Transcript of "Nonparametric hypothesis testing methods"
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Nonparametric Hypothesis Testing Guy Lion December 2005
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Nonparametric tests handle variables that are not normally distributed.
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When to use these methods <ul><li>With large samples (> 100), even if the variable is not normally distributed, the samples Mean is [ Central Limit Theorem ]. Use Parametric test. </li></ul><ul><li>Nonparametric tests can be superior with samples with less than 100 observations. </li></ul><ul><li>Before proceeding with a nonparametric test confirm that the variable does not have a normal distribution (Kurtosis and Skewness close to Zero using Excel). </li></ul>
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Sign Test Testing for Differences in Paired data. <ul><li>Count number of paired data values that are different. This is the Modified Sample Size (MSS). </li></ul><ul><li>Count how many outcome values have increased (or decreased). </li></ul><ul><li>Use a binomial distribution algorithm to figure out what is the probability that the two samples come from populations with identical distribution. </li></ul>
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Sign Test testing the creativity of an Ad campaign Classic situation where we need to use a nonparametric test. This is because the samples are small (17 observations), and the variables are not normally distributed (check Skewness and Kurtosis).
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Sign Test testing the creativity of an Ad campaign (continued)
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Mann-Whitney test for Unpaired testing Steps. <ul><li>Put both samples together. Rank values in ascending order. For repeated numbers (ties) across samples, use the average of their ranks so that identical numbers get identical ranks. </li></ul><ul><li>Find the average rank for each sample. </li></ul><ul><li>Calculate Difference in average rank. </li></ul><ul><li>Find the Standard Error for the average difference in the ranks: (n 1 + n 2 )[SQRT(n 1 + n 2 + 1)/(12n 1 n 2 )]. </li></ul><ul><li>Divide the Difference in avg. rank (step 3) by the Standard Error (step 4) to find the test statistic (a Z value). </li></ul><ul><li>Calculate P Value using NORMSDIST. </li></ul>Note Steps 2 through 6 are similar to the unpaired t Test except it uses Ranks instead of Values .
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Example: Testing income of mortg. applicants Testing if applicants for fixed-rate mortgages have higher income than applicants for variable-rate mortgages. The fixed-rate applicants have one high income value ($240,000). Kurtosis and Skewness of both samples confirm they are not normally distributed. The unpaired t test would not work well.
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Sorting and Ranking Ranked in ascending order. The figures in yellow are identical ($36,500). They originally ranked 12 th , 13 th , and 14 th . So, they all received the tied ranking of 13 th .
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P Value (probability difference is due to chance) Based on ranks (not values), there is an 18.3% probability the two samples come from same population. There is a 81.7% probability that the Variable Rate mortgage applicants have a higher income because they have a higher average rank (17.79 vs 13.50).
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Mann-Whitney U. Things to watch for <ul><li>Breaking off the ties may not have much impact. Having redone the last example without breaking the ties, depending on how the yellow figures got ranked you get P values of 17.0% or 19.8% not much different than the 18.3%. </li></ul><ul><li>Important caveat . You need at least 10 observations for each of the two unpaired samples you test for to obtain a valid Z variable to calculate a P value. </li></ul><ul><li>Mann-Whitney U is calculated differently than as shown that reflects calculations by Andrew Siegel that gets the same result faster. See Appendix on next slide. </li></ul>
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Appendix: The actual Mann-Whitney U Calculation
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