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Unformatted text preview: AN alysis O f VA riance (ANOVA) for a sequence of models Model comparison can be generalized to a sequence of models (not just one full and one reduced model) Context: usual nGM model: y = X β + , ∼ N ( , σ 2 I ) Let X 1 = 1 and X m = X . But now, we have a sequence of models “in between” 1 and X Suppose X 2 , . . . , X m 1 are design matrices satisfying C ( X 1 ) < C ( X 2 ) < . . . < C ( X m 1 ) < C ( X m ) . We’ll also define X m + 1 = I c 2011 Dept. of Statistics (Iowa State University) Stat 511, section 13 1 / 19 Some examples Multiple Regression 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 ) is the decrease in SSE that results when the explanatory variable x i is added to a model containing 1 , x 1 , . . . , x j 1 . Test for linear trend and test for lack of linear fit. X 1 = 1 1 1 1 1 1 1 1 , X 2 = 1 1 1 1 1 2 1 2 1 3 1 3 1 4 1 4 , X 3 = 1 1 1 1 1 1 1 1 c 2011 Dept. of Statistics (Iowa State University) Stat 511, section 13 2 / 19 Context for linear lack of fit Let μ i = mean surface smoothness for a piece of metal ground for i minutes ( i = 1 , 2 , 3 , 4 ) . MS ( 2  1 ) / MSE can be used to test H o : μ 1 = μ 2 = μ 3 ⇐⇒ μ i = β i = 1 , 2 , 3 , 4 for some β o ∈ IR vs. H A : μ i = β + β 1 i i = 1 , 2 , 3 , 4 for some β o ∈ IR , β 1 ∈ IR \{ } . This is the F test for a linear trend, β 1 = vs. β 1 6 = MSE ( 3  2 ) / MSE can be used to test H o : μ i = β + β 1 i i = 1 , 2 , 3 , 4 for some β o β 1 ∈ IR vs. H A : There does not exist β , β 1 ∈ IR such that μ i = β + β 1 i ∀ i = 1 , 2 , 3 , 4 . This is known as the F test for lack of linear fit. Compares fit of linear regression model C ( X 2 ) to fit of means model C ( X 3 ) c 2011 Dept. of Statistics (Iowa State University) Stat 511, section 13 3 / 19 All tests can be written as full vs. reduced model tests Which means they could be written as tests of C β = d But, what is C ? Especially when interpretation of β changes from model to model Example: Y i = β + β 1 X i + i slope is β 1 Y i = β + β 1 X i + β 2 X 2 i + i slope at X i is β 1 + 2 β 2 X i In grinding study, X i = outside range of X i in data What can we say about the collection of tests in the ANOVA table?...
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 Spring '08
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 Regression Analysis, Variance

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