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Nonparametric Methods
Week 13a
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9. Wilcoxon test

• W test is used in the same circumstances as the
  sign test (one sample two related samples).
• One of the main advantages of sign test is that
  the assumptions for its use are minimal. However,
  the sign test considers only whether each
  observation is above () or below (-) the median.
  It does not keep track of how far above or below
  median, each data value is. Consequently, a large
  amount of pertinent information is potentially
  ignored.
• In addition to the sign, W test takes into
  consideration the magnitude of the difference.
  This feature results in a more powerful test (see
  Table 1).
• However, there is an assumption for the W test
  that is not required for the sign test
• population being sampled has a symmetric
  distribution.

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Example 3

• Assuming that that populations are approximately
  symmetric but not normal, test whether there is
  a significant difference between the number of
  cigarettes smoked before and after pregnancy.
• Table 2 Number of Cigarettes Usually Smoked per
  Day - Before (B) and After (A) Pregnancy

W 7
4

• Null and alternative hypotheses are the same as
  for the sign test
• Ho MA MB. Ha MB ? MB.
• The test statistic is calculated as follows
• 1. Delete any observation equal to 0 before you
  start.
• 2. Arrange the observations from lowest to
  highest.
• disregarding the signs (take absolute
  values).
• 3. Rank the absolute differences in order
  from lowest (1)
• to highest (in our example 10).
• 4. Assign the sign of the difference to the
  ranks.
• 5. Test statistic W is the sum of the ranks
  of the positive
• differences.
• The idea behind the W test is that if the null
  hypothesis is true then the sum of ranks of the
  positive differences should be about the same as
  the sum of the ranks of the negative difference
  (48 in our example).

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Sign Test for Median Sign test of median
0.00000 versus not 0.00000
N Below Equal Above P
Median Difference 11 3 1
7 0.3437 5.000
Paired T-Test and Confidence Interval Paired T
for Before - After N
Mean StDev SE Mean Before 11
14.64 6.27 1.89 After 11
9.36 7.57 2.28 Difference 11
5.27 7.54 2.27 95 CI for mean
difference (0.21, 10.34) T-Test of mean
difference 0 (vs not 0) T-Value 2.32
P-Value 0.043
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Wilcoxon Signed Rank Test Test of median
0.000000 versus median not 0.000000
N for Wilcoxon
Estimated N Test
Statistic P Median Difference
11 10 7 0.041 4.500

• This result indicates that the difference between
  the observed and expected rank sums is
  significant (p lt 0.05).
• Thus, it leads us to reject H0.
• Can we conclude that there is a significant
  reduction in the smoking habit consequent to
  pregnancy?
• RECOMMENDATION If you are sure that the
  population is normal use the t test. However, if
  you believe that the underlying population is
  symmetric (but not normal) use W test.

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10. The Mann Whitney Wilcoxon test (MWW)

• MWW test is a nonparametric alternative to the
  unrelated (unpaired, pooled) t test for analysing
  data from a different subjects design with two
  groups.
• Assumptions Unlike the parametric t test, this
  nonparametric makes no assumptions about the form
  of the sampled distribution. The only assumption
  is that the population has a continuous
  distribution.
• We will state the hypotheses in terms of equality
  versus inequality of two population medians
• Ho M1 M2
• Ha M1 ? M2
  (two-sided alternative)

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10. The Mann Whitney Wilcoxon test (MWW)

• Conceptual Basis
• A real difference between two treatments should
  make observations in one sample generally larger
  than those in the other.
• If the null hypothesis is false, then when we
  combine the two samples together and rank all the
  combined observations, the data from one sample
  should be concentrated at one end of the scale,
  and the other samples data should be at the
  other end.
• However, if the hull hypothesis is true, then
  data from both samples should be randomly
  scattered throughout the ranking of the pooled
  data.

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Example 4

• The data in the following table depict sulfate
  concentrations in rainwater that were measured at
  two different locations.
• Location A
• 2.4 11.6 11.9 12 12 12.1 12.2 12.2 12.4 14
  14.7 14.8
• Location B
• 10.1 10.4 10.5 10.6 10.8 10.9 11 11.1 11.4
  11.5 13.9 25.1
• Using the provided normal plots test the
  hypothesis that there is no significant
  difference between two locations.

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p value 0 in both cases
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Example 4
State the null and the alternative hypotheses
and nominate the significance level, ?
STEP 1

• Ho M1 M2 (there is no difference between 2
  locations)
• Ha M1 ? M2

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Example 4

• The test statistic is calculated by
• joining the two samples into one and arranging
  them into increasing order. For our data we have
• 2.4 10.1 10.4 10.5 10.6 10.8 10.9 11 11.1 11.4
  11.5 11.6 11.9 12 12 12.1 12.2 12.2 12.4 13.9 14
  14.7 14.8 25.1
• Assigning ranks to all the observations in the
  combined sample (giving average ranks for ties).
  In our example
• 1 2 3 4 5 6 7 8 9 10 11 12 13 14.5 14.5 16 17.5
  17.5 19 20 21 22 23 24
• If you saw only these ranks, what would you guess
  about the null hypothesis?
• WWW test simply quantifies this pattern. The test
  statistic is the sum of ranks of the data in the
  first sample (191).

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Two Sample T-Test and Confidence Interval Two
sample T for A vs B N Mean StDev
SE Mean A 12 11.86 3.18 0.92 B 12
12.27 4.15 1.2 95 CI for mu A -
mu B ( -3.57, 2.7) T-Test mu A mu B (vs not
) T -0.28 P 0.79 DF 20
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Mann-Whitney Confidence Interval and Test A
N 12 Median 12.150 B N
12 Median 10.950 Point estimate for
ETA1-ETA2 is 1.200 95.4 Percent CI for
ETA1-ETA2 is (0.499,1.901) W 191.0 Test of
ETA1 ETA2 vs ETA1 not ETA2 is significant
at 0.0194 The test is significant at 0.0193
(adjusted for ties)
Test statistic
P-Value

• The terms ETA1 and ETA2 represent the medians of
  two groups. What is the conclusion?
• RECOMMENDATION If you are sure that the
  population is normal use the t test. However, if
  you believe that the underlying population is not
  normal, particularly if you have outliers, use
  WWW test. It is much less sensitive to outliers.