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Agresti/Franklin Statistics, 1 of 62 Chapter 14 Nonparametric Statistics Learn …. About Nonparametric Statistical Methods

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Agresti/Franklin Statistics, 2 of 62 Nonparametric Statistical Methods Nonparametric methods are used: When the data are ranks for the subjects, rather than quantitative measurements. When it’s inappropriate to assume normality.

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Agresti/Franklin Statistics, 3 of 62 Section 14.1 How Can We Compare Two Groups by Ranking?

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Agresti/Franklin Statistics, 4 of 62 Example: How to Get A Better Tan Experiment: A student wanted to compare ways of getting a tan without exposure to the sun. She decided to investigate which of two treatments would give a better tan: An “instant bronze sunless tanner” lotion A tanning studio

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Agresti/Franklin Statistics, 5 of 62 Example: How to Get A Better Tan Subjects: Five female students participated in the experiment. Three of the students were randomly selected to use the tanning lotion. The other two students used the tanning studio.

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Agresti/Franklin Statistics, 6 of 62 Example: How to Get A Better Tan Results: The girls’ tans were ranked from 1 to 5, with 1 representing the best tan. Possible Outcomes: Consider all possible rankings of the girls’ tans. A table of possibilities is displayed on the next page.

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Agresti/Franklin Statistics, 7 of 62 Example: How to Get A Better Tan

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Agresti/Franklin Statistics, 8 of 62 Example: How to Get A Better Tan For each possible outcome, a mean rank is calculated for the ‘lotion’ group and for the ‘studio’ group. The difference in the mean ranks is then calculated for each outcome.

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Agresti/Franklin Statistics, 9 of 62 Example: How to Get A Better Tan For this experiment, the samples were independent random samples – the responses for the girls using the tanning lotion were independent of the responses for the girls using the tanning studio.

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Agresti/Franklin Statistics, 10 of 62 Example: How to Get A Better Tan Suppose that the two treatments have identical effects. A girl’s tan would be the same regardless of which treatment she uses. Then, each of the ten possible outcomes is equally likely. So, each outcome has probability of 1/10.

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Agresti/Franklin Statistics, 11 of 62 Example: How to Get A Better Tan Using the ten possible outcomes, we can construct a sampling distribution for the difference between the sample mean ranks. The distribution is displayed on the next page.

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Agresti/Franklin Statistics, 12 of 62 Example: How to Get A Better Tan

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Agresti/Franklin Statistics, 13 of 62 Example: How to Get A Better Tan Graph of the Sampling Distribution:

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Agresti/Franklin Statistics, 14 of 62 Example: How to Get A Better Tan The student who planned the experiment hypothesized that the tanning studio would give a better tan than the tanning lotion.

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Agresti/Franklin Statistics, 15 of 62 Example: How to Get A Better Tan She wanted to test the null hypothesis: H 0 : The treatments are identical in tanning quality. Against H a : Better tanning quality results with the tanning studio.

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Agresti/Franklin Statistics, 16 of 62 Example: How to Get A Better Tan This alternative hypothesis is one-sided. If H a were true, we would expect the ranks to be smaller (better) for the tanning studio. Thus, if H a were true, we would expect the differences between the sample mean rank for the tanning lotion and the sample mean rank for the tanning studio to be positive.

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Agresti/Franklin Statistics, 17 of 62 Wilcoxon Test The test comparing two groups based on the sampling distribution of the difference between the sample mean ranks is called the Wilcoxon test.

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Agresti/Franklin Statistics, 18 of 62 Wilcoxon Nonparametric Test for Comparing Two Groups 1. Assumptions: Independent random samples from two groups.

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Agresti/Franklin Statistics, 19 of 62 Wilcoxon Nonparametric Test for Comparing Two Groups 2. Hypotheses: H 0 : Identical population distributions for the two groups (this implies equal expected values for the sample mean ranks). H a : Higher expected value for the sample mean rank for a specified group (one- sided).

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Agresti/Franklin Statistics, 20 of 62 Wilcoxon Nonparametric Test for Comparing Two Groups 3. Test Statistic: Difference between sample mean ranks for the two groups (Equivalently, can use sum of ranks for one sample).

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Agresti/Franklin Statistics, 21 of 62 Wilcoxon Nonparametric Test for Comparing Two Groups 4. P-value: One-tail or two-tail probability, depending on H a, that the difference between the sample mean ranks is as extreme or more extreme than observed. 5. Conclusion: Report the P-value and interpret it. If a decision is needed, reject H 0 if the P-value ≤ significance level such as 0.05.

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Agresti/Franklin Statistics, 22 of 62 Example: Does the UV Tanning Studio Give a Better Tan than the Tanning Lotion? For the actual experiment: the ranks were (2,4,5) for the girls using the tanning lotion the ranks were (1,3) for the girls using the tanning studio.

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Agresti/Franklin Statistics, 23 of 62 Example: Does the UV Tanning Studio Give a Better Tan than the Tanning Lotion? The mean rank for the tanning lotion is: (2+4+5)/3 = 3.7 The mean rank for the tanning studio is: (1+3)/2=2

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Agresti/Franklin Statistics, 24 of 62 Example: Does the UV Tanning Studio Give a Better Tan than the Tanning Lotion? The test statistic is the difference between the sample mean ranks: 3.7 – 2 = 1.7

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Agresti/Franklin Statistics, 25 of 62 Example: Does the UV Tanning Studio Give a Better Tan than the Tanning Lotion? The one-sided alternative hypothesis states that the tanning studio gives a better tan. This means that the expected mean rank would be larger for the tanning lotion than for the tanning studio, if H a is true. And, the difference between the mean ranks would be positive.

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Agresti/Franklin Statistics, 26 of 62 Example: Does the UV Tanning Studio Give a Better Tan than the Tanning Lotion? The test statistic we obtained from the data was: Difference between the sample mean ranks = 1.7. P-value = P(difference between sample mean ranks at least as large as 1.7)

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Agresti/Franklin Statistics, 27 of 62 Example: Does the UV Tanning Studio Give a Better Tan than the Tanning Lotion? The P-value can be obtained from the graph of the sampling distribution (as seen on a previous page and displayed again here):

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Agresti/Franklin Statistics, 28 of 62 Example: Does the UV Tanning Studio Give a Better Tan than the Tanning Lotion? P-value = This is not a very small P-value. The evidence does not strongly support the claim that the tanning studio gives a better tan.

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Agresti/Franklin Statistics, 29 of 62 The Wilcoxon Rank Sum The Wilcoxon test can, equivalently, use as the test statistic the sum of the ranks in just one of the samples. This statistic will have the same probabilities as the differences between the sample mean ranks. Some software reports the sum of ranks as the Wilcoxon rank sum statistic.

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Agresti/Franklin Statistics, 30 of 62 Example: Is there a treatment difference between the UV Tanning Studio and the Tanning Lotion? Suppose the experiment was designed with a two-sided alternative hypothesis: H 0 : The treatments are identical in tanning quality. H a : The treatments are different in tanning quality.

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Agresti/Franklin Statistics, 31 of 62 Example: Is there a treatment difference between the UV Tanning Studio and the Tanning Lotion?

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Agresti/Franklin Statistics, 32 of 62 Using the Wilcoxon Test with a Quantitative Response When the response variable is quantitative, the Wilcoxon test is applied by converting the observations to ranks. For the combined sample, the observations are ordered from smallest to largest. The test compares the mean ranks for the two samples.

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Agresti/Franklin Statistics, 33 of 62 Example: Do Drivers Using Cell Phones Have Slower Reaction Times? Experiment: A sample of 64 college students were randomly assigned to a cell phone group or a control group, 32 to each. On a machine that simulated driving situations, participants were instructed to press a “brake button” when they detected a red light.

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Agresti/Franklin Statistics, 34 of 62 Example: Do Drivers Using Cell Phones Have Slower Reaction Times? Experiment: The control group listened to the radio while they performed the simulated driving. The cell phone group carried out a conversation on a cell phone. Each subject’s response time to the red lights is recorded and averaged over all of his/her trials.

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Agresti/Franklin Statistics, 35 of 62 Example: Do Drivers Using Cell Phones Have Slower Reaction Times? Boxplots of the data:

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Agresti/Franklin Statistics, 36 of 62 Example: Do Drivers Using Cell Phones Have Slower Reaction Times? The box plots do not show any substantial skew, but there is an extreme outlier for the cell phone group. The t inferences that we have used previously assume normal population distributions. The Wilcoxon Test does not assume normality. This test can be used in place of the t test if the normality assumption is questioned.

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Agresti/Franklin Statistics, 37 of 62 Example: Do Drivers Using Cell Phones Have Slower Reaction Times? To use the Wilcoxon test, we need to rank the data (response times) from 1 (smallest reaction time) to 64 (largest reaction time). The test statistic is then calculated from the ranks.

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Agresti/Franklin Statistics, 38 of 62 Example: Do Drivers Using Cell Phones Have Slower Reaction Times? The next page shows the output for the hypothesis test: H 0: The distribution of reaction times is identical for the two groups. H a: The distribution of reaction times differs for the two groups.

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Agresti/Franklin Statistics, 39 of 62 Example: Do Drivers Using Cell Phones Have Slower Reaction Times?

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Agresti/Franklin Statistics, 40 of 62 Example: Do Drivers Using Cell Phones Have Slower Reaction Times? The small P-value (.019) shows strong evidence against the null hypothesis. The sample mean ranks suggest that reaction times tend to be slower for those using cell phones.

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Agresti/Franklin Statistics, 41 of 62 Example: Do Drivers Using Cell Phones Have Slower Reaction Times? Insight: The Wilcoxon test is not affected by outliers. No matter how far the largest observation falls from the next largest, it still gets the same rank.

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Agresti/Franklin Statistics, 42 of 62 Nonparametric Estimation Comparing Groups When the response variable is quantitative, we can compare a measure of center for the two groups. One way to do this is by comparing means. This method requires the assumption of normal population distributions.

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Agresti/Franklin Statistics, 43 of 62 Nonparametric Estimation Comparing Groups When the response distribution is highly skewed, nonparametric methods are preferred. For highly skewed distributions, a better measure of the center is the median. We can then estimate the difference between the population medians for the two groups.

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Agresti/Franklin Statistics, 44 of 62 Nonparametric Estimation Comparing Groups Most software for the Wilcoxon test reports point and interval estimates comparing medians. Some software refers to the equivalent Mann-Whitney test.

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Agresti/Franklin Statistics, 45 of 62 Nonparametric Estimation Comparing Groups The Wilcoxon test (and the Mann- Whitney test) does not require a normal population assumption. It does require an extra assumption: the population distributions for the two groups are symmetric and have the same shape.

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Agresti/Franklin Statistics, 46 of 62 For a study on the effects of hypnosis, subjects were divided into a control group and treatment group and a measure of respiratory ventilations was taken on each subject. Controls: Treated: What is the mean rank for the Control Group? a.8 b.49 c d.7.5

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Agresti/Franklin Statistics, 47 of 62 What is the mean rank for the Treatment Group? a.8 b c d.16 For a study on the effects of hypnosis, subjects were divided into a control group and treatment group and a measure of respiratory ventilations was taken on each subject. Controls: Treated:

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Agresti/Franklin Statistics, 48 of 62 A test of the hypothesis that subjects in the treatment group tended to ventilate more resulted in a P-value of Does this P- value support the claim that subjects in the treatment group ventilated more? a.yes b.no For a study on the effects of hypnosis, subjects were divided into a control group and treatment group and a measure of respiratory ventilations was taken on each subject. Controls: Treated:

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Agresti/Franklin Statistics, 49 of 62 Section 14.2 Nonparametric Methods for Several Groups and for Matched Pairs

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Agresti/Franklin Statistics, 50 of 62 Comparing Mean Ranks of Several Groups The Wilcoxon test for comparing mean ranks of two groups extends to a comparison of mean ranks for several groups. This test is called the Kruskal- Wallis test.

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Agresti/Franklin Statistics, 51 of 62 ANOVA test vs. Kruskal- Wallis test Both tests are used to compare many groups. The ANOVA F test assumes normal population distributions. The Kruskal-Wallis test does not make this assumption. The Kruskal-Wallis test is a “safer” method to use with small samples when not much information is available about the shape of the distributions.

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Agresti/Franklin Statistics, 52 of 62 ANOVA test vs. Kruskal- Wallis test The Kruskal-Wallis test is also useful when the data are merely ranks and we don’t have a quantitative measurement of the response variable.

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Agresti/Franklin Statistics, 53 of 62 Summary: Kruskal-Wallis Test 1. Assumptions: Independent random samples from several (g) groups. 2. Hypotheses: H 0 : Identical population distributions for the g groups H a : Population distributions not all identical.

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Agresti/Franklin Statistics, 54 of 62 Summary: Kruskal-Wallis Test 3. Test statistic: Uses between-groups variability of sample mean ranks. Software easily calculates this. 4. P-value: Right-tail probability above observed test statistic value from chi- squared distribution. 5. Conclusion: Report the P-value and interpret in context.

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Agresti/Franklin Statistics, 55 of 62 Example: Does Heavy Dating Affect College GPA? Experiment: A student in a statistics class (Tim) decided to study whether dating was associated with college GPA. He wondered whether students who data a lot tend to have poorer GPAs.

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Agresti/Franklin Statistics, 56 of 62 Example: Does Heavy Dating Affect College GPA? Experiment: He asked 17 students in the class to anonymously fill out a short questionnaire in which they were asked to give their college GPA and to indicate whether, during their college careers, they had dated regularly, occasionally, or rarely.

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Agresti/Franklin Statistics, 57 of 62 Example: Does Heavy Dating Affect College GPA? Dot plots of the GPA data for the 3 dating groups:

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Agresti/Franklin Statistics, 58 of 62 Example: Does Heavy Dating Affect College GPA? Since the dot plots showed evidence of severe skew to the left and since the sample size was small in each group, Tim felt safer analyzing the data with the Kruskal-Wallis test than with the ordinary ANOVA F test.

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Agresti/Franklin Statistics, 59 of 62 Example: Does Heavy Dating Affect College GPA? The hypotheses for the Kruskal-Wallis test: H 0 : Identical population distributions for the three dating groups H a : Population distributions for the three dating groups are not all identical.

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Agresti/Franklin Statistics, 60 of 62 Example: Does Heavy Dating Affect College GPA? This table shows the data with the GPA values ordered from smallest to largest for each dating group.

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Agresti/Franklin Statistics, 61 of 62 Example: Does Heavy Dating Affect College GPA? MINITAB output for the Kruskal-Wallis test:

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Agresti/Franklin Statistics, 62 of 62 Example: Does Heavy Dating Affect College GPA? The test statistic reported in the output is H = The corresponding P-value reported in the output is This large P-value does not give any evidence against H 0. It is plausible that GPA is independent of dating group.

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