1 Nonparametric Statistical Techniques Chapter 18

2 The statistical techniques introduced in this chapter deal with ordinal data. We test to determine whether the population locations differ. In testing the locations we will not refer to any parameter, thus the procedure’s name. Introduction

3 When comparing two populations the hypotheses generally are: 17.1 Introduction H 0 : The population locations are the same H 1 : (i) The locations differ, or (ii) Population 1 is located to the right (left) of population 2 The random variable X 1 is generally larger (smaller) than X 2.

4 Wilcoxon Rank Sum Test The problem characteristics of this test are: The problem objective is to compare two populations. The data are either ordinal or interval (but not normal). The samples are independent.

5 Wilcoxon Rank Sum Test – Example Example Based on the two samples shown below, can we infer at 5% significance level that the location of population 1 is to the left of the location of population 2? Sample 1: 22, 23, 20; Sample 2: 18, 27, 26; The hypotheses are: H 0 : The two population locations are the same. H 1 : The location of population 1 is to the left of the location of population 2.

6 Graphical Demonstration Why use the sum of ranks to test locations? Sum of ranks = 37Sum of ranks = If the locations of the two populations are about the same, (the null hypothesis is true) we would expect the ranks to be evenly spread between the samples. In this case the sum of ranks for the two samples will be close to one another. Two hypothetical populations and their corresponding samples are presented, the GREEN population and the PURPLE population. Populations Let us rank the observations of the two samples together

7 Allow the GREEN population to shift to the left of the PURPLE population. Graphical Demonstration Why use the sum of ranks to test locations?

Sum of ranks = 38Sum of ranks = The green sample is expected to shift to the left too. As a result, several observations exchange location. What happens to the sum of ranks? Click. Attention Sum of ranks = 37Sum of ranks = 41 Sum of ranks = 45 Sum of ranks = 33 Graphical Demonstration Why use the sum of ranks to test locations?

9 67 Sum of ranks = 38Sum of ranks = Sum of ranks = 37Sum of ranks = 41 Sum of ranks = 45 Sum of ranks = 33 The “green” sum decreases, and the “purple” sum increases. Changing the relative location of two populations affect the sum of ranks of the two samples combined. Graphical Demonstration Why use the sum of ranks to test locations?

10 Example – continued Test statistic 1. Rank all the six observations (1 for the smallest). Sample Sample Rank Calculate the sum of ranks: 9 2. Calculate the sum of ranks:12 3. Let T = 9 be the test statistic (We arbitrarily define the test statistic as the rank sum of sample 1. Wilcoxon Rank Sum Test – Example

11 Example continued If T is sufficiently small then most of the smaller observations are located in population 1. Reject the null hypothesis. Question: How small is sufficiently small? We need to look at the distribution of T. Wilcoxon Rank Sum Test – Rationale

12 1,2, ,2,41,2,51,2,6 1,3,4 1,3,6 1,3,51,4,5 1,4,61,5,6 2,3,42,3,5 2,3,6 2,4,5 2,4,6 2,5,6 3,4,5 3,4,6 3,5,64,5,6 T T is the rank sum of a sample of size 3. This sample received the ranks 3, 4, 5 If H 0 is true (the two populations have the same location), each ranking is equally likely, and each possible value of T has the same probability = 1/20 This sample received the ranks 1, 2, 3 The distribution of T under H 0 for two samples of size 3

13 The distribution of T under H 0 for two samples of size 3 1,2, ,2,41,2,51,2,6 1,3,4 1,3,6 1,3,51,4,5 1,4,61,5,6 2,3,42,3,5 2,3,6 2,4,5 2,4,6 2,5,6 3,4,5 3,4,6 3,5,64,5,6 T The significance level is 5%, and under H 0 P(T  6) =.05. Thus, the critical value of T is 6.

14 Example - continued Conclusion H 0 is rejected if T  Since T = 9, there is insufficient evidence to conclude that population 1 is located to the left of population 2, at the 5% significance level. Wilcoxon Rank Sum Test – Example

15 Critical values of the Wilcoxon Rank Sum Test  =.025 for two tail test, or  =.05 for one tail test Using the table: For given two samples of sizes n 1 and n 2, P(T T U )=  For a two tail test: P(T 25) =.025 if n 1 =4 and n 2 =4. For a one tail test: P(T 25) =.05 if n 1 =4 and n 2 = A similar table exists for  =.05 (one tail test) and  =.10 (two tail test) T L T U T L T U T L T U T L T U

16 Wilcoxon rank sum test for samples where n > 10 The test statistic is approximately normally distributed with the following parameters: n 1 (n 1 + n 2 + 1) 2 E(T) = Therefore, Z = T - E(T)  T

17 Example (using Wilcoxon rank sum test with ordinal data)Example A pharmaceutical company is planning to introduce a new painkiller. To determine the effectiveness of the drug, 30 people were randomly selected. 15 were given the tested drug (Sample 1). 15 were given aspirin (Sample 2). Each participant was asked to indicate which one of five statements best represented the effectiveness of the drug they took. Wilcoxon rank sum test for samples where n > 10, Example

18 Example – continued Summary of the experiment results. Solution The objective is to compare two populations of ordinal data. The two samples are independent. Wilcoxon rank test is the appropriate technique to apply. Wilcoxon test for samples where n > 10, Example

19 The hypotheses H 0 : The locations of population 1 and 2 are the same H 1 : The location of population 1 is to the right of the location of population 2. Note: A high score selected from among the five possible scores 1, 2, 3, 4, 5, indicates high effectiveness. Wilcoxon rank sum test for samples where n > 10, Example Received the new painkillerReceived Aspirin Solving by hand To reject the null hypothesis, we need to show that z is “large enough”. First we rank the observations, Secondly, we run a z-test, with rejection region of Z > Z .

20 Ranking the raw data There are three observations with an effectiveness score of 1. The original ranks for these observations are 1, 2, and 3. This tie is broken by giving each observation the average rank of 2. Sum of ranks: T 1 =276.5T 2 =188.5 These are the effectiveness scores provided by the experiment participants for each drug. Wilcoxon rank sum test for samples where n > 10, Example

21 To standardize the test statistic we need: E(T) = n 1 (n 1 +n 2 +1)/2= (15)(31)/2=232.5 Wilcoxon rank sum test for samples where n > 10, Example

22 For 5% significance level z= Since z = 1.83 > 1.645, there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. At 5% significance level, the new drugs is perceived as more effective than Aspirin. Wilcoxon rank sum test for samples where n > 10, Example

23 Excel solution Wilcoxon rank sum test for samples where n > 10, Example

24 Wilcoxon rank sum test for non- normal interval data, Example The human resource manager of a large company wanted to compare how long business and non-business graduates worked for the company before quitting. Two samples of 25 business graduates and 20 non-business graduates were randomly selected. The data representing their time with the company were recorded. Retaining WorkersWorkers

25 Can the personnel manager conclude at 5% significance level that a difference in duration of employment exists between business and non- business graduates? Retaining workers - continued Wilcoxon rank sum test for non- normal interval data, Example

26 Solution The problem objective is to compare two populations of interval data. The samples are independent. The non-normality of the two populations is apparent from the sample histograms: Non Business graduates Business graduates Wilcoxon rank sum test for non- normal interval data, Example

27 Solution – continued The Wilcoxon rank test is the correct procedure to run. H 0 : The two population locations are the same H 1 : The location of population 1(business graduates) is different from the location of population 2 (non- business graduates). Wilcoxon rank sum test for non- normal interval data, Example

28 Solution – continued Solving by hand The rejection region is After the ranking process is completed, we have: T = T business graduates = 463. E(T) = n 1 (n 1 +n 2 +1)/2=575;  T =[n 1 n 2 (n 1 +n 2 +1)/12] 1/2 =43.8 Reject the null hypothesis Wilcoxon rank sum test for non-normal interval data, Example

29 Excel solution (Workers.xls)Workers.xls There is a strong evidence to infer that the duration of employment is different for business and non-business graduates Wilcoxon rank sum test for non-normal interval data, Example

30 Required conditions for nonparametric tests A rejection of the null hypothesis when performing a nonparametric test can occur due to: different location different spread (variance) different shape (distribution). Since we are interested in the location, we require that the two distributions are identical, except for location.

31 This test is used when the problem objective is to compare two populations, the data are interval but not normal, the samples are matched pairs. The test statistic and sampling distribution T is based on rank sum of the absolute values of the positive and negative differences When n T U or T<T L (T L and T U tabulated values related to n). When n > 30, T is approximately normally distributed. Use a Z-test. Wilcoxon Signed Rank Sum Test

32 Example Does “flextime” work-schedule help reduce the travel time of workers to work? A random sample of 32 workers was selected, and workers recorded their travel time before and after the program was implemented. The hypotheses test are The two population locations are the same. The two population locations are different. Wilcoxon Signed Rank Sum Test, Example

33 Example Does “flextime” work-schedule help reduce the travel time of workers to work? A random sample of 32 workers was selected, and workers recorded their travel time before and after the program was implemented. The hypotheses are H 0 : The two population locations are the same. H 1 : The two population locations are different. The rejection region: |z| > z  The rejection region: |z| > z  Wilcoxon Signed Rank Sum Test, Example

34 This data were sorted by the absolute value of the differences Ties were broken by assigning the average rank to the tied observations Average rank = (1 + 8)/2 = 4.5

35 T is the rank sum of the positive differences. T = T + = E(T) = n(n+1)/4 = 32(33)/4 = 264  T = [n(n+1)(2n+1)/24].5 = The test statistic is: Z =  TT E(T)T  T  E(T) T = =

36 Excel – solution Wilcoxon Signed Rank Sum Test, Example

37 The rejection region for  =.05 is |z| > z.025 = 1.96 Conclusion: Since |1.94| < 1.96, There is insufficient evidence to infer that the flextime program was effective at 5% significance level. Solution – continued Wilcoxon Signed Rank Sum Test, Example

Kruskal-Wallis Test The problem characteristics for this test are: The problem objective is to compare two or more populations. The data are either ordinal or interval but not normal. The samples are independent. The hypotheses are H 0 : The location of all the k populations are the same. H 1 : At least two population locations differ.

39 Rank the data from 1(smallest) to n (largest). Calculate the rank sums T 1, T 2,…T k for all the k samples. Calculate the statistic H as follows: Kruskal-Wallis Test Statistic

40 Test Rationale and Rejection region If all the populations have the same location (H 0 is true)… The ranks should be evenly distributed among the k samples. The statistic H will be small. Uneven distribution of ranks T 1 =6T 2 =15T 3 =24 H = 7.2 Even distribution of ranks T 1 =14T 2 =15T 3 =16 H =.0888

41 Sampling distribution When the sample sizes  5, H is approximately chi-squared distributed with k-1 degrees of freedom. The rejection region: Since a large value of H justifies the rejection of H 0, we have: Test Rationale and Rejection Region

42 Example 17.5 How do customers rate three shifts with respect to speed of service in a certain restaurant? Three samples of 10 customer response-cards were randomly selected, one sample from each shift. Customer ratings were recorded. The Kruskal-Wallis Test Example

43 Can we conclude that customers perceive the speed of service to be different among the three shifts at 5% significance level? The Kruskal-Wallis Test Example

44 Solution The problem objective is to compare three populations. The data are ordinal. The hypotheses: H 0 : The locations of all three populations are the same. H 1 : At least two population locations differ. The Kruskal-Wallis Test Example

45 Solution - continued Test statistic: T 1 = T 2 = T 3 = n = n 1 + n 2 + n 3 = = 30 Ranking The Kruskal-Wallis Test Example

46 For  =.05,  2 ,k-1 =  2.05,2 = Solution - continued The critical value The Kruskal-Wallis Test Example

47 Conclusion: Since H=2.64 < , do not reject the null hypothesis. There is insufficient evidence to conclude at 5% significance level, that there is a difference in customers’ perception regarding service speed among the three shifts. The Kruskal-Wallis Test Example