16- Chapter Sixteen McGraw- © 2005 The McGraw-Hill Companies, Inc., All Chapter Sixteen 16- Nonparametric Methods: Analysis of Ranked Data GOALS When you have completed this chapter, you ONE will be able to: Conduct the sign test for dependent samples using the binomial and standard normal distributions as the test statistics TWO Conduct a test of hypothesis for dependent samples using the Wilcoxon signed-rank test THREE Conduct and interpret the Wilcoxon rank-sum test for independent samples FOUR Conduct and interpret the Kruskal-Wallis test for several independent samples 16- Chapter Sixteencontinued Nonparametric Methods: Analysis of Ranked Data GOALS When you have completed this chapter, you will FIVEbe able to: Compute and interpret Spearman’s coefficient of rank correlation SIX Conduct a test of hypothesis to determine whether the correlation among the ranks in the population is different from zero Goals The Sign Test Based on the sign of a difference between two related observations The test requires dependent (related) samples 16- No assumption is necessary regarding the shape of the population of differences The binomial distribution is the test statistic for small samples and the standard normal (z) for large samples 16- The Sign Test Determine the sign of the difference between related pairs Compare the number of positive (or negative) differences to the critical value Determine the number of usable pairs n is the number of usable pairs (without ties), X is the number of pluses or minuses, and the binomial probability p=.5 The Sign Test continued 16- Normal Approximation If both n and n(1- ) are greater than 5, the z distribution is appropriate If the number of pluses or minuses is more than n/2, then If the number of pluses or minuses is less than n/2, then ( X 5) 5n z n ( X .5) 5n z n Normal Approximation 16- The Gagliano Research Institute for Business Studies is comparing the research and development expense (R&D) as a percent of income for a sample of glass manufacturing firms for 2000 and 2001 At the 05 significance level has the R&D expense declined? Use the sign test Example 16- Company 2000 2001 Difference Sign Savoth Glass 20 16 + Ruisi Glass 14 13 + Rubin Inc 23 20 + Vaught 24 17 + Lambert Glass 31 22 + Pimental 22 20 + Olson Glass 14 20 -6 - Flynn Glass 18 11 + Example 16- Step 4: H0 is rejected We conclude that R&D expense as a percent of income declined from 2000 to 2001 Step 3: There is one negative difference That is there was an increase in the percent for one company Step 2: H0: is rejected if the number of negative signs is or Step H0: p >.5 Example H : p 1.96 or z is less than –1.96 n1 ( n1 n 1) z n1 n ( n1 n 1) 12 8(8 1) 81.5 8(9)(8 1) 12 0.914 W 16- 21 Step 3: The value of the test statistic is 0.914 Step 4: We not reject the null hypothesis We cannot conclude that there is a difference in the distributions of the repair costs of the two vehicles Example 16- 22 Kruskal-Wallis Test Analysis of Variance by Ranks Used to compare three or more samples to determine if they came from equal populations The ordinal scale of measurement is required The sample data is ranked from low to high as if it were a single group It is an alternative to the one-way ANOVA Each sample should have at least five observations The chi-square distribution is the test statistic Kruskal-Wallis Test: Analysis of Variance by Ranks 16- 23 Kruskal-Wallis Test Analysis of Variance by Ranks Test Statistic ( Rk ) 12 ( R1 ) ( R2 ) H 3(n 1) n(n 1) n1 n2 nk Kruskal-Wallis Test: Analysis of Variance by Ranks continued 16- 24 Keely Ambrose, director of Human Resources for Miller Industries, wishes to study the percent increase in salary for middle managers at the four manufacturing plants She gathers a sample of managers and determines the percent increase in salary from last year to this year At the 5% significance level can Keely conclude that there is a difference in the percent increases for the various plants? Example Ranked Increases in Managers’ Salaries 16- 25 M ilv ille Rank Cam den Rank E a to n Rank V in e la n d Rank 2 7 7 1 16 12 17 1 1 15 1 20 39 35 62 74 EXAMPLE continued 16- 26 Step 1: H0: The populations are the same H1: The populations are not the same Step 2: H0 is rejected if is greater than 7.185 There are degrees of freedom at the 05 significance level 12 (R1 ) (R ) (R3 ) (R ) H 3(n 1) n(n 1) n1 n2 n2 n k 2 2 12 39 35 62 74 3(20 1) 20(20 1) 5 5 5.949 There is no difference in the percent increases in manager salaries in the four plants The null hypothesis is not rejected 16- 27 Spearman’s Coefficient of Rank Correlation Reports the association between two sets of ranked observations 6d rs 1 n( n 1) d is the difference in the ranks and n is the number of observations Similar to Pearson’s coefficient of correlation, but is based on ranked data Ranges from –1.00 up to 1.00 Rank-Order Correlation 16- 28 Testing the Significance of rs Ho: Rank correlation in population is t rs H1: Rank correlation in population is not n 2 rs Testing the Significance of rs Preseason Football Rankings for the Atlantic Coast Conference by the coaches and sports writers School Maryland NC State NC Virginia Clemson Wake Forest Duke Florida State Coaches 16- 29 Writers Example Continued 16- 30 d2 School Coaches Writers Maryland NC State NC 6 Virginia 5 Clemson Wake Forest Duke Florida State 1 Total d d2 -1 -1 0 -1 1 0 1 Example continued 16- 31 Coefficient of Rank Correlation rs 1 1 6d 2 n(n 1) 6(8) 8(8 1) There is a strong correlation between the ranks of the coaches and the sports writers 0.905 Example Continued ... minuses is more than n/2, then If the number of pluses or minuses is less than n/2, then ( X 5) 5n z n ( X .5) 5n z n Normal Approximation 16- The Gagliano Research Institute for Business. .. by Ranks continued 16- 24 Keely Ambrose, director of Human Resources for Miller Industries, wishes to study the percent increase in salary for middle managers at the four manufacturing plants She... managers and determines the percent increase in salary from last year to this year At the 5% significance level can Keely conclude that there is a difference in the percent increases for the various