By David J. Olive
ISBN-10: 331955252X
ISBN-13: 9783319552521
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Additional info for Linear Regression
Sample text
2. a) Suppose that interest is in predicting a function of Z from functions of w1 , . . , wk . If Y = t(Z) = xT β + e where t is a function and each xi is some function of w1 , . . , wk , then there is an MLR model in Y and β. Similarly, Z = t(Y ) = wT β + e is an MLR model in Z and β. b) To see that Y = β1 + β2 x + β3 x2 + e is an MLR model in Y and β, take w1 = 1, w2 = x, and w3 = x2 . Then Y = wT β + e. c) If Y = β1 + β2 exp(β3 x) + e, then the model is a nonlinear regression model that is not an MLR model in Y and β.
The normal probability plot plots the e˜(i) versus r(i) where the e˜(i) are the expected values of the order statistics from a sample of size n from an N (0, 1) distribution. ) Rules of thumb: i) if the plotted points scatter about some straight line in the normal probability plot, then there is no evidence against the normal assumption. ii) if the plotted points have an “ess shape” (concave up then concave down), then the error distribution is symmetric with lighter tails than the normal distribution.
Hence the number of predictors p ≤ n. The ith row of X is xTi = (xi,1 , . . , xi,p ) where xi,k is the value of the ith observation on the kth predictor xk . We will denote the jth column of X by Xj ≡ v j which corresponds to the jth variable or predictor xj . 4. 5 ⎥ ⎢ 4261 ⎥ ⎥ ⎢ ⎥ ⎢ Y = ⎢ . ⎥, X = ⎢ . .. ⎥ = [v 1 v 2 v 3 ]. ⎣ .. ⎣ .. ⎦ . 5. After deleting observations with missing values, there were n = 267 cases (people measured on brain weight, age, and size), and x267 = (1, 19, 141)T . The second predictor x2 = age corresponds to the 2nd column of X and is X2 = v 2 = (39, 35, .
Linear Regression by David J. Olive
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