F16-MLR-InMatrices

F16-MLR-InMatrices - PubH 7405: REGRESSION ANALYSIS...

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PubH 7405: REGRESSION ANALYSIS MULTIPLE REGRESSION ANALYSIS
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THE DRAWBACKS OF SLR There are effect modifiers ; SLR does not allow us to study effect modifications Even without interactions, information provided by different factors may be redundant . There are confounders ; SLR does not allow us to investigate marginal contribution - contribution of a factor adjusted for other factors. SLR does not allow us to study or investigate non-linear relationships .
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) , 0 ( 2 2 2 1 1 0  N x x x Y k k NORMAL ERROR REGRESSION MODEL
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The terms involved could be continuous, binary, or categorical (several categories); they do not need to represent k different predictors ; some term could be the product of two predictors, some term could be the quadratic power of another predictor.
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In particular, Multiple Regression allows us to get into two new areas that were not possible with Simple Linear Regression : (i) Interaction or Effect Modification , and (ii) Non-linear Relationship .
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The primary reasons for emphasizing matrices are: (1) Simple Linear Regression and Multiple Linear Regression look the same in matrix terms; we do not have to prove some of the results again; and (2) It lead to the same computational tools/software and it allows more theoretical works.
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OBSERVATIONS & ERRORS 2 1 1 n nx Y Y Y Y n nx 2 1 1 ε
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X : the “Design Matrix” (a matrix of constants X’s) . . . 1 1 1 2 1 2 22 21 1 12 11 ) 1 ( kn k k n n k nx x x x x x x x x x X First subscript: Variable; Second subscript: Subject
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The dimension of “ Design Matrix ”X is changed to handle more predictors: one column for each predictor (the number of rows is still the sample size. The first column (filled with “1”) is still “optional”; not included when doing “Regression through the origin” (i.e. no intercept).
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: Regression Coefficient (a column vector of parameters) k x k 1 0 1 ) 1 ( β
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I (Y) σ X β E(Y) ε β X Y 2 2 1 1 1 ) 1 ( ) 1 ( 1 n by n by n by n by k k by n by n MLR MODEL IN MATRIX TERMS
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OPERATIONS ON BASIC DATA MATRICES i ki i i i n kn n k k k i n n y x y x y y y y x x x x x x x x y y y y y y y 1 2 1 1 3 13 2 1 12 11 2 2 1 2 1 1 . . . 1 1 1 ] [ ] [ Y X Y Y ' '
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2 1i ki 1i ki ki 1i 2 1i 1i ki 1i kn 1n k2 12 k1 11 kn k2 k1 1n 12 11 x x x x x x x x x x 1 x x 1 x x 1 x x 1 x x x x
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This note was uploaded on 11/21/2011 for the course PUBH 7405 taught by Professor Staff during the Fall '08 term at Minnesota.

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F16-MLR-InMatrices - PubH 7405: REGRESSION ANALYSIS...

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