Parametric versus Nonparametric Statistics

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Parametric versus Nonparametric Statistics When
to use them and which is more powerful?

• Angela Hebel
• Department of Natural Sciences
• University of Maryland Eastern Shore
• April 5, 2002

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Parametric Assumptions

• The observations must be independent
• The observations must be drawn from normally
  distributed populations
• These populations must have the same variances
• The means of these normal and homoscedastic
  populations must be linear combinations of
  effects due to columns and/or rows

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Nonparametric Assumptions

• Observations are independent
• Variable under study has underlying continuity

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Measurement

• What are the 4 levels of measurement discussed in
  Siegels chapter?
• 1. Nominal or Classificatory Scale
• Gender, ethnic background
• 2. Ordinal or Ranking Scale
• Hardness of rocks, beauty, military ranks
• 3. Interval Scale
• Celsius or Fahrenheit
• 4. Ratio Scale
• Kelvin temperature, speed, height, mass or weight

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Nonparametric Methods

• There is at least one nonparametric test
  equivalent to a parametric test
• These tests fall into several categories
• Tests of differences between groups (independent
  samples)
• Tests of differences between variables (dependent
  samples)
• Tests of relationships between variables

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Differences between independent groups

• Two samples compare mean value for some
  variable of interest

Parametric Nonparametric
t-test for independent samples Wald-Wolfowitz runs test
Mann-Whitney U test
Kolmogorov-Smirnov two sample test
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Mann-Whitney U Test

• Nonparametric alternative to two-sample t-test
• Actual measurements not used ranks of the
  measurements used
• Data can be ranked from highest to lowest or
  lowest to highest values
• Calculate Mann-Whitney U statistic
• U n1n2 n1(n11) R1
• 2

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Example of Mann-Whitney U test

• Two tailed null hypothesis that there is no
  difference between the heights of male and female
  students
• Ho Male and female students are the same height
• HA Male and female students are not the same
  height

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Heights of males (cm) Heights of females (cm) Ranks of male heights Ranks of female heights
193 175 1 7
188 173 2 8
185 168 3 10
183 165 4 11
180 163 5 12
178 6
170 9
n1 7 n2 5 R1 30 R2 48
U n1n2 n1(n11) R1
2 U(7)(5) (7)(8) 30
2 U 35 28 30 U 33 U n1n2 U U
(7)(5) 33 U 2 U 0.05(2),7,5 U
0.05(2),5,7 30 As 33 gt 30, Ho is rejected
Zar, 1996
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Differences between independent groups

• Multiple groups

Parametric Nonparametric
Analysis of variance (ANOVA/ MANOVA) Kruskal-Wallis analysis of ranks
Median test
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Differences between dependent groups

• Compare two variables measured in the same sample
• If more than two variables are measured in same
  sample

Parametric Nonparametric
t-test for dependent samples Sign test
Wilcoxons matched pairs test
Repeated measures ANOVA Friedmans two way analysis of variance
Cochran Q
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Relationships between variables

• Two variables of interest are categorical

Parametric Nonparametric
Correlation coefficient Spearman R
Kendall Tau
Coefficient Gamma
Chi square
Phi coefficient
Fisher exact test
Kendall coefficient of concordance
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Summary Table of Statistical Tests
 
(Plonskey, 2001)
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Advantages of Nonparametric Tests

• Probability statements obtained from most
  nonparametric statistics are exact probabilities,
  regardless of the shape of the population
  distribution from which the random sample was
  drawn
• If sample sizes as small as N6 are used, there
  is no alternative to using a nonparametric test

Siegel, 1956
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Advantages of Nonparametric Tests

• Treat samples made up of observations from
  several different populations.
• Can treat data which are inherently in ranks as
  well as data whose seemingly numerical scores
  have the strength in ranks
• They are available to treat data which are
  classificatory
• Easier to learn and apply than parametric tests

Siegel, 1956
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Criticisms of Nonparametric Procedures

• Losing precision/wasteful of data
• Low power
• False sense of security
• Lack of software
• Testing distributions only
• Higher-ordered interactions not dealt with

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Power of a Test

• Statistical power probability of rejecting the
  null hypothesis when it is in fact false and
  should be rejected
• Power of parametric tests calculated from
  formula, tables, and graphs based on their
  underlying distribution
• Power of nonparametric tests less
  straightforward calculated using Monte Carlo
  simulation methods (Mumby, 2002)

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Questions?