Independent samplesWilcoxon rank sum test
Example
The main outcome measure in MS is the
expanded disability status scale (EDSS)
 The EDSS is a 0-10 scale with steps of 0.5
 Ordinal scale

– Ordered, but magnitude between steps is uncertain

Dr. Kurtzke who developed the scale believes the
steps of scale are just a rank, not a measure of
magnitude
– This makes a t-test inappropriate
Pediatric vs. adult
Most MS patients develop the disease
between age 20-40, but a subset of
patients develop MS younger
 What is different about these patients?
 If we investigated patients at similar
disease duration, is there a significant
difference in EDSS?

Since we have two
independent samples,
we could have used
two-sample t-test
 Unfortunately, there
seem to be outliers in
the adult group
 Also, we know that
we have ordinal data
so a t-test is not
appropriate

Wilcoxon rank sum test
Since we have two independent samples and the
t-test is not appropriate, we need a
nonparametric test. The test for two
independent samples is Wilcoxon rank sum.
 Again, we are interested in the median rather
than the mean.
 The hypothesis test of interest is

– H0: medianadult = medianpediatric
– HA: medianadult != medianpediatric
Wilcoxon rank sum
Again, we use the rank
of the data points,
rather than the actual
values.
 An exact Wilcoxon
rank sum test can be
used, but we focus on
the approximate

Patient EDSS
Group
Rank
1
0
P
1
2
1.5
P
4.5
3
1.5
P
4.5
4
1
P
2.5
5
2
A
6
6
1
A
2.5
7
3
A
7
Approximate Wilcoxon test

If the sample size is large enough (rule of
thumb, n=20) an approximate Wilcoxon test
based on the normal approximation can be used
zW
W  mW

sW
– W=sum of ranks in smaller group
– mW =expected sum of ranks in smaller group under
null
– sW =standard deviation of sum of ranks in smaller
group under null
mT and sT

Under the null of no difference between
the groups, this expression is the expected
sum of ranks in the small group
n S (n S  n L  1)
mW 
2

The standard deviation is given by this
formula
n S n L (n S  n L  1)
sW 
12
Results

From our results,
– sum of the ranks in smaller group: W=1526
– expected value of sum of positive ranks:
mW 
21* (21  110  1)
 1386
2
– Standard deviation of sum of positive ranks
21*110 * (21  110  1)
sT 
 159.4
12

Our approximate test statistic is
1526  1386
Z
 0.88
159.4
Ties
In this example, we have many ties
 As with the Wilcoxon signed rank test, a
correction for ties can be made to the
variance (see Rosner or other text book)
 This correction is included in STATA and all
other computer packages

Hypothesis test
1)
2)
3)
4)
5)
6)
7)
H0: median difference=0
Continuous outcome from paired data
Wilcoxon signed rank test
Test statistic: z=0.91
p-value= 0.36
Since the p-value is more than 0.05, we fail to
reject the null hypothesis
We conclude that the there is no significant
difference in terms of EDSS in pediatric and
adult MS patients
z-statistic
p-value
Comments
Wilcoxon rank sum test is becoming more
prominent because computers allow this
statistic to be calculated very quickly
 There is not a large loss of power in using
a Wilcoxon rank sum test compared to a ttest even when the normality assumption
holds.
 If normality does not hold or ordinal data,
Wilcoxon test is better

Parametric tests-nonparametric
equivalent
Paired t-test – Wilcoxon signed rank
 Two sample t-test – Wilcoxon rank sum
 ANOVA – Kruskal-Wallis test

– When you have two or more independent
samples and the assumptions of ANOVA are
not met, you can use the Kruskal-Wallis test.
This is a rank based test.