672 CHAPTER 13 Nonparametric Tests Test Statistic H = 12 N1N + 12 a R2 1 n1 + R2 2 n2 + g+ R2 k nk b - 31N + 12 P-Values P-values are often provided by technology. By using the test statistic H and the number of degrees of freedom 1k - 12, Table A-4 can be used to find a range of values for the P-value. Critical Values 1. The test is right-tailed and critical values can be found from technology or from the chi-square distribution in Table A-4. 2. df = k - 1 (where df is the number of degrees of freedom and k is the number of different samples) Procedure for Finding the Value of the H Test Statistic To see how the following steps are applied, refer to the sample data in Table 13-6. Table 13-6 includes only some of the head injury data from Data Set 35 “Car Data.” This shortened data set is more suitable for illustrating the method of the Kruskal-Wallis test. Step1: Temporarily combine all samples into one big sample and assign a rank to each sample value. (Sort the values from lowest to highest, and in cases of ties, assign to each observation the mean of the ranks involved.) EXAMPLE: In Table 13-6, the numbers in parentheses are the ranks of the combined data set. The rank of 1 is assigned to the lowest value of 90, the rank of 2 is assigned to the next lowest value of 114, and so on. In the case of ties, each of the tied values is assigned the mean of the ranks involved in the tie. (The seventh and eighth values are tied at 178, so they are each assigned a rank of 7.5.) Step2: For each sample, find the sum of the ranks and find the sample size. EXAMPLE: In Table 13-6, the sum of the ranks from the first sample is 110, the sum of the ranks for the second sample is 47.5, and the sum of the ranks for the third sample is 32.5. Step3: Calculate H using the results of Step 2 and the notation and test statistic given in the preceding Key Elements box. EXAMPLE: The test statistic is computed in Example 1. TABLE 13-6 Head Injury Criterion (HIC) Measurements in Car Crash Tests (Ranks in parentheses) Small Midsize Large 253 (14) 117 (3) 249 (13) 143 (6) 121 (4) 90 (1) 124 (5) 204 (12) 178 (7.5) 301 (17) 195 (11) 114 (2) 422 (19) 186 (10) 183 (9) 324 (18) 178 (7.5) 258 (15) 271 (16) n1 = 8 n2 = 6 n3 = 5 R1 = 110 R2 = 47.5 R3 = 32.5
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