Parametric versus Parametric versus Nonparametric Nonparametric

Statistics – When to Statistics – When to use them and which use them and which is more powerful?is more powerful?

Angela HebelAngela HebelDepartment of Natural SciencesDepartment of Natural SciencesUniversity of Maryland Eastern University of Maryland Eastern

ShoreShoreApril 5, 2002April 5, 2002

Parametric AssumptionsParametric Assumptions

The observations must be The observations must be independentindependent

The observations must be drawn from The observations must be drawn from normally distributed populationsnormally distributed populations

These populations must have the These populations must have the same variancessame variances

The means of these normal and The means of these normal and homoscedastic populations must be homoscedastic populations must be linear combinations of effects due to linear combinations of effects due to columns and/or rows*columns and/or rows*

Nonparametric AssumptionsNonparametric Assumptions

Observations are independentObservations are independent Variable under study has underlying Variable under study has underlying

continuitycontinuity

MeasurementMeasurement

What are the 4 levels of measurement What are the 4 levels of measurement discussed in Siegel’s chapter?discussed in Siegel’s chapter?1. Nominal or Classificatory Scale1. Nominal or Classificatory Scale

Gender, ethnic backgroundGender, ethnic background

2. Ordinal or Ranking Scale2. Ordinal or Ranking Scale Hardness of rocks, beauty, military ranksHardness of rocks, beauty, military ranks

3. Interval Scale3. Interval Scale Celsius or FahrenheitCelsius or Fahrenheit

4. Ratio Scale4. Ratio Scale Kelvin temperature, speed, height, mass or Kelvin temperature, speed, height, mass or

weightweight

Nonparametric MethodsNonparametric Methods

There is at least one nonparametric There is at least one nonparametric test equivalent to a parametric testtest equivalent to a parametric test

These tests fall into several These tests fall into several categoriescategories

1.1. Tests of differences between groups Tests of differences between groups (independent samples)(independent samples)

2.2. Tests of differences between variables Tests of differences between variables (dependent samples)(dependent samples)

3.3. Tests of relationships between variablesTests of relationships between variables

Differences between Differences between independent groupsindependent groups

Two samples – Two samples – compare mean compare mean value for some value for some variable of interestvariable of interest

ParametricParametric NonparametriNonparametricc

t-test for t-test for independent independent samplessamples

Wald-Wald-Wolfowitz Wolfowitz runs testruns test

Mann-Mann-Whitney U Whitney U testtest

Kolmogorov-Kolmogorov-Smirnov two Smirnov two sample testsample test

Mann-Whitney U TestMann-Whitney U Test

Nonparametric alternative to two-Nonparametric alternative to two-sample t-testsample t-test

Actual measurements not used – Actual measurements not used – ranks of the measurements usedranks of the measurements used

Data can be ranked from highest to Data can be ranked from highest to lowest or lowest to highest valueslowest or lowest to highest values

Calculate Mann-Whitney U statisticCalculate Mann-Whitney U statisticU = nU = n11nn22 + + nn11(n(n11+1)+1) – R – R11

22

Example of Mann-Whitney U Example of Mann-Whitney U testtest

Two tailed null hypothesis that there Two tailed null hypothesis that there is no difference between the heights is no difference between the heights of male and female studentsof male and female students

HHoo: Male and female students are the : Male and female students are the same heightsame height

HHAA: Male and female students are not : Male and female students are not the same heightthe same height

HeightHeights of s of males males (cm)(cm)

HeightHeights of s of femalefemales (cm)s (cm)

Ranks Ranks of male of male heightsheights

Ranks Ranks of of female female heightheightss

193 193 175175 11 77

188188 173173 22 88

185185 168168 33 1010

183183 165165 44 1111

180180 163163 55 1212

178178 66

170170 99

nn11 = 7 = 7 nn22 = 5 = 5 RR11 = 30 = 30 RR22 = = 4848

U = nU = n11nn22 + + nn11(n(n11+1)+1) – R – R11

22

U=(7)(5) + U=(7)(5) + (7)(8)(7)(8) – 30 – 30 22

U = 35 + 28 – 30U = 35 + 28 – 30

U = 33U = 33

U’ = nU’ = n11nn22 – U – U

U’ = (7)(5) – 33U’ = (7)(5) – 33

U’ = 2U’ = 2

U U 0.05(2),7,50.05(2),7,5 = U = U 0.05(2),5,70.05(2),5,7 = 30 = 30

As 33 > 30, HAs 33 > 30, Hoo is rejected is rejected Zar, 1996

Differences between Differences between independent groupsindependent groups

Multiple groupsMultiple groupsParametriParametricc

NonparametrNonparametricic

Analysis Analysis of of variance variance (ANOVA/ (ANOVA/ MANOVA)MANOVA)

Kruskal-Kruskal-Wallis Wallis analysis of analysis of ranksranks

Median testMedian test

Differences between Differences between dependent groupsdependent groups

Compare two Compare two variables measured in variables measured in the same samplethe same sample

If more than two If more than two variables are variables are measured in same measured in same samplesample

ParametriParametricc

NonparametrNonparametricic

t-test for t-test for dependendependent samplest samples

Sign testSign test

Wilcoxon’s Wilcoxon’s matched matched pairs testpairs test

Repeated Repeated measures measures ANOVAANOVA

Friedman’s Friedman’s two way two way analysis of analysis of variancevariance

Cochran QCochran Q

Relationships between Relationships between variablesvariables

Two variables of Two variables of interest are interest are categoricalcategorical

ParametricParametric NonparametricNonparametric

Correlation Correlation coefficientcoefficient

Spearman RSpearman R

Kendall TauKendall Tau

Coefficient Coefficient GammaGamma

Chi squareChi square

Phi coefficientPhi coefficient

Fisher exact testFisher exact test

Kendall coefficient Kendall coefficient of concordanceof concordance

Summary Table of Statistical Summary Table of Statistical TestsTests

Level of Measurement

Sample Characteristics Correlation

1 Sample

2 Sample K Sample (i.e., >2)

Independent Dependent

Independent Dependent

Categorical or Nominal

Χ2 or bi-

nomial

Χ2 Macnarmar’s Χ2

Χ2 Cochran’s Q  

Rank or Ordinal

  Mann Whitney U

Wilcoxin Matched

Pairs Signed Ranks

Kruskal Wallis H

Friendman’s ANOVA

Spearman’s rho

Parametric (Interval &

Ratio)

z test or t test

t test between groups

t test within groups

1 way ANOVA between groups

1 way ANOVA (within or repeated measure)

Pearson’s r

Factorial (2 way) ANOVA  

 (Plonskey,

2001)

Advantages of Nonparametric Advantages of Nonparametric TestsTests

Probability statements obtained from Probability statements obtained from most nonparametric statistics are most nonparametric statistics are exact probabilities, regardless of the exact probabilities, regardless of the shape of the population distribution shape of the population distribution from which the random sample was from which the random sample was drawndrawn

If sample sizes as small as N=6 are If sample sizes as small as N=6 are used, there is no alternative to using used, there is no alternative to using a nonparametric testa nonparametric test

Siegel, 1956

Advantages of Nonparametric Advantages of Nonparametric TestsTests

Treat samples made up of observations Treat samples made up of observations from several different populations.from several different populations.

Can treat data which are inherently in Can treat data which are inherently in ranks as well as data whose seemingly ranks as well as data whose seemingly numerical scores have the strength in numerical scores have the strength in ranksranks

They are available to treat data which are They are available to treat data which are classificatoryclassificatory

Easier to learn and apply than parametric Easier to learn and apply than parametric teststests

Siegel, 1956

Criticisms of Nonparametric Criticisms of Nonparametric ProceduresProcedures

Losing precision/wasteful of dataLosing precision/wasteful of data Low powerLow power False sense of securityFalse sense of security Lack of softwareLack of software Testing distributions onlyTesting distributions only Higher-ordered interactions not dealt Higher-ordered interactions not dealt

withwith

Power of a TestPower of a Test

Statistical power – probability of Statistical power – probability of rejecting the null hypothesis when it is rejecting the null hypothesis when it is in fact false and should be rejectedin fact false and should be rejected– Power of parametric tests – calculated Power of parametric tests – calculated

from formula, tables, and graphs based from formula, tables, and graphs based on their underlying distributionon their underlying distribution

– Power of nonparametric tests – less Power of nonparametric tests – less straightforward; calculated using Monte straightforward; calculated using Monte Carlo simulation methods (Mumby, 2002) Carlo simulation methods (Mumby, 2002)

Questions?Questions?