1. Statistical Data Analysis 2011/2012 M. de Gunst Lecture 6

2. Statistical Data Analysis 2 Statistical Data Analysis: Introduction Topics Summarizing data Exploring distributions Bootstrap Robust methods Nonparametric tests Analysis of categorical data Multiple linear regression

3. Statistical Data Analysis 3 Today’s topic: Nonparametric methods for one sample problems (Chapter 6: 6.1, 6.2) 6. Nonparametric methods 6.1. One sample: nonparametric tests for location 6.1.1. Sign test 6.1.2. (Wilcoxon) signed rank test 6.2. Aymptotic efficiency

4. Statistical Data Analysis 4 Nonparametric methods: Introduction Nonparametric tests No assumption of parametric family for underlying distribution of data For problems with large class of distributions belonging to H 0 Distribution of test statistic same under every distribution that belongs to H 0 Why these tests? Robust w.r.t. confidence level: conf level α for large class of distributions More efficient than tests with more assumptions when these assumptions do not hold: fewer observations necessary for same power (= onderscheidend vermogen)

5. Statistical Data Analysis 5 6.1. One-sample problem Situation realizations of, independent, unknown distr. F Location of F? Which test do you know for this situation?

6. Statistical Data Analysis 6 One-sample problem: t-test t-test Assumptions: F ~ with unknown. composite which class? Test statistic: ~ t n-1 rejected if P (T ≤ t) ≤ ½ α or P (T ≥ t) ≤ ½ α, t observed value of T Class of all normal distributions with expectation m 0 If σ 2 would be known, say σ 2 =1, H 0 would contain only the N(m 0, 1) distribution and thus would be simple. In that case, no need for t-test, but use Gauss test!

7. Statistical Data Analysis 7 6.1.1. One-sample problem: sign test sign test Assumptions: F has unique median m composite: large class of families which class? Test statistic: ~ binom(n,0.5) rejected if P (T ≤ t) ≤ ½ α or P (T ≥ t) ≤ ½ α, t observed value of T Same for all distributions under : nonparametric test If k observations = m 0 : delete them and use conditional test binom(n-k,0.5))

8. Statistical Data Analysis 8 6.2.2. One-sample problem: (Wilcoxon) signed rank test (1) Signed rank test or rank sign test or symmetry test of Wilcoxon Assumptions: F continuous and symmetric with symmetry point m composite: large class of families which class? Test statistic: ~ R i rank of X i in ordered sequence of | | random permutations of 1, …, n random variables with Same for all distributions under : nonparametric test

9. Statistical Data Analysis 9 One-sample problem: (Wilcoxon) signed rank test (2) (Wilcoxon) signed rank test Test statistic: ~ Why? R i rank of X i in ordered sequence of | |, random permutations of 1, …, n random variables with

10. Statistical Data Analysis 10 One-sample problem: (Wilcoxon) signed rank test (3) (Wilcoxon) signed rank test Assumptions: F continuous and symmetric with symmetry point m Test statistic: ~ R i rank of X i in ordered sequence of | | n large: normal approximation Groups of equal values: adjust ranks R uses equivalent test

11. Statistical Data Analysis 11 One-sample problem: tests and confidence intervals Tests correspond to confidence intervals for location parameter of F What is the correspondence? Confidence intervals based on a test contain values of parameter for which H 0 is not rejected if they would have been in H 0 In R: for t-test and (Wilcoxon) signed rank test: confidence interval also in output for sign test: compute separately

12. Statistical Data Analysis 12 Illustration of one sample tests with R (1) Example Data of Example 6.1

13. Statistical Data Analysis 13 Illustration of one sample tests with R (2) Example Data of Example 6.1

14. Statistical Data Analysis 14 Illustration of one sample tests with R (3) Example Data of Example 6.1

15. Statistical Data Analysis 15 Illustration of one sample tests with R (4) Data of Example 6.1 Example

16. Statistical Data Analysis 16 Illustration of one sample tests with R (5) Data of Example 6.1 Example

17. Statistical Data Analysis 17 Illustration of one sample tests with R (6) Data of Example 6.1 Example

18. Statistical Data Analysis 18 Illustration of one sample tests with R (7) Data of Example 6.1 Example

19. Statistical Data Analysis 19 6.2. Asymptotic efficiency On blackboard

20. Statistical Data Analysis 20 Illustration of investigating efficiency with R (1) Example

21. Statistical Data Analysis 21 Illustration of investigating efficiency with R (2) Example

22. Statistical Data Analysis 22 Illustration of investigating efficiency with R (3) Example

23. Statistical Data Analysis 23 Illustration of investigating efficiency with R (4) Results last sheet: Example

24. Statistical Data Analysis 24 Recap 6. Nonparametric methods 6.1. One sample: nonparametric tests for location 6.1.1. Sign test 6.1.2. (Wilcoxon) signed rank test 6.2. Aymptotic efficiency

25. Statistical Data Analysis 25 Nonparametric methods for one sample problems The end