Non-parametric: Analysis of Ranked Data Chapter 18 McGraw-Hill/Irwin ©The McGraw-Hill Companies, Inc 2008 GOALS Conduct the sign test for dependent samples using the binomial and standard normal distributions as the test statistics Conduct a test of hypothesis for dependent samples using the Wilcoxon signed-rank test Conduct and interpret the Wilcoxon rank-sum test for independent samples Conduct and interpret the Kruskal-Wallis test for several independent samples Compute and interpret Spearman’s coefficient of rank correlation Conduct a test of hypothesis to determine whether the correlation among the ranks in the population is different from zero The Sign Test The Sign Test is 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 Procedure to conduct the test: Determine the sign (+ or -) of the difference between related pairs Determine the number of usable pairs Compare the number of positive (or negative) differences to the critical value n is the number of usable pairs (without ties), X is the number of pluses or minuses, and the binomial probability π = The Sign Test - Example The director of information systems at Samuelson Chemicals recommended that an in-plant training program be instituted for managers The objective is to improve the knowledge of database usage in accounting, procurement, production, and so on A sample of 15 managers was selected at random A panel of database experts determined the general level of competence of each manager with respect to using the database Their competence and understanding were rated as being either outstanding, excellent, good, fair, or poor After the three-month training program, the same panel of information systems experts rated each manager again The two ratings (before and after) are shown along with the sign of the difference A “+” sign indicates improvement, and a “-” sign indicates that the manager’s competence using databases had declined after the training program Did the in-plant training program effectively increase the competence of the managers using the company’s database? Step 1: State the Null and Alternative Hypotheses H0: π ≤.5 (There is no increase in competence as a result of the inplant training program.) H1: π >.5 (There is an increase in competence as a result of the inplant training program.) Step 2: Select a level of significance We chose the 10 level Step 3: Decide on the test statistic It is the number of plus signs resulting from the experiment Step 4: Formulate a decision rule In this example α is 10 The probability of or fewer successes is 029, found by 000 + 001 + 006 + 022 The probability of 11 or more successes is also 029 Adding the two probabilities gives 058 This is the closest we can come to 10 without exceeding it Hence, the decision rule for a two-tailed test would be to reject the null hypothesis if there are or fewer plus signs, or 11 or more plus signs Step 5: Make a decision regarding the null hypothesis Eleven out of the 14 managers in the training course increased their database competency The number 11 is in the rejection region, which starts at 10, so is rejected We conclude that the three-month training course was effective It increased the database competency of the managers Kruskal-Wallis Test: Analysis of Variance by Ranks This is 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 The chi-square distribution is the test statistic Each sample should have at least five observations The sample data is ranked from low to high as if it were a single group Kruskal-Wallis Test: Analysis of Variance by Ranks - Example A management seminar consists of executives from manufacturing, finance, and engineering Before scheduling the seminar sessions, the seminar leader is interested in whether the three groups are equally knowledgeable about management principles Plans are to take samples of the executives in manufacturing, in finance, and in engineering and to administer a test to each executive If there is no difference in the scores for the three distributions, the seminar leader will conduct just one session However, if there is a difference in the scores, separate sessions will be given We will use the Kruskal-Wallis test instead of ANOVA because the seminar leader is unwilling to assume that (1) the populations of management scores follow the normal distribution or (2) the population standard deviations are the same Kruskal-Wallis Test: Analysis of Variance by Ranks - Example Step 1: H0: The population distributions of the management scores for the populations of executives in manufacturing, finance, and engineering are the same H1: The population distributions of the management scores for the populations of executives in manufacturing, finance, and engineering are NOT the same Step 2: H0 is rejected if χ2 is greater than 7.185 There are degrees of freedom at the 05 significance level Kruskal-Wallis Test: Analysis of Variance by Ranks - Example The next step is to select random samples from the three populations A sample of seven manufacturing, eight finance, and six engineering executives was selected Their scores on the test are recorded below Kruskal-Wallis Test: Analysis of Variance by Ranks - Example Considering the scores as a single population, the engineering executive with a score of 35 is the lowest, so it is ranked There are two scores of 38 To resolve this tie, each score is given a rank of 2.5, found by (2+3)/2 This process is continued for all scores The highest score is 107, and that finance executive is given a rank of 21 The scores, the ranks, and the sum of the ranks for each of the three samples are given in the table below Kruskal-Wallis Test: Analysis of Variance by Ranks - Example Because the computed value of H (5.736) is less than the critical value of 5.991, the null hypothesis is not rejected There is not enough evidence to conclude there is a difference among the executives from manufacturing, finance, and engineering with respect to their typical knowledge of management principles From a practical standpoint, the seminar leader should consider offering only one session including executives from all areas Kruskal-Wallis Test: Analysis of Variance by Ranks - Example Rank-Order Correlation Spearman’s coefficient of rank correlation reports the association between two sets of ranked observations The features are: It can range from –1.00 up to 1.00 It is similar to Pearson’s coefficient of correlation, but is based on ranked data It computed using the formula: Rank-Order Correlation - Example Lorrenger Plastics, Inc., recruits management trainees at colleges and universities throughout the United States Each trainee is given a rating by the recruiter during the oncampus interview This rating is an expression of future potential and may range from to 15, with the higher score indicating more potential The recent college graduate then enters an in-plant training program and is given another composite rating based on tests, opinions of group leaders, training officers, and so on The on-campus rating and the in-plant training ratings are given in the table on the right Rank-Order Correlation - Example Rank-Order Correlation - Example Testing the Significance of rs State the null hypothesis: Rank correlation in population is State the alternate hypothesis: Rank correlation in population is not For a sample of 10 or more, the significance of is determined by computing t using the following formula The sampling distribution of follows the t distribution with n - degrees of freedom Testing the Significance of rs - Example End of Chapter 18 ... H0: π ≤.5 (There is no increase in competence as a result of the inplant training program.) H1: π >.5 (There is an increase in competence as a result of the inplant training program.) Step 2:... seminar leader is interested in whether the three groups are equally knowledgeable about management principles Plans are to take samples of the executives in manufacturing, in finance, and in. .. single group Kruskal-Wallis Test: Analysis of Variance by Ranks - Example A management seminar consists of executives from manufacturing, finance, and engineering Before scheduling the seminar