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Nonparametric Statistics Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Nonparametric Statistics • Many of the nonparametric statistical tests answer the same sorts of questions as the parametric tests. With nonparametric tests the assumptions can be relaxed considerably. Consequently, nonparametric methods are used for situations that violate the assumptions of parametric procedures. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Test for Randomness; The Runs Test Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Test for Randomness; The Runs Test Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing n! number of arrangement A n1 !n2 ! The Runs Test (Small Samples) • Sequence 1: HHHHH TTTTT Run 1 Run 2 • Sequence 3: T HHH T H T H TT Run 1 Run 7 • R = number of Runs Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing The Runs Test (Small Samples) Ho: The sequence was generated in a random manner Ha: The sequence was not generated in a random manner P (R k1 ) .025 2 P (R k2 ) .025 2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Test for Randomness; The Runs Test Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Runs Test (Large Samples) Ho: The sequence was generated in a random manner Ha: The sequence was not generated in a random manner R R Z R R 2n1 n2 1 n1 n2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing R 2n1 n2 (2n1 n 2 n1 n2 ) ( n1 n2 ) 2 (n1 n 2 1) Reject Ho if |Z| > Z/2 Nonparametric Test Central Tendency: Two Populations Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Mann-Whitney U (Independent Samples) Ho: The two populations have identical probability distributions Ha: The two populations differ in location T1 = sum of the ranks of the observations from the first sample T2 = sum of the ranks of the observations from the second sample Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Mann-Whitney U Small Samples n1 (n1 1) U1 n1 n 2 T1 2 n2 (n2 1) U 2 n1 n2 T2 2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Mann-Whitney U Small Samples Procedure: 1. Assume that n1 < n2 (reverse the samples if necessary) 2. Determine U1 and U2 3. Use the value from Table A.10 to test Ho vs. Ha Two-Sided Test Ha: the two populations differ in location Reject Ho if Table A.10 value for U is < /2, where U is the minimum of U1 and U2. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Mann-Whitney U Small Samples Procedure: 1. Assume that n1 < n2 (reverse the samples if necessary) 2. Determine U1 and U2 3. Use the value from Table A.10 to test Ho vs. Ha One-Sided Test Ha: Population 1 is shifted to the right of population2 Reject Ho if Table A.10 value for U is less than where U = U1. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Ha: Population 1 is shifted to the left of population 2 Reject Ho if Table A.10 value for U is less than Where U = U2 Mann-Whitney U Large Samples U 2 U 2 Z U2 U2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing n1 n 2 2 U2 n1n2 (n1 n2 1) 12 Mann-Whitney U Large Samples Ho: The two populations have identical probability distribution Determine U2 Two-Sided Test Ha: The two populations differ in location Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Reject Ho if |Z| > Z /2 Mann-Whitney U Large Samples Ho: The two populations have identical probability distribution Determine U2 One-Sided Test Ha: Population 1 is shifted to the right of population 2 Ha: Population 1 is shifted to the left of population 2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur Reject Ho if Z > Z (c)2000 South-Western College Publishing Reject Ho if Z < - Z Wilcoxon Signed Rank Test for Paired Samples When small samples from suspected nonnormal populations are used, a nonparametric technique is required. The Wilcoxon test is used for such situations Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Wilcoxon Signed Rank Test for Paired Samples • Determine the difference for each sample. • Arrange the absolute value of these differences in order, assigning a rank to each. • T+ = sum of ranks having a positive value and T- = sum of ranks having a negative value Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Wilcoxon Signed Rank Test for Paired Samples (small n) Ho: The population differences are centered at 0 Two-Sided Test Ha: the population differences are not centered at 0 Using the two-sided value from Table A.11, Reject Ho if T table value, where T = the minimum of T+ and T-. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Wilcoxon Signed Rank Test for Paired Samples (small n) Ho: The population differences are centered at 0 One-Sided Test Ha:the population differences are centered at a value >0 Using the one-sided value from Table A.11, Reject Ho if T- table value. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Ha:the population differences are centered at a value < 0 Using the one-sided value from Table A.11, Reject Ho if T- table value. Wilcoxon Signed Rank Test for Paired Samples (Large n) T T Z T T Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing n(n 1) 4 T n(n 1)(2n 1) 24 Wilcoxon Signed Rank Test for Paired Samples (Large n) Ho: The population differences are centered at 0 Two-Sided Test Ha: the population differences are not centered at 0 Reject Ho if |Z| > Z/2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Wilcoxon Signed Rank Test for Paired Samples (Large n) Ho: The population differences are centered at 0 One-Sided Test Ha:the population differences are centered at a value >0 Reject Ho if Z > Z Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Ha:the population differences are centered at a value < 0 Reject Ho if Z < -Z Wilcoxon Signed Rank Test for Paired Samples Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Kruskal-Wallis Test The nonparametric counterpart to the oneway ANOVA test. Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Kruskal-Wallis Test Ho: the k populations have identical probability distributions Ha: at least two of the populations differ in location k T2 12 i KW 3(n 1) n(n 1) i1 n i Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Reject Ho if KW > 2.df p-value for KW Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Kruskal-Wallis Test Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing The Friedman Test The nonparametric counterpart to the randomized block ANOVA test. k 12 FR Ti 2 3b( k 1) bk( k 1) i 1 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Reject Ho if FR > 2.df The Friedman Test Ho: the k populations have identical probability distributions Ha: at least two of the populations differ in location k 12 FR Ti 2 3b( k 1) bk( k 1) i 1 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Spearman’s Rank Correlation Spearman’s is the nonparametric counterpart to the Pearson Correlation rs R( x)R( y) [ R( x)][ R( y)]/ n R2 ( x) [ R( x)]2 / n R( y)] [ R( y)]2 / n Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Spearman’s Rank Correlation rs R( x)R( y) [ R( x)][ R( y)]/ n R2 ( x) [ R( x)]2 / n R( y)] [ R( y)]2 / n If there are no ties then a shortcut equation may be used.0 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing 6d 2 r s 1 n(n2 1) Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Spearman’s Rank Correlation Ho: No association between X and Y exists Ha: An association between X and Y exists Two-Sided Test Determine rs Use Table A.12 for /2 Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Reject Ho if |rs| > table value Spearman’s Rank Correlation Ho: No association between X and Y exists One-Sided Test Ha: A positive association between X and Y exists Determine rs Use Table A.12 for Reject Ho if rs > table value Introduction to Business Statistics, 5e Kvanli/Guynes/Pavur (c)2000 South-Western College Publishing Ha: A negative association between X and Y exists Determine rs Use Table A.12 for Reject Ho if rs < table value
