Chapter 12

CHAPTER 12 NONPARAMETRIC STATISTICS   

The hypothesis tests discussed so far in this text are called parametric tests. This chapter discusses a few nonparametric tests. These tests do not require the same kinds of assumptions, and hence, they are called distribution-free tests.

12.1 The Sign Test  

The sign test is used to male hypothesis tests about preferences, a single median, and the median of paired differences for two dependent populations We use only plus and minus signs perform these tests.

12.1.1 Single Sample Sign Test Assumptions a) The sample available for analysis is a random variable from population with unknown median, M. b) The variable of interest is measured on at least an ordinal scale. c) The variable of interest is continuous. The n sample values are designated by X 1 , X 2 ,, X n . Hypotheses H 0 : M  M 0 , H1 : M  M 0 (two sided) or H1 : M  M 0 (one sided) or H1 : M  M 0 (one sided) Test Statistic  Record the sign of the difference obtained by subtracting the hypothesis median M 0 from each of the sample value: X i  M 0 , i = 1, 2,  , n. 

If the data, X i value above the M 0 value, it is assigned a plus sign.

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If the data, X i value below the M 0 value, it is assigned a minus sign.

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If X i is exactly the same as M 0 , it is assigned a 0 and we discard the observations

 

from the sample (reduce n). The test value is the smaller number of plus or minus signs. If the test value is less than or equal to the critical value obtained from the Table of Critical Values for the Sign test, the H 0 should be rejected.

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When the sample size n > 25 , normal approximation can be used to find the test value by computing k  0.5  n / 2 z n /2 where k = smaller number of plus or minus signs n = sample size.

Example 12.1: A past study claims that adults in Malaysia spend a median of 18 hours a week on leisure activities. A researcher took a sample of 10 adults and asked them how many hour they spend per week on leisure activities. She obtained the following data: 14 25 22 38 16 26 19 23 41 33 Using α = 0.05 can you conclude that the median amount of time spent per week on leisure activities by all adults is more than 18 hours?

12.1.2 Test About the median Median Difference Between Paired Data 

We can use the sign test to perform a test of hypothesis about the difference between the mesians of two dependent populations using the data obtained from paired samples.

Hypotheses H 0 : M D  0 , H1 : M D  0 (two sided) or H1 : M D  0 (one sided) or H1 : M D  0 (one sided) Test Statistic Same as the single sample sign test. Example 12.2: A researcher wanted to find the effects of a special diet on systolic blood pressure in adults. She selected a sample of 12 adults and out them on this dietary plan for three months. The following table gives the systolic blood pressyre of each adults before and after the completion of the plan. Before After

210 196

185 192

215 204

198 193

187 181

225 233

234 208

217 211

212 190

191 186

226 218

238 236

Using the 2.5% significance level, can we conclude that the dietary plan reduces the median systolic blood pressure of adults?

12.2 Wilcoxon Signed-Rank Test  

Similar to the sign test. Test for median, M.

12.2.1 Single Sample Test Assumptions a) The sample has been randomly selected from the population it represents. b) The original scores obtained for each of the subjects/objects are in the format of interval/ratio data. c) The underlying population distribution is symmetrical. Hypotheses H 0 : M  M 0 , H1 : M  M 0 (two sided) or H1 : M  M 0 (one sided) or H1 : M  M 0 (one sided) Test Statistic  Record the sign of the difference obtained by subtracting the hypothesis median M 0 from each of the sample value: X i  M 0 , i = 1, 2,  , n. 

If the data, X i value above the M 0 value, it is assigned a plus sign.



If the data, X i value below the M 0 value, it is assigned a minus sign.

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If X i is exactly the same as M 0 , it is assigned a 0 and we discard the observations

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from the sample (reduce n). Without consider the sign of the differences, we rank the differnces X i  M 0 , i = 1, 2,

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 , n in an ascending order. Now give the ranks of the digns for their corresponding differences.

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Let w  be the sum of the positive ranks, w  be the absolute value of the sum of the negative ranks.

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The smaller value of w  or w  is the Wilcoxon test value. If the test value is less than or equal to the critical value obtained from the Table of Critical Values for Wilcoxon’s Signed-Ranks test, H 0 should be rejected.

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Generally when the samples sizes larger than those documented in the Table of Critical Values for Wilcoxon’s Signed-Ranked test, normal approximation can be used to find the test value by computing. W  W Z

W

where W 

n(n  1) and  W  4

n(n  1)(2n  1) 24

Example 12.3: A physician states that the median number of times he sees each of his patients during the year is five. In order to evaluate the validity of this statement, he randomly selects ten of his patients and determines the number of office visits each of them made during the past year. He obtains the following values for the ten patients in his sample: 9, 10, 8, 4, 8, 3, 0, 10, 15, 9. Do the data support his contention that the median number of times he sees a patient is five?

12.2.2 Paired Sample Test (two dependent samples)  The Wilcoxon Signed-rank test can also be employed in a hypothesis testing situation involving a design with two dependent samples.  Recall when two dependent samples taken from normally distributed, the t-test is used.  But when the condition of normality cannot be met, the nonparametric Wilcoxon Signed-rank test can be use. Hypotheses H 0 : M D  0 , H1 : M D  0 (two sided) or H1 : M D  0 (one sided) or H1 : M D  0 (one sided) Test Statistic Same as the single sample Wilcoxon Signed-Rank test. Example 12.4: Eight couples are given a questionnaire designed to measure marital compatibility. After completing a workshop, they are given a second questionnaire to see whether there is a change in their attitudes toward each other. The data are shown below. At α = 0.10 is there any difference in the scores of the couples? Before After

43 48

52 59

37 36

29 29

51 60

62 68

57 59

61 72

12.3 Mann-Whitney Test  

An alternative test for testing the hypothesis about the difference between the means of two independent populations. Recall when two independent samples taken from two normally distributed populations, the z-test or the t-test is used.

Assumptions a) The two independent random samples are independent within each sample as well as between samples. b) The random variable are ordinal or numerical. Hypotheses H 0 : 1   2 , H1 : 1   2 (two sided) or H1 : 1   2 (one sided) or H1 : 1   2 (one sided) Test Statistic  Let X 11, X 12 ,, X 1n1 and X 21, X 22 ,, X 2 n2 be two independent random samples of size n1  n2 from two populations X 1 and X 2 .

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Arrange all n1  n2 observations in ascending order (from the lowest to the highest) of magnitude and assign rank to them. If two or more observations are tied (identical), then use the mean of the ranks that would have been assigned if the observation differed. Calculate the sum of ranks for sample 1 and sample 2 ( R1 and R2 ). Now calculate the Mann-Whitney test statistic using the two formulas for sample 1 and sample 2 n n  1  R2  U1  n1n2  2 1 2 n n  1  R1 .  U 2  n1n2  1 2 2 The test value is the smaller of U 1 and U 2 .

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If the test value is less than or equal to the critical value obtained from the Table of Critical Values for Mann-Whitney U Statistic, the H 0 should be rejected.

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If the sample size employed in a study is relatively large, n1 > 20, n2 > 20 the normal distribution can be employed to approximate the Mann-Whitney U statistic. Equation below provides the normal approximation of the Mann-Whitney U test statistic nn U 1 2 2 z n1n2 n1  n2  1 12 where U = smaller value between U 1 and U 2 .

Example 12.5: An electrical engineer must design a circuit to deliver the maximum amount of current to a display tube to achieve sufficient image brightness. Within his allowable design constraint, he has developed two candidate circuits and tests prototypes of each. The resulting data (in microamperes) are as follows: Circuit 1 Circuit 2

251 250

255 253

258 249

257 256

250 259

251 252

254 260

250 251

248

Use the Mann-Whitney test to test H 0 : 1   2 against alternative H1 : 1   2 . Use α = 0.05.

12.4 THE RUNS TEST    

A nonparametric test to determine randomness of data. A run is a sequence of one or more consecutive occurances of the same outcome in a sequence of occeirences in which there sre onlt two outcomes. The number of runs in a sequence is denored by R. The value of R obtained for a sequence of outcomes for a sample gives the observed value of the test statistic for the runs test for randomness.

Hypotheses H 0 : Tenants with and without children are randomly mixed among the 10 units H 1 : These tenats are not randomly mixed

Test Statistics  The test value is the number of runs R.  If the test value is less than or equal to the left-hand critical value and bigger than or equal to the right-hand critical value obtained from the Table of Critical Values for Total Number of Runs, the H 0 should be rejected. Example 12.6: A college admissions office is interested in knowing whether applications for admission arrive randomly with respect to gender. The gender of 25 consecutively arriving applications were found to arrive in the following order (here M denoted a male applicant and F a female applicant). M F M M F F F M F M M M F F F F M M M F F M F M M Can you conclude that the applicantions for admission arrive randomly with respect to gender? User α = 0.05.