SIGN TEST SLIDE.ppt

Masters in biostatics coarse

1. Statistical Methods for Non parametric
Continuous Variables
Yilma ch, ass.t prof bio HI
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2. Objective
At the end of the presentation you will able to:
list non parametric statistical tests
describe sign test
test hypothess using sign test
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3. •Wilcoxon Sign test
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4. Introduction
•Whenyourdatadonotsatisfythedistributional
assumptionsrequiredbyparametricprocedures,
otherstatisticalmethodsareneededthatisNon
parametricstatistics.
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5. •Thedistributionalassumptionsrequiredfornon-parametric
proceduresareusuallylessspecificthanthoserequiredfor
parametricprocedures.
•Manynon-parametrictestshavelesspowerthanthe
correspondingparametrictests.
•Becausepowershouldneverbegivenupunlessabsolutely
necessary,non-parametricmethodsshouldnotbeusedwhen
parametricmethodsareappropriate.
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6. Cont...
•Thesigntestisanexampleofoneofthesenonparametric
tests.
•DonotrushtousetheNPT
•Ifyouroutcomevariableisnotnormal,trytonormalizeusing
logorln.
•Ifitisstillnotnormalized,gotononparametrictests
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7. What is the Sign Test?
Thesigntestcomparesthesizesoftwogroups.
Itisanon-parametricor“distribution-free”test,whichmeans
thetestdoesn’tassumethedatacomesfromaparticular
distribution,likethenormaldistribution.
Thesigntestisanalternativetoaone-samplet-test.
Itcanalsobeusedforordered(ranked)categoricaldata.
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8. Cont...
Thesigntestisusedtotestthenullhypothesisthatthemedian
ofadistributionisequaltosomehypothetical(standard)value.
Itcanbeused;
inplaceofaone-samplet-test
inplaceofapairedt-testor
fororderedcategoricaldatawhereanumericalscaleis
inappropriatebutwhereitispossibletoranktheobservations.
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9. Assumptions of sign test
1.dataisnonnormallydistributed
2.arandomsampleofindependentmeasurementfora
populationwithunknownmedian
3.thevariableofinterestiscontinuousorranked
ordinalscaleofmeasurement
4.theonesampletesthandlenonsymmetricdataset
(skewedeithertorightorleft)
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10. procedure
•LetAandBrepresenttwomaterialsortreatmentstobe
compared.
•LetxandyrepresentmeasurementsmadeonAandB.
•Letthenumberofpairsofobservationsben.
•Thenpairsofobservationsandtheirdifferencesmaybe
denotedby:
(X
1,Y
1),(X
2,y
2),.....,(X
n,Y
n)and
X
1-Y
1,X
2-Y
2..............X
n-Y
n.
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11. Cont...
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12. Cont...
•Thesigntestisbasedonthesignsofthesedifferences.
•TheletterBswillbeusedtodenotethenumberoftimesthe
maximumsignhasoccurred.
•Ifsomeofthedifferencesarezero,wecancancelthe
observation.
•BS=Max｛N
+,N
-｝
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13. •Asanexampleofthetypeofdataforwhichthesigntestis
appropriate,wemayconsiderthefollowingyieldsoftwo
hybridlinesofcornobtainedfromseveraldifferent
experiments.
•InthisexampleN=28
N
+=7
N
-=21
BS=Max｛N
+,N
-｝
BS=21
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14. •wecanfindthecriticalvaluesandp-valuesofBsfromthesign
testtable.
•ifthep-valueislessthansignificanceleveloralphavalue,
rejectthenullhypothesis
P<α→rejectthenullhypothesiswhichstatesnomedian
difference.
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16. Example Hypothesis testing using sign test
N
o
Driver
injury(x)
passenger
injury(y)
sign(+,-)
(x-y)
Driver
injury(x)
passenger
injury(y)
sign(+,-)
(x-y)
1.42 35 + 36 37 -
2.42 35 + 36 37 -
3.34 45 - 43 58 -
434 45 - 40 42 -
5.45 45 0 43 58 -
6.40 42 - 37 41 -
7.42 46 - 37 41 -
8.43 58 - 44 57 -
9.45 43 + 42 42 0
Test at 95% confidence interval that the driver injury is equal to
passengers injury.
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17. Step 1. Ho : driver injury = passenger injury
HA : driver injury ≠ passenger injury
Step2.N=18-2=16(twoobservationscanceled)
N
+=3
N
-=13
BS=Max｛N
+,N
-｝
BS=13
Step3.appropriatetestissigntestα=5%
step 4. find p value from the table
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18. p-value = 0.021
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19. Step5.Interpretation
0.021<0.05(0.021isobtainedfromsigntesttableatn=16,BS
13andalpha0.05,whichis0.021
P<α→rejectthenullhypothesis
Thereforthedriversinjuryisnotequaltothepassengersinjury.
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20. Exercise
Thetablebelowshowsthehoursofreliefprovidedbytwo
analgesicdrugsin12patientssufferingfromarthritis.Isthere
anyevidencethatonedrugprovideslongerreliefthantheother?
Test at 95% confidence interval.
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21. Solution
Inthiscaseournullhypothesisisthatthemediandifferenceis
zero.
Ouractualdifferences(DrugB-DrugA)are:+1.5,+2.1,
+0.3,−0.2,+2.6,−0.1,+1.8,−0.6,+1.5,+2.0,+2.3,+12.4
Ouractualmediandifferenceis1.65hours.
N+=9,N−=3,n=12,
Bs=max(N−,N+)=9
Ourp-valueatn=12,Bs=9andalpha=0.05
(fromtables)isp=0.146
Wewouldconcludethatthereisnoevidenceforadifference
betweenthetwotreatmentsonreliefofpain.
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22. Sign test with large sample size
largesamplesize=N>30weuseZtest.
Z=(X±0.5)-N/2
0.5*√N
where:X=nooffewersign
N=totalpairofsample
wecanuseZ=(X+0.5)-N/2ifN/2>X
0.5*√N
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23. find the p value of Z from the table then:
P <α → reject the null hypothesis
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24. Example
Aresearcherhastaken50pairofstudentsforthestudyand
obtainedthedata.
canyouconcludefromthedatabyusingsigntestthatthe
trainingofthetwogroupsdiffersignificantly?
Givennoof
+vesign=37
 -vesign=12
 of0=1
solution
HO=thetrainingofonegroup=trainingoftheother.
N=37+12=49
N/2=49/2=24.5
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25. x= 12
As 24.5 > 12 we use the formula
Z=(X+0.5)-N/2
1/2√N
= (12 +0.5)-24.5
1/2 √49 Z= -3.43
3.43 > 1.96 or -3.43 < -1.96
•p-value of Z= -3.43 = 0.0003.
•Since the hypothesis is two sided, multiply 0.0003*2= 0.0006
0.0006 < 0.05 and 0.01
so, reject the null hypothesis at 5% as well as at 1%.
•Hence, training of two groups differ significantly.
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