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Eastern Mediterranean University Faculty of Medicine Biostatistics course
Non-parametric methods March 4&7, 2016
Instructor: Dr. Nimet İlke Akçay ([email protected])
Learning Objectives
1. Distinguish Parametric & Nonparametric Test Methods
2. Explain commonly used Nonparametric Test Methods
3. Perform Hypothesis Tests Using Nonparametric Procedures
Nonparametric Statistics Introduction to nonparametric methods
• Parametric tests (t-test, z-test, etc.)
involve estimating population parameters such as the mean
are based on the assumptions of normality or known variances (stringent assumptions)
• What if data does not follow a Normal distribution?
• Non-parametric tests
were developed for these situations where no (or fewer) assumptions have to be made
are also called as Distribution-free tests
Still have assumptions but they are less stringent
Can be applied for a normally distributed data, but Parametric tests have greater power IF the assumptions met
Parametric vs. Nonparametric test procedures Parametric
• Involve population parameters Example: Population mean
• Require interval scale or ratio scale Whole numbers or fractions
Example: height in inches
(72, 60.5, 54.7)
• Have stringent assumptions Example: Normal dist.
• Examples: z-test, t-test, F-test
Nonparametric
• Do not involve population parameters Example: Probability
distributions, indepedence
• Data measured on any scale Ratio or Interval
Ordinal Example: good-better-best
Nominal Example: male-female
• Have no stringent assumptions
• Examples: Sign test, Mann-Whitney U test, Wilcoxon test
Advantages and Disadvantages of Non-parametric tests Advantages
• Used with all measurement scales Analysis possible for ranked
or categorical data
• Distribution-free: may be used when the form of the sampled population is unknown
• Easy to compute
• Make fewer assumptions
• No need to involve population parameters
• Results may be as exact as parametric procedures
Disadvantages
• May waste information, if data and assumptions permit using parametric procedures Example: converting data
from ratio to ordinal scale
E.g. Height values -> short-average-tall
• Difficult to compute by hand for large samples
• Tables not widely available
Determination: Parametric or Non-parametric?
Hypothesis Testing Procedures
Type of Design Parametric
Test
Nonparametric
Test
One sample One-sample t-
test
Sign Test
Two indepent
samples
Independent-
samples t-test
Wilcoxon Rank
Sum test
Mann Whitney U
test
Two paired
sample
Paired-samples
t-test
Wilcoxon Signed
Ranks test
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One sample hypothesis testing design: Sign Test
Sign Test • This test is used as an alternative to one sample t-test,
when normality assumption is not met
• The only assumption is that the distribution of the underlying variable (data) is continuous.
• Test focuses on median rather than mean.
• The test is based on signs, plus and minuses
• Test is used for one sample as well as for two samples
Test Statistic in Sign Test • Test Statistic: The test statistic for the sign test is
either the observed number of plus signs or the observed number of minus signs.
• The nature of the alternative hypothesis determines which of these test statistics is appropriate. In a given test, any one of the following alternative hypotheses is possible:
• HA: P(+) > P(-) one-sided alternative
• HA: P(+) < P(-) one-sided alternative
• HA: P(+) ≠ P(-) two-sided alternative
Test Statistic in Sign Test • If the alternative hypothesis is HA: P(+) > P(-) a
sufficiently small number of minus signs causes rejection of H0. The test statistic is the number of minus signs.
• If the alternative hypothesis is HA: P(+) < P(-) a sufficiently small number of plus signs causes rejection of H0. The test statistic is the number of plus signs.
• If the alternative hypothesis is HA: P(+) ≠ P(-) either a sufficiently small number of plus signs or a sufficiently small number of minus signs causes rejection of the null hypothesis. We may take as the test statistic the less frequently occurring sign.
• Calculation of test statistic:
P (X ≤ k | n, p) =
qpCxnxk
x
n
x
0
Sign Test – Example • 7 patients were asked to rate the
services in a private hospital on a 5-point scale (1=terrible,..., 5=excellent)
• The obtained ratings are given in the table
• At the α=0.05 level, is there evidence that the median rating is at least 3?
Patient Rating Sign
1
2
3
4
5
6
7
2
5
3
4
1
4
5
-
+
+
+
-
+
+
Sign Test – Example (cont’d) Hypotheses:
H0: = 3
Ha: < 3
Significance Level:
= .05
Test Statistic:
P-Value:
Decision:
Conclusion:
Do Not Reject at = .05
There is No evidence
for Median < 3
P(x 2) = 1 - P(x 1)
= 0.9375
(Binomial Table, n = 7, p = 0.50)
Or calculate from the formula
S = 2
(Ratings 1 & 2
are < = 3:
2, 5, 3, 4, 1, 4, 5)
Comparing Two Populations: Independent Samples Wilcoxon Rank Sum test
Wilcoxon Rank Sum test
• Tests two independent population probability distributions
• Corresponds to t-test for two independent means
• Assumptions — Independent, random samples
— Populations are continuous
• Can use normal approximation if ni 30
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Wilcoxon Rank Sum Test: Independent Samples Let D1 and D2 represent the probability distributions for populations 1 and 2, respectively.
One-Tailed Test
H0: D1 and D2 are identical
Ha: D1 is shifted to the right of D2
[or D1 is shifted to the left of D2]
Test statistic:
T1, if n1 < n2; T2, if n2 < n1
(Either rank sum can be used if n1 = n2.)
Wilcoxon Rank Sum Test: Independent Samples Let D1 and D2 represent the probability distributions for populations 1 and 2, respectively.
Two-Tailed Test
H0: D1 and D2 are identical
Ha: D1 is shifted to the left or to the right of D2
Test statistic:
T1, if n1 < n2; T2, if n2 < n1
(Either rank sum can be used if n1 = n2.) We will denote this rank sum as T.
Conditions Required for Valid Wilcoxon Rank Sum Test
1. The two samples are random and independent.
2. The two probability distributions from which the samples are drawn are continuous.
Comparing Two Populations: Paired Differences Experiment Wilcoxon Signed Ranks test
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Wilcoxon Signed Rank Test
• Tests probability distributions of two related populations
• Corresponds to t-test for dependent (paired) means
• Assumptions — Random samples
— Both populations are continuous
• Can use normal approximation if n 30
© 2011 Pearson Education, Inc
Wilcoxon Signed Rank Test for a Paired Difference Experiment
Let D1 and D2 represent the probability distributions for populations 1 and 2, respectively.
One-Tailed Test
H0: D1 and D2 are identical
Ha: D1 is shifted to the right of D2
[or D1 is shifted to the left of D2]
© 2011 Pearson Education, Inc
Wilcoxon Signed Rank Test for a Paired Difference Experiment
Let D1 and D2 represent the probability distributions for populations 1 and 2, respectively.
Two-Tailed Test
H0: D1 and D2 are identical
Ha: D1 is shifted to the left or to the right of D2
© 2011 Pearson Education, Inc
Conditions Required for a Valid Signed Rank Test
1. The sample of differences is randomly selected from the population of differences.
2. The probability distribution from which the sample of paired differences is drawn is continuous.