Chapter Sixteen McGraw-Hill/Irwin © 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

Chapter Sixteen Nonparametric Methods: Analysis of Ranked Data GOALS When you have completed this chapter, you will be able to: ONE 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.

Chapter Sixteen continued Nonparametric Methods: Analysis of Ranked Data Chapter Sixteen GOALS When you have completed this chapter, you will be able to: FIVE 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. 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. The test requires dependent (related) samples.

The Sign Test continued Determine the number of usable pairs. Determine the sign of the difference between related pairs. n is the number of usable pairs (without ties), X is the number of pluses or minuses, and the binomial probability p=.5. Compare the number of positive (or negative) differences to the critical value The Sign Test continued

Normal Approximation If both np and n(1-p) 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 Normal Approximation

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 1

Company 2000 2001 Difference Sign Savoth Glass 20 16 4 + Ruisi Glass 14 13 1 + Rubin Inc. 23 20 3 + Vaught 24 17 7 + Lambert Glass 31 22 9 + Pimental 22 20 2 + Olson Glass 14 20 -6 - Flynn Glass 18 11 7 + Example 1

Step 2: H0: is rejected if the number of negative signs is 0 or 1. 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 0 or 1. Step 1 H0: p >.5 H1: p <.5 Example 1

Testing a Hypothesis About a Median When testing the value of the median, we use the normal approximation to the binomial distribution. The z distribution is used as the test statistic Testing a Hypothesis About a Median

The Gordon Travel Agency claims that their median airfare for all their clients to all destinations is $450. This claim is being challenged by a competing agency, who believe the median is different from $450. A random sample of 300 tickets revealed 170 tickets were below $450. Use the 0.05 level of significance. Example 2

H0 is rejected if z is less than –1.96 or greater than 1.96. The value of z is 2.252. H0 is rejected. We conclude that the median is not $450. Example 2 Continued

Wilcoxon Signed-Rank Test Use if assumption of normality is violated for the paired-t test Requires the ordinal scale of measurement The observations must be related or dependent. Wilcoxon Signed-Rank Test

Wilcoxon Signed-Rank Test Compute the differences between related observations. Rank the absolute differences from low to high. Compare the smaller of the two rank sums with the T value, obtained from Appendix H. Return the signs to the ranks and sum positive and negative ranks. Wilcoxon Signed-Rank Test

Step 1: H0: The percent stayed the same. H1: The percent declined. From Example 1 Have R&D expenses declined as a percent of income? Use .05 significance level. Step 1: H0: The percent stayed the same. H1: The percent declined. Step 2 H0 is rejected if the smaller of the rank sums is less than or equal to 5. See Appendix H. Example 3

Company 2000 2001 Difference ABS-Diff Rank R+ R- Savoth Glass 20 16 4 4 4 4 * Ruisi Glass 14 13 1 1 1 1 * Rubin Inc. 23 20 3 3 3 3 * Vaught 24 17 7 7 7 7 * Lambert Glass 31 22 9 9 8 8 * Pimental 22 20 2 2 2 2 * Olson Glass 14 20 -6 6 5 * 5 Flynn Glass 18 11 7 7 6 6 * The smaller rank sum is 5, which is equal to the critical value of T. H0 is rejected. The percent has declined from one year to the next. Example 3 Continued

Wilcoxon Rank-Sum Test Used to determine if two independent samples came from the same or equal populations No assumption about the shape of the population is required The data must be at least ordinal scale Each sample must contain at least eight observations Wilcoxon Rank-Sum Test

Wilcoxon Rank-Sum Test Rank all data values from low to high as if they were from a single population Determine the sum of ranks for each of the two samples Use the smaller of the two sums W to compute the test statistic Wilcoxon Rank-Sum Test

Hills Community College purchased two vehicles, a Ford and a Chevy, for the administration’s use when traveling. The repair costs for the two cars over the last three years is shown on the next slide. At the .05 significance level is there a difference in the two distributions? Example 4

Ford ($) Rank Chevy($) Rank 25.31 3.0 14.89 1.0 33.68 5.5 20.31 2.0 25.31 3.0 14.89 1.0 33.68 5.5 20.31 2.0 46.89 7.0 25.97 4.0 51.83 8.0 33.68 5.5 87.65 13.0 68.98 9.0 87.90 14.0 78.23 10.0 90.89 15.0 80.31 11.0 120.67 16.0 81.75 12.0 157.90 17.0 81.50 71.5 Example 4

Step 1: H0: The populations are the same. H1: The populations are not the same. Step 2: H0 is rejected if z >1.96 or z is less than –1.96 Step 3: The value of the test statistic is 0.914. Step 4: We do 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 4

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 It is an alternative to the one-way ANOVA Each sample should have at least five observations The sample data is ranked from low to high as if it were a single group The chi-square distribution is the test statistic Kruskal-Wallis Test: Analysis of Variance by Ranks

Kruskal-Wallis Test: Analysis of Variance by Ranks continued Test Statistic Kruskal-Wallis Test: Analysis of Variance by Ranks continued

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 5

Ranked Increases in Managers’ Salaries EXAMPLE 5 continued

Step 1: H0: The populations are the same. H1: The populations are not the same. Step 2: H0 is rejected if H is greater than 7.185. There are 3 degrees of freedom at the .05 significance level. The null hypothesis is not rejected. There is no difference in the percent increases in manager salaries in the four plants.

Rank-Order Correlation Spearman’s Coefficient of Rank Correlation Reports the association between two sets of ranked observations Similar to Pearson’s coefficient of correlation, but is based on ranked data. Ranges from –1.00 up to 1.00 d is the difference in the ranks and n is the number of observations Rank-Order Correlation

Testing the Significance of rs H1: Rank correlation in population is not 0. Ho: Rank correlation in population is 0 Testing the Significance of rs

Preseason Football Rankings for the Atlantic Coast Conference by the coaches and sports writers School Coaches Writers Maryland 2 3 NC State 3 4 NC 6 6 Virginia 5 5 Clemson 4 2 Wake Forest 7 8 Duke 8 7 Florida State 1 1 Example 6 Continued

S d 2 School Coaches Writers d d2 Maryland 2 3 -1 1 NC State 3 4 -1 1 Virginia 5 5 0 0 Clemson 4 2 2 4 Wake Forest 7 8 -1 1 Duke 8 7 1 1 Florida State 1 1 0 0 Total 8 Example 6 continued

Coefficient of Rank Correlation There is a strong correlation between the ranks of the coaches and the sports writers. Example 6 Continued