Inferences about the Differences between Two Medians: Dependent Samples

1
Lesson 15 - 4

• Inferences about the Differences between Two
  Medians Dependent Samples

2
Objectives

• Test a claim about the difference between the
  medians of two dependent samples

3
Vocabulary

• Wilcoxon Matched-Pairs Signed-Ranks Test -- a
  nonparametric procedure that is used to test the
  equality of two population medians by dependent
  sampling.
• Ranks 1 through n with ties award the sum of
  the tied ranks divided by the number tied. For
  example, if 4th and 5th observation was tied then
  they would both receive a 4.5 ranking.
• Signed-Ranks ranks are constructed with
  absolute values and then the sign of the value
  (positive or negative) is applied to the ranking.

4
Parametric vs Nonparametric

• For our parametric test for matched-pairs
  (dependent samples), we
• Compared the corresponding observations by
  subtracting one from the other
• Performed a test of whether the mean is 0
• For our nonparametric case for matched-pairs
  (dependent samples), we will
• Compare the corresponding observations by
  subtracting one from the other
• Perform a test of whether the median is 0

5
Hypothesis Tests Using Wilcoxon Test
Step 0 Compute the differences in the
matched-pairs observations. Rank the absolute
value of all sample differences from smallest to
largest after discarding those differences that
equal 0. Handle ties by finding the mean of the
ranks for tied values. Assign negative values to
the ranks where the differences are negative and
positive values to the ranks where the
differences are positive. Step 1 Hypotheses
Step 2 Box Plot Draw a boxplot of the
differences to compare the sample data from the
two populations. This helps to visualize the
difference in the medians. Step 3 Level of
Significance (level of significance determines
the critical value) Determine a level of
significance, based on the seriousness of making
a Type I error Small-sample case Use Table XI.
Large-sample case Use Table IV. Step 4
Compute Test Statistic Step 5 Critical Value
Comparison Reject H0 if test statistic lt
critical value Step 6 Conclusion Reject or
Fail to Reject
Left-Tailed Two-Tailed Right-Tailed
H0 MD 0H1 MD lt 0 H0 MD 0H1 MD ? 0 H0 MD 0H1 MD gt 0
6
Test Statistic for the Wilcoxon Matched-Pairs
Signed-Ranks Test
Small-Sample Case (n 30) Large-Sampl
e Case (n gt 30)
Left-Tailed Two-Tailed Right-Tailed
H0 MD 0H1 MD lt 0 H0 MD 0H1 MD ? 0 H0 MD 0H1 MD gt 0
T T T smaller of T or T- T T-
Note MD is the median of the differences of
matched pairs. T is the sum of the ranks of
the positive differences T- is the sum of the
ranks of the negative differences
where T is the test statistic from the
small-sample case.
7
Critical Value for Wilcoxon Matched-Pairs
Signed-Ranks Test
Small-Sample Case (n 30) Using a as the level
of significance, the critical value is obtained
from Table XI in Appendix A. Large-Sample
Case (n gt 30) Using a as the level of
significance, the critical value is obtained from
Table IV in Appendix A. The critical value is
always in the left tail of the standard normal
distribution.
Left-Tailed Two-Tailed Right-Tailed
-Ta -Ta/2 -Ta
Left-Tailed Two-Tailed Right-Tailed
-za -za/2 -za
8
Example 1 from 15.4
X Y D Y-X D Signed Rank
11.5 26.0 14.5 14.5 7.5
14.1 26.2 12.1 12.1 5
19.3 24.6 5.3 5.3 3
35.0 30.8 -4.2 4.2 -2
15.9 37.5 21.6 21.6 11
21.5 36.0 14.5 14.5 7.5
11.7 25.9 14.2 14.2 6
17.1 16.9 -0.2 0.2 -1
27.3 50.2 22.9 22.9 12
13.8 33.1 19.3 19.3 10
43.2 99.9 56.7 56.7 14
11.2 26.1 14.9 14.9 9
34.2 43.8 9.6 9.6 4
26.7 63.8 37.1 37.1 13
T 88 T- -3 3
9
Example continued

• H0 Meddif 0
• Ha Meddif gt 0 (Right tailed test)
• Test Statistic T- 3
• From Table XI Tcritcal T0.05 25
• The test statistic is less than the critical
  value (3 lt 25) so we reject the null hypothesis.

10
Summary and Homework

• Summary
• The Wilcoxon sign test is a nonparametric test
  for comparing the median of two dependent samples
• This test is a weighted count of the differences
  in signs between the paired observations
• The critical values for small samples are given
  in tables
• The critical values for large samples can be
  approximated by a calculation with the normal
  distribution
• Homework
• problems 2, 4, 5, 8, 9, 15 from the CD