parametric versus nonparametric statistics – when to use them and which is more powerful? angela...
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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?