Post on 10-Feb-2018
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Parametric versus
NonparametricStatistics When to usethem and which is more
powerful?
Angela HebelDepartment of Natural Sciences
University of Maryland Eastern ShoreApril 5, 2002
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Parametric Assumptions
The observations must be independent
The observations must be drawn fromnormally distributed populations
These populations must have the samevariances
The means of these normal and
homoscedastic populations must be linearcombinations of effects due to columnsand/or rows*
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Nonparametric Assumptions
Observations are independent
Variable under study has underlyingcontinuity
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Measurement
What are the 4 levels of measurementdiscussed in Siegels chapter?1. Nominal or Classificatory Scale
Gender, ethnic background2. 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 testequivalent to a parametric test
These tests fall into several categories
1. Tests of differences between groups(independent samples)
2. Tests of differences between variables
(dependent samples)3. Tests of relationships between variables
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Differences between independentgroups
Two samplescompare mean valuefor some variable of
interest
Parametric Nonparametric
t-test forindependent
samples
Wald-Wolfowitzruns test
Mann-WhitneyU test
Kolmogorov-Smirnov twosample test
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Mann-Whitney U Test
Nonparametric alternative to two-samplet-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 statisticU = n1n2 + n1(n1+1) R1
2
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Example of Mann-Whitney U test
Two tailed null hypothesis that there is nodifference between the heights of maleand female students
Ho: Male and female students are thesame height
HA
: Male and female students are not thesame height
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Heightsofmales(cm)
Heightsoffemales(cm)
Ranks ofmaleheights
Ranksoffemaleheights
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(n1+1) R12
U=(7)(5) + (7)(8) 302
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 > 30, Ho
is rejected Zar, 1996
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Differences between independentgroups
Multiple groupsParametric Nonparametric
Analysis ofvariance(ANOVA/MANOVA)
Kruskal-Wallisanalysis ofranks
Median test
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Differences between dependentgroups
Compare two variablesmeasured in the samesample
If more than twovariables are measured insame sample
Parametric Nonparametric
t-test fordependent
samples
Sign test
Wilcoxonsmatched pairstest
Repeatedmeasures
ANOVA
Friedmans twoway analysis ofvariance
Cochran Q
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Relationships between variables
Two variables ofinterest arecategorical
Parametric Nonparametric
Correlationcoefficient
Spearman R
Kendall Tau
Coefficient Gamma
Chi squarePhi coefficient
Fisher exact test
Kendall coefficient ofconcordance
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Summary Table of Statistical TestsLevel of
Measurement
Sample Characteristics Correlation
1
Sample
2 Sample K Sample (i.e., >2)
Independent Dependent Independent Dependent
Categorical
or Nominal
2 or
bi-
nomial
2 Macnarmar
s 2
2 Cochrans Q
Rank or
Ordinal
Mann
Whitney U
Wilcoxin
Matched
Pairs Signed
Ranks
Kruskal Wallis
H
Friendmans
ANOVA
Spearmans
rho
Parametric
(Interval &Ratio)
z test
or t test
t test
betweengroups
t test within
groups
1 way ANOVA
betweengroups
1 way
ANOVA(within or
repeated
measure)
Pearsons r
Factorial (2 way) ANOVA
(Plonskey, 2001)
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Advantages of Nonparametric Tests
Probability statements obtained from mostnonparametric statistics are exactprobabilities, regardless of the shape of
the population distribution from which therandom sample was drawn
If sample sizes as small as N=6 are used,
there is no alternative to using anonparametric test
Siegel, 1956
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Advantages of Nonparametric Tests
Treat samples made up of observations fromseveral different populations.
Can treat data which are inherently in ranks as
well as data whose seemingly numerical scoreshave the strength in ranks
They are available to treat data which areclassificatory
Easier to learn and apply than parametric tests
Siegel, 1956
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Criticisms of NonparametricProcedures
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 rejectingthe null hypothesis when it is in fact falseand should be rejected
Power of parametric tests calculated fromformula, tables, and graphs based on theirunderlying distribution
Power of nonparametric tests lessstraightforward; calculated using Monte Carlosimulation methods (Mumby, 2002)
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Questions?