The Wilcoxon Signed Ranks Test
Douglas Whitaker
February 25, 2012
These notes should help clarify how the Wilcoxon Signed Ranks Test works for all three possible alter
native hypotheses.
1
Assumptions
The Wilcoxon Signed Ranks Test is the nonparametric alternative to the matchedpairs ttest, meaning that
if the assumptions for the matchedpairs ttest are violated, this should be the next method we look at. Its
assumptions are:
•
Data obtained from randomization
•
Data are matched pairs
•
Quantitative response
Note the lack of a sample size or normality assumption (this lack is what makes it ‘nonparametric’).
2
Hypotheses
Our null hypothesis for this test is
H
0
: the median of the differences = 0
There are three possible alternative hypotheses, and the one you choose will be based on the research question
we’re trying to answer. I’ll give examples of all three below.
H
A
: the median of the differences
>
0
H
A
: the median of the differences
<
0
H
A
: the median of the differences
6
= 0
Remember, we’re using the
median
rather than the mean because we will rank the data. It isn’t appropriate
to use the mean with ranked data.
3
The Problem
(This problem is from A&F Chapter 15.) A diet (one where participants aren’t allowed drinks containing
high fructose corn syrup) claims to promote weight loss. To that end, a pilot study is performed with three
participants. The data collected are:
1
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Subject
PreDiet
PostDiet
Difference
1
172
160
12
2
210
195
15
3
168
170
2
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 Spring '08
 TA
 Null hypothesis, Douglas Whitaker

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