11_ReducedVsFull(1)

11_ReducedVsFull(1) - REDUCED vs FULL MODEL F-TEST Tests of...

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REDUCED vs. FULL MODEL F -TEST I Tests of C β - d = 0 lead to F tests, but that’s not the only way to an F test I 500/402: big emphasis on model comparison I SS in ANOVA tables usually explained in terms of model comparison I e.g. SS for AB interaction is the difference in error SS between two models: Y ijk = μ + α i + β j + αβ ij + ijk and Y ijk = μ + α i + β j + ijk I Also leads to F tests I The test of no treatment effect in the dairy cow study can be viewed as a model comparison I Full model: Y ij = μ + α i + ij (or Y ij = μ i + ij I Reduced model (expresses Ho): Y ij = μ + ij Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1 / 1
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Questions about model comparison I Why does model comparison lead to F tests? I If you test Ho using a C β test or using a model comparison test, do you get the same answer? I Again, will answer using a general setup (Normal GM model) y = X β + , N ( 0 , σ 2 I ) I Each model (Full and Reduced) corresponds to an X matrix Two different column spaces for E ( y ) : C ( X o ) for reduced model, C ( X ) for full model I Key requirement: C ( X o ) ⊂ C ( X ) In words: Reduced model X matrix can be obtained by imposing constraints on full model. Ho describes those constraints. I Model comparison corresponds to testing H 0 : E ( y ) C ( X o ) vs. H A : E ( y ) C ( X ) / C ( X o ) . Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 2 / 1
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I The ”reduced” model corresponds to the null hypothesis and says that E ( y ) ∈ C ( X o ) , a specified subspace of C ( X ) . I The ”full” model says that E ( y ) can be anywhere in C ( X ) that is not in C ( X o ) . I For example, 1-way ANOVA with 3 groups. X o = 1 1 1 1 1 1 and X = 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 Copyright c 2011 Dept. of Statistics (Iowa State University)
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