Chapter 12 - Chapter 12 Multiple Regression 1 We can extend...

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1 Chapter 12 Multiple Regression
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2 We can extend the method of least squares estimation to analyze the relationship the response variable has with more than 1 predictor variable. General Form: Method of least squares says that we are to minimize To minimize this, we would differentiate with respect to each b i , set each derivative equal to 0 and solve for each b i to get a fitted model. This is most easily accomplished via software (Minitab, SAS, SPSS) or the use of matrices. Fitted Model: i i i e y + + + + = k k i 1 1 0 X B X B B 2 1 1 0 2 ) ... ( ki k i i i x b x b b y e - - - - = k k 1 1 0 X B X B B y + + + =
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3 12.2 Matrix Approach to Estimation Our model is Y = X Β +E Solving for Β, we get b=(X T X) -1 X T y where = = = - - - - - k k nk n n k n n n k k n n b b b b b x x x x x x x x x x x x X y y y y y 1 1 0 2 1 , 1 2 , 1 1 , 1 2 22 21 1 12 11 1 2 1 ... , ... 1 ... 1 ... ... ... ... ... ... 1 ... 1 , ... Most calculators have a matrix entry area so you can easily multiply the matrices to obtain the vector containing the estimates of B .
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4 Example. Estimate the regression coefficients for the following data where x 1 =oil viscosity, x 2 =load and y=wear of a bearing: Y X 1 X 2 193 16 851 230 15.5 816 172 22 1058 91 43 1201 113 33 3571 125 40 1115 Identify the matrices y, X, and b. b =
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5 y = X =
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6 (X T X) = (X T X) -1 = (X T X) -1 X T = 17947 263626 8612 26362 25 . 5518 5 . 169 8612 5 . 169 6 - - - - - - - - - - - 7 6 4 6 4 992296281 . 1 551269664 . 5 291375582 . 1 551269664 . 5 0015247764 . 0351070108 . 291375582 . 1 0351070108 . 34375162 . 1 e e e e e - - - - - - - - - - - - - - - - - 4 4 4 5 5 5 290473094 . 1 0196943794 . 2044736453 . 991195448 . 3 0046129737 . 2758864128 . 285673704 . 1 0237912993 . 3209005076 . 048054426 . 4 261086144 . 5 841345929 . 4 0074351734 . 0160028127 . 0154347189 . 4348133892 . 6942602482 . 6721869283 . e e e e e e
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7 b = (X T X) -1 X T y= So, our regression model is
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8 Partitioning the Variation (ANOVA) We can use either matrices or statistical software, such as JMP, to find SS Total , SS Regression. SS Error . SS total = y[I-1(1’1) -1 1]y= SS Regression = y[X(X’X) -1 X’-1(1’1) -1 1]y= SS Error = y[I-X(X’X) -1 X’]y=SS Total -SS Regression 2 ) ( y y i - 2 ) ( y y i -
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9 Source of Variation SS df MS F Regression k= Error N-(k+1)= Total N-1=
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Section 12.5 Hypothesis Testing Test for the significance of the regression model (F test) Tests for the model parameters (t test) 10
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11 Hypothesis Test for the Model H 0 : B 1 =B 2 =…. = B k =0 H 1 : At least 1 B i is different from 0. ( This means
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This note was uploaded on 01/17/2011 for the course STAT 4714 at Virginia Tech.

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Chapter 12 - Chapter 12 Multiple Regression 1 We can extend...

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