Lec18.Review2 - Re w I I vie Re MultipleRe ssion KNN C...

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Review II Review II Multiple Regression KNN Chps 6,(7.6),8,9,10,11 Multiple Regression KNN Chps 6,(7.6),8,9,10,11 Design of Experiments, and Mixed-Effects Models Design of Experiments, and Mixed-Effects Models
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Multiple Regression Multiple Regression Polynomial Regression Polynomial Regression
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MultiplePredictors SingleResponse ε Y + = + + = + + + + + = - = i p k ki k i i i i i i i x y x x x x y ε β β ε β β β β β 1 1 0 4 4 3 3 2 2 1 1 0
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Percent Coal Ash x y 5 10 15 5 10 15 20 10.21 9.92 11.17 10.01 11.15 11.31 9.92 10.73 10.82 10.14 9.97 9.93 11.65 9.46 12.5 11.05 9.41 9.91 10.82 8.23 10.39 10.41 9.76 10.93 9.79 10.74 11.21 9.7 10.27 8.96 9.35 9.63 10.11 9.37 10.17 11.75 11.04 11.11 10.82 11.1 10.94 9.64 9.29 10.59 10.43 8.75 9.52 9.53 10.8 17.61 10.96 10.28 9.78 10.55 11.21 11.46 10.82 9.78 9.88 10.21 9.84 9.89 12.8 9.06 10.39 9.32 8.96 10.06 10.61 8.86 10.87 10.83 13.07 11 11.61 9.93 10.41 10.12 10.38 8.9 11.09 10.29 10.34 10.03 10.7 10.65 8.27 12.65 10.27 9.48 10.09 10.47 9.79 9.16 10.7 8.45 9.4 9.79 10.18 10.63 9.84 8.2 9.36 11.21 10.36 11.62 8.14 9.63 9.59 9.22 13.06 8.69 11.58 10.19 10.04 9.27 8.9 9.48 8.91 9.34 8.82 10.01 9.82 8.57 8.98 9.58 10.91 9.82 9.61 11.41 11.17 9.46 9.15 11.19 9.28 8.07 10.99 9.22 10.56 10.18 9.01 10.06 9.01 9.27 10.66 8.76 7.81 8.2 9.96 9.39 8.54 8.15 8.1 10.13 7.96 9.92 11.43 9.06 9.34 7.68 8.58 9.04 8.19 8.92 8.89 8.59 9.15 9.56 10.87 9.2 11.3 8.61 7 7.85 8.61 9.25 8.89 7.28 7.88 7.8 9.1 9 8.78 7.9 8.21 7.83 8.64 9.58 7.61 7.84 7.62 11.86 9.14 7.04 9.69 8.2 9.03 9.65 8.91 8.81 9.96 8.77 8.6 9.99 7.63 7.95 9.91 9.07
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x coal 5 10 15 8 10 12 14 16 18 Coal vs. X
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Coal vs. Y y coal 5 10 15 20 8 10 12 14 16 18
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Multiple Regression on Coal Ash NorthSouth EastWest 2 2 1 1 0 i i i i x x y ε β β β + + + =
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Splus/R Linear Model coal.lm = lm(coal~x+y,data=coal.ash) # summary(coal.lm) Coefficients:                Value Std. Error  t value Pr(>|t|)  (Intercept)  11.2468   0.2250    49.9818   0.0000           x  -0.1771   0.0252    -7.0332   0.0000           y  -0.0104   0.0141    -0.7368   0.4621
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Coal vs. X and Y
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Higher Order Effects Multiple Regression Polynomial Regression i i i i i i i i i i i i x x x x y x x x x y ε β β β β β ε β β β β β + + + + + = + + + + + = 4 4 3 3 2 2 1 0 4 4 3 3 2 2 1 1 0
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Linear vs. Nonlinear Linear Models Nonlinear Models i i i i i i i i i i i i i i i i i i i i i x x y x y x y x x x y x x y x y ε β β ε β β ε β β ε β β β β ε β β β β β + + = + = + = + + + + = + + + = + = ) 1 /( ) exp( ) exp( ) log( 1 0 1 0 1 0 2 2 2 1 0 2 2 1 0 1 0
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Linear If you can write it in this form: 1 1 1 where 1 0 2 1 1 12 11 = = m mn m m n X X X X X X β β β β X
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Colinearity x y ) ( 2 1 0 β β β + + =
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Interaction Multiple Regression 2 1 3 2 2 1 1 0 i i i i i i x x x x y ε β β β β + + + + =
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Splus/R Linear Model w/ Interactions # coal2.lm = lm(coal~x*y,data=coal.ash) # summary(coal2.lm) Coefficients:                Value Std. Error  t value Pr(>|t|)  (Intercept)  11.3290   0.5374    21.0806   0.0000           x  -0.1887   0.0734    -2.5701   0.0109           y  -0.0164   0.0383    -0.4278   0.6693         x:y   0.0008   0.0047     0.1685   0.8663
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F-test Proposed Proposed Proposed Prior Proposed Prior df SSR df df SSR SSR F - - =
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F-test Coal-Ash Model coal1.lm = lm(coal ~ x, data = coal.ash) anova(coal1.lm,coal2.lm) Coefficients:                Value Std. Error  t value Pr(>|t|)  (Intercept)  11.1624   0.1935    57.6989   0.0000           x  -0.1837   0.0235    -7.8151   0.0000 Response: coal   Terms Resid. Df      RSS   Test Df Sum of Sq   F Value     Pr(F)  1     x       206 260.1363                                        
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