F16-MLR-InMatrices

# F16-MLR-InMatrices - PubH 7405 REGRESSION ANALYSIS MULTIPLE...

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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 .
) , 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.
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.
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
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
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 ' '
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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F16-MLR-InMatrices - PubH 7405 REGRESSION ANALYSIS MULTIPLE...

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