Discuss the assumptions of parametric statistical testing versus the assumptions of nonparametric tests.
Discuss why a researcher would select a nonparametric approach based on the data and when they
would select parametric tests for their data set. Does it matter what type of variables have been collected
in the dataset?
Embed course material concepts, principles, and theories (which require supporting citations) in your
initial response along with at least one scholarly, peer-reviewed journal article. Keep in mind that these
scholarly references can be found in the Saudi Digital Library by conducting an advanced search specific
to scholarly references. Use Saudi Electronic University academic writing standards and APA style
guidelines.
Module 10
Chapter 10
Nonparametric
Tests
Learning Objectives
• Compare and contrast parametric and
nonparametric tests
• Identify multiple applications where
nonparametric approaches are appropriate
• Compare and interpret the t-test and
Wilcoxon Signed Rank test
• Assess and interpret the Kruskal-Wallis test
• Identify the appropriate nonparametric
hypothesis testing procedure based on type
of outcome variable and 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)
Parametric and Nonparametric
Test Assumptions
• Parametric
–
–
–
–
Normal distribution
Representative sample
Powerful
More flexible
• Nonparametric
– Data not normally
distributed
– Sample too small to
meet confidence
interval
– Not as powerful as
parametric tests
– Based on ranks
General Approach
• Rank data
• Perform analysis on ranks
• Follow same 5-step procedure for
hypothesis testing
Ranking Data
• Raw data
7
5
• Ordered data
0
2
9
3
0
2
3
5
7
9
3
4
5
6
• Ranked data
1
2
Ranking Data with Ties
• Raw data
7
7
• Ordered data
0
2
9
3
0
2
3
7
7
9
• Ranked data
1
2
3
4.5 4.5
Assign mean rank to ties,
Sum of ranks = n(n + 1)/2
6
Tests with Matched Samples:
Wilcoxon Signed Rank Test (1 of 2)
• 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 >, 25. We do
not have significant evidence to show that
the median difference in SBP is not zero.
Example 10.5.
Sign Test (1 of 5)
• 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 (2 of 5)
• Observed data
Example 10.5.
Sign Test (3 of 5)
• Difference scores
Example 10.5.
Sign Test (4 of 5)
• Signs of the difference scores
NOTE: Randomly assign “+” or “–” when there are zeros.
Example 10.5.
Sign Test (5 of 5)
• 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 More Than Two Independent
Samples: Kruskal–Wallis Test (1 of 2)
• 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 (2 of 2)
• Test statistic is H,
k R2
12
j
− 3(N + 1)
H=
N(N + 1) j=1 n
j
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.
Example 10.8.
Kruskal–Wallis Test (1 of 4)
• A clinical study is run to assess
differences in albumin levels in patients
following 5%, 10%, and 15% protein diets.
Example 10.8.
Kruskal–Wallis Test (2 of 4)
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 (3 of 4)
Example 10.8.
Kruskal–Wallis Test (4 of 4)
• R1 = 7.5, R2 = 30.5, R3 = 40.
• Test statistic is H.
k R2
12
j
− 3(N + 1)
H=
N(N + 1) j=1 n
j
12 7.52 30.52 40 2
− 3(13) = 7.52
=
+
+
12(13) 3
5
4
• Reject H0 if H ≥ 5.656 (Table 8).
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.
Module 11
Chapter 10
Distribution Free
Methods
Learning Objectives
• Identify the statistical tests used for
distribution-free data
• Identify the assumptions for a nonparametric
test
• Interpret the Mann-Whitney U test
• Interpret the test for independence
• Interpret hypothesis testing with
nonparametric tests
Tests with Two Independent Samples:
Mann–Whitney U Test (1 of 2)
• Continuous outcome that is not assumed
to follow a normal distribution
• Two independent samples
H0: Two populations are equal
H1: Two populations are not equal
Tests with Two Independent Samples:
Mann–Whitney U Test (2 of 2)
• Test statistic is U = min(U1, U2),
n 2 (n 2 + 1)
n1 (n 1 + 1)
U1 = n1n 2 −
− R2
− R 1 U 2 = n1n 2 −
2
2
where R1 and R2 are the sums of the ranks in
groups 1 and 2.
• Reject H0 if U ≤ critical value in Table 5.
Example 10.1.
Mann–Whitney U Test (1 of 4)
• 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
Drug
7
3
5
6
6
4
4
2
12
1
Example 10.1.
Mann–Whitney U Test (2 of 4)
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 (3 of 4)
Example 10.1.
Mann–Whitney U Test (4 of 4)
n1 (n1 + 1)
5(6)
U1 = n1n 2 −
− R1 = 5(5) −
− 37 = 3
2
2
n 2 (n 2 + 1)
5(6)
U 2 = n 1n 2 −
− R 2 = 5(5) −
− 18 = 22
2
2
• Test statistic is 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.
Tests with Matched Samples:
Sign Test (1 of 2)
• 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 >,
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