ps233.lecture5 2

ps233.lecture5 2 - PS 233 Intermediate Statistical Methods...

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PS 233 Intermediate Statistical Methods Lecture 5 Multiple Regression & Matrix Format
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What If Y has more than just ONE cause? We have found an estimator for the relationship between X and Y We have developed methods to use the estimator to test hypotheses derived from theories about X and Y But we have only 1 X (and only 1 b) The world is more complicated than that!
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Multiple Regression Analysis We can make a simple extension of the bivariate model to the multivariate case Instead of a two dimensional space (X and Y axes) we move into multi- dimensional space If we have X1 and X2, then we are fitting a two dimensional plane through points in space.
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Multiple Regression Analysis Above 3 dimensions this becomes difficult to visualize. Logic of the process is the same. We are fitting SETS of X’s to each point on a Y dimension. The basic equation in scalar notation is: Y a b X b X b X e i i i k k i i = + + + + + 1 1 2 2 . . .
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Shifting to Matrix Notation Writing out these terms and multiplying them in scalar notation is clumsy. Represented in simpler terms through linear algebra The basic equation becomes: Y X B e = +
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The Multiple Regression Equation The vectors and matrices in are represented by Note that we postmultiply X by B since this order makes them conformable. Y X B e = + Y Y Y Y n = 1 2 X X X X X X X X X X i k k n n n k = 1 1 1 2 2 1 2 2 2 1 2 . . . . . . . . . . . . . . . . . . . . . B B B B k = 1 2 e e e e n = 1 2
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Math Tools With Matrices To derive our vector of coefficients b, we will need to do some math with matrices Multiplying matrices Taking the transpose of a matrix Inverting a matrix
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We Can Multiply Matrices Multiplication Where = 32 31 22 21 12 11 32 31 22 21 12 11 33 32 31 23 22 21 13 12 11 c c c c c c b b b b b b x a a a a a a a a a 32 33 22 32 12 31 32 31 33 21 32 11 31 31 32 23 22 22 12 21 22 31 23
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This note was uploaded on 10/21/2011 for the course PS 233 taught by Professor Staff during the Spring '11 term at Duke.

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ps233.lecture5 2 - PS 233 Intermediate Statistical Methods...

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