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Statistics 07 Nonparametric Hypothesis Testing

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Parametric testing such as Z test, t test and F test is suitable for the test of range variables or ratio variables and assumes that the population measured is in normal distribution.

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Nonparametric Hypothesis Testing Non-Parametric tests are often used in place of their parametric counterparts when certain assumptions about the underlying population are questionable. For example, when comparing two independent samples, the Wilcoxon Mann-Whitney test does not assume that the difference between the samples is normally distributed whereas its parametric counterpart, the two sample t-test does. Non-Parametric tests may be, and often are, more powerful in detecting population differences when certain assumptions are not satisfied.

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Nonparametric Hypothesis Testing All tests involving ranked data, i.e. data that can be put in order, are non-parametric Disadvantages: Not sensitive to data.

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Actual rank test (Mann-Whitney U Test) The Mann-Whitney Test is one of the most powerful of the non-parametric tests for comparing two populations. It is used to test the null hypothesis that two populations have identical distribution functions against the alternative hypothesis that the two distribution functions differ only with respect to location (median), if at all. This test can also be applied when the observations in a sample of data are ranks, that is, ordinal data rather than direct measurements.

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Type of Data Two sets of data from two populations Scores of female and male students Scores of two classes of different levels FemaleMale 1512 1718 1310

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In computation of U, we need to know The actual ranks The maximum ranks U is the difference between the sum of actual ranks and that of maximum ranks. Maximum rank sum: n 1 *n 2 +n 1 *(n 1 +1)/2 Actual rank sum: ΣR

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Procedure of U Test 1. Lump two sets of data into one group. 2. Rank the pooled data 3. Compute actual rank sum: ΣR 4. Compute U U 1 =n 1 *n 2 +n 1 *(n 1 +1)/2-ΣR 1 U 2 = n 1 *n 2 +n 2 *(n 2 +1)/2-ΣR 2 5. Determine the nature of the test and look up for the critical value U(n 1,n 2,) from Mann- Whitney Test Table.

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6.If U smaller > U(n 1,n 2,), accept H 0 : There is no significant difference between the two samples. If U smaller < U(n 1,n 2,), reject H 0 : There is significant difference between the two samples. Procedure of U Test

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Wilcoxon signed-ranked test The Wilcoxon Signed Ranks test is designed to test a hypothesis about the location (median) of a population distribution. It often involves the use of matched pairs, for example, before and after data, in which case it tests for a median difference of zero.

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Type of Data Matched pairs Two scores given by a group of judges Scores before and after the program Scores given to a sample of students ’ writing JudgeCandidate 1Candidate 2 185 257 398

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Procedure 1. Compute the absolute difference between the two scores. 2. Sign the absolute difference by + for a positive difference, － for a negative difference, and no sign for no difference. 3. Rearrange the differences in an ascending order, i.e. from the smallest to the largest. 4. Rank the absolute differences with 0 for none. 5. Group the ranks into positive group and negative group respectively

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6. Compute positive rank sumΣR + and the negative rank sumΣR - 7. Let the smaller of the two sums be the value T 8. Determine n: n=total number — the number of zero difference 9. Look up in the Signed Rank Table for the critical value T n Procedure

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10. If T< T n, H 0 is rejected: There is significant difference between the two scores. If T>T n, H 0 is accepted: There is no significant difference between the two scores. Procedure

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