABLE 12.4.5
A 2 2 Contingency Table
First Criterion of Classification
Second Criterion
of Classification 1 2 Total
1 a b
2 c d
a + b
c + d
Total
a + c
b + d
n