Unit-III Non-Parametric Tests BSRM.pdf

Unit-III Non Parametric tests: Wilcoxon Rank Sum Test, Mann-Whitney U test, Kruskal-Wallis
test, Friedman Test. BP801T. BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory)

1. Unit-III
Non-Parametric tests
✓Wilcoxon Rank Sum Test
✓Mann-Whitney U test
✓Kruskal-Wallis test
✓Friedman Test
Ravinandan A P 1
Ravinandan A P
Assistant Professor,
Department ofPharmacyPractice,
Sree Siddaganga College of Pharmacy
In association with
Siddaganga Hospital,
Tumkur-02

2. Level of significance
(Non-parametric data)-
1.
Wilcoxan’s signed rank test
2.
Wilcoxanrank sum test
3.
Mann Whitney U test
4.
Kruskal -Wallis test (one way ANOVA)
5.Friedman Test
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4. Why non-parametric methods?
••Certainstatisticaltestslikethet-testrequireassumptionsofthe
distributionofthestudyvariablesinthepopulation
•–t-testrequirestheunderlyingassumptionofanormaldistribution
•–Suchtestsareknownasparametrictests
••Therearesituationswhenitisobviousthatthestudyvariable
cannotbenormallydistributed,e.g.,
#ofhospitaladmissionsperpersonperyear
#ofsurgicaloperationsperperson
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5. •Thestudyvariablegeneratesdatawhicharescores&soshouldbe
treatedasacategoricalvariablewithdatameasuredonordinalscale
–E.g.,scoringsystemfordegreeofskinreactiontoachemicalagent:
•1:intenseskinreaction
•2:lessintensereaction
•3:Noreaction
•Forsuchtypeofdata,theassumptionrequiredforparametrictestsseeminvalid=>non-
parametricmethodsshouldbeused
••Akadistribution-freetests,becausetheymakenoassumptionabouttheunderlying
distributionofthestudyvariables
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7. Advantages of Non-Parametric tests
1.Distributionfree&hencenoassumptionaboutthepopulationis
required.
2.Whensamplesizeissmallitissimpleto&understand&easyto
apply.
3.Itislesstimeconsuming&forsignificantresultnofurtherworkis
necessary
4.Applicabletoalltypeofdata
5.Helpfultoresearcherscollectingpilotstudyorthemedical
researchersworkingwithararedisease
6.Makefewerassumptionsthanclassicalprocedures.
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11. Wilcoxontest.
•Usedtotestwhetherornotthedifferencebetweentwopaired
populationmediansiszero.
•Thenullassumptionisthatitis,i.e.thetwomediansareequal.
•Variablescanbeeithermetricorordinal.
•Distributionsanyshape,butthedifferencesshouldbe
distributedsymmetrically.
•Thisisthenon-parametricequivalentofthematched-pairst
test.
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12. WILCOXON SIGNED RANK TEST
•For the comparison of two treatments in a paired design, a more
sensitive non-parametric test is Wilcoxon signed rank test
•Ex: paired data obtained from bioavailability experiment: Time to
Peak concentration
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14. WILCOXON SIGNED RANK TEST
Subject Time to Peak Difference
A B B-A
1 2.5 3.5 1
2 3 4 1
3 1.25 2.5 1.25
4 1.75 2 0.25
5 3.5 3.5 0
6 2.5 4 1.5
7 1.75 1.5 -0.25
8 2.25 2.5 0.25
9 3.5 3 -0.5
10 2.5 3 0.5
11 2 3.5 1.5
12 3.5 4 0.5 14Ravinandan A P

15. Ranks with +ve Sign Ranks with -Ve sign
2 2
2 5
5
5
7.5
7.5
9
10.5
10.5
59 7
If the Smaller Rank sum is Less than or Equal to the
table value , the comparative groups are different at the
Indicated level of significance
At 5% and 1%
For N= 11 table Value = 13 at 5% - Level
Cal value < Table Value
Difference is Significant Ravinandan A P

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17. WILCOXON RANK SUM TEST
•Thewilcoxonsignedranktestfornon-parametrictestforthe
comparisonofpairedsample
•Iftwotreatmentsaretocompared,wheretheobservationshave
beenobtainedfromtwoindependentgroups,thenon-parametric
wilcoxonranksumtestorMann–whitneyU-testisalternative
forPooledt-test
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18. Wilcoxon rank sum test
(aka Mann-Whitney U test)
•Non-parametric equivalent of parametric t-test for 2
independent samples (unpaired t-test)
•Suppose the waiting time (in days) for cataract surgery at
two eye clinics are as follows:
Patients at clinic A (nA=18) 1, 5, 15, 7, 42, 13, 8, 35, 21,
12, 12, 22, 3, 14, 4, 2, 7, 2
Patients at clinic B (nB=15) 4, 9, 6, 2, 10, 11, 16, 18, 6,
0, 9, 11, 7, 11, 10
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20. Amount Amount Rank(old) Rank(New)
DissolvedDissolved
53 58 3 11
61 55 14 5.5
57 67 9 21
50 62 1 15.5
63 55 17 5.5
62 64 15.5 18.5
54 66 4 20
59 59 2 12.5
59 68 12.5 12
57 57 9 9
64 69 18.5 23
56 7
105.5 160.5
Z = (| T - N1(N1+N2+1)/2 |) / (sqrt(N1N2(N1+N2+1)/12)
T= Sum of the ranks for the smaller sample size
N1 = Size of sample1 11
N2 = Size of sample2 12 20Ravinandan A P

21. Mann-Whitney test
•Usedtotestwhetherornotthedifferencebetweentwoindependent
populationmediansiszero.
•Thenullassumptionisthatitis,i.e.thetwomediansareequal.
•Variablescanbeeithermetricorordinal.
•Norequirementastoshapeofthedistributions,buttheyneedtobe
similar.
•Thisisthenon-parametricequivalentofthetwo-samplettest.
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22. Kruskal-Wallis test.
•Usedtotestwhetherthemediansofthreeofmore
independentgroupsarethesame.
•Variablescanbeeitherordinalormetric.Distributionsany
shape,butallneedtobesimilar.
•Thisnon-parametrictestisanextensionoftheMann-
Whitneytest.
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23. Friedman Test
•TheFriedmantestisanon-parametricstatistical
testdevelopedbyMiltonFriedman.
•Similartotheparametricrepeated
measuresANOVA,itisusedtodetectdifferences
intreatmentsacrossmultipletestattempts.
•Theprocedureinvolvesrankingeachrow
(orblock)together,thenconsideringthevaluesof
ranksbycolumns.
•Applicabletocompleteblockdesigns.
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24. Friedman Test
•The Friedman test is a non-parametric alternative to ANOVA
with repeated measures.
•No normality assumption is required.
•The test is similar to the Kruskal-Wallis Test.
•We will use the terminology from Kruskal-Wallis Test and Two
Factor ANOVA without Replication.
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28. Non-parametric vs. parametric methods
Advantages:
1.Donotrequirestheassumptionneededforparametrictests.
Thereforeusefulfordatawhicharemarkedlyskewed
2.Goodfordatageneratedfromsmallsamples.Forsuchsmall
samples,parametrictestsarenotrecommendedunlessthenature
ofpopulationdistributionisknown
3.Goodforobservationswhicharescores,i.e.measuredon
ordinalscale
4.Quickandeasytoapplyandyetcomparequitewellwith
parametricmethods
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29. Non-parametric vs. parametric methods
•Disadvantages
1.Notsuitableforestimationpurposesasconfidence
intervalsaredifficulttoconstruct
2.Noequivalentmethodsformorecomplicatedparametric
methodsliketestingforinteractionsinANOVAmodels
3.Notquiteasstatisticallyefficientasparametricmethods
iftheassumptionsneededfortheparametricmethods
havebeenmet
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31. THANK YOU
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