This limitation does not allow us under the procedures of hypothesis testing to address the basic question Does the length the reaction rate the fraction displaying a particular behavior or having a particular opinion the temperature the kinetic ene ID: 33182
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IntroductiontoStatisticalMethodologyCompositeHypotheses Inreality,incorrectdecisionsaremade.Thus,for20,()istheprobabilityofmakingatypeIerror;i.e.,rejectingthenullhypothesiswhenitisindeedtrue,For21,1�()istheprobabilityofmakingatypeIIerror;i.e.,failingtorejectthenullhypothesiswhenitisfalse.Thegoalistomakethechanceforerrorsmall.ThetraditionalmethodisanalogoustothatemployedintheNeyman-Pearsonlemma.Fixa(signicance)level ,nowdenedtobethelargestvalueof()intheregion0denedbythenullhypothesis.Inotherwords,byfocusingonthevalueoftheparameterin0thatismostlikelytoresultinanerror,weinsurethattheprobabilityofatypeIerrorisnomorethat irrespectiveofthevaluefor20.Then,welookforacriticalregionthatmakesthepowerfunctionaslargeaspossibleforvaluesoftheparameter21Example1.LetX1;X2;:::;XnbeindependentN(;20)randomvariableswith20knownandunknown.Forthecompositehypothesisfortheone-sidedtestH0:0versusH1:�0:WeusetheteststatisticfromthelikelihoodratiotestandrejectH0ifXistoolarge.Thus,thecriticalregionC=fx;xk(0)g:Ifisthetruemean,thenthepowerfunction()=PfX2Cg=PfXk(0)g:Asweshallseesoon,thevalueofk(0)dependsonthelevelofthetest.Notethat()increaseswith.Astheactualmeanincreases,thentheprobabilitythatthesamplemeanXexceedsaparticularvaluek(0)alsoincreases.Toobtainlevel forthehypothesistest,weusethatfactthat()increaseswithtoconcludethatthemaximumvalueofontheset0=f;0gtakesplaceforthevalue0,i.e., =(0)=P0fXk(0)g:Wenowusethistondthevaluek(0).When0isthevalueofthemean,westandardizetogiveastandardnormalrandomvariableZ=X�0 0=p n:Choosez sothatPfZz g= .Inthiscase,(z )=1� whereisthedistributionfunctionforthestandardnormal,thusP0fZz g=P0fX0+0 p nz gandk(0)=0+(0=p n)z .Ifisthetruestateofnature,thenZ=X� 0=p nisastandardnormalrandomvariable.Weusethisfacttodeterminethepowerfunctionforthistest.()=PfX0 p nz +0g=PfX�0 p nz �(�0)g(1)=PX� 0=p nz ��0 0=p n=1�z ��0 0=p n(2)227 IntroductiontoStatisticalMethodologyCompositeHypotheses Nbethetotalpopulation.IfN0isthelevelthatawildlifebiologistsayisdangerouslylow,thenthenaturalhypothesisisone-sided.H0:NN0versusH1:NN0:Thedataarethehenumberinthesecondcapturethataretagged,r.ThelikelihoodfunctionforNisthehypergeo-metricdistribution.L(Njr)=�tr�N�tk�r �Nkandthemaximumlikelihoodestimateis^N=[tk=r]:Thus,highervaluesforrleadustolowerestimatesforN.LetRbethe(random)numberinthesecondcapturethataretagged,then,foran leveltest,welookfortheminimumvaluer sothatPNfRr g forallNN0:(3)AsNincreases,thenrecapturesbecomelesslikelyandtheprobabilityin(3)decreases.Thus,weshouldsetthevalueofr accordingtotheparametervalueN0,theminimumvalueunderthenullhypothesis.Let'sdeterminer forseveralvaluesof usingtheexamplefromthetopic,MaximumLikelihoodEstimation,andconsiderthecaseinwhichthecriticalpopulationisN0=2000.�N0&#x-200;�]TJ;&#x 0 -;.9;U T; [0;t&#x-200;&#x]TJ ;� -1;.95; Td;&#x [00;k&#x-400;&#x]TJ ;� -1;.95; Td;&#x [00;alpha()&#x-c0.;,0;&#x.02,;�.01;&#x]TJ ;� -1;.95; Td;&#x [00;ralpha()&#x-qhy;&#xper1;&#x-alp;&#xha,t;&#x,N0-;&#xt,k];&#xTJ 0;&#x -11;&#x.955;&#x Td ;&#x[000;data.frame(alpha,ralpha)alpharalpha10.054920.025130.0153Forexample,wemustcapturealleast49thatweretaggedinordertorejectH0atthe =0:05level.InthiscasetheestimateforNis^N=[kt=r ]=1632.Asanticipated,r increasesandthecriticalregionsshrinksasthevalueof decreases.Usingthelevelr determinedusingthevalueN0forN,weseethatthepowerfunction(N)=PNfRr g:RisahypergeometricrandomvariablewithmassfunctionfR(r)=PNfR=rg=�tr�N�tk�r �Nk:Theplotforthecase =0:05isgivenusingtheRcommands�N()&#x-c13;�:2;Ā];&#xTJ 0;&#x -11;&#x.955;&#x Td ;&#x[000;pi()&#x-1-p;&#xhype;&#xr49,;&#xt,N-;&#xt,k];&#xTJ 0;&#x -11;&#x.955;&#x Td ;&#x[000;plot(N,pi,type="l",ylim=c(0,1))Wecanincreasepowerbyincreasingthesizeofk,thenumberthevalueinthesecondcapture.Thisincreasesthevalueofr .For =0:05,wehavethetable.229 IntroductiontoStatisticalMethodologyCompositeHypotheses Figure3:Powerfunctionforthetwo-sidedtest.0=10,0=3.(left)n=16, =0:05(black),0.02(red),and0.01(blue).Noticethatlowersignicancelevel reducespower.(right) =0:05,n=15(black),40(red),and100(blue).Asbefore,decreasedsignicancelevelreducespowerandincreasedsamplesizenincreasespower.Weusethisfacttodeterminethepowerfunctionforthistest()=PfX2Cg=1�PfX=2Cg=1�PX�0 0=p nz =2=1�P�z =2X�0 0=p nz =2=1�P�z =2��0 0=p nX� 0=p nz =2��0 0=p n=1�z =2��0 0=p n+�z =2��0 0=p nIfwedonotknowifthemimicislargerorsmallerthatthemodel,thenweuseatwo-sidedtest.BelowistheRcommandsforthepowerfunctionwith =0:05andn=16observations.�zalpha=qnorm(.975)�mu0&#x-10];&#xTJ 0;&#x -11;&#x.955;&#x Td ;&#x[000;sigma0&#x-3]T;&#xJ 0 ;&#x-11.;ॕ ;&#xTd [;mu()&#x-600;&#x:140;�/10;�]TJ;&#x 0 -;.9;V T; [0;n&#x-16];&#xTJ 0;&#x -11;&#x.955;&#x Td ;&#x[000;pi(()(()))(()(()))&#x-1-p;&#xnorm;&#xzalp;&#xha-m;&#xu-mu;�/si;&#xgma0;&#x/sqr;&#xtn+p;&#xnorm;&#x-zal;&#xpha-;&#xmu-m;&#xu0/s;&#xigma;�/sq;&#xrtn];&#xTJ 0;&#x -11;&#x.955;&#x Td ;&#x[000;plot(mu,pi,type="l")Weshallseeinthethenexttopichowthesetestsfollowfromextensionsofthelikelihoodratiotestforsimplehypotheses.Thenextexampleisunlikelytooccurinanygenuinescienticsituation.Itisincludedbecauseitallowsustocomputethepowerfunctionexplicitlyfromthedistributionoftheteststatistic.Webeginwithanexercise.Exercise6.ForX1;X2;:::;XnindependentU(0;)randomvariables,2=(0;1).ThedensityfX(xj)=1 if0x;0otherwise:231 IntroductiontoStatisticalMethodologyCompositeHypotheses LetX(n)denotethemaximumofX1;X2;:::;Xn,thenX(n)hasdistributionfunctionFX(n)(x)=PfX(n)xg=x n:Example7.ForX1;X2;:::;XnindependentU(0;)randomvariables,takethenullhypothesisthatlandsinsomenormalrangeofvalues[L;R].Thealternativeisthatliesoutsidethenormalrange.H0:LRversusH1:Lor&#x-277;R:IfanyofourobservationsXiaregreaterthanR,thenwearecertain&#x-277;RandweshouldrejectH0.Ontheotherhand,alloftheobservationscouldbebelowLandthestateofnaturemightstilllandinthenormalrange.Consequently,wewilltrytobaseatestbasedonthestatisticX(n)=max1inXiandrejectH0ifX(n)&#x-277;RandtoomuchsmallerthanL,say~.Weshallsoonseethatthechoiceof~willdependonnthenumberofobservationsandon ,thesizeofthetest.Thepowerfunction()=PfX(n)~g+PfX(n)RgWecomputethepowerfunctioninthreecases.Thesecondcasehasthevaluesofunderthenullhypothesis.Therstandthethirdcaseshavethevaluesforunderthealternativehypothesis.AnexampleofthepowerfunctionisshowninFigure3. Figure4:PowerfunctionforthetestabovewithL=1;R=3;~=0:9,andn=10.Thesizeofthetestis(1)=0:3487.Case1.~.InthiscasealloftheobservationsXimustbelessthanwhichisinturnlessthan~.Thus,X(n)iscertainlylessthan~andPfX(n)~g=1andPfX(n)Rg=0andtherefore()=1.Case2.~R.232 IntroductiontoStatisticalMethodologyCompositeHypotheses PfX(n)~g= ~ !nandPfX(n)Rg=0andtherefore()=(~=)n.Case3.�R.RepeattheargumentinCase2toconcludethatPfX(n)~g= ~ !nandthatPfX(n)Rg=1�PfX(n)Rg=1�R nandtherefore()=(~=)n+1�(R=)n.Thesizeofthetestisthemaximumvalueofthepowerfunctionunderthenullhypothesis.Thisiscase2.Here,thepowerfunction()= ~ !ndecreasesasafunctionof.Thus,itsmaximumvaluetakesplaceatLand =(L)= ~L !nToachievethislevel,wesolveof~andtake~=Lnp :Notethat~increaseswith .Consequently,wemustreducethecriticalregioninordertoreducethesignicancelevel.Also,~increaseswithnandwecanreducethecriticalregionwhilemaintainingsignicanceifweincreasethesamplesize.2Thep-valueThereportofrejectthenullhypothesisdoesnotdescribethestrengthoftheevidencebecauseitfailstogiveusthesenseofwhetherornotasmallchangeinthevaluesinthedatacouldhaveresultedinadifferentdecision.Consequently,onecommonmethodisnottochoose,inadvance,asignicancelevel ofthetestandthenreport“reject”or“failtoreject”,butrathertoreportthevalueoftheteststatisticandtogiveallthevaluesfor thatwouldleadtotherejectionofH0.Thep-valueistheprobabilityofobtainingaresultatleastasextremeastheonethatwasactuallyobserved,assumingthatthenullhypothesisistrueandmeasuresthestrengthofevidenceagainstH0.Consequently,averylowp-valueindicatesstrongevidenceagainstthenullhypothesis.Ifthep-valueisbelowagivensignicancelevel ,thenwesaythattheresultisstatisticallysignicantatthelevel .Forexample,ifthetestisbasedonhavingateststatisticS(X)exceedalevelk,i.e.,wehavedecisionrejectH0ifandonlyifS(X)k:andifthevalueS(X)=k0isobserved,thenthep-valueequalstothelowestvalueofsignicancelevelthatwoldresultinrejectionofthenullhypothesis.maxf();20g=maxfPfS(X)k0g;20g:233 IntroductiontoStatisticalMethodologyCompositeHypotheses data:88outof112,nullprobability0.7X-squared=3.5208,df=1,p-value=0.0303alternativehypothesis:truepisgreaterthan0.795percentconfidenceinterval:0.71078071.0000000sampleestimates:p0.7857143Exercise9.Isthehypothesistestabovesignicantatthe5%level?the1%level?3AnswerstoSelectedExercises2.InthiscasethecriticalregionsisC=fx;xk(0)gforsomevaluek(0).Tondthisvalue,notethatP0fZ�z g=P0fX�0 p nz +0gandk(0)=�(0=p n)z +0.Thepowerfunction()=PfX�0 p nz +0g=PfX��0 p nz �(�0)g=PX� 0=p n�z ��0 0=p n=�z ��0 0=p n:5.ThetypeIIerrorrateis1�(1600)=P1600fRr g:ThisisthedistributionfunctionofahypergeometricrandomvariableandthustheseprobabilitiescanbecomputedusingthephypercommandForvaryingsignicance,wehavetheRcommands:&#x-278;t&#x-160;�]TJ;&#x 0 -;.9;U T; [0;k&#x-400;&#x]TJ ;� -1;.95; Td;&#x [00;alpha()&#x-c0.;,0;&#x.02,;�.01;&#x]TJ ;� -1;.95; Td;&#x [00;ralpha()&#x-c49;&#x,51,;S]T;&#xJ 0 ;&#x-11.;ॕ ;&#xTd [;beta()&#x-phy;&#xperr; lph; ,t,;&#xN-t,;&#xk]TJ;&#x 0 -;.9;U T; [0;data.frame(alpha,beta)alphabeta10.050.469559120.020.607165630.010.7316226NoticethatthetypeIIerrorprobabilityishighfor =0:05andincreasesas decreases.Forvaryingrecapturesize,wecontinuewiththeRcommands:&#x-phy;&#xperr; lph; ,t,;&#xN-t,;&#xk]TJ;&#x 0 -;.9;U T; [0;k()&#x-c40;�,60;�,80;�]TJ;&#x 0 -;.9;U T; [0;ralpha()&#x-c49;&#x,70,;‘]T;&#xJ 0 ;&#x-11.;ॖ ;&#xTd [;beta()&#x-phy;&#xperr; lph; ,t,;&#xN-t,;&#xk]TJ;&#x 0 -;.9;U T; [0;data.frame(k,beta)kbeta14000.4695591326000.2419490538000.09933596235