 # Mann-Whitney and Wilcoxon Tests.

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Mann-Whitney and Wilcoxon Tests

Types of Inferential Statistics
Inferential Statistics: estimate the value of a population parameter from the characteristics of a sample Parametric Statistics: Assumes the values in a sample are normally distributed Interval/Ratio level data required Nonparametric Statistics: No assumptions about the underlying distribution of the sample Used when the data do not meet the assumption for a nonparametric test (ordinal and nominal data)

Choosing Statistical Procedures

Mann Whitney U Test Nonparametric equivalent of the independent t test Two independent groups Ordinal measurement of the DV The sampling distribution of U is known and is used to test hypotheses in the same way as the t distribution. U

Mann Whitney U Test To compute the Mann Whitney U:
Rank the scores in both groups (together) from highest to lowest. Sum the ranks of the scores for each group. The sum of ranks for each group are used to make the statistical comparison. 1. The null hypothesis states that there is no difference in the scores of the populations from which the samples were drawn. 2. The Mann Whitney U is sensitive to both the central tendency of the scores and the distribution of the scores. 3. The Mann Whitney U statistic is defined as the smaller of U1 and U2. U1 = n1n2 + [n1(n1 + 1) / 2] - R1 U2 = n1n2 + [n2(n2 + 1) / 2] - R2 Where: n1 = number of observations in group 1 n2 = number of observations in group 2 R1 = sum of the ranks assigned to group 1 R2 = sum of the ranks assigned to group 2 4. The critical values for the U statistic are found in table C.14. The computed U value must be less than the critical value found in table C.14.

Non-Directional Hypotheses
Null Hypothesis: There is no difference in scores of the two groups (i.e. the sum of ranks for group 1 is no different than the sum of ranks for group 2). Alternative Hypothesis: There is a difference between the scores of the two groups (i.e. the sum of ranks for group 1 is significantly different from the sum of ranks for group 2).

Computing the Mann Whitney U Using SPSS
Enter data into SPSS spreadsheet; two columns  1st column: groups; 2nd column: scores (ratings) Analyze  Nonparametric  2 Independent Samples Select the independent variable and move it to the Grouping Variable box  Click Define Groups  Enter 1 for group 1 and 2 for group 2 Select the dependent variable and move it to the Test Variable box  Make sure Mann Whitney is selected  Click OK

Interpreting the Output
The output provides a z score equivalent of the Mann Whitney U statistic. It also gives significance levels for both a one-tailed and a two-tailed hypothesis.

Generating Descriptives for Both Groups
Analyze  Descriptive Statistics  Explore Independent variable  Factors box Dependent variable  Dependent box Click Statistics  Make sure Descriptives is checked  Click OK

Wilcoxon Signed-Rank Test
Nonparametric equivalent of the dependent (paired-samples) t test Two dependent groups (within design) Ordinal level measurement of the DV. The test statistic is T, and the sampling distribution is the T distribution. T The Wilcoxon matched-pairs signed-ranks test 1. The nonparametric analog of the two-sample case with dependent samples 2. The null hypothesis states that there is no difference on an identified variable before and after treatment or between two matched groups. 3. The test statistic for the Wilcoxon test is T. 4. The sampling distribution is the T distribution. Critical values are found in Table C.15.

Wilcoxon Test To compute the Wilcoxon T:
Determine the differences between scores. Rank the absolute values of the differences. Place the appropriate sign with the rank (each rank retains the positive or negative value of its corresponding difference) T = the sum of the ranks with the less frequent sign 5. The Wilcoxon test is computed as follows: a. Determine the difference between the pretest and the posttest score for each individual or between the scores for each matched pair. b. Rank the absolute values of the difference scores, and then place the appropriate sign with the rank. c. Sum the ranks with the less frequent sign.

Non-Directional Hypotheses
Null Hypothesis: There is no difference in scores before and after an intervention (i.e. the sums of the positive and negative ranks will be similar). Non-Directional Research Hypothesis: There is a difference in scores before and after an intervention (i.e. the sums of the positive and negative ranks will be different).

Computing the Wilcoxon Test Using SPSS
Enter data into SPSS spreadsheet; two columns  1st column: pretest scores; 2nd column: posttest scores Analyze  Nonparametric  2 Related Samples Highlight both variables  move to the Test Pair(s) List  Click OK To Generate Descriptives: Analyze  Descriptive Statistics  Explore Both variables go in the Dependent box Click Statistics  Make sure Descriptives is checked  Click OK

Interpreting the Output
The T test statistic is the sum of the ranks with the less frequent sign. The output provides the equivalent z score for the test statistic. Two-Tailed significance is given.