Non-Parametric Two-Sample Analysis: The Mann-Whitney U Test

Transcription

1 Non-Parametric Two-Sample Analysis: The Mann-Whitney U Test

2 When samples do not meet the assumption of normality parametric tests should not be used. To overcome this problem, non-parametric tests can be used. These tests are distribution-free (do not assume normality. They are fairly robust and nearly as powerful as parametric tests. They often use RANKS rather than observed values.

3 Earthquake Depth

4 Chilean earthquakes Kolmogorov-Smirnov(a) Tests of Normality (May) Shapiro-Wilk Statistic df Sig. Statistic df Sig. Mag Tests of Normality (June) Kolmogorov-Smirnov(a) Shapiro-Wilk Statistic df Sig. Statistic df Sig. Mag

5 Magnitude Equal variances assumed Equal variances not assumed Levene's Test for Equality of Variances F Sig. Independent Samples Test t df Sig. (2-tailed) t-test for Equality of Means Me an Difference 95% Confidence Interval of the Std. Error Difference Difference Lower Upper Using a t test gives the result that the magnitude of the earthquakes between May and June were significantly different.

6 Magnitude Mo nth 5 6 Total Ranks N Mean Rank Sum of Ranks Test Statistics b Mann-Whitney U Wilcoxon W Z As ymp. Sig. (2-tailed) Exact Sig. [2*(1-tailed Sig.)] a. Not corrected for ties. Magnitude b. Groupi ng Variable: Month a Using a non-parametric test gives the result that the magnitude of the earthquakes between May and June was not significantly different.

7 When the distribution of the data sets deviate substantially from normal, it is better to use non-parametric (distribution free) tests. There are no assumptions made concerning the sample distributions. Tied ranks are assigned the average rank of the tied observations. The Mann-Whitney U test is approximately 95% as powerful as the t test. If the data are severely non-normal, the Mann-Whitney U test is substantially more powerful than the t test.

8 The Mann-Whitney U test (2-tailed) U = U ' = n n 1 n n n1 ( n1 + 1) + 2 U R 1 where R 1 is the sum of the ranks for group1 Compare the critical U value to either U or U, whichever is larger.

9 The sample space The theoretical sum of all ranks for group 1 U U = ' = n n 1 n n n 1 U ( n1 + 1) 2 R 1 The actual sum of all ranks for group 1 This equation is essentially comparing the theoretical sum of the ranks from group 1 to the actual sum of the ranks for group 1 while taking into account the sample space. If the group samples get smaller the test gets more conservative.

10 Observations are first sorted. Tied ranks are dealt with by assigning the average rank to the tied observations: Obs Value Rank (with Ties) Rank (tied) (tied) (tied (tied) (tied) 7.5 ( ) / 3 = 4 (7 + 8) / 2 = 7.5

11 The U test uses the rank of the pooled observations. For a 2- tailed test, ranks can be from highest to lowest or lowest to highest. Earthquake Magnitudes in Chile Oceanic Rank Oceanic Continental Rank Continental Σ 147

12 H o : There is no significant difference magnitude of oceanic versus continental earthquakes in Chile. H a : There is a significant difference magnitude of oceanic versus continental earthquakes in Chile. α = 0.05 n1 = 12 n2 = 11 df = n1, n2 = 12, 11 Note that we are performing a 2-tailed test, so we will use the larger of the test statistics either U or U.

13 U U 12(12 + 1) = (12)(11) + 2 = U = 63 U ' = (12)(11) 63 U ' = U is larger, so it will be used. df = 12,11 U critical = 99 IMPORTANT: This Mann-Whitney table is 1-tailed. Our α level is For a 2- tailed test using a 1-tailed table, you MUST divide the α level between each of the 2 tails So the α level we look up on the table is 0.025, or ½ of 0.05.

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15 69 < 99 Since U is less than U Critical, Accept H o. There is no significant difference in the magnitudes of oceanic versus continental earthquakes in Chile (U 69, p > 0.10). SPSS Test Statistics a Magnitude Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed).853 Exact Sig. [2*(1-tailed Sig.)].880 b a. Grouping Variable: Location Note that SPSS calculates the exact probability b. Not corrected for ties.

16 Mann-Whitney U test (1-tailed) Performing a 1-tailed Mann-Whitney test is somewhat different than other methods. The appropriate test statistic is determined using the following method: This technique simply forces one to declare in which tail the difference will be found in advance since U is to the right of the mean (greater than) and U is to the left of the mean (less than).

17 Using depth of epicenter data for the same Chilean earthquakes, a 1-tailed test is performed with the data ranked from low to high and continental earthquakes as group 1. H o : Continental earthquake depths are not significantly deeper than oceanic earthquakes in Chile. H a : Continental earthquake depths are significantly deeper than oceanic earthquakes in Chile.

18 Using depth of epicenter data for the same Chilean earthquakes, a 1-tailed test is performed with the data ranked from low to high and continental earthquakes as group 1. H o : Continental earthquake depths are not significantly deeper than oceanic earthquakes in Chile. H a : Continental earthquake depths are significantly deeper than oceanic earthquakes in Chile. Therefore the test statistic will be:

19 Earthquake Depths (km) in Chile Oceanic Rank Oceanic Continental Rank Continental Σ 180.5

20 U U 11(11+ 1) = (11)(12) + 2 = U = 17.5 U U ' ' = (12)(11) 17.5 = df = 12,11 = 94 U critical > 94 Since U is greater than U Critical, reject H o. Continental earthquake depths are significantly deeper than oceanic earthquakes in Chile (U 114.5, > p > ).

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22 Therefore, this table is used for 2 purposes: 1. Declaring which group is 1 and which is Declaring which group is greater (or less then) which. It is important to do this because we could easily restate the direction of H o and H a as: H o : Continental earthquake depths are not significantly deeper than oceanic earthquakes in Chile. H a : Oceanic earthquake depths are significantly shallower than continental earthquakes in Chile.

23 The table of which U value to choose helps keep things straight, regardless of how the null and alternate hypotheses are framed (either Group 1 > Group 2 or Group 2 < Group 1. This is clearly demonstrated on the next slide. No matter how we frame the H o and H a, we will use the appropriate statistic.

24 Continental = Group 1, Ranked Low to High Ho: Continental = Oceanic Ha: Continental < Oceanic Oceanic = Group 1, Ranked Low to High Ho: Continental = Oceanic Ha: Continental > Oceanic U U ' ' 11(11+ 1) = (11)(12) + 2 = U U ' ' 12(12 + 1) = (12)(11) + 2 = U ' = 17.5 U = U = U ' = U = U = 17.5 Continental = Group 1, Ranked High to Low Ho: Continental = Oceanic Ha: Continental < Oceanic Oceanic = Group 1, Ranked High to Low Ho: Continental = Oceanic Ha: Continental > Oceanic U U ' ' 11(11+ 1) = (11)(12) + 2 = U U ' ' 12(12 + 1) = (12)(11) + 2 = U ' = U = U = 17.5 U ' = 17.5 U = U = 114.5

25 SPSS uses a different technique that reports the smaller calculated value, regardless of how you arrange the groups. Note that the sum of the ranks does change.

26 Paired Sample t Test

27 Paired Sample t Test This t test is used ONLY when the data are repeat measurements (e.g. measurement at time 1 and time 2 ) or when samples are paired in some manner. The equation is: t = s d d n where d is the mean difference between paired observations, and is the standard deviation of the paired differences. s d Let s test the null hypothesis that unemployment rates in 2007 were lower than in 2008 for selected cities.

28 Determining which data column to subtract from which depends on your hypothesis: Test only difference: does not matter. Testing pair 1 > pair 2: subtract pair 1 from pair 2. Testing pair 1 < pair 2: subtract pair 2 from pair 1. So in this example subtract the 2008 (pair 2) from the 2007 (pair 1) unemployment rate.

29 Unemployment Rate for Selected Cities 2007 (pair 1) 2008 (pair2) d Los Angeles San Francisco Washington DC Bethesda Fort Lauderdale Miami Chicago Boston Detroit Long Island Newark Camden Philadelphia Wilmington Dallas-Fort Worth Seattle Tacoma t = t critical(1) = = = t= > 1.746, reject H o. Remember, the sign just tells us direction, not magnitude. Therefore: Unemployment in 2007 was significantly lower than in 2008 for selected cities (t , p < ). n=17 d v=17-1=16 S d 0.583

30 The paired t test does not have the assumption of normality of the groups or of equality of variances. This is because we are using the paired differences rather than the actual observations. The only assumption is that the paired differences are normally distributed. This test is considered to be fairly robust.

31 Non-Parametric Paired Sample Test Wilcoxon T

32 The Wilcoxon paired-sample test is used when the paired differences are non-normal. The paired t test is fairly robust for slightly non-normal paired differences are not typically a problem If the differences are very non-normal, especially if there is activity in the tails, this test is more appropriate. As with the Mann-Whitney U test, two values are calculated: T + : the sum of the positive ranked differences. T - : the sum of the negative ranked differences.

33 The same subtraction rules for the paired t test apply here. 1. First determined the paired differences. 2. Rank the differences from lowest to highest, ignoring the sign. 3. Apply the signs of the differences to the ranks (called the signed-ranks.).

34 For a 1-tailed test If Ha: Pair1 > Pair 2, reject Ho if T- < the critical value. If Ha: Pair1 < Pair 2, reject Ho if T+ < the critical value. For a 2-tailed test Use the smaller value.

35 H o : Monthly precipitation in 2013 was not greater than in H a : Monthly precipitation in 2013 was greater than in Since our Ha is 2012 (Pair 1) < 2013 (Pair 2) we use T+. Total Precipitation for Shippensburg d Rank d Ranks d Signed January February March April May June July August September October November December The signs are simply used to create 2 groups whose values are summed. n = 12 T + = = 33.5 T - = = 44.5 T critical = 17 Assign the ranks the signs.

36 Calculated value is about here. The Wilcoxon table is one of the few where larger statistics result in accepting the null hypothesis. Make sure you note this.

37 Since we are testing for a positive difference between 2012 and 2013 use the T + statistic. 33 > 13 accept the null hypothesis. Monthly precipitation in 2013 was not greater than in 2012 (Wilcoxon Matched-Pairs T 33, p > 0.25).