Independent samples- Wilcoxon rank sum test

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Independent samples-Wilcoxon rank sum test
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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

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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?

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• 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

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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

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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
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Approximate Wilcoxon test

• If the sample size is large enough (rule of
  thumb, n20) an approximate Wilcoxon test based
  on the normal approximation can be used
• Wsum 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

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mT and sT

• Under the null of no difference between the
  groups, this expression is the expected sum of
  ranks in the small group
• The standard deviation is given by this formula

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Results

• From our results,
• sum of the ranks in smaller group W1526
• expected value of sum of positive ranks
• Standard deviation of sum of positive ranks
• Our approximate test statistic is

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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

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Hypothesis test

1. H0 median difference0
2. Continuous outcome from paired data
3. Wilcoxon signed rank test
4. Test statistic z0.91
5. p-value 0.36
6. Since the p-value is more than 0.05, we fail to
   reject the null hypothesis
7. We conclude that the there is no significant
   difference in terms of EDSS in pediatric and
   adult MS patients

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z-statistic
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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 t-test even
  when the normality assumption holds.
• If normality does not hold or ordinal data,
  Wilcoxon test is better

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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.