 # 1 Pertemuan 25 Metode Non Parametrik-1 Matakuliah: A0064 / Statistik Ekonomi Tahun: 2005 Versi: 1/1.

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1 Pertemuan 25 Metode Non Parametrik-1 Matakuliah: A0064 / Statistik Ekonomi Tahun: 2005 Versi: 1/1

2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Menyimpulkan hasil pengujian hipotesis dengan menggunakan uji tanda (the sign test), runtunan (the run test), dan Mainn- Whitney

3 Outline Materi Uji tanda (the sign test) Uji Runtunan (the runs test) Uji U-Mainn-Whitney

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-4 Using Statistics The Sign Test The Runs Test - A Test for Randomness The Mann-Whitney U Test The Wilcoxon Signed-Rank Test The Kruskal-Wallis Test - A Nonparametric Alternative to One-Way ANOVA Nonparametric Methods and Chi-Square Tests (1) 14

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-5 The Friedman Test for a Randomized Block Design The Spearman Rank Correlation Coefficient A Chi-Square Test for Goodness of Fit Contingency Table Analysis - A Chi-Square Test for Independence A Chi-Square Test for Equality of Proportions Summary and Review of Terms Nonparametric Methods and Chi-Square Tests (2) 14

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-6 Parametric Methods Inferences based on assumptions about the nature of the population distribution Usually: population is normal Types of tests z-test or t-test » Comparing two population means or proportions » Testing value of population mean or proportion ANOVA » Testing equality of several population means Parametric Methods Inferences based on assumptions about the nature of the population distribution Usually: population is normal Types of tests z-test or t-test » Comparing two population means or proportions » Testing value of population mean or proportion ANOVA » Testing equality of several population means 14-1 Using Statistics (Parametric Tests)

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-7 Nonparametric Tests Distribution-free methods making no assumptions about the population distribution Types of tests Sign tests » Sign Test: Comparing paired observations » McNemar Test: Comparing qualitative variables » Cox and Stuart Test: Detecting trend Runs tests » Runs Test: Detecting randomness » Wald-Wolfowitz Test: Comparing two distributions Nonparametric Tests Distribution-free methods making no assumptions about the population distribution Types of tests Sign tests » Sign Test: Comparing paired observations » McNemar Test: Comparing qualitative variables » Cox and Stuart Test: Detecting trend Runs tests » Runs Test: Detecting randomness » Wald-Wolfowitz Test: Comparing two distributions Nonparametric Tests

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-8 Nonparametric Tests Ranks tests Mann-Whitney U Test: Comparing two populations Wilcoxon Signed-Rank Test: Paired comparisons Comparing several populations: ANOVA with ranks » Kruskal-Wallis Test » Friedman Test: Repeated measures Spearman Rank Correlation Coefficient Chi-Square Tests Goodness of Fit Testing for independence: Contingency Table Analysis Equality of Proportions Nonparametric Tests Ranks tests Mann-Whitney U Test: Comparing two populations Wilcoxon Signed-Rank Test: Paired comparisons Comparing several populations: ANOVA with ranks » Kruskal-Wallis Test » Friedman Test: Repeated measures Spearman Rank Correlation Coefficient Chi-Square Tests Goodness of Fit Testing for independence: Contingency Table Analysis Equality of Proportions Nonparametric Tests (Continued)

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-9 Deal with enumerative (frequency counts) data. Do not deal with specific population parameters, such as the mean or standard deviation. Do not require assumptions about specific population distributions (in particular, the normality assumption). Deal with enumerative (frequency counts) data. Do not deal with specific population parameters, such as the mean or standard deviation. Do not require assumptions about specific population distributions (in particular, the normality assumption). Nonparametric Tests (Continued)

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-10 Comparing paired observations Paired observations: X and Y p = P(X>Y) Two-tailed test H 0 : p = 0.50 H 1 : p  0.50 Right-tailed testH 0 : p  0.50 H 1 : p  0.50 Left-tailed testH 0 : p  0.50 H 1 : p  0.50 Test statistic: T = Number of + signs Comparing paired observations Paired observations: X and Y p = P(X>Y) Two-tailed test H 0 : p = 0.50 H 1 : p  0.50 Right-tailed testH 0 : p  0.50 H 1 : p  0.50 Left-tailed testH 0 : p  0.50 H 1 : p  0.50 Test statistic: T = Number of + signs 14-2 Sign Test

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-11 Small Sample: Binomial Test For a two-tailed test, find a critical point corresponding as closely as possible to  /2 (C 1 ) and define C 2 as n-C 1. Reject null hypothesis if T  C 1 or T  C 2. For a right-tailed test, reject H 0 if T  C, where C is the value of the binomial distribution with parameters n and p = 0.50 such that the sum of the probabilities of all values less than or equal to C is as close as possible to the chosen level of significance, . For a left-tailed test, reject H 0 if T  C, where C is defined as above. Small Sample: Binomial Test For a two-tailed test, find a critical point corresponding as closely as possible to  /2 (C 1 ) and define C 2 as n-C 1. Reject null hypothesis if T  C 1 or T  C 2. For a right-tailed test, reject H 0 if T  C, where C is the value of the binomial distribution with parameters n and p = 0.50 such that the sum of the probabilities of all values less than or equal to C is as close as possible to the chosen level of significance, . For a left-tailed test, reject H 0 if T  C, where C is defined as above. Sign Test Decision Rule

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-12 Cumulative Binomial Probabilities (n=15, p=0.5) x F(x) 00.00003 10.00049 20.00369 30.01758 40.05923 50.15088 60.30362 70.50000 80.69638 90.84912 100.94077 110.98242 120.99631 130.99951 140.99997 151.00000 CEO Before After Sign 1 34 1+ 2 55 0 3 2 3 1+ 4 24 1+ 5 44 0 6 23 1+ 7 12 1+ 8 54 -1- 9 45 1+ 10 54 -1- 11 34 1+ 12 25 1+ 13 25 1+ 14 23 1+ 15 1 2 1+ 16 32 -1- 17 45 1+ CEO Before After Sign 1 34 1+ 2 55 0 3 2 3 1+ 4 24 1+ 5 44 0 6 23 1+ 7 12 1+ 8 54 -1- 9 45 1+ 10 54 -1- 11 34 1+ 12 25 1+ 13 25 1+ 14 23 1+ 15 1 2 1+ 16 32 -1- 17 45 1+ n = 15 T = 12  0.025 C1=3 C2 = 15-3 = 12 H 0 rejected, since T  C2 C1 Example 14-1

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-13 Example 14-1- Using the Template H 0 : p = 0.5 H 1 : p  Test Statistic: T = 12 p-value = 0.0352. For  = 0.05, the null hypothesis is rejected since 0.0352 < 0.05. Thus one can conclude that there is a change in attitude toward a CEO following the award of an MBA degree.

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-14 A run is a sequence of like elements that are preceded and followed by different elements or no element at all. Case 1: S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E : R = 20 Apparently nonrandom Case 2: SSSSSSSSSS|EEEEEEEEEE : R = 2 Apparently nonrandom Case 3: S|EE|SS|EEE|S|E|SS|E|S|EE|SSS|E : R = 12 Perhaps random Case 1: S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E|S|E : R = 20 Apparently nonrandom Case 2: SSSSSSSSSS|EEEEEEEEEE : R = 2 Apparently nonrandom Case 3: S|EE|SS|EEE|S|E|SS|E|S|EE|SSS|E : R = 12 Perhaps random A two-tailed hypothesis test for randomness: H 0 : Observations are generated randomly H 1 : Observations are not generated randomly Test Statistic: R=Number of Runs Reject H 0 at level  if R  C1 or R  C2, as given in Table 8, with total tail probability P(R  C 1 ) + P(R  C 2 ) =  A two-tailed hypothesis test for randomness: H 0 : Observations are generated randomly H 1 : Observations are not generated randomly Test Statistic: R=Number of Runs Reject H 0 at level  if R  C1 or R  C2, as given in Table 8, with total tail probability P(R  C 1 ) + P(R  C 2 ) =  14-3 The Runs Test - A Test for Randomness

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-15 Table 8: Number of Runs (r) (n 1,n 2 )11121314151617181920. (10,10) 0.5860.7580.8720.9490.9810.9960.9991.0001.0001.000 Case 1: n 1 = 10 n 2 = 10 R= 20 p-value  0 Case 2: n 1 = 10 n 2 = 10 R = 2 p-value  0 Case 3: n 1 = 10 n 2 = 10 R= 12 p-value  P  R  F(11)] = (2)(1-0.586) = (2)(0.414) = 0.828 H 0 not rejected Case 1: n 1 = 10 n 2 = 10 R= 20 p-value  0 Case 2: n 1 = 10 n 2 = 10 R = 2 p-value  0 Case 3: n 1 = 10 n 2 = 10 R= 12 p-value  P  R  F(11)] = (2)(1-0.586) = (2)(0.414) = 0.828 H 0 not rejected Runs Test: Examples

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-16 Large-Sample Runs Test: Using the Normal Approximation

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-17 Example 14-2: n 1 = 27 n 2 = 26 R = 15 H 0 should be rejected at any common level of significance. Large-Sample Runs Test: Example 14-2

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-18 Large-Sample Runs Test: Example 14-2 – Using the Template Note:The computed p-value using the template is 0.0005 as compared to the manually computed value of 0.0006. The value of 0.0005 is more accurate. Note: The computed p-value using the template is 0.0005 as compared to the manually computed value of 0.0006. The value of 0.0005 is more accurate. Reject the null hypothesis that the residuals are random.

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-19 The null and alternative hypotheses for the Wald-Wolfowitz test: H 0 : The two populations have the same distribution H 1 : The two populations have different distributions The test statistic: R = Number of Runs in the sequence of samples, when the data from both samples have been sorted The null and alternative hypotheses for the Wald-Wolfowitz test: H 0 : The two populations have the same distribution H 1 : The two populations have different distributions The test statistic: R = Number of Runs in the sequence of samples, when the data from both samples have been sorted Salesperson A:35443950482960754966 Salesperson B:172313243321181632 Using the Runs Test to Compare Two Population Distributions (Means): the Wald-Wolfowitz Test Example 14-3:

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-20 Table Number of Runs (r) (n 1,n 2 )2345. (9,10) 0.0000.0000.0020.004... Sales SalesSalesPerson SalesPerson(Sorted)(Sorted)Runs 35A13 B 44A16 B 39A17 B 48A21 B 60A24 B 1 75A29 A 2 49A32 B 66A33 B 3 17B35 A 23B39 A 13B44 A 24B48 A 33B49 A 21B50 A 18B60 A 16B66 A 32B75 A 4 Sales SalesSalesPerson SalesPerson(Sorted)(Sorted)Runs 35A13 B 44A16 B 39A17 B 48A21 B 60A24 B 1 75A29 A 2 49A32 B 66A33 B 3 17B35 A 23B39 A 13B44 A 24B48 A 33B49 A 21B50 A 18B60 A 16B66 A 32B75 A 4 n 1 = 10 n 2 = 9 R= 4 p-value  P  R  H 0 may be rejected n 1 = 10 n 2 = 9 R= 4 p-value  P  R  H 0 may be rejected The Wald-Wolfowitz Test: Example 14-3

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-21 Ranks tests Mann-Whitney U Test: Comparing two populations Wilcoxon Signed-Rank Test: Paired comparisons Comparing several populations: ANOVA with ranks Kruskal-Wallis Test Friedman Test: Repeated measures Ranks tests Mann-Whitney U Test: Comparing two populations Wilcoxon Signed-Rank Test: Paired comparisons Comparing several populations: ANOVA with ranks Kruskal-Wallis Test Friedman Test: Repeated measures Ranks Tests

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-22 The null and alternative hypotheses: H 0 : The distributions of two populations are identical H 1 : The two population distributions are not identical The Mann-Whitney U statistic: where n 1 is the sample size from population 1 and n 2 is the sample size from population 2. The null and alternative hypotheses: H 0 : The distributions of two populations are identical H 1 : The two population distributions are not identical The Mann-Whitney U statistic: where n 1 is the sample size from population 1 and n 2 is the sample size from population 2. 14-4 The Mann-Whitney U Test (Comparing Two Populations)

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-23 Cumulative Distribution Function of the Mann-Whitney U Statistic n 2 =6 n 1 =6 u. 40.0130 50.0206 60.0325. Rank ModelTimeRankSum A35 5 A38 8 A4010 A4212 A4111 A36 652 B29 2 B27 1 B30 3 B33 4 B39 9 B37 726 Rank ModelTimeRankSum A35 5 A38 8 A4010 A4212 A4111 A36 652 B29 2 B27 1 B30 3 B33 4 B39 9 B37 726 P(u  5) The Mann-Whitney U Test: Example 14-4

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-24 Example 14-5: Large-Sample Mann-Whitney U Test Score Rank Score Program Rank Sum 85120.020.0 87121.041.0 92127.068.0 98130.098.0 90126.0124.0 88123.0147.0 75117.0164.0 72113.5177.5 6016.5184.0 93128.0212.0 88123.0235.0 89125.0260.0 96129.0289.0 73115.0304.0 6218.5312.5 Score Rank Score Program Rank Sum 85120.020.0 87121.041.0 92127.068.0 98130.098.0 90126.0124.0 88123.0147.0 75117.0164.0 72113.5177.5 6016.5184.0 93128.0212.0 88123.0235.0 89125.0260.0 96129.0289.0 73115.0304.0 6218.5312.5 Score Rank Score Program Rank Sum 65210.0 10.0 5724.0 14.0 74216.0 30.0 4322.0 32.0 3921.0 33.0 88223.0 56.0 6228.5 64.5 69211.0 75.5 70212.0 87.5 72213.5101.0 5925.0106.0 6026.5112.5 80218.0130.5 83219.0149.5 5023.0152.5 Score Rank Score Program Rank Sum 65210.0 10.0 5724.0 14.0 74216.0 30.0 4322.0 32.0 3921.0 33.0 88223.0 56.0 6228.5 64.5 69211.0 75.5 70212.0 87.5 72213.5101.0 5925.0106.0 6026.5112.5 80218.0130.5 83219.0149.5 5023.0152.5 Since the test statistic is z = -3.32, the p-value  0.0005, and H 0 is rejected. Since the test statistic is z = -3.32, the p-value  0.0005, and H 0 is rejected.

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 14-25 Example 14-5: Large-Sample Mann-Whitney U Test – Using the Template Since the test statistic is z = -3.32, the p-value  0.0005, and H 0 is rejected. That is, the LC (Learning Curve) program is more effective.

26 Penutup Pembahasan materi dilanjutkan dengan Materi Pokok 24 (Metode Non Parametrik- 2)

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