Nonparametric statistics ppt @ bec doms
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Nonparametric statistics ppt @ bec doms

Nonparametric statistics ppt @ bec doms

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Nonparametric statistics ppt @ bec doms Nonparametric statistics ppt @ bec doms Presentation Transcript

  • Nonparametric Statistics
  • Chapter Goals
    • After completing this chapter, you should be able to:
    • Recognize when and how to use the Wilcoxon signed rank test for a population median
    • Recognize the situations for which the Wilcoxon signed rank test applies and be able to use it for decision-making
    • Know when and how to perform a Mann-Whitney U-test
    • Perform nonparametric analysis of variance using the Kruskal-Wallis one-way ANOVA
  • Nonparametric Statistics
    • Nonparametric Statistics
      • Fewer restrictive assumptions about data levels and underlying probability distributions
        • Population distributions may be skewed
        • The level of data measurement may only be ordinal or nominal
  • Wilcoxon Signed Rank Test
    • Used to test a hypothesis about one population median
      • the median is the midpoint of the distribution: 50% below, 50% above
    • A hypothesized median is rejected if sample results vary too much from expectations
      • no highly restrictive assumptions about the shape of the population distribution are needed
  • The W Test Statistic
    • Performing the Wilcoxon Signed Rank Test
    • Calculate the test statistic W using these steps:
    • Step 1: collect sample data
    • Step 2: compute d i = difference between each value and the hypothesized median
    • Step 3: convert d i values to absolute differences
  • The W Test Statistic
    • Performing the Wilcoxon Signed Rank Test
    • Step 4: determine the ranks for each d i value
      • eliminate zero d i values
      • Lowest d i value = 1
      • For ties, assign each the average rank of the tied observations
    (continued)
  • The W Test Statistic
    • Performing the Wilcoxon Signed Rank Test
    • Step 5: Create R+ and R- columns
      • for data values greater than the hypothesized median, put the rank in an R+ column
      • for data values less than the hypothesized median, put the rank in an R- column
    (continued)
  • The W Test Statistic
    • Performing the Wilcoxon Signed Rank Test
    • Step 6: the test statistic W is the sum of the ranks in the R+ column
    • Test the hypothesis by comparing the calculated W to the critical value from the table in appendix P
      • Note that n = the number of non-zero d i values
    (continued)
  • Example
    • The median class size is claimed to be 40
    • Sample data for 8 classes is randomly obtained
    • Compare each value to the hypothesized median to find difference
    Class size = x i Difference d i = x i – 40 | d i | 23 45 34 78 34 66 61 95 -17 5 -6 38 -6 26 21 55 17 5 6 38 6 26 21 55
  • Example
    • Rank the absolute differences:
    tied (continued) | d i | Rank 5 6 6 17 21 26 38 55 1 2.5 2.5 4 5 6 7 8
  • Example
    • Put ranks in R+ and R- columns
    • and find sums:
    (continued) These three are below the claimed median, the others are above Class size = x i Difference d i = x i – 40 | d i | Rank R+ R- 23 45 34 78 34 66 61 95 -17 5 -6 38 -6 26 21 55 17 5 6 38 6 26 21 55 4 1 2.5 7 2.5 6 5 8 1 7 6 5 8 4 2.5 2.5  = 27  = 9
  • Completing the Test
    • H 0 : Median = 40
    • H A : Median ≠ 40
    Test at the  = .05 level: This is a two-tailed test and n = 8, so find W L and W U in appendix P: W L = 3 and W U = 33 The calculated test statistic is W =  R+ = 27
  • Completing the Test
    • H 0 : Median = 40
    • H A : Median ≠ 40
    W L = 3 and W U = 33 W L < W < W U so do not reject H 0 (there is not sufficient evidence to conclude that the median class size is different than 40) (continued) W L = 3 do not reject H 0 reject H 0 W =  R+ = 27 W U = 33 reject H 0
  • If the Sample Size is Large
    • The W test statistic approaches a normal distribution as n increases
    • For n > 20, W can be approximated by
    where W = sum of the R+ ranks d = number of non-zero d i values
  • Nonparametric Tests for Two Population Centers
    • Nonparametric
    • Tests for Two
    • Population Centers
    Wilcoxon Matched-Pairs Signed Rank Test Mann-Whitney U-test Large Samples Small Samples Large Samples Small Samples
  • Mann-Whitney U-Test Used to compare two samples from two populations Assumptions: The two samples are independent and random The value measured is a continuous variable The measurement scale used is at least ordinal If they differ, the distributions of the two populations will differ only with respect to the central location
    • Consider two samples
      • combine into a singe list, but keep track of which sample each value came from
      • rank the values in the combined list from low to high
        • For ties, assign each the average rank of the tied values
      • separate back into two samples, each value keeping its assigned ranking
      • sum the rankings for each sample
    Mann-Whitney U-Test (continued)
    • If the sum of rankings from one sample differs enough from the sum of rankings from the other sample, we conclude there is a difference in the population medians
    Mann-Whitney U-Test (continued)
  • Mann-Whitney U-Test (continued) Mann-Whitney U-Statistics where: n 1 and n 2 are the two sample sizes  R 1 and  R 2 = sum of ranks for samples 1 and 2
  • Mann-Whitney U-Test (continued) Claim: Median class size for Math is larger than the median class size for English A random sample of 9 Math and 9 English classes is selected (samples do not have to be of equal size) Rank the combined values and then split them back into the separate samples
    • Suppose the results are:
    (continued) Mann-Whitney U-Test Class size (Math, M) Class size (English, E) 23 45 34 78 34 66 62 95 81 30 47 18 34 44 61 54 28 40
  • Mann-Whitney U-Test Ranking for combined samples tied (continued) Size Rank 18 1 23 2 28 3 30 4 34 6 34 6 34 6 40 8 44 9 Size Rank 45 10 47 11 54 12 61 13 62 14 66 15 78 16 81 17 95 18
    • Split back into the original samples:
    Mann-Whitney U-Test (continued) Class size (Math, M) Rank Class size (English, E) Rank 23 45 34 78 34 66 62 95 81 2 10 6 16 6 15 14 18 17 30 47 18 34 44 61 54 28 40 4 11 1 6 9 13 12 3 8  = 104  = 67
  • Mann-Whitney U-Test H 0 : Median M ≤ Median E H A : Median M > Median E Claim: Median class size for Math is larger than the median class size for English Note: U 1 + U 2 = n 1 n 2 (continued) Math: English:
    • The Mann-Whitney U tables in Appendices L and M give the lower tail of the U-distribution
    • For one-tailed tests like this one, check the alternative hypothesis to see if U 1 or U 2 should be used as the test statistic
    • Since the alternative hypothesis indicates that population 1 (Math) has a higher median, use U 1 as the test statistic
    Mann-Whitney U-Test (continued)
    • Use U 1 as the test statistic: U = 22
    • Compare U = 22 to the critical value U  from the appropriate table
      • For sample sizes less than 9, use Appendix L
      • For samples sizes from 9 to 20, use Appendix M
    • If U < U  , reject H 0
    Mann-Whitney U-Test (continued)
    • Use U 1 as the test statistic: U = 19
    • U  from Appendix M for  = .05, n 1 = 9 and n 2 = 9 is U  = 7
    Mann-Whitney U-Test Since U  U  , do not reject H 0 (continued) U  = 7 U = 19 do not reject H 0 reject H 0
  • Mann-Whitney U-Test for Large Samples
    • The table in Appendix M includes U  values only for sample sizes between 9 and 20
    • The U statistic approaches a normal distribution as sample sizes increase
    • If samples are larger than 20, a normal approximation can be used
  • Mann-Whitney U-Test for Large Samples
    • The mean and standard deviation for Mann-Whitney U Test Statistic:
    (continued) Where n 1 and n 2 are sample sizes from populations 1 and 2
  • Mann-Whitney U-Test for Large Samples
    • Normal approximation for Mann-Whitney U Test Statistic:
    (continued)
  • Large Sample Example
    • We wish to test
    • Suppose two samples are obtained:
    • n 1 = 40 , n 2 = 50
    • When rankings are completed, the sum of ranks for sample 1 is  R 1 = 1475
    • When rankings are completed, the sum of ranks for sample 2 is  R 2 = 2620
    H 0 : Median 1  Median 2 H A : Median 1 < Median 2
    • U statistic is found to be U = 655
    Large Sample Example Since the alternative hypothesis indicates that population 2 has a higher median, use U 2 as the test statistic Compute the U statistics: (continued)
  • Large Sample Example Since z = -2.80 < -1.645, we reject H 0 Reject H 0  = .05 Do not reject H 0 0 (continued)
  • Wilcoxon Matched-Pairs Signed Rank Test
    • The Mann-Whitney U-Test is used when samples from two populations are independent
    • If samples are paired, they are not independent
    • Use Wilcoxon Matched-Pairs Signed Rank Test with paired samples
  • The Wilcoxon T Test Statistic
    • Performing the Small-Sample Wilcoxon Matched Pairs Test (for n < 25)
    • Calculate the test statistic T using these steps:
    • Step 1: collect sample data
    • Step 2: compute d i = difference between the sample 1 value and its paired sample 2 value
    • Step 3: rank the differences, and give each rank the same sign as the sign of the difference value
  • The Wilcoxon T Test Statistic
    • Performing the Small-Sample Wilcoxon Matched Pairs Test (for n < 25)
    • Step 4: The test statistic is the sum of the absolute values of the ranks for the group with the smaller expected sum
      • Look at the alternative hypothesis to determine the group with the smaller expected sum
      • For two tailed tests, just choose the smaller sum
    (continued)
  • Small Sample Example
    • Paired samples, n = 9:
    Claim: Median value is smaller after than before Value (before) Value (after) 38 45 34 58 30 46 42 55 41 30 47 18 34 34 31 24 38 40
  • Small Sample Example
    • Paired samples, n = 9:
    (continued) Value (before) Value (after) Difference d Rank of d Ranks with smaller expected sum 36 45 34 58 30 46 42 55 41 30 47 18 54 38 31 24 62 40 6 -2 16 4 -8 15 18 -7 1 4 -2 8 3 -6 7 9 -5 1 2 6 5  = T = 13
    • The calculated T value is T = 13
    • Complete the test by comparing the calculated T value to the critical T-value from Appendix N
    • For n = 9 and  = .025 for a one-tailed test,
    • T  = 6
    Small Sample Example Since T  T  , do not reject H 0 T  = 6 T = 13 do not reject H 0 reject H 0 (continued)
  • Wilcoxon Matched Pairs Test for Large Samples
    • The table in Appendix N includes T  values only for sample sizes from 6 to 25
    • The T statistic approaches a normal distribution as sample size increases
    • If the number of paired values is larger than 25, a normal approximation can be used
    • The mean and standard deviation for Wilcoxon T :
    Wilcoxon Matched Pairs Test for Large Samples (continued) where n is the number of paired values
  • Mann-Whitney U-Test for Large Samples
    • Normal approximation for the Wilcoxon T Test Statistic:
    (continued)
    • Tests the equality of more than 2 population medians
    • Assumptions:
      • variables have a continuous distribution.
      • the data are at least ordinal.
      • samples are independent.
      • samples come from populations whose only possible difference is that at least one may have a different central location than the others.
    Kruskal-Wallis One-Way ANOVA
  • Kruskal-Wallis Test Procedure
    • Obtain relative rankings for each value
      • In event of tie, each of the tied values gets the average rank
    • Sum the rankings for data from each of the k groups
      • Compute the H test statistic
  • Kruskal-Wallis Test Procedure
    • The Kruskal-Wallis H test statistic:
    • (with k – 1 degrees of freedom)
    where: N = Sum of sample sizes in all samples k = Number of samples R i = Sum of ranks in the i th sample n i = Size of the i th sample (continued)
    • Complete the test by comparing the calculated H value to a critical  2 value from the chi-square distribution with k – 1 degrees of freedom
    • (The chi-square distribution is Appendix G)
    • Decision rule
      • Reject H 0 if test statistic H >  2 
      • Otherwise do not reject H 0
    Kruskal-Wallis Test Procedure (continued)
    • Do different departments have different class sizes?
    Kruskal-Wallis Example Class size (Math, M) Class size (English, E) Class size (History, H) 23 45 54 78 66 55 60 72 45 70 30 40 18 34 44
    • Do different departments have different class sizes?
    Kruskal-Wallis Example Class size (Math, M) Ranking Class size (English, E) Ranking Class size (History, H) Ranking 23 41 54 78 66 2 6 9 15 12 55 60 72 45 70 10 11 14 8 13 30 40 18 34 44 3 5 1 4 7  = 44  = 56  = 20
    • The H statistic is
    Kruskal-Wallis Example (continued)
    • Since H = 6.72 <
    • do not reject H 0
    Kruskal-Wallis Example (continued) Compare H = 6.72 to the critical value from the chi-square distribution for 5 – 1 = 4 degrees of freedom and  = .05: There is not sufficient evidence to reject that the population medians are all equal
  • Kruskal-Wallis Correction
    • If tied rankings occur, give each observation the mean rank for which it is tied
    • The H statistic is influenced by ties, and should be corrected
    • Correction for tied rankings:
    where: g = Number of different groups of ties t i = Number of tied observations in the i th tied group of scores N = Total number of observations
  • H Statistic Corrected for Tied Rankings
    • Corrected H statistic: