Parametric vs Nonparametric

7/22/2019 Parametric vs Nonparametric

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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


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


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?

Documents

Parametric vs Nonparametric