This does not happen often in the regression setting

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when the three independent variables are orthogonal to each other. This does not happen often in the regression setting, but designed experiments are usually set up to take advantage of this. Note that if X 1 and X 2 are uncorrelated, then SSR( X 1 ) = R( X 1 | X 2 ) and SSR( X 2 ) = R( X 2 | X 1 ). PAGE 32
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2.2 General linear tests - REDUCED vs. FULL MODEL c circlecopyrt HYON-JUNG KIM, 2017 Example: DATA Y X 0 X 1 X 2 -2 1 -2 1 -3 1 1 0 0 1 2 0 2 1 1 3 Y Y = 17 , SST = 17 9 / 4 = 14 . 75 (X X) (X X) 1 X Y ˆ β Regress Y on X 0 only 4 1/4 -3 -3/4 (SSE= 14 . 75) Regress Y on X 0 , X 1 bracketleftBigg 4 2 2 10 bracketrightBigg bracketleftBigg 10 / 36 2 / 36 2 / 36 4 / 36 bracketrightBigg bracketleftBigg 3 3 bracketrightBigg bracketleftBigg 1 . 0 0 . 5 bracketrightBigg (SSE= 12 . 50) Regress Y on X 0 , X 1 , X 2 4 2 4 2 10 1 4 1 10 0 . 467 0 . 076 0 . 179 0 . 076 0 . 113 0 . 019 0 . 179 0 . 019 0 . 170 3 3 4 2 . 344 0 . 642 1 . 274 (SSE = 2 . 9481) SSE( X 2 ) =6.5833 R( X 2 | X 1 ) = 9 . 552 = SSR( X 1 , X 2 ) SSR( X 1 ) SOURCE TYPE I SSq(sequential) TYPE II SSq(partial) X1 R( X 1 ) = 2 . 25 R ( X 1 | X 2 ) =? X2 R( X 2 | X 1 ) = 9 . 552 R ( X 2 | X 1 ) = 9 . 552 ? = 3 . 6353 (last row: type I and II are equal) PAGE 33
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2.2 General linear tests - REDUCED vs. FULL MODEL c circlecopyrt HYON-JUNG KIM, 2017 Example: (cheese data) a) We want to know whether or not the variables X 1 (ACETIC) and X 2 (H2S) should be added to the model. Does the smaller model do just as well at describing the data as the full model? Analysis of Variance: Reduced Model Source DF Sum of Squares Mean Squares F Pr > F Regression 1 3800.4 3800.4 27.55 < 0.0001 Error 28 3862.5 137.9 Total 29 7662.887 Analysis of Variance: Full Model Source DF Sum of Squares Mean Squares F Pr > F Regression 3 4994.509 1664.836 16.22 < 0.0001 Error 26 2668.378 102.629 Total 29 7662.887 b) The ‘lm’ in R produces the following sequential sums of squares for the cheese data: > ch.lm=lm(TASTE ACETIC+H2S+LACTIC, data= cheese) Obtain the following sequential sums of squares to check whether or not each variable should be added to the model (given the preceding terms): R( X 1 ), R( X 2 | X 1 ), R( X 3 | X 1 , X 2 ) State the appropriate test hypotheses for each F-value computed in the ANOVA table. Source DF Sum of Squares Mean Squares F Pr > F ACETIC 1 2314.14 2314.14 22.55 < 0.0001 H2S 1 2147.11 2147.11 20.92 0.0001 LACTIC 1 533.26 533.26 5.20 0.0311 PAGE 34
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2.2 General linear tests - REDUCED vs. FULL MODEL c circlecopyrt HYON-JUNG KIM, 2017 c) Suppose that we had used the different ordering of model: > ch.lm1=lm(TASTE H2S+LACTIC+ACETIC, data= cheese) Compute R( X 1 ), R( X 2 | X 1 ), R( X 3 | X 1 , X 2 ). What is the major difference between results of b) and c) due to different ordering? Source DF Sum of Squares Mean Squares F Pr > F H2S 1 4376.8 4376.8 42.65 < 0.0001 LACTIC 1 617.1 617.1 6.01 0.02123 ACETIC 1 0.6 0.6 0.0054 0.94193 d) In R, > drop1(ch.lm, test = ”F”) gives the partial sums of squares. After adjusting for the effects of acetic and lactic concentrations, do we have significant evidence that the hydrogen sulfide concentration is important in describing taste?
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