1. Copyright © Cengage Learning. All rights reserved. 15 Distribution-Free Procedures

2. Copyright © Cengage Learning. All rights reserved. 15.4 Distribution-Free ANOVA

3. 3 The single-factor ANOVA model for comparing I population or treatment means assumed that for i = 1, 2,..., I, a random sample of size J i was drawn from a normal population with mean  i and variance  2. This can be written as X ij =  I +  ij j = 1,..., J i ; i = 1,..., I where the  ij ’s are independent and normally distributed with mean zero and variance  2. The next procedure for testing equality of the  i ’s requires only that the  ij ’s have the same continuous distribution. (15.14)

4. 4 The Kruskal-Wallis Test

5. 5 Let N =  J i, the total number of observations in the data set, and suppose we rank all N observations from 1 (the smallest X ij ) to N (the largest X ij ). When H 0 :  1 =  2 = · · · =  1 is true, the N observations all come from the same distribution, in which case all possible assignments of the ranks 1, 2,..., N to the I samples are equally likely and we expect ranks to be intermingled in these samples.

6. 6 The Kruskal-Wallis Test If, however, H 0 is false, then some samples will consist mostly of observations having small ranks in the combined sample, whereas others will consist mostly of observations having large ranks. More specifically, if R ij denotes the rank of X ij among the N observations, and R i  and R i  denote, respectively, the total and average of the ranks in the ith sample, then when H 0 is true,

7. 7 The Kruskal-Wallis Test The Kruskal-Wallis test statistic is a measure of the extent to which the R i  ’s deviate from their common expected value (N + 1)/2, and H 0 is rejected if the computed value of the statistic indicates too great a discrepancy between observed and expected rank averages.

8. 8 The Kruskal-Wallis Test Test Statistic K The second expression for K is the computational formula; it involves the rank totals (R i  ’s) rather than the averages and requires only one subtraction. If H 0 is rejected when k  c, then c should be chosen so that the test has level . (15.15)

9. 9 The Kruskal-Wallis Test That is, c should be the upper-tail critical value of the distribution of K when H 0 is true. Under H 0, each possible assignment of the ranks to the I samples is equally likely, so in theory all such assignments can be enumerated, the value of K determined for each one, and the null distribution obtained by counting the number of times each value of K occurs. Clearly, this computation is tedious, so even though there are tables of the exact null distribution and critical values for small values of the J i ’s, we will use the following “large-sample” approximation.

10. 10 The Kruskal-Wallis Test Proposition When H 0 is true and either I = 3 J i  6 (i = 1, 2, 3) or I > 3 J i  5 (i = 1,..., I ) then K has approximately a chi-squared distribution with I – 1 df. This implies that a test with approximate significance level  rejects H 0 if k 

11. 11 Example 9 The accompanying observations (Table 15.6) on axial stiffness index resulted from a study of metal-plate connected trusses in which five different plate lengths—4 in., 6 in., 8 in., 10 in., and 12 in.—were used (“Modeling Joints Made with Light-Gauge Metal Connector Plates,” Forest Products J., 1979: 39–44). Table 15.6 Data and Ranks for Example 9

12. 12 Example 9 The computed value of K is K At level.01, = 13.277, and since 20.12  13.277, H 0 is rejected and we conclude that expected axial stiffness does depend on plate length. cont’d

13. 13 Friedman’s Test for a Randomized Block Experiment

14. 14 Friedman’s Test for a Randomized Block Experiment Suppose X ij =  +  i +  j +  ij, where  i is the ith treatment effect,  j is the jth block effect, and the  ij ’s are drawn independently from the same continuous (but not necessarily normal) distribution. Then to test H 0 :  1 =  2 = · · · =  1 = 0, the null hypothesis of no treatment effects, the observations are first ranked separately from 1 to I within each block, and then the rank average r i  is computed for each of the I treatments.

15. 15 Friedman’s Test for a Randomized Block Experiment When H 0 is true, the r i  ’s should be close to one another, since within each block all I! assignments of ranks to treatments are equally likely. Friedman’s test statistic measures the discrepancy between the expected value (I + 1)/2 of each rank average and the r i  ’s.

16. 16 Friedman’s Test for a Randomized Block Experiment Test Statistic As with the Kruskal-Wallis test, Friedman’s test rejects H 0 when the computed value of the test statistic is too large. For the cases I = 3, J = 2,..., 15 and I = 4, J = 2,..., 8, Lehmann’s book gives the upper-tail critical values for the test.

17. 17 Friedman’s Test for a Randomized Block Experiment Alternatively, for even moderate values of J, the test statistic F r has approximately a chi-squared distribution with I – 1 df when H 0 is true, so H 0 can be rejected if f r 

18. 18 Example 10 The article “Physiological Effects During Hypnotically Requested Emotions” (Psychosomatic Med., 1963: 334–343) reports the following data (Table 15.7) on skin potential (mV) when the emotions of fear, happiness, depression, and calmness were requested from each of eight subjects. Table 15.7 Data and Ranks for Example 10

19. 19 Example 10 Thus f r = (1686) – 3(8)(5) = 6.45 At level.05, = 7.815, and because 6.45 < 7.815, H 0 is not rejected. There is no evidence that average skin potential depends on which emotion is requested. cont’d