{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

13-AnalysisOfVariance(1)

# 13-AnalysisOfVariance(1) - AN alysisO fVA riance(ANOVA (...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: AN alysisO fVA riance(ANOVA) forasequenceofmodels Modelcomparisoncanbegeneralizedtoasequenceofmodels (notjustonefullandonereducedmodel) Context:usualnGMmodel: y = X β + , ∼ N ( ,σ 2 I ) Let X 1 = 1 and X m = X . Butnow,wehaveasequenceofmodels“inbetween” 1 and X Suppose X 2 ,..., X m − 1 aredesignmatricessatisfying C ( X 1 ) < C ( X 2 ) <...< C ( X m − 1 ) < C ( X m ) . We’llalsodefine X m + 1 = I c 2011Dept.ofStatistics(IowaStateUniversity) Stat511,section13 1/1 Someexamples MultipleRegression X 1 = 1 , X 2 =[ 1 , x 1 ] , X 3 =[ 1 , x 1 , x 2 ] ,... X m =[ 1 1 x 1 ,..., x m − 1 ] . SS ( j + 1 | j ) isthedecreaseinSSEthatresultswhentheexplanatory variable x i isaddedtoamodelcontaining 1 , x 1 ,..., x j − 1 . Testforlineartrendandtestforlackoflinearfit. X 1 = ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣ 1 1 1 1 1 1 1 1 ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦ , X 2 = ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣ 11 11 12 12 13 13 14 14 ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦ , X 3 = ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣ 1000 1000 0100 0100 0010 0010 0001 0001 ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦ c 2011Dept.ofStatistics(IowaStateUniversity) Stat511,section13 2/1 Contextforlinearlackoffit Let μ i = meansurfacesmoothnessforapieceofmetalgroundfor i minutes ( i = 1 , 2 , 3 , 4 ) . MS ( 2 | 1 ) / MSEcanbeusedtotest H o : μ 1 = μ 2 = μ 3 ⇐⇒ μ i = β i = 1 , 2 , 3 , 4 forsome β o ∈ IR vs. H A : μ i = β + β 1 ii = 1 , 2 , 3 , 4 forsome β o ∈ IR ,β 1 ∈ IR \{ } . ThisistheFtestforalineartrend, β 1 = vs. β 1 = MSE ( 3 | 2 ) / MSEcanbeusedtotest H o : μ i = β + β 1 ii = 1 , 2 , 3 , 4 forsome β o β 1 ∈ IR vs. H A : Theredoesnotexist β ,β 1 ∈ IR suchthat μ i = β + β 1 i ∀ i = 1 , 2 , 3 , 4 . ThisisknownastheFtestforlackoflinearfit. Comparesfitoflinearregressionmodel C ( X 2 ) tofitofmeansmodel C ( X 3 ) c 2011Dept.ofStatistics(IowaStateUniversity) Stat511,section13 3/1 Alltestscanbewrittenasfullvs.reducedmodeltests Whichmeanstheycouldbewrittenastestsof C β = d But,whatis C ? Especiallywheninterpretationof β changesfrommodeltomodel Example: Y i = β + β 1 X i + i slopeis β 1 Y i = β + β 1 X i + β 2 X 2 i + i slopeat X i is β 1 + 2 β 2 X i Ingrindingstudy, X i = outsiderangeof X i indata WhatcanwesayaboutthecollectionoftestsintheANOVAtable? c 2011Dept.ofStatistics(IowaStateUniversity) Stat511,section13 4/1 Generalframework Context:usualnGMmodel: y = X β + , ∼ N ( ,σ 2 I ) Let X 1 = 1 and X m = X . Suppose X 2 ,..., X m − 1 aredesignmatricessatisfying C ( X 1 ) < C ( X 2 ) <...<<....
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

13-AnalysisOfVariance(1) - AN alysisO fVA riance(ANOVA (...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online