128 CHAPTER 8 Contingency Tablesmedicine versus alternative medicine between men and women.
Suppose we have a questionnaire where we ask the respondents if they
prefer western medicine or alternative medicine, and we tell them that
they must choose one over the other. We also keep track of the gender
of all the respondents. Table 8.1 illustrates the 2 × 2 table that corre-
sponds to the results of such a survey.
The table above presents the observed frequencies from a survey
of men and women. Out of 100 men, 49 favored western medicine, and
51 favored alternative medicine. Out of 100 women, 30 favored western
medicine, and 70 favored alternative medicine. The expected frequen-
cies represent the total (not necessarily an integer because it is an
average) in each of the four cells under the assumption of no associa-
tion. The cell counts are represented algebraically as A for men favoring
western medicine, B for men favoring alternative medicine, and C for
women favoring western medicine, and D for women favoring alterna-
tive medicine.
The expected value under the null hypothesis of no difference or
independence for the cell containing A , we shall denote as E ( A ), and
similarly for B , C , and D. E ( A ) is the product of the two proportions
involving A multiplied by N (the grand total). So E ( A ) = N [( A + B )/ N ]
[( A + C )/ N ] = N ( A + B )( A + C )/ N 2 = ( A + B )(A + C)/ N. By the same
argument, E (B) = ( B + A)( B + D )/ N , and E ( C ) = ( C + D )( C + A )/ N ,
and E ( D ) = ( D + C )( D + B )/ N. Applying these formula in the example
in Table 8.1 , the values in parentheses are obtained using A = 49,
B = 51, C = 30, and D = 70.
Table 8.1
Preference for Type of Medical Care by Gender
Type of medical care preference
Gender Western medicine Alternative medicine Row total
Men 49 (39.5) A 51 (60.5) B 100 (100) A + B
Women 30 (39.5) C 70 (60.5) D 100 (100) C + D
Column total 79 (79) A + C 121 (121) B + D 200 (200)
N = A + B + C + D
Grand total
Expected frequencies are shown in parentheses.