01Introduction(1)

01Introduction(1) - Introduction - Asimplestudy:...

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Unformatted text preview: Introduction ThefirstpartofStat511re-examinesmethodsfromStat500from alinearmodelsperspective. Asimplestudy:doesaproprietaryfoodadditiveincreasemilk productionindairycows? 6cows,housedoneperstall.Randomlychoose3togetfoodwith theadditive;other3without. response:averagedailymilkproductiontwoweeksafterstartof treatment Most(all?)woulduseat-testandrelatedmethodsto: estimatethedifferenceoftreatmentmeans estimatethestandarderrorofthatdifference constructa95%confidenceintervalforthatdifference testwhetherthedifferenceis0 Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 1/36 Introduction(continued) Onepossiblemodel: y ij = μ i + ij y ij istheresponsefromcow j intreatment i , i =1,2; j =1 ··· 3 μ i isthepopulationmeanfortreatment i ij isthedeviationfromthepopulationmeanforcow j intrt i Assume: ij ∼ N ( ,σ 2 ) Standardmethodsuse: ¯ y 1 − ¯ y 2 toestimate μ 1 − μ 2 s 2 = ∑ i , j ( y ij − ¯ y i ) 2 / ( 2 n − 2 ) toestimate σ 2 at-distributiontoconstructtheconfidenceintervalortesta hypothesis Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 2/36 Introduction(continued) Somequestions: Whataboutt-testsandrelatedestimationmethodsgeneralizeto morecomplicatedmodels? i.e.,howisthet-testanexampleof y = X β + Doesitmatterwhetheryouuseacellmeansoreffectsmodel? i.e.,doesitmatterwhich X youuse? Is ¯ y 1 − ¯ y 2 agoodestimateof μ 1 − μ 2 ? Isthereamorepreciseestimateof μ 1 − μ 2 ? Whendoes [( ¯ y 1 − ¯ y 2 ) − ( μ 1 − μ 2 )] / s 2 ∗ 2 / n followatdistribution? We’llanswertheseandmoreusingaverygeneralframework Basedonlinearmodelstheory. Providesguidetowhenmethodsworkand whenyouhaveaproblem. Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 3/36 Introduction(continued) Noteshavelotsofequationsand lots ofmatrices Willusematricestopresentresults Skim(orread)theKoehlernotesonmatrices http://www.public.iastate.edu/ ∼ kkoehler/stat511/sect2.4page.pdf Skipmaterialoneigenvaluesandotherdecompositionsfornow Willshowproofs.Wantyoutounderstandconceptsandresults. Willnotexpectyoutoproveresultsonexams. StatPhDstudentstake611:fullsemesteroflinearmodeltheory withproofs. Willtrytobecarefulaboutnotation.Askifyoudon’tunderstand something. Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 4/36 TheGauss-MarkovLinearModel y = X β + y isan n × 1 randomvectorofresponses. X isan n × p matrixofconstantswithcolumnscorrespondingto explanatoryvariables. X issometimesreferredtoasthe design matrix . β isanunknownparametervectorin IR p . isan n × 1 randomvectoroferrors. E ()= and Var ()= σ 2 I ,where σ 2 isanunknownparameterin IR + . Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 5/36 TheGauss-MarkovLinearModel Notethatthemodelisnotcompletelyspecifiedbecausethe distributionof y isnotcompletelyspecified. y = X β + , E ()= , Var ()= σ 2 I = ⇒ E ( y )= X β , Var ( y )= σ 2 I = ⇒ y ∼ ( X β ,σ 2 I ) “ y hasadistributionwithmean X β andvariance...
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.

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01Introduction(1) - Introduction - Asimplestudy:...

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