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Unformatted text preview: CHAPTER 3 MULTIPLE REGRESSION 1 Let be the set of statistical subjects (units). Examples of : { 1 , 2 , 3 ,...,n } { All patients in a clinical trial } { All the samples tested in an experiment } On subject i , we observe y i ,x i 1 ,...,x ip . 2 The basic equation of multiple regression is: y i = p X j =1 x ij j + i , 1 i n, where the i satisfy one of the same two con ditions as in Chapter 2: (a) i uncorrelated, mean 0, common variance 2 , (b) i independent N [0 , 2 ]. Often assume x i 1 = 1 for all i , so that 1 is an intercept. 3 Method of least squares: choose estimates 1 ,..., p , to minimize S = n X i =1 y i p X j =1 x ij j 2 . Differentiating with respect to 1 ,..., p leads to the set of p simultaneous linear equations in p unknowns: n X i =1 x ik y i p X j =1 x ij j = 0 , k = 1 , 2 ,...,p. These equations are known as the normal equa tions and they are a fundamental starting point for the analysis of linear models. 4 Rewrite in vector notation: Define Y = y 1 . . . y n , = 1 . . . p , = 1 . . . n , X = x 11 x 12 ... x 1 p x 21 x 22 ... x 2 p . . . . . . . . . . . . x n 1 x n 2 ... x np . Then Y = X + . The normal equations may also be written in vector notation as X T Y = X T X . Usually assume X T X invertible: then = ( X T X ) 1 X T Y. 5 Geometrical Explanation Let x i = ( x 1 i ,...,x ni ) . Both Y and x i s are points in R n . We can define inner product on R n as < x,y > = x y, thus  x y  2 = < x y,x y > = ( x y ) ( x y ) , and SSE=  Y X  2 . Method of least squares: choose estimate such that  Y X  2 is minimized. 6 Geometrical Explanation...
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This note was uploaded on 11/17/2011 for the course STOR 664 taught by Professor Staff during the Fall '11 term at UNC.
 Fall '11
 Staff

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