EPHE591

PostHocComparisonsandContrastsField,Chapter10

BasicLogic

RecallthatANOVAteststhefollowingnullhypothesis:

μ1 =μ2 =μ3

Thus,ifitisrejected,alltheANOVAtellsyouisthatthereisadifference.WeuseCONTRASTSorPOSTHOCcomparisonstodeterminewhichgroups

actuallydiffer.

HowTheyDiffer

- PostHoccomparisonscomparegroups- Contrastspartitionthevarianceinthemodel

- Contrastsareusedtotesthypotheses- PostHoctestsareusedwhenyoudonothaveahypothesis

Posthoc Comparisons

BasicLogic

Sowhynotjustrunaseriesoft-tests?

μ1 vs μ2,μ2=μ3,μ1 vs μ3

Becausewehavetotalkaboutthe“familywiseerrorrate”becausewearemakingaseriesofcomparisons.

I’veheardthatIamsafewiththreecomparisons,butnotwithfour…

μ1 =μ2 =μ3 weonlyhaveonecasewherewecanmakeaTypeIerror

Thisisalsotruefor

μ1 ≠μ2 =μ3

μ1 =μ2 ≠μ3

μ1 ≠μ2 ≠μ3

BUT…

I’veheardthatIamsafewiththreegroups,butnotfour…

μ1 =μ2 ANDμ3=μ4

HerewehaveTWOchancestomakeaTypeIerror,thusourfamilywiseerrorrateisactually0.1,not0.05

Thus,weshouldadjustfortheINFLATIONofTypeIerror.

Fisher’sLeastSignificantDifferenceTest

€

t =X1 − X 2

MSerror (1n1

+1n2)

Alsoknownasaprotectedt-test(wehavereplacedtheestimateofpooledvariancewithMSerror)

But,thisisinrealitystilljustat-testsoitisveryliberal(i.e.,notyourbestchoice).

OtherCommonPosthoc Tests

Bonferroni:Essentiallyadjustsalphatocorrectformultiplecomparisonsandthent-testsareused.

alphanew =alpha/numberoftests

OtherCommonPosthoc Tests

Sidak’s:SimilartoBonferroni butismoreconservative(hardertofinddifferences)

OtherCommonPosthoc Tests

Tukey’s:Reflectsanadjustmenttotheactualformulaeusedtogeneratethetstatistic,andcorrectsformultiplecomparisonsthisway.

Foralargenumberofcomparisons,Tukey’s TestismorepowerfulthanBonferonni.Forasmallnumberofcomparisons,viceversa.

OtherCommonPosthoc Tests

Hochberg’sGT2andGabriel’sPairwiseComparisonsTestsaresimilartoTukey’s Test.

Hochberg’sTestsisgenerallymoreliberal.Gabriel’sPairwiseTestsaregoodwhencellsizesareunequal.

OtherCommonPosthoc Tests

Dunnett’s PairwiseTestscompareallofthegroupmeansagainstacontrolmean(thefirstofthelast).Mostlyusedwhencomparingclinicalpopulationstoacontrolgroup.

OtherCommonPosthoc Tests

Tamhane’s T2,Dunnett’s T3,Games-Howell,andDunnett’s Careallteststobeusedwhenthevariancesareunequal.

OtherCommonPosthoc Tests

Waller-DuncanTest:ABayesian(probability)approach.

Scheffe Test:Generallythemostconservative,examinesalllinearcombinationsofvariables.

SimpleEffectsAnalysis

SimpleEffectsAnalysisdenotesonlyusingt-tests,butusingalogicalcomparison.IE,lookattheplotanddonotcomparethingsthatdonotmakesense.IfGroup2>Group1,andGroup3>Group2,whycompareGroup1andGroup3?

ComplexProcedures

PermutationTests

FDRMethodsBenjamini Algorithms

SowhatdoIdo…

1.Decideifyouaregoingtouseacontrastapproachoraposthocapproach.

2.Ifyouuseacontrastapproach,thekeyconcepttogetisthatyourcontrastsMUSTbelogicallydefined.

3.Ifyouuseaposthocapproach,thekeyconcepttogetisthatifyouaremakingmanycomparisons(morethan3)youneedtodosomething,anythingreally,toadjustforthis.

Contrasts

Essentiallycontrastsbreakdownthevarianceaccountedforbythemodeltoseewherethe

differenceslie.

RulesWhenUsingContrasts

1. Choosesensiblecomparisons,youonlywanttocomparetwo“chunks”ofvariancewitheachcomparison.Further,ifyousingleoutasinglegroupinacomparison,onlyuseitinonecomparisonandnosubsequentcomparisons.

RulesWhenUsingContrasts

2.Weightsareusedtosetupcontrasts.Ifasked,youmustdefinetheweightstosumtozero.

ContrastGroupA: +1ContrastGroupB: - 1

RulesWhenUsingContrasts

3.Groupsnotinacontrastshouldbegivenaweightof0.Why?0xGroupVariance=0soitisnegatedfromthecontrast.

TypesofContrastsDeviation:

Deviationfromoverallmeanwith1groupomitted.(generallynotused)

Simple:Level2vs Level1,Level3vs Level1,etc…(orlast)

Difference:Level2vs Level1,Level3vs Previous,etc…

Helmert:Level1vs Later,Level2vs Later,etc…

Repeated:Contrasts1to2,2to3,3to4,etc…

Polynomial:Looksforalinear,quadratic,orcubictrend.