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**Non-parametric statistics**

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**Non-parametric methods**

Also known as “distribution free” methods Parametric methods assume the data is normally distributed Distribution free methods do not rely on the data conforming to any particular distribution

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**Books Neave & Worthington “Distribution free methods”**

Siegel, S “Non-parametric statistics for the Behavioural Sciences”

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**Methods Comparison of two samples Comparing more than two samples**

Mann-Witney Test Wilcoxon’s signed ranks test Comparing more than two samples Kruskall-Wallis test Friedman test Large samples

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**Mann-Witney test Used to compare two independent samples**

Involves putting the samples into a common ranking The locations of the samples within the common ranking is a measure of similarity or difference of the samples.

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**Mann-Witney Test Procedure (label the two samples A & B)**

Arrange data in a common list in rank order. Compute the sums of the ranks of the two samples Calculate the test statistic U from UA = SA – ½nA(nA + 1) UB= nA nB - UA The test statistic U is the smaller of UA and UB for a two tailed test and the value that should have the smaller median for a one-tailed test

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**Mann-Witney test example**

Two groups were asked to assess their perception of the creaminess of a chocolate milk, by marking a point on a line marked thus 1______________10. The results from the two groups are given below. Is there any evidence to suggest that the two groups differ in their perception of creaminess? Group A Group B

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**Mann-Witney test example**

Null Hypothesis H0: Median A = Median B H1: Median A Median B Decision Rule Reject Ho if test statistic, U Critical value for a two-tailed test at a = 0.05

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**Mann-Witney test example**

UA = (5 x 6) = 19 UB = 5 x = 6 U = 6 From tables, critical value of U is 2. So fail to reject the null hypothesis and conclude that there is insufficient evidence to conclude the two groups differ in their perception of the chocolate milk.

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**Wilcoxon’s signed ranks test**

Used to compare two samples that are related in some way (same location, same tester etc.) The differences between each pair is calculate, preserving the sign The differences are ranked regardless of sign and sums of the ranks calculated. The Test statistic, T is taken as; For a two-tailed test, the smaller sum of the ranks For a one-tailed test, the sum of the ranks expected to be smaller

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**Wilcoxon’s signed ranks test**

Example A water company sought evidence the measures taken to clean up a river were effective. Biological oxygen demand (BOD) at 12 sites on the river were compared before and after cleanup.

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**Wilcoxon’s signed ranks test**

Null hypothesis H0: BOD after treatment BOD before treatment H1: The BOD after treatment < BOD before treatment Decision rule: Reject H0 if the value of the test statistic, T is less than the critical value at =0.05.

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**Wilcoxon’s signed ranks test**

Sum of negative ranks = 7 Critical value = 17 Reject Ho and conclude there is evidence for significant cleanup of the river.

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Kruskall-Wallis test Essentially an extension of the Mann-Witney test to more than two samples The data is grouped into a single list and ranked overall. The mean rank for each group is compared with the overall mean rank for the whole sample {(Ri-R)2} A test statistic H is calculated from Reject Ho if H critical value.

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**Friedman test The Friedman test compares more than two related samples**

The data is arranged in a table of k columns (“treatments”) and n rows (“blocks”) and is ranked across each row. The sum of the ranks for each column is calculated and used to calculate a test statistic M from; Reject the null hypothesis if M critical value

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Large samples The method of calculating the test statistics is based on small samples For large samples, the statistic may have to be modified in some way to enable the result to be looked up in standard tables; usually either normal distribution or -squared distribution. The methods of calculation may be found in the books listed.

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© The McGraw-Hill Companies, Inc., 2000 10-1 Chapter 10 Testing the Difference between Means and Variances.

© The McGraw-Hill Companies, Inc., 2000 10-1 Chapter 10 Testing the Difference between Means and Variances.

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