Testing utility of model -F-test

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1、Testingutilityofmodel-F-test•Testingtheutilityofthemodeltopredictybyconductingindividualt−testsoneachoftheβ’sisnotagoodidea.•Why?Evenifweconducteachtestattheα=0.05level,theoverallprobabilityofincorrectlyrejectingH0(theprobabilityofaTypeIerror)islargerthan0.05.

2、•Evenifwebeginwithanαlevelof,forexample,0.05oneachindividualtestforβ’s,theoverallprobabilityofanerrorwillalwaysbelargerthantheprobabilityofTypeIerroronindividualtests.•Thelargerthenumberofpredictorsinthemodel,thehighertheprobabilitythatatleastoneofthosehypothe

3、sestestswillleadtothewrongconclusion.Stat328-Fall20041Testingutilityofmodel-F-test(cont’d)•Illustration:supposeweconducttestsontwoparametersβ1,β2andweuseα=0.05oneachtest.Seewhatmayhappen:Correctdecisionforβ1Incorrectdecisionforβ1Correctdecisionforβ20.95×0.950.

4、95×0.05Incorrectdecisionforβ20.95×0.050.05×0.05•Theprobabilityofacorrectdecisiononbothparametersisonly0.95×0.95=0.90!SoourindividualTypeIerrorrate(α)iserodedinmultipletests.•Ingeneral,foranindividualTypeIerrorrateofα,theoverallerrorrateonktestsis(1−α)k.Forexam

5、ple,fork=4andα=0.05,theprobabilitythatwewillreachthecorrectconclusionforallfourβ’sisonly0.954=0.81,sotheexperiment-wiseαisreally0.19.Stat328-Fall20042Testingutilityofmodel-F-test(cont’d)•Ratherthantestingeachβindividually,weuseaglobaltestthatencompassesallβ’sa

6、ndtestthefollowingoverallhypothesis:H0:β1=β2=...=βk=0Ha:atleastoneβj6=0.•TheteststatistictotestthishypothesisiscalledF−statisticandiscalculatedas:(SS−SSE)/kR2/kyyF==.MSE(1−R2)/[n−(k+1)]Stat328-Fall20043Testingutilityofmodel-F-test(cont’d)•TheFstatisticistherat

7、ioof–theexplainedvariability(asreflectedbyR2)and–theunexplainedvariability(asreflectedby1−R2),eachdividedbythecorrespondingdegreesoffreedom.•ThelargertheFstatistic,themoreusefulthemodel.Stat328-Fall20044Testingutilityofmodel-F-test(cont’d)•Criticalvalueforthetes

8、t:Fα,numdf,denomdf(threesubscripts:α,numeratordegreesoffreedomanddenominatordegreesoffreedom).•WeuseanF−table(pages763-770inbook).•Eachtablecorrespondstoadifferentα:0.1,0.05,0.025an