F14-SLRinMatrices

F14-SLRinMatrices - PubH 7405: REGRESSION ANALYSIS SL...

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PubH 7405: REGRESSION ANALYSIS SL REGRESSION IN MATRIX TERMS
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BASIC REGRESSION MATRICES n n x x x Y Y Y 1 1 1 2 1 2 1 X Y X is special, called “ Design Matrix
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The dimension of “ Design Matrix ”X could be changed to handle more than one predictor: more columns ; the first column (filled with “1”) is not included when doing “Regression through the origin” (i.e. no intercept).
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X is called the Design Matrix . There are two reasons for the name: (1) By the model, the values of X are under the controlled of investigators: entries are fixed/designed , (2) The design/choice is consequential : the larger the variation in x’s the more precise the estimate of the slope.
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REGRESSION OPERATIONS i i i n n i n n y x y y y y x x x y y y y y y y 2 1 2 1 2 2 1 2 1 1 1 1 ] [ ] [ Y X Y Y ' ' Order is important ; cannot form YX’
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2 2 1 2 1 1 1 1 1 1 1 i i i n n x x x n x x x x x x X X' Here, we can form XX’ but it is a different n-by-n matrix which we do not need.
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If Y is a (column) vector, Y’Y is like a “Sum of Squares” – a scalar. We’ll use it to form SSE Y’Y, X’Y, and X’X form “ sufficient statistics
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INVERSE OF A 2x2 MATRICE | A | or A; of " " the is bc) ad ( D 1 t determinan A A D a D c D b D d d c b a
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A singular matrix does not have an inverse because its “determinants” is zero: 0 (6)(7) - (2)(21) D : and 0 0 21 6 ) 1 ( 7 2 ) 3 ( : have We 21 7 6 2 A
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REGRESSION EXAMPLE 0 ) ( ) )( ( 2 _ 2 2 x x n x x x n D x x x n X X'
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2 _ 2 _ _ 2 _ _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 _ 2 1 ) ( 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( x x x x x x x x x x x n x x n n x x n x x x n x x x n x X X' Entries are functions of mean and variance of X
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SIMPLE LINEAR REGRESSION MODEL ε X β Y ε β X Y or ; 1 1 1 ,..., 2 , 1 ; 1 1 2 2 1 2 1 1 0 2 1 2 1 1 0 nx x nx nx n n n i i i X X X Y Y Y n i X Y
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