is based on the distribution of . Exact Tests reports both the asymptotic and exact p values. The formulas for computing the three contingency coefficients are given below. The formula for each measure involves taking the square root of a function of . The positive root is always selected. For a more detailed discussion of these measures of as- sociation, see Liebetrau (1983). The phi contingency coefficient is given by the formula Equation 15.1 The minimum value assumed by is 0, signifying no association. However, its upper bound is not fixed but depends on the dimensions of the contingency table. Therefore, it is not a very suitable measure for arbitrary tables. For the special case of the table, Gibbons (1985) shows that is identical to the absolute value of Kendall’s co- efficient and is evaluated by the formula Equation 15.2 Notice from Equation 15.2 that, for the contingency table, could be either positive or negative, which implies a positive or negative association in the table. The Pearson contingency coefficient is given by the formula Equation 15.3 This contingency coefficient assumes a minimum value of 0, signifying no association. It is bounded from above by 1, signifying perfect association. However, the maximum value attainable by CC is , where . Thus, the range of this contingency coefficient still depends on the dimensions of the table. Cramér’s V coefficient ranges between 0 and 1, with 0 signifying no association and 1 signifying perfect association. It is given by Equation 15.4 Exact Tests reports the point estimate of the contingency coefficient. The formulas for these asymptotic standard errors are fairly complicated. These formulas are described in the algorithms manual available on the Manuals CD and also available by selecting Algorithms on the Help menu. CH y ( ) CH x ( ) φ CH x ( ) N --------------- = φ r c × 2 2 × φ τ b φ x 11 x 22 x 12 x 21 – m 1 m 2 n 1 n 2 ------------------------------------ = 2 2 × φ 2 2 × CC CH x ( ) CH x ( ) N + -------------------------- = q 1 – ( ) q ⁄ q min r c , ( ) = r c × V CH x ( ) N q 1 – ( ) -------------------- =
Measures of Association for Nominal Data 187 These measures may be used to analyze an unordered contingency table given in Sie- gel and Castellan (1988). The data consist of a crosstabulation of three possible responses ( completed , declined , no response ) to a questionnaire concerning the financial account- ing standards used by six different organizations responsible for maintaining such stan- dards. These organizations are identified only by their initials ( AAA, AICPA, FAF, FASB, FEI, and NAA ). The crosstabulated data are shown in Figure 15.1. First, these data are analyzed using only the first three columns of Figure 15.1. For this subset of the data, Figure 15.2 shows the results for the contingency coefficients. The exact two-sided p value for testing the null hypothesis that there is no association is also reported. Its value is 0.090, slightly lower than the asymptotic p value of 0.092.
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- Spring '08
- Statistical tests, Pearson's chi-square test, Non-parametric statistics, Pearson chi-square test, exact tests