Chapter 16 Nonparametric Statistics
Introduction: Distribution-Free Tests Distribution-free tests – statistical tests that don’t rely on assumptions about the probability distribution of the sampled population Nonparametrics – branch of inferential statistics devoted to distribution-free tests Rank statistics (Rank tests) – nonparametric statistics based on the ranks of measurements
Single Population Inferences: The Sign Test The Sign test is used to make inferences about the central tendency of a single population Test is based on the median η Test involves hypothesizing a value for the population median, then testing to see if the distribution of sample values around the hypothesized median value reaches significance
Single Population Inferences: The Sign Test Sign Test for a Population Median η One-Tailed Test Two-Tailed Test H 0 :η 1 = η 0 H 0 : η 1 = η 0 H a : η 1 η 0 H a : η 1 < η 0 {or H a : η 1 > η 0 ] Test Statistic S = Number of sample S = Larger of S 1 and S 2 , where S 1 measurements greater than η 0 [or is the number of measurements less than η 0 and S 2 is the number S = number of measurements less than η 0] of measurements greater than η 0 Observed Significance Level p- value = P(x ≥ S) p- value = 2P(x ≥ S) Where x has a binomial distribution with parameters n and p = .5 Rejection region : Reject H 0 if p -va lue ≤ .05 Conditions required for sign test – sample must be randomly selected from a continuous probability distribution
Single Population Inferences: The Sign Test Large-Sample Sign Test for a Population Median η One-Tailed Test Two-Tailed Test H 0 :η 1 = η 0 H 0 : η 1 = η 0 H a : η 1 η 0 H a : η 1 < η 0 {or H a : η 1 > η 0 ] Test Statistic S .5 .5 n z .5 n Observed Significance Level p- value = P(x ≥ S) p- value = 2P(x ≥ S) Where x has a binomial distribution with parameters n and p = .5 Rejection region : z Rejection region : z z z / 2 Conditions required for sign test – sample must be randomly selected from a continuous probability distribution
Comparing Two Populations: The Wilcoxon Rank Sum Test for Independent Samples The Wilcoxon Rank Sum Test is used when two independent random samples are being used to compare two populations, and the t- test is not appropriate It tests the hypothesis that the probability distributions associated with the two populations are equivalent
Comparing Two Populations: The Wilcoxon Rank Sum Test for Independent Samples Rank Data from both samples from smallest to largest If populations are the same, ranks should be randomly mixed between the samples Percentage Cost of Living Change, as Predicted by Government and University Economists Government Economist (1) University Economist (2) Prediction Rank Prediction Rank 3.1 4 4.4 6 4.8 7 5.8 9 2.3 2 3.9 5 5.6 8 8.7 11 0.0 1 6.3 10 2.9 3 10.5 12 10.8 13 Test statistic is based on the rank sums – the totals of the ranks for each of the samples. T 1 is the sum for sample 1, T 2 is the sum for sample 2
Comparing Two Populations: The Wilcoxon Rank Sum Test for Independent Samples Wilcoxon Rank Sum Test: Independent Samples One-Tailed Test Two-Tailed Test H 0 :D 1 and D 2 are identical H 0 :D 1 and D 2 are identical H a :D 1 is shifted to the right of D 2 H a :D 1 is shifted either to the left {or H a : D 1 is shifted to the left of or to the right of D 2 D 2 ] Test Statistic T 1 , if n 1 <n 2 ; T 2 , if n 2 < n 1 T 1 , if n 1 <n 2 ; T 2 , if n 2 < n 1 (Either rank sum can be used if n 1 (Either rank sum can be used if n 1 = n 2 ) = n 2 ) We will denote this rank sum as T Rejection region : Rejection region: T 1 : T 1 ≥ T U [or T 1 ≤ T L ] T ≤ T L or T ≥ T U T 1 : T 1 ≤ T L [or T 1 ≥ T U ] Where T L and T U are obtained from table Required Conditions: random, independent samples Probability distributions samples drawn from are continuous
Comparing Two Populations: The Wilcoxon Rank Sum Test for Independent Samples Wilcoxon Rank Sum Test for Large Samples(n1 and n2 ≥ 10) One-Tailed Test Two-Tailed Test H 0 :D 1 and D 2 are identical H 0 :D 1 and D 2 are identical H a :D 1 is shifted to the right of D 2 H a :D 1 is shifted either to the left {or H a : D 1 is shifted to the left of or to the right of D 2 D 2 ] Test Statistic ( 1) n n n 1 1 2 T 1 2 T e s t s ta tis tic : z n n ( n n 1) 1 2 1 2 1 2 Rejection region : Rejection region: z>z (or z<-z ) |z|>z /2
Comparing Two Populations: The Wilcoxon Signed Rank Test for the Paired Differences Experiment An alternative test to the paired difference of means procedure Analysis is of the differences between ranks Softness Ratings of Paper Product Difference Judge A B (A-B) Absolute Value of Difference Rank of Absolute Value 1 6 4 2 2 5 2 8 5 3 3 7.5 3 4 5 -1 1 2 4 9 8 1 1 2 5 4 1 3 3 7.5 6 7 9 -2 2 5 7 6 2 4 4 9 8 5 3 2 2 5 9 6 7 -1 1 2 10 8 2 6 6 10 T + = Sum of positive ranks = 46 T - = Sum of negative ranks = 9 Any differences of 0 are eliminated, and n is reduced accordingly
Comparing Two Populations: The Wilcoxon Signed Rank Test for the Paired Differences Experiment Wilcoxon Signed Rank Test for a Paired Difference Experiment Let D1 and D2 represent the probability distributions for populations 1 and 2, respectively One-Tailed Test Two-Tailed Test H 0 :D 1 and D 2 are identical H 0 :D 1 and D 2 are identical Required Conditions H a :D 1 is shifted to the right of D 2 H a :D 1 is shifted either to the left [or H a : D 1 is shifted to the left of or to the right of D 2 Sample of differences D 2 ] is randomly selected Test Statistic T-, the rank sum of the negative T, the smaller of T + or T - Probability distribution distances from which sample is (or T + , the rank sum of the positive distances) drawn is continuous Rejection region : Rejection region: T - : ≤ T 0 [or T + : ≤ T 0 ] T ≤ T 0 Where T 0 is from table
The Kruskal-Wallis H-Test for a Completely Randomized Design An alternative to the completely randomized ANOVA Based on comparison of rank sums Number of Available Beds Hospital 1 Hospital 2 Hospital 3 Beds Rank Beds Rank Beds Rank 6 5 34 25 13 9.5 38 27 28 19 35 26 3 2 42 30 19 15 17 13 13 9.5 4 3 11 8 40 29 29 20 30 21 31 22 0 1 15 11 9 7 7 6 16 12 32 23 33 24 25 17 39 28 18 14 5 4 27 18 24 16 R 1 = 120 R 2 = 210.5 R 3 = 134.5
The Kruskal-Wallis H-Test for a Completely Randomized Design Kruskal-Wallis H-Test for Comparing p Probability Distributions H 0 : The p probability distributions are identical H a : At least two of the p probability distributions differ in location 2 R 1 2 Test statistic : j H 3( n 1) n n 1 n j Where N j = Number of measurements in sample j R j = Rank sum for sample j , where the rank of each measurement is computed according to its relative magnitude in the totality of data for the p samples n = Total Sample Size = n 1 +n 2 + ….+ n p Rejection region : with (p-1) degrees of freedom 2 H Required Conditions: • The p samples are random and independent • 5 or more measurements per sample • Probability distributions samples drawn from are continuous
The Friedman F r -Test for a Randomized Block Design A nonparametric method for the randomized block design Based on comparison of rank sums Reaction Time for Three Drugs Subject Drug A Rank Drug B Rank Drug C Rank 1 1.21 1 1.48 2 1.56 3 2 1.63 1 1.85 2 2.01 3 3 1.42 1 2.06 3 1.70 2 4 2.43 2 1.98 1 2.64 3 5 1.16 1 1.27 2 1.48 3 6 1.94 1 2.44 2 2.81 3 R1 = 7 R2 = 12 R3 = 17
The Friedman F r -Test for a Randomized Block Design H 0 : The probability distributions for the p treatments are identical H a : At least two of the p probability distributions differ in location 1 2 Test statistic : 2 F R 3 b ( p 1) r j b p p 1 Where b = Number of blocks p = number of treatments R j = Rank sum of the j th treatment; where the rank of each measurement is computed relative to its position within its own block Rejection region : with (p-1) degrees of freedom 2 F r Required Conditions: • Random assignment of treatments to units within blocks • Measurements can be ranked within blocks • Probability distributions samples within each block drawn from are continuous
Spearman’s Rank Correlation Coefficient Provides a measure of correlation between ranks Brake Rankings of New Car Models: Less than Perfect Agreement Magazine Difference between Rank 1 and Rank 2 D 2 Car Model 1 2 D 1 4 5 -1 1 2 1 2 -1 1 3 9 10 -1 1 4 5 6 -1 1 5 2 1 1 1 6 10 9 1 1 7 7 7 0 0 8 3 3 0 0 9 6 4 2 4 10 8 8 0 0 2 d 1 0
Spearman’s Rank Correlation Coefficient One-Tailed Test Two-Tailed Test H 0 :p = 0 H 0 : p = 0 H a : p 0 H a :p < 0 {or H a : p> 0] Test Statistic 2 6 d i r 1 s 2 n n ( 1) Where di = ui – vi (difference in ranks of i th observations for samples 1 and 2 Rejection region : Rejection region : r r r r s s , s s , / 2 (or when H a : p> 0) r r s s , Conditions Required: Sample of experimental units is randomly selected Probability distributions of two variables are continuous
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