STAT E-150: Week 12 Bootstrapping and Non-parametric tests

1. STAT E-150: Week 12Bootstrapping and Non-parametric tests By Kela Roberts

2. Section Overview • Questions • Proposal Questions Review • Key Concepts from Non-parametric tests and Bootstrapping • Sample Project • Additional Questions

3. Proposal Review • Population • Concepts from the first half of the course • Interaction • Multicollinearity

4. Nonparametric Tests for Comparison •  Don’t rely on data coming from a normal distribution •  Essentially, they compare medians instead of means Use ranks to do so •  Wilcoxon-­‐Mann-­‐Whitney is to compare two medians •  Kruskal-­‐Wallis Test: alternative to a one-­way ANOVA

5. Nonparametric Tests for Comparison Non-Parametric Tests - Don’t rely on data coming from a normal distribution Still some assumptions about errors –  Have a median of zero –  Be independent –  Distribution of each group has a similar shape (verify using a histogram)

6. Nonparametric Tests for Comparison • H0: θ1= θ2 • Ha: θ1 ≠ θ2 • Since the p<.01, we reject the null hypothesis. This suggests that the dependent variable differs amongst the predictors. Wilcoxon-Mann-Whitney: is to compare two medians using rank to do so, use on small sample size data with outliers •  If there is no difference between the groups, the sum of the ranks for the two groups should be similar –  Group data will be intermixed •  Alternatively, if there is a difference, one group will tend to have higher ranks than the other

7. Nonparametric Tests for Comparison Kruskal-­‐Wallis test: alternative to a one-­way ANOVA •  Same assumptions: –  Errors have median of 0 –  Errors are independent –  Distributions are continuous, similar shape •  When to use: –  More than two groups –  When don’t meet normality condition –  Have outliers –  Have ordinal data – like a rating scale, instead of a quantitative dependent variable

8. Bootstrap Summary A way to get confidence intervals for our betas •  Including a p-value •  Useful when conditions might not be completely met, like normality of residuals or with small sample sizes •  Based on the idea of investigating the variability due to our sample, to get at the variability in the population.

9. Bootstrap Summary

10. Sample Question Which non-parametric test would you perform in the following scenarios? • A medical researcher has heard anecdotal evidence that certain anti-depressive drugs can have the positive side-effect of lowering neurological pain in those individuals with chronic, neurological back pain, when administered in doses lower than those prescribed for depression. The medical researcher would like to investigate this anecdotal evidence with a study. The researcher identifies 3 well-known, anti-depressive drugs which might have this positive side-effect, and labels them Drug A, Drug B and Drug C. The researcher then recruits a group of 60 individuals with a similar level of back pain and randomly assigns them to one of three groups - Drug A, Drug B or Drug C treatment groups - and prescribes the relevant drug for a 4 week period. At the end of the 4 week period, the researcher asks the participants to rate their back pain on a scale of 1 to 10, with 10 indicating the greatest level of pain. The researcher wishes to compare the levels of pain experienced by the different groups at the end of the drug treatment period. After comparison, it was found that the p-value is 0.001. • [https://statistics.laerd.com/spss-tutorials/kruskal-wallis-h-test-using-spss-statistics.php

11. Sample Question Which non-parametric test would you perform in the following scenarios? The researcher wishes to compare the levels of pain experienced by the different groups at the end of the drug treatment period. After comparison, it was found that the p-value is 0.001. Kruskal-Wallis test to compare pain score with type of drug (between the three drug treatments) H0: θ1= θ2=θ3 Ha: The medians are not all equal Since the p<.01, we reject the null hypothesis. This suggests that the side effects differs amongst the treatment groups. • [https://statistics.laerd.com/spss-tutorials/kruskal-wallis-h-test-using-spss-statistics.php

12. Sample Question Which non-parametric test would you perform in the following scenarios? The pain researcher is interested in finding methods to reduce lower back pain in individuals without having to use drugs. The researcher thinks that having acupuncture in the lower back might reduce back pain. To investigate this, the researcher recruits 25 participants to their study. At the beginning of the study, the researcher asks the participants to rate their back pain on a scale of 1 to 10, with 10 indicating the greatest level of pain. After 4 weeks of twice weekly acupuncture the participants are asked again to indicate their level of back pain on a scale of 1 to 10, with 10 indicating the greatest level of pain. The researcher wishes to understand whether the participants' pain levels changed after they had undergone the acupuncture. • [https://statistics.laerd.com/spss-tutorials/wilcoxon-signed-rank-test-using-spss-statistics.php

13. Sample Question Which non-parametric test would you perform in the following scenarios? The researcher wishes to understand whether the participants' pain levels changed after they had undergone the acupuncture. Wilcoxon signed-rank test will be used to compare pain levels before and after acupuncture treatment. H0: θ1= θ2 Ha: θ1 ≠ θ2

14. Sample Question Which non-parametric test would you perform in the following scenarios? The researcher wishes to understand whether the participants' pain levels changed after they had undergone the acupuncture. Wilcoxon signed-rank test will be used to compare pain levels before and after acupuncture treatment. The null and alternative hypotheses written in words: H0: The pain level of the groups before and after acupuncture treatment have the same distributions (aka the medians would be the same) or the pain level of the groups are the same size- i.e. - there is no difference between the pain level of the subjects before and after the acupuncture treatment Ha: The pain level of the groups before and after acupuncture treatment do not have the same distributions (aka the medians would not be the same for the different groups) or the pain level of the groups are not the same size- i.e. - there is a difference between the pain level of the subjects before and after the acupuncture treatment

15. Sample Project

16. Additional Questions