Nonparametric Methods
Overview
All of what we have done thus far assumed that we were working with data that came from some underlying
distribution. Most of the time we assumed that the data were normally distributed (or that the Central Limit Theorem
allowed us to treat data as normally distributed) or that it came from a binomial distribution. The methods we used
(constructing confidence intervals, hypothesis testing) were based on these assumptions. Sometimes it is difficult to
make such assumptions and we need to rely on other methods of analysis. Chapter 9 of Rosner covers the subject
of Nonparametric Methods, a class of statistical methods that require far fewer assumptions. There are many such
nonparametric methods and there are entire texts and entire courses that cover this subject area. In this course we
will introduce and describe some of the more common nonparametric methods.
While we have recommended that you read the Cartoon Guide throughout this course as a supplemental text,
please note that the Cartoon Guide does not cover the topic of non parametric statistics.
Learning Objectives
1. Understand when nonparametric statistical methods are appropriate.
2. Understand the difference between cardinal and ordinal data.
3. Know how to perform the Sign Test and when it should be used.
4. Know how to perform the Wilcoxon Signed-Rank Test and when it should be used.
5. Know how to perform the Wilcoxon Rank-Sum Test and when it should be used.
Readings
Chapter 9: 5/4
Problems
9.1- 9.6, 9.19- 9.23,9.43: 5/9 (NOTE: Use STATCRUNCH when you have real data.)
What You Should Know
Throughout this course we have been making assumptions about the underlying shape or distribution of
the data. These assumptions were needed in order to apply the statistical methods discussed thus far.
We always seemed to be beginning our problem solving by assuming the data were normal or assuming it
came from a binomial distribution. What should we do if we cannot make these assumptions? What do
we do when we have the type of data that is not measurable but which is rather in a form where we cannot
perform the usual arithmetic functions? When these type of situations arise, we rely on a class of
statistical methods that are called nonparametric. They require far fewer assumptions in order to be used.
The introduction to Chapter 9 defines cardinal data and ordinal data. If you read the examples provided in
this section, you should get a pretty good idea of what these terms mean and what types of data fall into
these categories. If you have data on a cardinal scale, nonparametric methods are most useful if you
cannot or are not willing to make the required assumptions about the normality of the underlying data to
justify the use of the traditional parametric methods thus far covered. If you have ordinal data to begin
with, then non parametric methods are the techniques of choice.
Section 9.2 covers the Sign Test. This test is useful when we have ordinal data where we only know that
one score or value is greater than, less than, or equal to another score or value. It is analogous to the
paired t-test covered in Chapter 8 of Rosner. Example 9.7 should enable you to understand this type of
situation. Two methods for performing the sign test are given: the large sample method which assumes
that the normal approximation to the binomial is valid and the small sample method for n less than or
equal 20. Note that this method is called the sign test because it depends on the sign of the differences
and not on the actual magnitude of the differences. If you follow the examples provided in the text, you
should be able to perform these tests without too much difficulty.
The Wilcoxon Signed-Rank Test is covered in Section 9.3. This also is a nonparametric analogue to the
paired t-test, covered in Chapter 8 or Rosner. The example provided in Section 9.3 should give you a
good idea of the type of situation where the use of the Wilcoxon Signed-Rank Test is appropriate.
Equations 9.4 and 9.5 outline the procedure to be followed to use this test. Please note that this
procedure is to be followed when n (number of pairs with non zero differences) is greater than 15. For
cases where n is less than or equal to 15, the normal approximation to the binomial is no longer valid.
Table 11 in Rosner can be used for these small sample cases. Example 9.14 shows you how to use this
table.
The last nonparametric method we will consider is the Wilcoxon Rank-Sum Test which is the analogue of
the previously discussed t-test for two independent samples. If you carefully read Section 9.4 and follow
the examples provided you should be able to identify those situations where the Wilcoxon Rank-Sum Test
is appropriate. The procedure is outlined in Equations 9.6 and 9.7 for cases where both sample sizes are
at least 10. If either sample size is less than 10, the normal approximation is not valid and you can use
Table 12 in Rosner to find the exact significance levels.