1. Nonparametric Methods III Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University hslu@stat.nctu.edu.tw http://tigpbp.iis.sinica.edu.tw/courses.htm

2. PART 4: Bootstrap and Permutation Tests • Introduction • References • Bootstrap Tests • Permutation Tests • Cross-validation • Bootstrap Regression • ANOVA

3. References • Efron, B.; Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman & Hall/CRC. • http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-bootstrapping.pdf • http://cran.r-project.org/bin/macosx/2.1/check/bootstrap-check.ex • http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf

4. Hypothesis Testing (1) • A statistical hypothesis test is a method of making statistical decisions from and about experimental data. • Null-hypothesis testing just answers the question of “how well the findings fit the possibility that chance factors alone might be responsible.” • This is done by asking and answering a hypothetical question. • http://en.wikipedia.org/wiki/Statistical_hypothesis_testing

5. Hypothesis Testing (2) • Hypothesis testing is largely the product of Ronald Fisher, Jerzy Neyman, Karl Pearson and (son) Egon Pearson. Fisher was an agricultural statistician who emphasized rigorous experimental design and methods to extract a result from few samples assuming Gaussian distributions.

6. Hypothesis Testing (3) Neyman (who teamed with the younger Pearson) emphasized mathematical rigor and methods to obtain more results from many samples and a wider range of distributions. Modern hypothesis testing is an (extended) hybrid of the Fisher vs. Neyman/Pearson formulation, methods and terminology developed in the early 20th century.

7. Hypothesis Testing (4)

8. Hypothesis Testing (5)

9. Hypothesis Testing (6)

10. Hypothesis Testing (7) • Parametric Tests: • Nonparametric Tests: • Bootstrap Tests • Permutation Tests

11. Confidence Intervals vs. Hypothesis Testing (1) • Interval estimation ("Confidence Intervals") and point estimation ("Hypothesis Testing") are two different ways of expressing the same information. • http://www.une.edu.au/WebStat/unit_materials/c5_inferential_statistics/confidence_interv_hypo.html

12. Confidence Intervals vs. Hypothesis Testing (2) • If the exact p-value is reported, then the relationship between confidence intervals and hypothesis testing is very close.  However, the objective of the two methods is different: • Hypothesis testing relates to a single conclusion of statistical significance vs. no statistical significance.  • Confidence intervals provide a range of plausible values for your population.

13. Confidence Intervals vs. Hypothesis Testing (3) • Which one? • Use hypothesis testing when you want to do a strict comparison with a pre-specified hypothesis and significance level. • Use confidence intervals to describe the magnitude of an effect (e.g., mean difference, odds ratio, etc.) or when you want to describe a single sample. • http://www.nedarc.org/nedarc/analyzingData/advancedStatistics/convidenceVsHypothesis.html

14. P-value • http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf

15. Achieved Significance Level (ASL) • Definition: A hypothesis test is a way of deciding whether or not the data decisively reject the hypothesis . The archived significance level of the test (ASL) is defined as: . The smaller ASL, the stronger is the evidence of false. The ASL is an estimate of the p-value by permutation and bootstrap methods. • https://www.cs.tcd.ie/Rozenn.Dahyot/453Bootstrap/05_Permutation.pdf

16. Bootstrap Tests • Methodology • Flowchart • R code

17. Bootstrap Tests • Beran (1988) showed that bootstrap inference is refined when the quantity bootstrapped is asymptotically pivotal. • It is often used as a robust alternative to inference based on parametric assumptions. • http://socserv.mcmaster.ca/jfox/Books/Companion/appendix-bootstrapping.pdf

18. Hypothesis Testing by a Pivot (1) • Pivot or pivotal quantity: a function of observations whose distribution does not depend on unknown parameters. • http://en.wikipedia.org/wiki/Pivotal_quantity • Examples: • A pivot: when and is known

19. Hypothesis Testing by a Pivot (2) • An asymptotic pivot: when where , is unknown, and

20. One Sample Bootstrap Tests • T statistics can be regarded as a pivot or an asymptotic pivotal when the data are normally distributed. • Bootstrap T tests can be applied when the data are not normally distributed.

21. Bootstrap T tests • Flowchart • R code

22. Flowchart of Bootstrap T Tests Bootstrap B times

23. Bootstrap T Tests by R • Output

24. Bootstrap Tests by The “Bca” • The BCa percentile method is an efficient method to generate bootstrap confidence intervals. • There is a correspondence between confidence intervals and hypothesis testing. • So, we can use the BCa percentile method to test whether H0 is true. • Example: use BCa to calculate p-value

25. BCa Confidence Intervals: • Use R package “boot.ci(boot)” • Use R package “bcanon(bootstrap)” • http://qualopt.eivd.ch/stats/?page=bootstrap • http://www.stata.com/capabilities/boot.html

26. R package "boot.ci(boot)" • http://finzi.psych.upenn.edu/R/library/boot/DESCRIPTION

27. An Example of "boot.ci" in R • Output

28. R package "bcanon(bootstrap)" • http://finzi.psych.upenn.edu/R/library/bootstrap/DESCRIPTION

29. An example of "bcanon" in R • Output

30. BCa • http://qualopt.eivd.ch/stats/?page=bootstrap

31. Two Sample Bootstrap Tests • Flowchart • R code

32. Flowchart of Two-Sample Bootstrap Tests combine m+n=N Bootstrap B times

33. Two-Sample Bootstrap Tests by R • Output

34. Permutation Tests • Methodology • Flowchart • R code

35. Permutation • In several fields of mathematics, the term permutation is used with different but closely related meanings. They all relate to the notion of (re-)arranging elements from a given finite set into a sequence. • http://en.wikipedia.org/wiki/Permutation

36. Permutation Tests (1) • Permutation test is also called a randomization test, re-randomization test, or an exact test. • If the labels are exchangeable under the null hypothesis, then the resulting tests yield exact significance levels.

37. Permutation Tests (2) • Confidence intervals can then be derived from the tests. • The theory has evolved from the works of R.A. Fisher and E.J.G. Pitman in the 1930s. • http://en.wikipedia.org/wiki/Pitman_permutation_test

38. Applications of Permutation Tests (1) • We can use a permutation test only when we can see how to resample in a way that is consistent with the study design and with the null hypothesis. • http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf

39. Applications of Permutation Tests (2) • Two-sample problems when the null hypothesis says that the two populations are identical. We may wish to compare population means, proportions, standard deviations, or other statistics. • Matched pairs designs when the null hypothesis says that there are only random differences within pairs. A variety of comparisons is again possible. • Relationships between two quantitative variables when the null hypothesis says that the variables are not related. The correlation is the most common measure of association, but not the only one.

40. Inference by Permutation Tests (1) • A traditional way is to consider some hypotheses: and , and the null hypothesis becomes . Under , the statistic can be modeled as a normal distribution with mean 0 and variance . • https://www.cs.tcd.ie/Rozenn.Dahyot/453Bootstrap/05_Permutation.pdf

41. Inference by Permutation Tests (2) • The ASL is then computed by when is unknown and has to be estimated from the data by We will reject if .

42. Flowchart of The Permutation Test for Mean Shift in One Sample Partition 2 subset B times (treatment group) (treatment group) (control group) (control group)

43. An Example for One Sample Permutation Test by R (1)

44. An Example for One Sample Permutation Test by R (2) • http://mason.gmu.edu/~csutton/EandTCh15a.txt

45. An Example for One Sample Permutation Test by R (3) • Output

46. Flowchart of The Permutation Test for Mean Shift in Two Samples combine m+n=N Partition subset B times treatment subgroup control subgroup treatment subgroup control subgroup

47. Bootstrap Tests vs. Permutation Tests • Very similar results between the permutation test and the bootstrap test. • is the exact probability when . • is not an exact probability but is guaranteed to be accurate as an estimate of the ASL, as the sample size B goes to infinity. • https://www.cs.tcd.ie/Rozenn.Dahyot/453Bootstrap/05_Permutation.pdf

48. Cross-validation • Methodology • R code

49. Cross-validation • Cross-validation, sometimes called rotation estimation, is the statistical practice of partitioning a sample of data into subsets such that the analysis is initially performed on a single subset, while the other subset(s) are retained for subsequent use in confirming and validating the initial analysis. • The initial subset of data is called the training set. • The other subset(s) are called validation or testing sets. • http://en.wikipedia.org/wiki/Cross-validation

50. Overfitting Problems (1) • In statistics, overfitting is fitting a statistical model that has too many parameters. • When the degrees of freedom in parameter selection exceed the information content of the data, this leads to arbitrariness in the final (fitted) model parameters which reduces or destroys the ability of the model to generalize beyond the fitting data.