2. 1 QNT 531 Advanced Problems in Statistics and Research Methods WORKSHOP 5 By Dr. Serhat Eren University OF PHOENIX

3. 2 SECTION 5 NONPARAMETRIC METHODS  One consideration in determining whether a parametric or a nonparametric method is appropriate is the scale of measurement used to generate the data.  All data are generated by one of four scales of measurement: nominal, ordinal, interval, and ratio.

4. 3 SECTION 5 NONPARAMETRIC METHODS 1. Nominal scale  The scale of measurement is nominal if the data are labels or categories used to define an attribute of an element. Nominal data may be numeric or nonnumeric. Examples  The exchange where a stock is listed (NYSE, NASDAQ, or AMEX) is nonnumeric nominal data. An individual’s social security number is numeric nominal data.

5. 4 SECTION 5 NONPARAMETRIC METHODS 2. Ordinal scale.  The scale of measurement is ordinal if the data can be used to rank, or order, the observations. Ordinal data may be numeric or nonumeric. Examples  The measures small, medium, and large for the size of an item are nonnumeric ordinal data. The class ranks of individuals measured as 1, 2, 3,...are numeric ordinal data.

6. 5 SECTION 5 NONPARAMETRIC METHODS 3. Interval scale  The scale of measurement is interval if the data have the properties of ordinal data and the interval between observations is expressed in terms of a fixed unit of measure. Interval data must be numeric. Examples  Measures of temperature are interval data. Suppose it is 70 degrees in one location and 40 degrees in another.

7. 6 SECTION 5 NONPARAMETRIC METHODS 4. Ratio scale  The scale of measurement is ratio if the data have the properties of interval data and the ratio of measures is meaningful. Ratio data must be numeric. Examples  Variables such as distance, height, weight, and time are measured on a ratio scale. Temperature measures are not ratio data because there is no inherently defined zero point.

8. 7 SECTION 5 NONPARAMETRIC METHODS  In general, for a statistical method to be classified as nonparametric, it must satisfy at least one of the following conditions. 1.The method can be used with nominal data. 2.The method can be used with ordinal data. 3.The method can be used with interval or ratio data when no assumption can be made about the population probability distribution.

9. 8 5-1 SIGN TEST  A common market-research application of the sign test involves using a sample of n potential customers to identify a preference for one of two brands of a product such as coffee, soft drinks, or detergents.  The n expressions of preference are nominal data because the consumer simply names, or labels, a preference. Given these data, our objective is to determine whether a difference in preference exists between the two items being compared.

10. 9 5-1 SIGN TEST 5.1.1 Small-Sample Case  The small-sample case for the sign test should be used whenever n> 20.  For example, Sun Coast produces an orange juice product marketed under the name Citrus Valley. A competitor of Sun Coast Farms has begun producing a new orange juice product known as Tropical Orange.  In a study of consumer preferences for the two brands, 12 individuals were given unmarked samples of each product.

11. 10 5-1 SIGN TEST  The brand each individual tasted first was selected randomly.  After tasting the two products, the individuals were asked to state a preference for one of the two brands.  The purpose of the study is to determine whether consumers prefer one product over the other.

12. 11 5-1 SIGN TEST  Letting p indicate the proportion of the population of consumers favoring Citrus Valley, we want to test the following hypotheses.  If H 0 cannot be rejected, we will have no evidence indicating a difference in preference for the two brands of orange juice. However, if H 0 can be rejected, we can conclude that the consumer preferences are different for the two brands.

13. 12 5-1 SIGN TEST  To record the preference data for the 12 individuals participating in the study, we use a plus sign if the individual expresses a preference for Citrus Valley and a minus sign if the individual expresses a preference for Tropical Orange.  Because the data are recorded in terms of plus or minus signs, this nonparametric test is called the sign test.

14. 13 5-1 SIGN TEST  Under the assumption that H 0 is true ( p = 0.50), the number of plus signs follows a binomial probability distribution with p ≠ 0.50.  With a sample size of n = 12, Table 5 in Appendix B shows the probabilities for the binomial probability distribution with p = 0.50 as displayed in Table 5-1.  For example, using  = 0.05, we would place a rejection region or area of approximately 0.025 in each tail of the distribution in Figure 5-1.

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17. 16 5-1 SIGN TEST  Starting at the lower end of the distribution, we see that the probability of obtaining zero, one, or two plus signs is 0.0002 + 0.0029 + 0.0161 = 0.0192  Note that we stop at 2 plus signs because adding the probability of three plus signs would make the area in the lower tail equal to 0.0192 + 0.0537 = 0.0729  which substantially exceeds the desired area of 0.025.

18. 17 5-1 SIGN TEST  At the upper end of the distribution, we find the same probability of 0.0192 corresponding to 10, 11, or 12 plus signs. Thus, the closest we can come to  = 0.05 without exceeding it is 0.0192 + 0.0192 = 0.0384  We therefore adopt the following rejection rule. The preference data obtained for the Sun Coast Farms example are reported in Table 5-2. Because only two plus signs are observed, the null hypothesis is rejected.

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20. 19 5-1 SIGN TEST  The study provides evidence that consumer preference differs for the two brands of orange juice.  Reject H 0 if p -value <  can also be used for nonparametric tests. With two plus signs, the p - value for this two-tailed test is 2(0.0161 + 0.0029 + 0.0002) = 0.0384  In many situations, one or more individuals in the sample are not able to state a definite preference.

21. 20 5-1 SIGN TEST  In such cases, the individual’s response of no preference can be dropped and the analysis conducted with a smaller sample size.  Appendix B does not provide binomial probability distribution tables for sample sizes greater than 20.  In such cases, we can use the large-sample normal approximation of binomial probabilities to determine the appropriate rejection rule for the sign test.

22. 21 5-1 SIGN TEST 5.1.2 Large-Sample Case  Using the null hypothesis H 0 : p = 0.50 and a sample size of n > 20, the sampling distribution for the number of plus signs can be approximated by a normal probability distribution.  Normal Approximation of the Sampling Distribution of the Number of Plus Signs When No Preference Is Stated

23. 22 5-1 SIGN TEST  Distribution form: approximately normal provided n > 20.  For example, a poll taken during a recent presidential election campaign asked 200 registered voters to rate the Democratic and Republican candidates in terms of best overall foreign policy. Results of the poll showed 72 rated the Democratic candidate higher, 103 rated the Republican candidate higher, and 25 indicated no difference between the candidates.

24. 23 5-1 SIGN TEST  Does the poll indicate a significant difference between the two candidates in terms of public opinion about their foreign policies?  n = 200 - 25 = 175 individuals were able to indicate the candidate they believed had the best overall foreign policy. We find that the sampling distribution of the number of plus signs has the following properties.

25. 24 5-1 SIGN TEST  In addition, with n =175 we can assume that the sampling distribution is approximately normal. This distribution is shown in Figure 5-2.  With  = 0.05, the rejection rule for this two- tailed test can be written as follows.  Reject H 0 if z +1.96  Using the number of times the Democratic candidate received the higher foreign policy rating as the number of plus signs ( x = 72), we have the following value of the test statistic.

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27. 26 5-1 SIGN TEST  Because z = -2.35 is less than -1.96, the hypothesis of no difference in foreign policy for the two candidates should be rejected at the 0.05 level of significance.  With z = -2.35 the standard normal distribution can be used to show that the p -value is 2(0.5000 -0.4906) = 0.0188  This study indicates that the candidates are perceived to have different foreign policy ratings.

28. 27 5-1 SIGN TEST 5.1.3 Hypothesis Test about a Median  Recall that the median splits a population in such a way that 50% of the values are at the median or above and 50% are at the median or below.  We can apply the sign test by using a plus sign whenever the data in the sample are above the hypothesized value of the median and a minus sign whenever the data in the sample are below the hypothesized value of the median. Any data exactly equal to the hypothesized value of the median should be discarded.

29. 28 5-1 SIGN TEST  For example, the following hypothesis test is being conducted about the median price of new homes in St. Louis, Missouri.  In a sample of 62 new homes, 34 have prices above $130,000, 26 have prices below $130,000 and two have prices of exactly $130,000. n= 60 homes with prices different from $130,000, we obtain

30. 29 5-1 SIGN TEST  With x = 34 as the number of plus signs, the test statistic becomes  Using a two-tailed test and a level of significance of  = 0.05, we reject H 0 if z is less than -1.96 or greater than +1.96. The test statistic is z =1.03, therefore we cannot reject H 0.  The p -value is 2(0.5000 - 0.3485) = 0.303.

31. 30 5-2 WILCOXON SIGNED-RANK TEST  The Wilcoxon signed-rank test is the nonparametric alternative to the parametric matched-sample test presented earlier.  In the matched-sample situation, each experimental unit generates two paired or matched observations, one from population 1 and one from population 2. The differences between the matched observations provide insight about the differences between the two populations.

32. 31 5-2 WILCOXON SIGNED-RANK TEST  For example, a manufacturing firm is attempting to determine whether two production methods differ in task completion time. A sample of 11 workers was selected, and each worker completed a production task using each of the production methods.  The production method that each worker used first was selected randomly. Thus, each worker in the sample provided a pair of observations, as shown in Table 5-3.

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34. 33 5-2 WILCOXON SIGNED-RANK TEST  A positive difference in task completion times indicates that method 1 required more time, and a negative difference in times indicates that method 2 required more time.  Do the data indicate that the methods are significantly different in terms of task completion times?

35. 34 5-2 WILCOXON SIGNED-RANK TEST  We have two populations of task completion times, one population associated with each method. The following hypotheses will be tested.  If H 0 cannot be rejected, we will not have evidence to conclude that the task completion times differ for the two methods. However, if H 0 can be rejected, we will conclude that the two methods differ in task completion time.

36. 35 5-2 WILCOXON SIGNED-RANK TEST  The first step of the Wilcoxon signed-rank test requires a ranking of the absolute value of the differences between the two methods. We discard any differences of zero and then rank the remaining absolute differences from lowest to highest.  Tied differences are assigned the average ranking of their positions in the combined data set. The ranking of the absolute values of differences is shown in the fourth column of Table 5-4.

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38. 37 5-2 WILCOXON SIGNED-RANK TEST  Once the ranks of the absolute differences have been determined, the ranks are given the sign of the original difference in the data. For example, the 0.1 difference for worker 7, which was assigned the rank of 1, is given the value of +1 because the observed difference between the two methods was positive.  The test for significance under the Wilcoxon signed-rank test involves determining whether the computed sum of signed ranks (+44 in our example) is significantly different from zero.

39. 38 5-2 WILCOXON SIGNED-RANK TEST  Let T denote the sum of the signed-rank values in a Wilcoxon signed-rank test.  It can be shown that if the two populations are identical and the number of matched pairs of data is 10 or more, the sampling distribution of T can be approximated by a normal probability distribution as follows.

40. 39 5-2 WILCOXON SIGNED-RANK TEST  Sampling Distribution of T for Identical Populations  Distribution form: approximately normal provided n ≥ 10.

41. 40 5-2 WILCOXON SIGNED-RANK TEST  For the example, we have n =10 after discarding the observation with the difference of zero (worker 8).  Thus, using the formula above, we have  Figure 5-3 is the sampling distribution of T under the assumption of identical populations.

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43. 42 5-2 WILCOXON SIGNED-RANK TEST  The value of the test statistic z is:  Testing the null hypothesis of no difference using a level of significance of  = 0.05, we reject H 0 if z 1.96. With the value of z = 2.24, we reject H 0 and conclude that the two populations are not identical and that the methods differ in task completion time.

44. 43 5-3 MANN-WHITNEY-WILCOXON TEST  This test, unlike the signed-rank test, is not based on a matched sample. Two independent samples, one from each population, are used. The test was developed jointly by Mann, Whitney, and Wilcoxon.  It is sometimes called the Mann-Whitney test and sometimes the Wilcoxon rank-sum test. Both the Mann-Whitney and Wilcoxon versions of this test are equivalent; we refer to it as the Mann-Whitney-Wilcoxon (MWW) test.

45. 44 5-3 MANN-WHITNEY-WILCOXON TEST  The only requirement of the MWW test is that the measurement scale for the data is at least ordinal.  Then, instead of testing for the difference between the means of the two populations, the MWW test determines whether the two populations are identical.The hypotheses for the MWW test are as follows.

46. 45 5-3 MANN-WHITNEY-WILCOXON TEST 5.3.1 Small-Sample Case  The small-sample case for the MWW test should be used whenever the sample sizes for both populations are less than or equal to 10.  For example, the majority of students attending Johnston High School previously attended either Garfield Junior High School or Mulberry Junior High School. The question raised by school administrators was whether the population of students who had attended Garfield was identical to the population of students who had attended Mulberry in terms of academic potential.

47. 46 5-3 MANN-WHITNEY-WILCOXON TEST  The following hypotheses were considered.  Johnston High School administrators selected a random sample of four high school students who had attended Garfield Junior High and another random sample of five students who had attended Mulberry Junior High. The ordinal class standings for the nine students are listed in Table 5-5.

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49. 48 5-3 MANN-WHITNEY-WILCOXON TEST  The first step in the MWW procedure is to rank the combined data from the two samples from low to high. The lowest value (class standing 8) receives a rank of 1 and the highest value (class standing 202) receives a rank of 9. The ranking of the nine students is given in Table 5- 6.  The next step is to sum the ranks for each sample separately, Table 5-7. We use the sum of the ranks for the sample of four students from Garfield and we denote this sum by the symbol T. Thus, for our example, T = 11.

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52. 51 5-3 MANN-WHITNEY-WILCOXON TEST  With four students in the sample, Garfield could have the top four students in the study. If this were the case, T = 1 + 2 + 3 + 4 = 10 would be the smallest value possible for the rank sum T.  Conversely, Garfield could have the bottom four students, in which case T = 6 + 7 + 8 + 9 = 30 would be the largest value possible for T. Hence, T for the Garfield sample must take a value between 10 and 30.

53. 52 5-3 MANN-WHITNEY-WILCOXON TEST  Note that values of T near 10 imply that Garfield has the significantly better or higher ranking, students, whereas values of T near 30 imply that Garfield has the significantly weaker or lower ranking, students.  Thus, if the two populations of students were identical in terms of academic potential, we would expect the value of T to be near the average of the two values, or (10 + 30)/2 = 20.

54. 53 5-3 MANN-WHITNEY-WILCOXON TEST  Critical values of the MWW T statistic are provided in Table 9 of Appendix B for cases in which both sample sizes are less than or equal to 10. n 1 refers to the sample size corresponding to the sample whose rank sum is being used in the test.  The value of T L is read directly from the table and the value of T U is computed from the following formula

55. 54 5-3 MANN-WHITNEY-WILCOXON TEST  The null hypothesis of identical populations should be rejected only if T is strictly less than T L or strictly greater than T U.  For example, using Table 9 of Appendix B with a 0.05 level of significance, we see that the lower-tail critical value for the MWW statistic with n 1 =4 (Garfield) and n 2 =5 (Mulberry) is T L =12. The upper-tail critical value for the MWW statistic computed by using the formula above is

56. 55 5-3 MANN-WHITNEY-WILCOXON TEST  The rejection rule can be written as: Reject H 0 if T 28  Referring to Table 5-7, we see that T = 11. Hence, the null hypothesis H 0 is rejected, and we can conclude that the population of students at Garfield differs from the population of students at Mulberry in terms of academic potential.

57. 56 5-3 MANN-WHITNEY-WILCOXON TEST 5.3.2 Large-Sample Case  When both sample sizes are greater than or equal to 10,a normal approximation of the distribution of T can be used to conduct the analysis for the MWW test.  For example, Third National Bank has two branch offices. Data collected from two independent simple random samples, one from each branch, are given in Table 5-8.

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59. 58 5-3 MANN-WHITNEY-WILCOXON TEST  Do the data indicate whether the populations of checking account balances at the two branch banks are identical?  The first step in the MWW test is to rank the combined data from the lowest to the highest values. Using the combined set of 22 observations in Table 5-8,we find the lowest data value of $750 (sixth item of sample 2) and assign to it a rank of 1.Continuing the ranking gives us the following list.

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61. 60 5-3 MANN-WHITNEY-WILCOXON TEST  Table 5-9 is the entire data set with the assigned rank of each observation.  The next step in the MWW test is to sum the ranks for each sample. The sums are given in Table 5-9.  The test procedure can be based on the sum of the ranks for either sample. We use the sum of the ranks for the sample from branch 1. Thus, for this example, T =169.5.

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63. 62 5-3 MANN-WHITNEY-WILCOXON TEST  Sampling Distribution of T for Identical Populations  Distribution form: approximately normal provided n 1 ≥ 10 and n 2 ≥10.

64. 63 5-3 MANN-WHITNEY-WILCOXON TEST  For branch 1, we have  Figure 5-4 is the sampling distribution of T. We compute the test statistic z to determine whether the observed value of T appears to be from the sampling distribution of Figure 5-4.

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66. 65 5-3 MANN-WHITNEY-WILCOXON TEST  If T does not appear to be from that distribution, we will reject the null hypothesis and conclude that the populations are not identical. Computing the test statistic, we have  At a 0.05 level of significance, we know that, to reject H 0, z must be less than -1.96 or greater than +1.96. With z =2.08, we reject H 0. Thus, we conclude that the two populations are not identical.

67. 66 5-3 MANN-WHITNEY-WILCOXON TEST  In summary, the Mann-Whitney-Wilcoxon rank- sum test consists of the following steps to determine whether two independent random samples are selected from identical populations. 1.Rank the combined sample observations from lowest to highest, with tied values being assigned the average of the tied rankings. 2.Compute T, the sum of the ranks for the first sample.

68. 67 5-3 MANN-WHITNEY-WILCOXON TEST 3.In the large-sample case, make the test for significant differences between the two populations by using the observed value of T and comparing it to the sampling distribution of T for identical populations. The value of the standardized test statistic z or the p -value will provide the basis for deciding whether to reject H 0. In the small-sample case, use Table 9 in Appendix B to find the critical values for the test.

69. 68 5-4 KRUSKAL-WALLIS TEST  The MWW test can be used to test whether two populations are identical. It has been extended to the case of three or more populations by Kruskal and Wallis. The hypotheses for the Kruskal-Wallis test with k ≥3 populations can be written as follows.

70. 69 5-4 KRUSKAL-WALLIS TEST  The nonparametric Kruskal-Wallis test can be used with ordinal data as well as with interval or ratio data. In addition, the Kruskal-Wallis test does not require the assumption of normally distributed populations.  Hence, whenever the data from k ≥ 3 populations are ordinal, or whenever the assumption of normally distributed populations is questionable, the Kruskal-Wallis test provides an alternate statistical procedure for testing whether the populations are identical.

71. 70 5-4 KRUSKAL-WALLIS TEST  For example, Williams Manufacturing Company hires employees for its management staff from three local colleges.  Recently, the company’s personnel department has been collecting and reviewing annual performance ratings in an attempt to determine whether there are differences in performance among the managers hired from these colleges and data are summarized in Table 5-11.

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73. 72 5-4 KRUSKAL-WALLIS TEST  The overall performance rating of each manager is given on a 0 –100 scale, with 100 being the highest possible performance rating.  Suppose we want to test whether the three populations are identical in terms of performance evaluations.  The Kruskal-Wallis test statistic, which is based on the sum of ranks for each of the samples, can be computed as follows:

74. 73 5-4 KRUSKAL-WALLIS TEST  k= the number of populations  n i = the number of items in sample i  n T = Σni = total number of items in all samples  R i = sum of the ranks for sample i

75. 74 5-4 KRUSKAL-WALLIS TEST  Kruskal and Wallis were able to show that, under the null hypothesis in which the populations are identical, the sampling distribution of W can be approximated by a chi- square distribution with k -1 degrees of freedom. This approximation is generally acceptable if each of the sample sizes is greater than or equal to five.  To compute the W statistic for our example, we must first rank all 20 data items. The data values, their associated ranks, and the sum of the ranks for the three samples are given in Table 5-12.

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77. 76 5-4 KRUSKAL-WALLIS TEST  The chi-square distribution table (Table 3 of Appendix B) shows that with k -1 = 2 degrees of freedom and  = 0.05 in the upper tail of the distribution, the critical chi-square value is X² = 5.99147.

78. 77 5-4 KRUSKAL-WALLIS TEST  Because the test statistic W = 8.92 is greater than 5.99147, we reject the null hypothesis that the three populations are identical.  As a result, we conclude that manager performance differs significantly depending on the college attended.

79. 78 5-5 RANK CORRELATION  We consider measures of association between two variables when only ordinal data are available.  The Spearman rank-correlation coefficient (r s ) has been developed for this purpose.

80. 79 n = the number of items or individuals being ranked x i = the rank of item i with respect to one variable y i = the rank of item i with respect to a second variable d i = x i - y i  For example, a company wants to determine whether individuals who were expected at the time of employment to be better salespersons actually turn out to have better sales records. 5-5 RANK CORRELATION

81. 80  To investigate this question, the vice president in charge of personnel carefully reviewed the original job interview summaries, academic records, and letters of recommendation for 10 current members of the firm’s sales force.  After the review, the vice president ranked the 10 individuals in terms of their potential for success, basing the assessment solely on the information available at the time of employment. Then a list was obtained of the number of units sold by each salesperson over the first two years. 5-5 RANK CORRELATION

82. 81  On the basis of actual sales performance, a second ranking of the 10 salespersons was carried out. Table 5-13 gives the relevant data and the two rankings.  The statistical question is whether there is agreement between the ranking of potential at the time of employment and the ranking based on the actual sales performance over the first two years. The computations are summarized in Table 5-14. 5-5 RANK CORRELATION

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85. 84  We see that the rank-correlation coefficient is a positive 0.73. The Spearman rank-correlation coefficient ranges from -1.0 to 1.0 and its interpretation is similar to that of the sample correlation coefficient in that positive values near 1.0 indicate a strong association between the rankings; as one rank increases, the other rank increases.  Rank correlations near -1.0 indicate a strong negative association between the rankings; as one rank increases, the other rank decreases. 5-5 RANK CORRELATION

86. 85 5.5.1 Test for Significant Rank Correlation  We may want to use the sample results to make an inference about the population rank correlation r s.To make an inference about the population rank correlation, we must test the following hypotheses. 5-5 RANK CORRELATION

87. 86  Sampling Distribution of r s  Distribution form: approximately normal provided n ≥ 10. 5-5 RANK CORRELATION

88. 87  For our example, 5-5 RANK CORRELATION

89. 88  At a 0.05 level of significance, we see that the null hypothesis of no correlation will be rejected if z 1.96. Because z = 2.21, we reject the hypothesis of no rank correlation.  Thus, we can conclude that there is a significant rank correlation between sales potential and sales performance. 5-5 RANK CORRELATION