Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics S eventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College Chapter Twelve Nonparametric Statistics

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Nonparametric Statistics Require no assumptions about the population distributions from which samples are drawn

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 Advantages of Nonparametric Tests They are quite general. They are not difficult to apply.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 Disadvantages of Nonparametric Tests They tend to waste information. They are less sensitive than other tests. They tend not to reject the null hypothesis as often as they should.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 The Sign Test Compare sample distributions from two populations that are not independent. Data values come from dependent pairs. Often used in “before-and-after” studies.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 A drug used to lower cholesterol has the following effects on sixteen subjects:

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 Find the sign of difference by subtracting the pre- from the post- treatment result

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 To use the sign test Compute the proportion (x) of plus signs to all signs. Ignore the pairs with no differences.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 In our example so far: There are four minus signs. There are two plus signs. One pair had no difference.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 Assumption for Sign Test Samples are sufficiently large to permit normal approximation to binomial distribution.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 The Null Hypothesis If the medication is not effective we would expect the number of plus and minus signs to be about equal. Let p = population proportion of plus signs. H 0 : p = 0.5

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 The Alternate Hypothesis If we wish to determine if the medication decreases cholesterol, the alternate hypothesis would be: H 1 : p < 0.5 Fewer than half of the signs would be plus signs.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Using the Normal Distribution for Sign Test When the total number of plus and minus signs is 12 or more, the sample statistic x (proportion of plus signs) has a distribution that is approximately normal.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Critical Values for the Sign Test

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Sample Test Statistic We are assuming that the population proportion of plus signs p = 0.5 The z value corresponding to the sample test statistic x (the total number of plus signs divided by n) is:

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 Back to Our Example: If we wish to determine if a medication decreases cholesterol, the null and alternate hypotheses are: H 0 : p = 0.5 H 1 : p < 0.5 where p = proportion of plus signs (indicating increases) in the population. Use  = 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 Suppose that in our example : There are nine minus signs. There are four plus signs. Three pairs had no difference.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Critical z Value for a Left- Tailed Test z 0 = – for  = 0.05

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Calculate the Test Statistic

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 Conclusion Our critical value of z is Our test statistic is We note that the z value corresponding to x = does not fall in the critical region. Therefore we cannot reject the null hypothesis at 0.05 level of significance.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 Conclusion We are not able to conclude that the medication lowers cholesterol.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 P Value Approach Since the area falling to the left of is , we would be able to reject the null hypothesis only for   We would,therefore, not be able to reject the hypothesis for  = 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 Rank-Sum Test Independent samples are drawn from two populations. Test the difference between sample means.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 The Rank-Sum Test The rank-sum test is also called the Mann-Whitney test. It can be used when the assumptions about normal populations are not satisfied. It can be used when the assumptions about equal population variances are not satisfied.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 We wish to examine the difference in sample mean times for two groups to complete a task.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Arrange the samples jointly in increasing order and note the ranks.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 Sum the ranks for the group with the smaller sample size. Group B has the smaller sample size. Sum of ranks for group B = = 73 This sum of the ranks is called R. R = 73

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 What if there were ties in any ranks? Each of the tied observations would be given the mean of the ranks that they occupy.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 Assumptions and Symbols for Rank-Sum Test n 1 = smaller sample size n 2 = larger sample size When n 1 and n 2 are both eight or more, the random variable R (the sum of the ranks from the smaller sample) is approximately normal.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 Mean and Standard Deviation of R

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 Mean and Standard Deviation of R for our example

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 Hypothesis Test for R We are testing to see if the difference in the sample means is significant.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 Null Hypothesis H 0 : The distributions are the same.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 Alternate Hypothesis H 1 : The distributions are different.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 The Test Statistic Use  = 0.05.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 Critical Values for the Rank- Sum Test (Each Sample Size  8)

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 Critical z For  = 0.05, the critical values of z are  1.96

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 Calculating z for our samples R = 73  R = 72  R = 10.39

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 Conclusion Since z = 0.10 does not fall within the critical region, we are not able to reject the null hypothesis that the distributions are the same. We conclude that the time to complete the task for the two groups are essentially the same.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40 Spearman Rank Correlation A method used to study correlation of data in ranked form

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 41 The Spearman test checks for the existence of a monotone relationship between variables As x increases, y also increases. or As x increases, y decreases.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 42 Monotone Increasing y x

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 43 Monotone Increasing y x

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 44 Monotone Decreasing y x

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 45 Not Monotone y x

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 46 Assumptions We have n randomly obtained ordered pairs (x, y). The x and y values are from ranked variables. There are no ties in the ranks.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 47 Spearman Rank Correlation Coefficient

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 48 Properties of r S -1  r S  1 If r S = -1, the relationship between x and y is perfectly monotone decreasing. If r S = + 1, the relationship between x and y is perfectly monotone increasing. If r S = 0, there is no monotone relationship between x and y.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 49 Properties of r S If r S is close to 1 or -1 there is a strong tendency for a monotone relationship between x and y. Values of r S close to zero indicate a weak or nonexistent monotone relationship. Table 9 in Appendix II gives critical values for some tests of r S.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 50 The variables x and y are not necessarily normally distributed. The relationship between x and y is not necessarily linear. r S is our sample estimate for  S, the population Spearman rank correlation coefficient. Properties of r S

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 51 Null Hypothesis There is no monotone relationship between the variables H 0 :  S = 0

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 52 Alternate Hypotheses There is a monotone relationship between the variables x and y.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 53 Alternate Hypothesis for Left- Tailed Test There is a monotone-decreasing relationship between x and y. H 1 :  S < 0

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 54 Alternate Hypothesis for Right- Tailed Test There is a monotone-increasing relationship between x and y. H 1 :  S > 0

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 55 Alternate Hypothesis for Two- Tailed Test There is a monotone relationship(either increasing or decreasing) between x and y. H 1 :  S  0

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 56 Eight brands of cookies are ranked according to price and to taste.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 57 Suppose we wish to determine if there is a monotone- increasing relationship between price and taste.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 58 Null Hypothesis There is no monotone relationship between price and taste. H 0 :  S = 0

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 59 Alternate Hypothesis There is a monotone-increasing relationship between price and taste. H 0 :  S > 0

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 60 Critical Value for  S Use Table 9 in Appendix II. Use  = For a right-tailed test with n = 8, the critical value for  S is

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 61 Calculate d = x - y and d 2.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 62 Calculate the Spearman Rank Correlation Coefficient

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 63 Conclusion Our calculated value of  S is Using Table 9 with  = 0.05 and n = 8, the critical value for  S is Our value of  S does not fall within the critical region. We cannot reject the null hypothesis that there is no monotone relationship.

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 64 Conclusion We conclude that there is not significant evidence of a monotone relationship between the price of cookies and their taste.