Common Nonparametric Statistical Techniques in Behavioral Sciences

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Common Nonparametric Statistical Techniques in
Behavioral Sciences

• Chi Zhang, Ph.D.
• University of Miami
• June, 2005

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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

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Assumptions about Parametric Statistics

• A normally distributed population
• Equal variances among the population
• The observation must be independent
• The variables must be measured at interval or
  ratio scale

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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

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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)

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Disadvantages of Nonparametric Statistical Tests

• They are less powerful
• They are unfamiliar to many researchers and
  editors

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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)

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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) gt 5 when df gt 1, 20 freq (exp) gt 5)

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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

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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

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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

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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
  t-test

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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)

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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

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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

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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)

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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)

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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 Mann-Whitney U Friedman 2-way ANOVA Median Test Kruskal-Wallis
Interval or Ratio t-test Paired t-test Indept t-test ANOVA with repeated measures ANOVA
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Reference

• Siegel, Sidney Castellan, N. John, Jr (1988).
  Nonparametric statistics for the behavioral
  sciences (2nd edition). New York McGraw-Hill,
  1988.