PBHL 5313
Nonparametric Methods
Module III
Zoran Bursac
Biostatistics
UAMS
Overview
1. Logistics
2. Chapter 5
a)
b)
c)
d)
e)
f)
g)
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
h) Sections 5.5,5.9-5.11 reading only
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
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)
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 ordinaltype 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%).
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.
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.
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.
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 X and Y mutually
independent?”.
• Exact quantiles are give in Tables A10
(Spearman) and A11 (Kendall) while large
sample approximation is given in Table A1.
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.
5.7 The One-sample or Matchedpairs 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.
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.