Download Nonparametric Statistics in Behavioral Sciences

Common Nonparametric Statistical
Techniques in Behavioral Sciences
Chi Zhang, Ph.D.
University of Miami
June, 2005
Objectives
• Assumptions for nonparametric statistics
• Scales of measurement
• Advantages and disadvantages of
nonparametric statistics
• Nonparametric tests for the single-sample case
• Nonparametric tests for two related samples
• Nonparametric tests for two independent
samples
• The case of k related samples
• The case of k independent samples
Assumptions about Parametric
Statistics
•
•
•
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A normally distributed population
Equal variances among the population
The observation must be independent
The variables must be measured at
interval or ratio scale
Assumptions about Nonparametric
Statistics
• Nonparametric (distribution free)
techniques make no assumptions bout the
population
• The fewer the assumptions, the more
general are the conclusions
• The more powerful tests are those that
have the strongest or most extensive
assumptions
Advantages of Nonparametric
Statistical Tests
• May be the only test when the sample size
is small
• Require fewer assumptions
• The only choice when the measurement
scales are nominal or ordinal (e.g. using
categories, rankings, medians)
Disadvantages of Nonparametric
Statistical Tests
• They are less powerful
• They are unfamiliar to many researchers
and editors
Scales of Measurement
• Nominal or categorical (e.g. gender,
nationality)
• Ordinal (e.g. ranking, ratings)
• Interval (e.g. temperature)
• Ratio (e.g. age, distance, weight)
The Chi-square Goodness of Fit
(Single-sample Case)
• It assesses the degree of correspondence
between the observed and expected
observations in each category
• Measurement scale: nominal or categorical
• Small expected frequencies (when df = 1, freq
(exp) => 5; when df > 1, 20%+ freq (exp) => 5)
The Kolmogorov-Smirnov One-sample Test
(Single-sample Case)
• It tests the goodness of fit for variable which are
measured on at least ordinal scale
• It involves specifying the cumulative frequency
distribution which would occur given the
theoretical distribution ( e.g. normal distribution)
and comparing that with the observed
cumulative frequency distribution
The NcNemar Test
(Two related samples)
• It is particularly applicable to “before and after”
designs in which each subject is used as its own
control
• The measurements are made on either a
nominal or ordinal scale
The Sign Test
(Two related samples)
• For research in which quantitative measurement
is impossible or infeasible
• It is possible to determine, for each pair of
observations, which is the “greater”
• The only assumption is that the variable has a
continuous distribution
The Wilcoxon Signed Rank Test
(Two related samples)
• All the observations must be measured at
ordinal scale
• Ranking the differences observed for the various
matched pairs
• Power-efficiency is about 95% of that of paired ttest
The Chi-square Test for Two
Independent Samples
• Suitable for nominal or stronger data
• Determining whether the two samples are
from populations that differ in any respect
at all (e.g. location, dispersion, skewness)
The Kolmogorov-Smirnov Two
Sample Tests
• Ordinal or stronger data
• It tests whether two independent samples
have been drawn from populations with
the same distribution
• The test is concerned with the agreement
between two cumulative distributions
The Man-Whitney U Test
(two independent samples)
• It tests whether two samples represent
populations that differ in central tendency
• Variables measured at least at ordinal
scale
• One of the most powerful of the
nonparametric tests
• A useful alternative to t-test
The Case of k Related Samples
• The Friedman two-way ANOVA by ranks is
appropriate when the measurements of
the variables are at least ordinal
• The Friedman two-way ANOVA by ranks
tests the probability that the k related
samples could have come from the same
population with respect the mean
rankings.
• The Cochran Q test (nominal data)
The Case of k Independent Samples
• The Kruskal-Wallis one-way ANOVA by ranks
tests tha null hypothesis that the k samples
come from the same population or from identical
populations with the same median
• The Kruskal-Wallis one-way ANOVA by ranks
requires at ordinal measurement of the variable
• The Chi-square test (nominal data) and the
Median test (ordinal data)
Choice of Statistical Tests
1-sample
2 Related
Samples
2 Indept
Samples
k Related
Samples
k Indept
Samples
Nominal
Chi-square
McNemar
Chi-square
Cochran Q
Chi-square
Ordinal
K-S
Sigh Test
Wilcoxon
Signed Rank
K-S
Median Test
MannWhitney U
Friedman 2way ANOVA
Median Test
KruskalWallis
Interval or
Ratio
t-test
Paired t-test
Indept t-test
ANOVA with
repeated
measures
ANOVA
Reference
• Siegel, Sidney & Castellan, N. John, Jr
(1988). Nonparametric statistics for the
behavioral sciences (2nd edition). New
York: McGraw-Hill, 1988.