PBHL 5313 Nonparametric Methods Module III

1. PBHL 5313Nonparametric MethodsModule III Zoran Bursac Biostatistics UAMS

2. Overview • Logistics • Chapter 5 • 5.1 - Mann-Whitney Test • 5.2 - Kruskal-Wallis Test • 5.3 - Squared Ranks Test for Variance • 5.4 - Spearman’s Rho, Kendall’s Tau • 5.6 - Monotonic Regression • 5.7 - Wilcoxon Signed Ranks Test • 5.8 - Friedman Test, Quade Test, Page Test for Ordered Alternatives • Sections 5.5,5.9-5.11 reading only

3. Logistics • Add section 5.6 to readings; add Quade test to section 5.8 readings; delete exc 1 section 5.5 from the syllabus (however you have to be able to apply methods of section 5.5 in order to do 5.6) • SAS code for several tests covered in this module will be available on the course web site www.uams.edu/biostat/PBHL5313.htm • SAS code for some nonparametric tests can also be found on the UCLA web site www.ats.ucla.edu/stat/mult_pkg/whatstat/default.htm • You can download a free 30 day trial version of StatXact software at www.cytel.com/Downloads/Default.asp

4. Change of Dynamic • Due to a lack of e-mail communication everyone is required to post weekly e-mail to the entire group (content is not to be jokes but related to the class matter, unless the joke is about the rank based method we are currently covering)

5. Chapter 5 • Previous chapters introduced methods that can be applied to data that follows dichotomous or nominal scale of measurement or that can be classified according to multiple criteria in multiple classes. • This chapter will introduce rank based methods. • If data are nonnumeric but ranked like ordinal-type data, methods of this chapter are often the most powerful ones. • If the data are numeric observations of random variables and meets the assumptions of usual parametric tests the loss of efficiency by applying the methods of this chapter are relatively small (~5%).

6. 5.1 Two Independent Samples • While there are many nonparametric tests available for this scenario we are going to focus on the Mann-Whitney test also known as the Wilcoxon-Mann-Whitney test. • For two random variables X and Y that are at least ordinal this test answers the question “Is distribution F(x) equal to the distribution G(y)?”. • For large sample approximation use Table A.1. • For the exact test use Table A.7.

7. 5.2 Several Independent Samples • Extension of Mann-Whitney test to k independent samples, where k>2, is called Kruskal-Wallis test. • For k random samples of possibly different sizes this test answers the question “Are distribution functions of k populations identical?”. • The exact distribution is given by Table A8. Use chi-squared distribution with k-1 degrees of freedom (Table A2) to approximate the null distribution of T for large samples.

8. 5.3 A Test for Equal Variances • In some situations the variances of populations may be the quantity of interest hence we could apply the squared ranks test for variances. • For two random samples X and Y, this test answers the question “Are X and Y identically distributed, except for possibly different means?”, or in other words Var(X)=Var(Y). • This test can easily be extended to more than two samples. • The exact distribution is given in Table A9, and large-sample approximation in Table A1. • Sometimes this test is referred to as Conover’s test.

9. 5.4 Measures of Rank Correlation • While there are several tests that measure correlation between bivariate pairs (X, Y) we are going to focus on Spearman’s Rho and Kendall’s Tau. • Both test the null hypothesis of mutual independence between two random variables and answer the question “Are Xand Y mutually independent?”. • Exact quantiles are give in Tables A10 (Spearman) and A11 (Kendall) while large sample approximation is given in Table A1.

10. 5.6 Methods for Monotonic Regression • The procedures for monotonic or rank based regression are based on the fact that if two variables have a monotonic relationship their ranks will have a linear relationship. • This test answers the question similar to rank correlation and in fact Spearman’s Rho can be adopted to test the null hypothesis of independence or equality of slope to 0 (beta=0). • Other nonparametric regression methods are available like robust regression, kernel density estimation, nonparametric curve smoothing etc.

11. 5.7 The One-sample or Matched-pairs Case • The rank test of this section looks at random sample of matched pairs. • The Wilcoxon signed ranks test will answer the question whether the difference between two matched random variables X and Y is equal to zero. • For n<=50 quantiles can be obtained from the Table A12, otherwise they can be approximated based on the Table A1.

12. 5.8 Several Related Samples • This section covers rank tests for multiple related samples namely Friedman’s test (extension of Sign test), Quade test (extension of Wilcoxon signed ranks test) and Page test (related to Kendall’s and Spearman’s measures of association). • All three tests of this section answer the question “Is each ranking of the random variable within the block equally likely?” or in other words do the treatments have identical effects? • Friedman’s and Quade’s quantiles can be found in Table A22 while Page’s use large sample approximation of Table A1.