LearningObjectives
Parametricvs.NonparametricStatistics
ParametricStatisticsarestatisticaltechniquesbased
Recognizetheadvantagesanddisadvantagesof
onassumptionsaboutthepopulationfromwhichthe
nonparametricstatistics.
sampledataarecollected.
Understandhowtousetherunstesttotestfor
Assumptionthatdatabeinganalyzedarerandomly randomness.
selectedfromanormallydistributedpopulation.
KnowwhenandhowtousetheMann‐WhitneyU test,
Requiresquantitativemeasurementthatyieldinterval
theWilcoxonmatched‐pairssignedranktest,the
orratioleveldata.
Kruskal‐Wallistest,andtheFriedmantest.
NonparametricStatisticsarebasedonfewer
assumptionsaboutthepopulationandthe parameters.
Sometimescalled“distribution‐free”statistics.
Avarietyofnonparametricstatisticsareavailablefor
usewithnominalorordinaldata.
AdvantagesofNonparametricTechniques
DisadvantagesofNonparametricStatistics
Sometimesthereisnoparametricalternativetothe
Nonparametrictestscanbewastefulofdataif
useofnonparametricstatistics.
parametrictestsareavailableforusewiththedata.
Certainnonparametrictestcanbeusedtoanalyze
Nonparametrictestsareusuallynotaswidely nominaldata.
availableandwellknowasparametrictests.
Certainnonparametrictestcanbeusedtoanalyze
Forlargesamples,thecalculationsfo  r many ordinaldata.
nonparametricstatisticscanbetedious.
Thecomputationsonnonparametricstatisticsare
usuallylesscomplicatedthanthoseforparametric
statistics,particularlyforsmallsamples.
Probabilitystatementsobtainedfrommost
nonparametrictestsareexactprobabilities.
Mann‐WhitneyU Test
Mann‐WhitneyU Test:
SampleSizeConsideration
Mann‐WhitneyUtest‐ anonparametriccounterpart
Sizeofsampleone:n1
ofthe ttestusedtocomparethemeansoftwo
Sizeofsampletwo:n independent 2 populations.
Ifbothn and are 1 n2
 10,thesmallsampleprocedure
Nonparametriccounterpartofthet testfor isappropriate. independentsamples If either
eithe n or
n o n is greater than 10 the large sample 1
n2 greaterthan10,thelargesampl 
Doesnotrequirenormallydistributedpopulations
procedureisappropriate.
Maybeappliedtoordinaldata Assumptions IndependentSamples
AtLeastOrdinalData
Mann‐WhitneyUTest:SmallSample
Mann‐WhitneyUTest:SmallSample Example‐Demonstration Example‐Demonstration H :  = 0 µ1=µ2 .05 Compensation Rank Group H : π Drug 18.75 1 H a A DrugB 19.80 2 H 20.10 26.19
Ifthefinalp‐value<.05,rejectH . 0 20.10 3 H 19.80 23.88 20.75 4 H 21.64 5 E 22.36 25.50 21.90 6 H 18.75 21.64
W =1+2+3+4+6 + 7 + 8 = 31 1       22.36 7 H 22.96 8 H 21.90 24.85 23.45 9 E
W =5+9+10+11+12+13+14+15= 22.96 25.30 2 89 23.88 10 E 24.12 11 E 20.75 24.12 24.85 12 E 23.45 25.30 13 E 25.50 14 E 26.19 15 E
Mann‐WhitneyU Test:Small
Mann‐WhitneyU Test: SampleExample
FormulasforLargeSampleCase 
SinceU <U ,U =3.    2 1
p‐value=.0011*2        (for   
atwo‐tailedtest)=.022 
<.05,rejectH . 0                      
Example– MannWhitneyUforlarge
RanksofIncomefromCombined samples
GroupsofPBSandNon‐PBSViewers PBS Non‐PBS Datavalue Rank Group Datavalue Rank Group 24,500 41,000 π 16,000 1 Non‐PBS 39,500 15 Non‐PBS 39,400 32,500 21,000 2 Non‐PBS 40,500 16 Non‐PBS 36,800 33,000 21,500 3 Non‐PBS 41,000 17 Non‐PBS 44,300 21,000 24,500 4 PBS 43,000 18 PBS 57 5 9 ,9 6 6 0 0 40 4 , 5 5 0 0 0 0 27,600 5 Non‐PBS 43,500 19.5 PBS n = 1 14 32,000 32,400 27,800 6 Non    ‐PBS 43,500 19.5 Non‐PBS 61,000 16,000 32,000 7 PBS 51,900 21 Non‐PBS    34,000 21,500 32,400 8 Non‐PBS 53,000 22 PBS n = 2 13 43,500 39,500 32,500 9 Non‐PBS 55,000 23 PBS 33,000 10 Non‐PBS 57,960 24 PBS 55,000 27,600 34,000 11 PBS 61,000 25 PBS 39,000 43,500 36,800 12 PBS 61,400 26 PBS 62,500 51,900 39,000 13 PBS 62,500 27 PBS 61,400 27,800 39,400 14 PBS 53,000
PBSandNon‐PBS:CalculationofU
PBSandNon‐PBS:Conclusion                                                                
WilcoxonMatched‐PairsSignedRankTest
WilcoxonMatched‐PairsSignedRankTest
Differencesofthescoresofthetwomatchedsamples
Differencesareranked,ignoringthesign
Ranksaregiventhesignofthedifference
Positiveranksaresummed Negative v ranks are summed ranksaresumme
T isthesmallersumofranks
WilcoxonMatched‐PairsSignedRankTest:
WilcoxonMatched‐PairsSignedRankTest:
SampleSizeConsideration
SmallSampleExample
n isthenumberofmatchedpairs
Considerthesurveyb 
y AmericanDemographicsthat
Ifn >15,T isapproximatelynormallydistributed,
estimatedtheaverageannualhouseholdspending
andaZ testisused.
onhealthcare.TheU.S.metropolitanaveragewas
Ifn  15,aspecial“smallsample”procedureis $1,800. followed.
SupposesixfamiliesinPittsburgh ,Pennsylvania,are
Thepaireddataarerandomlyselected.
matcheddemographicallywithsixfamiliesin
Theunderlyingdistributionsaresymmetric.
Oakland,California,andtheiramountso fhousehold
spendingonhealthcareforlastyearareobtained.
WilcoxonMatched‐PairsSignedRankTest:
WilcoxonMatched‐PairsSignedRankTest:
SmallSampleExample
SmallSampleExample Family H : =0 0 Md Pair SampleA SampleB d Rank H :M  0 a d Pair SampleA SampleB 1 1,950 1,760 190 +4 1 1,950 1,760 2 1,840 1,870 ‐30 ‐1 n=6 2 1,840 1,870 3 2,015 1,810 205 +5 3 2,015 1,810 4 1 5 , 8 5 0 8 1 6 , 6 6 0 6 8 ‐ 0 2  ‐ =0.05 4 1,580 1,660 5 1,790 1,340 450 +6 6 1,925 1,765 160 +3 5 1,790 1,340 IfT  1,rejectH 6 1,925 1,765 observed 0.
WilcoxonMatched‐PairsSignedRankTest:
WilcoxonMatched‐PairsSignedRankTest:
LargeSampleFormulas
LargeSampleFormulas
Forlargesamples,theTstatisticisapproximatelynormally
distributedandazscoreca  n b 
e usedastheteststatistic.
WilcoxonMatched‐PairsSignedRankTest: Example
LargeSampleFormulas                    City 1979 2011 d Rank City 1979 2011 d Rank 1 20 2 3 . 22 2 .8 8 2 ‐ .5 ‐ 8 10 1 20 2 .3 3 20 2 .9 9 ‐ 0.6 1 ‐   2 19.5 12.7 6.8 17 11 19.2 22.6 ‐3.4 ‐11.5  3 18.6 14.1 4.5 13 12 19.5 16.9 2.6 9  4 20.9 16.1 4.8 15 13 18.7 20.6 ‐1.9 ‐6.5 5 19.9 25.2 ‐5.3 ‐16 14 17.7 18.5 ‐0.8 ‐2 6 18.6 20.2 ‐1.6 ‐4 15 21.6 23.4 ‐1.8 ‐5 7 19.6 14.9 4.7 14 16 22.4 21.3 1.1 3 8 23.2 21.3 1.9 6.5 17 20.8 17.4 3.4 11.5 9 21.8 18.7 3.1 10
T Calculation Conclusion 
                                                             Kruskal‐WallisTest
Kruskal‐WallisK Statistic
Kruskal‐WallisTest‐ Anonparametricalternative
toone‐wayanalysisofvariance          
Mayusedtoanalyzeordinaldata       
Noassumedpopulationshape As A s s u s m u e m s e that the
that th C groups are independent
C groupsareindependen
Assumesrandomselectionofindividualitems  
NumberofPatientsperDayperPhysician
NumberofPatientsperDayperPhysician
inThreeOrganizationalCategories
inThreeOrganizationalCategories
Supposearesearcherwantstodeterminewhetherthe H :
o Thethreepopulationsareidentical
numberofphysiciansinanofficeproducessignificant H :
a Atleastoneofthethreepopulationsisdifferent
differencesinthenumberofoffic  e patientssee  n b  y eac  h  Threeor
physicianperday.   Two More Sh S et k a esara d n omsam l p e f o phy i s i
c ansfrompractices Partners P artners HMO     
inwhich(1)thereareonlytwopartners,(2)thereare 13 24 26  three 15 16 22
ormorepartners,or(3)theofficeisahealth  20 19 31
maintenanceorganization(HMO).  18 22 27 23 25 28 14 33 17
PatientsperDayData:Kruskal‐Wallis
PatientsperDayData:Kruskal‐Wallis
PreliminaryCalculations
CalculationsandConclusion Threeor Two More   Partners Partners HMO       Patients Rank
Patients Rank Patients Rank        13 1 24 12 26 14 15 3 16 4 22 9.5   20 8 19 7 31 17            18 6 22 9 5 . 27 15      23 11 25 13 28 16 14 2 33 18        17 5    T =29 T = = 1 2 52.5 T3 89.5  n = = = 1 5 n2 7 n3 6 n=n + + = + + = 1
n2 n3 5 7 6 18     FriedmanTest FriedmanTest
FriedmanTest‐ Anonparametricalternativetothe
randomizedblockdesign       Assumptions 
Theblocksareindependent. There 
isnointeractionbetweenblocksandtreatments.
Observationswithineachblockcanberanked. Hypotheses H :
o Thetreatmentpopulationsareequal H :
a Atleastonetreatmentpopulationyieldslargervalues   
thanatleastoneothertreatmentpopulation FriedmanTest: FriedmanTest:
H : The supplier equal o  populationsare
H : At least one supplier s larger a   
populationyield  values
thanatleastoneothersupplierpopulation        Supplier1 Supplier2 Supplier3 Supplier4 Monday 62 63 57 61 Tuesday 63 61 59 65   Wednesday 61 62 56 63 Thursday 62 60 57 64 Friday 64 63 58 66   FriedmanTest: FriedmanTest: Supplier1 Supplier2 Supplier3 Supplier4      Monday 3 4 1 2   Tuesday 3 2 1 4 Wednesday 2 3 1 4    Thursday 3 2 1 4  Friday 3 2 1 4 R  j 14 13 5 18 R2 196 169 25 324 j         