Statistical tests for data involving quantitative data

Statistical tests for quantitative dataDr. S. A. Rizwan, M.D.

PublicHealthSpecialistSBCM, JointProgram– Riyadh

MinistryofHealth,KingdomofSaudiArabia

Learningobjectives

Demystifying statistics! SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• Describetheapproachtostatisticaltestinginvolvingquantitativevariables

Revise:Categoricalvariables


• Numerical(Quantitative)

• Continuous• Interval• Ratio

• Answers“howmany?”• Quantitativedataismeasured

Revise:Categoricalvariables


Revise:Prerequisitesforatest


• Howmanyvariablesarethere?

• Whatisthenatureofdependentandindependentvariable?

• Howmanycategoriesarethereinthecategoricalvariable?

• Doesthecontinuousvariablefollownormaldistribution?

• Isthereanypairinginthedata/variables?

Statisticaltests:Bivariate

SBCM, Joint Program – RiyadhSBCM, Joint Program – RiyadhDemystifying statistics!

Forunpaireddata:Parametric Forunpaireddata:Non-parametric

• Categorical(2levels)vs.Quantitative• Independent samplesttest

• Categorical(>2levels)vs.Quantitative• OnewayANOVA

• Categorical(2levels)vs.Quantitative• MannWhitneyUtest(akaWilcoxonranksumtest)

• Categorical(>2levels)vs.Quantitative• Kruskal Wallis



Forpaireddata:Parametric Forpaireddata:Non-parametric

• Categorical(2levels)vs.Quantitative• Pairedttest

• Categorical(>2levels)vs.Quantitative• RepeatedmeasuresANOVA

• Categorical(2levels)vs.Quantitative• Wilcoxonsignranktest

• Categorical(>2levels)vs.Quantitative• Friedmantest



Parametric Non-parametric

• Quantitativevs.Quantitative• Pearson’scorrelation

• Quantitativevs.Quantitative• Simplelinearregression

• Quantitativevs.Quantitative• Spearman’s rankcorrelation

Statisticaltests:Multivariate

SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

Forunpaireddata ForpairedorRMdata

Demystifying statistics!

• IfDVisquantitativeand>1IV• Multiplelinearregression

• IfDVisquantitativeand>1IV• GLM

Measuresofassociation


• Correlationcoefficient• Regressioncoefficient

Someselectedtopics


• Coveredinotherclasses• Ttestsanditstypes• Correlation• Regression

• Inthisclasswewillcoverbasicsof:• Mann-WitneyUtest• KruskalWallistest• Wilcoxonsignedranktest• Friedman’stest

Thoughtexercise1


• Atotalofn=10participantsarerandomizedtoreceiveeitherthenewdrugoraplacebo.Participantsareaskedtorecordthenumberofepisodesofshortnessofbreathovera1weekperiod.Isthereadifferenceinthenumberofepisodesinparticipantsreceivingthenewdrugascomparedtothosereceivingtheplacebo?

Thoughtexercise2


• Threedietsarecompared,5%,10%,15%protein

• Thealbuminlevelsofparticipantsfollowingeachdietareshown

• Isthereisadifferenceinserumalbuminlevelsamongsubjectsonthethreedifferentdiets?

Thoughtexercise3


• Atotalof8childrenwithautismenroll inthestudyandtheamountoftimethateachchildisengagedinrepetitivebehavior duringthreehourobservationperiodsaremeasuredbothbeforetreatmentandthenagainaftertakingthenewmedicationforaperiodof1week. Thedataareshown.

• Isthereanyimprovedinthechildren?

Thoughtexercise4


• Fourresidentsaregiventhreesamplesofkahwa preparedbythreedifferentpersons.Theyscorethequalityofthekahwa innumericalterms.Thedataisshown.

• Isthereisadifferencebetweenthe3typesofkahwa?

Mann–WhitneyUtest

Demystifying statistics! – Lecture 10 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh

• AlsoknownasWilcoxonRankSumTest• Anonparametricprocedurethatdeterminesifrankedscores(i.e.,ordinaldata)intwoindependentgroupsdiffer

• Alsousedtoanalyze intervalorratioscalevariablesthatarenotnormallydistributed

• Someinterpretthistestascomparingthemediansbetweenthetwopopulations

• H0:Thetwopopulationsareequalversus• H1:Thetwopopulationsarenotequal

Mann–WhitneyUtest


• Atotalofn=10participantsarerandomizedtoreceiveeitherthenewdrugoraplacebo.Participantsareaskedtorecordthenumberofepisodesofshortnessofbreathovera1weekperiodfollowingreceiptoftheassignedtreatment.Thedataareshownbelow.

• Isthereadifferenceinthenumberofepisodesinparticipantsreceivingthenewdrugascomparedtothosereceivingtheplacebo?

Mann–WhitneyUtest


where R1 = sum of the ranks for group 1 and R2 = sum of the ranks for group 2.

Mann–WhitneyUtest


• Inourexample,U=3(smallerofthetwoUs).Isthisevidenceinsupportofthenullorresearchhypothesis?

• ThecriticalvalueofUcanbefoundinthetable.Todeterminetheappropriatecriticalvalueweneedsamplesizes(n1=n2=5)andourtwo-sidedlevelofsignificance(α=0.05).

• Thecriticalvalueis2,andthedecisionruleistorejectH0 ifU < 2.• WedonotrejectH0 because3>2.

Kruskal Wallistest


• NonparametrictesttocompareoutcomesamongmorethantwoindependentgroupsistheKruskal Wallistest.

• Usedtocomparemediansamongkcomparisongroups(k>2)• DescribedasanANOVAwiththedatareplacedbytheirranks.

• Thenullandresearchhypothesesarestatedasfollows:• H0:Thekpopulationmediansareequalversus• H1:Thekpopulationmediansarenotallequal

Kruskal Wallistest


• Threedietsarecompared,rangingfrom5%to15%protein,andthe15%proteindietrepresentsatypicalAmericandiet.Thealbuminlevelsofparticipantsfollowingeachdietareshownbelow

• Isthereisadifferenceinserumalbuminlevelsamongsubjectsonthethreedifferentdiets?

Kruskal Wallistest


• Wefirstorderthedatainthecombinedtotalsampleof12subjectsfromsmallesttolargest.

• Wealsoneedtokeeptrackofthegroupassignmentsinthetotalsample

Kruskal Wallistest


• TheteststatisticisdenotedHandisdefinedasfollows:

• wherek=thenumberofcomparisongroups,N=thetotalsamplesize,nj isthesamplesizeinthejth groupandRj isthesumoftheranksinthejth group

• InthisexampleR1 =7.5,R2 =30.5,andR3 =40

Kruskal Wallistest


• Thecriticalvalueis5.656,thuswerejectH0 because7.52 > 5.656,andweconcludethatthereisadifferenceinmedianalbuminlevelsamongthethreedifferentdiets.

Wilcoxonsignedranktest


• Nonparametrictestformatchedorpaireddata• Itisbasedondifferencescores,butinadditiontoanalyzing thesignsofthedifferences(unlikethesigntest),italsotakesintoaccountthemagnitudeoftheobserveddifferences



• Atotalof8childrenwithautismenroll inthestudyandtheamountoftimethateachchildisengagedinrepetitivebehavior duringthreehourobservationperiodsaremeasuredbothbeforetreatmentandthenagainaftertakingthenewmedicationforaperiodof1week. Thedataareshown



• First,wecomputedifferencescoresforeachchild.



• Wefirstorderthe absolutevaluesofthedifferencescores andassignrankfrom1throughntothesmallestthroughlargestabsolutevaluesofthedifferencescores,

• Assignthemeanrankwhentherearetiesintheabsolutevaluesofthedifferencescores.



• Thefinalstepistoattachthesigns("+"or"-")oftheobserveddifferencestoeachrankasshownbelow.



• TheteststatisticfortheWilcoxonSignedRankTestisW,definedasthesmallerofW+(sumofthepositiveranks)andW- (sumofthenegativeranks).

• Inthisexample,W+=32andW- =4.• ThecriticalvalueofWis6andthedecisionruleistorejectH0 ifW < 6.Thus,werejectH0,because4 < 6.

Friedmantest


• Non-parametricalternativetotheone-wayANOVAwithrepeatedmeasures

• Itisusedtotestfordifferencesbetweengroupswhenthedependentvariableisquantitativebutcorrelated

• Itcanalsobeusedforcontinuousdatathathasviolatedtheassumptionsnecessaryofone-wayANOVAwithrepeatedmeasures;forexample,markeddeviationsfromnormality

Friedmantest


• distributedaschi-squarewithdf =k– 1,• chqr =4.5• P=0.1054

Advancedlearning


• RepeatedmeasuresANOVA• GeneralisedLinearModels• Coxregression• Factoranalysis• Longitudinaldataanalysis• Othertypesofregressionmodels• Meta-analysis

Takehomemessages


• Manyapproachesareavailableforanalysingquantitativedata• Chooseamethodappropriateforyourproblem• Checkthattheassumptionsofthemethodarevalid• Makeconclusionsbasedontheresultsofthetest

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Health & Medicine

Statistical tests for data involving quantitative data