Let sse x 1 x 2 x 3 be the error sum of squares for

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Let SSE( X 1 , X 2 , X 3 ) be the error sum of squares for full model, i.e. SSE(FULL). Define SSE( X 1 ) = error sum of squares for model Y i = β 0 + β 1 X i 1 + ǫ i SSE( X 2 ) = error sum of squares for model Y i = β 0 + β 2 X i 2 + ǫ i SSE( X 3 ) = error sum of squares for model Y i = β 0 + β 2 X i 3 + ǫ i SSE( X 1 , X 2 ) = error sum of squares for model Y i = β 0 + β 1 X i 1 + β 2 X i 2 + ǫ i SSE( X 1 , X 3 ) = error sum of squares for model Y i = β 0 + β 1 X i 1 + β 2 X i 3 + ǫ i and SSE( X 2 , X 3 ) = error sum of squares for model Y i = β 0 + β 1 X i 2 + β 2 X i 3 + ǫ i PAGE 30
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2.2 General linear tests - REDUCED vs. FULL MODEL c circlecopyrt HYON-JUNG KIM, 2017 If you want to test whether β 3 is needed, given that β 1 and β 2 are already in the model, then the extra sums of squares are defined as R( X 3 | X 1 , X 2 ) = SSE( X 1 , X 2 ) SSE( X 1 , X 2 , X 3 ). The “R” stands for the reduction in the error sums of squares due to fitting the additional term. The F test for β 3 = 0 is F = R( X 3 | X 1 , X 2 ) / ( df red df full ) MSE(full) = (SSE(reduced) SSE(full)) / ( df red df full ) MSE(full) which is the same test as we had before but expressed in different notation. The extra sums of squares can also be written as a difference of regression sums of squares. For example, R( X 3 | X 1 , X 2 ) = SSE( X 1 , X 2 ) SSE( X 1 , X 2 , X 3 ) = [SST SSR( X 1 , X 2 )] [SST SSE( X 1 , X 2 , X 3 ) ] = SSR( X 1 , X 2 , X 3 ) SSR( X 1 , X 2 ) Sequential Sums of Squares or “Type I sums of squares” (in SAS) They are extra sums of squares for adding each term given that preceding terms in the model statement are already in the model. For the model with 3 independent variables: the Type I sums of squares are (in SAS MODEL Y= X 1 X 2 X 3 ;) These sequential sums of squares SOURCE df SS X 1 1 R( X 1 ) X 2 1 R( X 2 | X 1 ) X 3 1 R( X 3 | X 1 , X 2 ) error n 4 SSE( X 1 , X 2 , X 3 ) add up to SSR( X 1 , X 2 , X 3 ). R( X 1 )+ R( X 2 | X 1 )+ R( X 3 | X 1 , X 2 ) =SSR( X 1 )+ SSR( X 1 , X 2 ) SSR( X 1 )+ SSR( X 1 , X 2 , X 3 ) SSR( X 1 , X 2 ) =SSR( X 1 , X 2 , X 3 ) PAGE 31
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2.2 General linear tests - REDUCED vs. FULL MODEL c circlecopyrt HYON-JUNG KIM, 2017 The sequential sums of squares depend on the order that you write the independent variables in the model statement. For example, the model statement: MODEL Y= X 3 X 2 X 1 would give SOURCE df SS X 3 1 R( X 3 ) X 2 1 R( X 2 | X 3 ) X 1 1 R( X 1 | X 2 , X 3 ) error n 4 SSE( X 1 , X 2 , X 3 ) Partial Sums of Squares or “Type II” of sums of squares (in SAS) These are extra sums of squares for adding a term to the model given that all the other independent variables in the model statement are already in the model. The order of terms in the model makes no difference. For either of the following model statements MODEL Y= X 1 X 2 X 3 MODEL Y= X 3 X 2 X 1 the partial sums of squares are the same; But these sums of squares do not add up to the SOURCE df SS X 1 1 R( X 1 | X 2 , X 3 ) X 2 1 R( X 2 | X 1 , X 3 ) X 3 1 R( X 3 | X 1 , X 2 ) error n 4 SSE( X 1 , X 2 , X 3 ) total regression sum of squares. In other words, this is not a partition of the regression sum of squares. The only case that the Type II sums of squares do add up to SSR( X 1 , X 2 , X 3 ) is when the three independent variables are orthogonal to each other. This does not happen
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