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Non-parametric statistics http://sst.tees.ac.uk/external/U0000504/

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2 Non-parametric methods Also known as distribution free methods Also known as distribution free methods Parametric methods assume the data is normally distributed Parametric methods assume the data is normally distributed Distribution free methods do not rely on the data conforming to any particular distribution Distribution free methods do not rely on the data conforming to any particular distribution

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3 Books Neave & Worthington Distribution free methods Neave & Worthington Distribution free methods Siegel, S Non-parametric statistics for the Behavioural Sciences Siegel, S Non-parametric statistics for the Behavioural Sciences

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4 Methods Comparison of two samples Comparison of two samples Mann-Witney Test Mann-Witney Test Wilcoxons signed ranks test Wilcoxons signed ranks test Comparing more than two samples Comparing more than two samples Kruskall-Wallis test Kruskall-Wallis test Friedman test Friedman test Large samples Large samples

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5 Mann-Witney test Used to compare two independent samples Used to compare two independent samples Involves putting the samples into a common ranking 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. The locations of the samples within the common ranking is a measure of similarity or difference of the samples.

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6 Mann-Witney Test Procedure (label the two samples A & B) Procedure (label the two samples A & B) Arrange data in a common list in rank order. Arrange data in a common list in rank order. Compute the sums of the ranks of the two samples Compute the sums of the ranks of the two samples Calculate the test statistic U from Calculate the test statistic U from U A = S A – ½n A (n A + 1) U B = n A n B - U A The test statistic U is the smaller of U A and U B for a two tailed test and the value that should have the smaller median for a one-tailed test The test statistic U is the smaller of U A and U B for a two tailed test and the value that should have the smaller median for a one-tailed test

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7 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 A95.5688.5 Group B6.57.554.57

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8 Mann-Witney test example Null Hypothesis Null Hypothesis H 0 : Median A = Median B H 0 : Median A = Median B H 1 : Median A Median B H 1 : Median A Median B Decision Rule Decision Rule Reject Ho if test statistic, U Critical value for a two-tailed test at = 0.05 Reject Ho if test statistic, U Critical value for a two-tailed test at = 0.05

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9 Mann-Witney test example 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. 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. UA = 34-0.5(5 x 6) = 19 UB = 5 x 5 - 19 = 6 U = 6

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10 Wilcoxons signed ranks test Used to compare two samples that are related in some way (same location, same tester etc.) 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 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; 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 two-tailed test, the smaller sum of the ranks For a one-tailed test, the sum of the ranks expected to be smaller For a one-tailed test, the sum of the ranks expected to be smaller

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11 Wilcoxons signed ranks test Example 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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12 Wilcoxons signed ranks test Null hypothesis H 0 : BOD after treatment BOD before treatment H 1 : The BOD after treatment < BOD before treatment Decision rule: Reject H 0 if the value of the test statistic, T is less than the critical value at =0.05.

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13 Wilcoxons 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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14 Kruskall-Wallis test Essentially an extension of the Mann-Witney test to more than two samples 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 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 } 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 A test statistic H is calculated from Reject Ho if H critical value.

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15 Friedman test The Friedman test compares more than two related samples 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 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; 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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16 Large samples The method of calculating the test statistics is based on small 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. 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. The methods of calculation may be found in the books listed.

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