Data were collected on the following variables y

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Data were collected on the following variables: Y = taste test score (TASTE) X 1 = concentration of acetic acid (ACETIC) X 2 = concentration of hydrogen sulfide (H2S) X 3 = concentration of lactic acid (LACTIC) Variables ACETIC and H2S are both on the (natural) log scale. The variable LACTIC has not been transformed. Table below contains concentrations of the various chemicals in n = 30 specimens of mature cheddar cheese and the observed taste score. Suppose that the researchers postulate that each of the three chemical composition co- variates X 1 , X 2 , and X 3 are important in describing the taste. In this case, they might PAGE 21
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c circlecopyrt HYON-JUNG KIM, 2017 TASTE ACETIC H2S LACTIC TASTE ACETIC H2S LACTIC 12.3 4.543 3.135 0.86 40.9 6.365 9.588 1.74 20.9 5.159 5.043 1.53 15.9 4.787 3.912 1.16 39.0 5.366 5.438 1.57 6.4 5.412 4.700 1.49 47.9 5.759 7.496 1.81 18.0 5.247 6.174 1.63 5.6 4.663 3.807 0.99 38.9 5.438 9.064 1.99 25.9 5.697 7.601 1.09 14.0 4.564 4.949 1.15 37.3 5.892 8.726 1.29 15.2 5.298 5.220 1.33 21.9 6.078 7.966 1.78 32.0 5.455 9.242 1.44 18.1 4.898 3.850 1.29 56.7 5.855 10.20 2.01 21.0 5.242 4.174 1.58 16.8 5.366 3.664 1.31 34.9 5.740 6.142 1.68 11.6 6.043 3.219 1.46 57.2 6.446 7.908 1.90 26.5 6.458 6.962 1.72 0.7 4.477 2.996 1.06 0.7 5.328 3.912 1.25 25.9 5.236 4.942 1.30 13.4 5.802 6.685 1.08 54.9 6.151 6.752 1.52 5.5 6.176 4.787 1.25 initially consider the following regression model Y i = β 0 + β 1 X i 1 + β 2 X i 2 + β 3 X i 3 + ǫ i for i = 1 , 2 , ..., 30 . Are there other predictor variables that influence taste not considered here? Alternatively, what if not all of X 1 , X 2 , and X 3 are needed in the model? For example, if the acetic acid concentration ( X 1 ) is not helpful in describing taste, then we might consider a smaller model which excludes it; i.e., Y i = β 0 + β 2 X i 2 + β 3 X i 3 + ǫ i for i = 1 , 2 , ..., 30 . The goal of any regression modeling problem should be to identify each of the important predictors, and then find the simplest model that does the best job. The full model, Y i = β 0 + β 1 X i 1 + β 2 X i 2 + β 3 X i 3 + ǫ i in matrix notation is Y = + ǫ , PAGE 22
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c circlecopyrt HYON-JUNG KIM, 2017 where Y = 12 . 3 20 . 9 39 . 0 . . . 5 . 5 , X = 1 4 . 543 3 . 135 0 . 86 1 5 . 159 5 . 043 1 . 53 1 5 . 366 5 . 438 1 . 57 . . . . . . . . . . . . 1 6 . 176 4 . 787 1 . 25 and β = β 0 β 1 β 2 β 3 . We compute X X = 30 164 . 941 178 . 254 43 . 260 164 . 941 916 . 302 1001 . 806 240 . 879 178 . 254 1001 . 806 1190 . 343 269 . 113 43 . 26 240 . 879 269 . 113 65 . 052 and X Y = 736 . 000 4194 . 442 5130 . 932 1162 . 065 . Thus, the least squares estimate of β for these data is given by hatwide β = ( X X ) 1 X Y = ( 28 . 877 0 . 328 3 . 912 19 . 670) . The least-squares regression equation becomes ˆ Y i = 28 . 877 + 0 . 328 X i 1 + 3 . 912 X i 2 + 19 . 670 X i 3 or, in terms of the variable names, hatwider TASTE i = 28 . 877 + 0 . 328 ACETIC i + 3 . 912 H2S i + 19 . 670 LACTIC i . Using the first model, the ANOVA table for the cheese data is shown below. The F 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 statistic is used to test H 0 : β 1 = β 2 = β 3 = 0 vs. H 1 : β 1 , β 2 , β 3 not all zero. Since the P -value for the test is so small, we would conclude that at least one of X 1 , X 2 , or X 3 is important in describing taste. The coefficient of determination is R 2 0 . 652. Thus, about
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