Statistic in Health Care Management
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chapter10.pdf
Chapter 10
Nonparametric Tests
Multiple Logistic Regression Model for
Hypertension (Y/N)
Predictor b p OR (95% CI for OR)
Intercept -5.407 0.0001
Age 0.052 0.0001 1.053 (1.044-1.062)
Male -0.250 0.0007 0.779 (0.674-0.900)
BMI 0.158 0.0001 1.171 (1.146-1.198)
Learning Objectives
• Compare and contrast parametric and nonparametric
tests
• Identify multiple applications where nonparametric
approaches are appropriate
• Perform and interpret the Mann- Whitney U test
Learning Objectives
• Perform and interpret, compare and contrast the
Sign test and Wilcoxon Signed Rank test
• Perform and interpret the Kruskal Wallis test
• Identify the appropriate nonparametric hypothesis
testing procedure based on type of outcome and
number of samples
Nonparametric Tests
• Appropriate when outcome is continuous but
not normally distributed
– Rank scores (e.g., disease stage)
– Continuous but subject to extremes
– Continuous but there are limits of detection (on
high or low end of scale)
General Approach
• Rank data
• Perform analysis on ranks
• Follow same 5-step procedure for hypothesis
testing
Ranking Data
• Raw data
7 5 9 3 0 2
• Ordered data
0 2 3 5 7 9
• Ranked data
1 2 3 4 5 6
Ranking Data with Ties
• Raw data
7 7 9 3 0 2
• Ordered data
0 2 3 7 7 9
• Ranked data
1 2 3 4.5 4.5 6
Assign mean rank to ties,
Sum of ranks = n(n+1)/2
Tests with Two Independent
Samples: Mann-Whitney U Test • Continuous outcome that is not assumed to follow a normal
distribution
• 2 Independent Samples
H0: Two populations are equal
H1: Two populations are not equal
Tests with Two Independent
Samples: Mann-Whitney U Test
• Test Statistic is U = min(U1, U2)
where R1 and R2 are the sums of the ranks
in groups 1 and 2.
• Reject H0 if U < critical value in Table 5
1 11
211 R
2
1)(nn nnU
2 22
212 R
2
1)(nn nnU
Example 10.1.
Mann-Whitney U Test
A Phase II clinical trial is run to investigate
efficacy of a new drug for asthma in children.
Outcome is number of episodes of shortness of
breath over a 1-week period.
Placebo 7 5 6 4 12
Drug 3 6 4 2 1
Example 10.1.
Mann-Whitney U Test
H0: The two populations are equal
H1: The two populations are not equal
a=0.05
Test statistic is U.
Rank data in pooled sample (n=10), and compute
R1 and R2.
Example 10.1.
Mann-Whitney U Test
Example 10.1.
Mann-Whitney U Test
• Test Statistic U = 3
• Reject H0 if U < 2 (Table 5)
• Do not reject H0 because 3>2. We do not have significant
evidence to show that the two populations are not equal.
337 2
5(6) 5(5)R
2
1)(nn nnU
1 11
211
2218 2
5(6) 5(5)R
2
1)(nn nnU
2 22
212
Tests with Matched Samples: Sign
Test
• Continuous outcome measured in matched or paired
samples, differences are not assumed to follow a
normal distribution
• Matched or Paired Samples
H0: Median difference is zero
H1: Median difference >, < or ≠ 0
Tests with Matched Samples: Sign
Test
• Test Statistic is the smaller of the number of positive
or negative signs (of differences)
• Reject H0 if the smaller of the number of positive of
negative signs < critical value in Table 6
Example 10.5.
Sign Test
A new chemotherapy treatment is proposed for
patients with breast cancer. Investigators want to
assess tolerability of treatment.
Outcome is quality of life (QOL) measured on an
ordinal scale (1=poor, 2=fair, 3=good, 4=very good,
5=excellent) both before and after treatment.
Example
10.5.
Sign
Test
Patient QOL Before
Treatment
QOL After
Treatment
1 3 2
2 2 3
3 3 4
4 2 4
5 1 1
6 3 4
7 2 4
8 3 3
9 2 1
10 1 3
11 3 4
12 2 3
Observed Data
Example 10.5. Sign Test
Patient QOL Before
Treatment
QOL After
Treatment
Differences
1 3 2 -1
2 2 3 1
3 3 4 1
4 2 4 2
5 1 1 0
6 3 4 1
7 2 4 2
8 3 3 0
9 2 1 -1
10 1 3 2
11 3 4 1
12 2 3 1
Difference Scores
Example 10.5. Sign Test
Patient QOL Before
Treatment
QOL After
Treatment
Differences Signs
1 3 2 -1 -
2 2 3 1 +
3 3 4 1 +
4 2 4 2 +
5 1 1 0 -
6 3 4 1 +
7 2 4 2 +
8 3 3 0 +
9 2 1 -1 -
10 1 3 2 +
11 3 4 1 +
12 2 3 1 +
Signs of the Difference Scores
NOTE: Randomly assign “+” or “-” when there are zeros
Example 10.5.
Sign Test
• Test Statistic is 3
• Reject H0 if the smaller of the number of positive or
negative signs < 2 (Table 6)
• Do not reject H0 because 3>2. We do not have
significant evidence to show that there is a difference
in QOL measured before versus after chemotherapy
treatment.
Tests with Matched Samples:
Wilcoxon Signed Rank Test
• Continuous outcome measured in matched or paired
samples, differences are not assumed to follow a
normal distribution
• Matched or Paired Samples
H0: Median difference is zero
H1: Median difference >, < or ≠ 0
Tests with Matched Samples:
Wilcoxon Signed Rank Test
• Test Statistic is W, the smaller of W+ and W-, the
sums of the positive and negative ranks of the
differences scores
• Reject H0 if W < critical value in Table 7
Example 10.7.
Wilcoxon Signed Rank Test
A study is run to evaluate the effectiveness of a
new exercise program to reduce systolic blood
pressure (SBP) in patients with pre-hypertension.
n=15 patients participate and have SBP
measured before and after 6 weeks on the
program. Is there a significant difference in SBP
after participating in the program?
Example
10.7.
Wilcoxon
Signed
Rank Test
Patient SBP Before SBP After
1 125 118
2 132 134
3 138 130
4 120 124
5 125 105
6 127 130
7 136 130
8 139 132
9 131 123
10 132 128
11 135 126
12 136 140
13 128 135
14 127 126
15 130 132
Observed Data
Example 10.7. Wilcoxon Signed Rank Test
Patient SBP Before SBP After Differences
1 125 118 7
2 132 134 -2
3 138 130 8
4 120 124 -4
5 125 105 20
6 127 130 -3
7 136 130 6
8 139 132 7
9 131 123 9
10 132 128 4
11 135 126 9
12 136 140 -4
13 128 135 -7
14 127 126 1
15 130 132 -2
Difference Scores
Example 10.7. Wilcoxon Signed Rank Test Patient Differences Ordered Absolute Values of
Differences
Ranks
1 7 1 1
2 -2 -2 2.5
3 8 -2 2.5
4 -4 -3 4
5 20 -4 6
6 -3 -4 6
7 6 4 6
8 7 6 8
9 9 -7 10
10 4 7 10
11 9 7 10
12 -4 8 12.5
13 -7 8 12.5
14 1 9 14
15 -2 20 15
Ranks of the Difference Scores
Example 10.7. Wilcoxon Signed Rank Test
Differences Ordered Absolute Values of
Differences
Ranks Signed Ranks
7 1 1 1
-2 -2 2.5 -2.5
8 -2 2.5 -2.5
-4 -3 4 -4
20 -4 6 -6
-3 -4 6 -6
6 4 6 6
7 6 8 8
9 -7 10 -10
4 7 10 10
9 7 10 10
-4 8 12.5 12.5
-7 8 12.5 12.5
1 9 14 14
-2 20 15 15
Signed Ranks of the Difference Scores
W+ = 89 W-=31
Example 10.7.
Wilcoxon Signed Rank Test
• Test Statistic is W=31
• Reject H0 if W < 25 (Table 7)
• Do not reject H0 because 31>25. We do not have
significant evidence to show that the median
difference in SBP is not zero.
Tests with More Than Two Independent Samples:
Kruskal Wallis Test
• Continuous outcome that is not assumed to follow a
normal distribution
• k (k>2) Independent Samples
H0: k population medians are equal
H1: k population medians are not all
equal
Tests with More Than Two Independent Samples:
Kruskal Wallis Test
• Test Statistic is H
where k=number of groups, N=total sample size, nj =
sample size in jth group, Rj= sum of the ranks in jth group.
• Reject H0 if H > critical value in Table 8
1)3(N n
R
1)N(N
12 H
k
1j j
2
j
Example 10.8.
Kruskal Wallis Test
A clinical study is run to assess differences in
albumin levels in patients following 5%, 10%
and 15% protein diets.
5% Protein 10% Protein 15% Protein
3.1 3.8 4.0
2.6 4.1 5.5
2.9 2.9 5.0
3.4 4.8
4.2
Example 10.8.
Kruskal Wallis Test
H0: The three population medians are equal
H1: The three population medians are not
equal
a=0.05
Test statistic is H.
Rank data in pooled sample (n=12), and compute R1, R2
and R3.
Example 10.8.
Kruskal Wallis Test Observed Data Ordered Data (Pooled) Ranks
5%
Protein
10%
Protein
15%
Protein
5%
Protein
10%
Protein
15%
Protein
5%
Protein
10%
Protein
15%
Protein
3.1 3.8 4.0 2.6 1
2.6 4.1 5.5 2.9 2.9 2.5 2.5
2.9 2.9 5.0 3.1 4
3.4 4.8 3.4 5
4.2 3.8 6
4.0 7
4.1 8
4.2 9
4.8 10
5.0 11
5.5 12
Example 10.8.
Kruskal Wallis Test
• R1=7.5, R2=30.5, R3=40.
• Test Statistic is H
• Reject H0 if H > 5.656 (Table 8)
52.7)13(3 4
40
5
5.30
3
5.7
12(13)
12
1)3(N n
R
1)N(N
12 H
222
k
1j j
2
j
Tests with More Than Two Independent Samples:
Kruskal Wallis Test
• Reject H0 because 7.52>5.656. We have statistically
significant evidence to show that there is a difference
in median albumin levels among the three diets.
