week10 - Analysis of Variance in Matrix form The ANOVA...

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STA302/1001 - week 10 1 Analysis of Variance in Matrix form The ANOVA table sums of squares, SSTO, SSR and SSE can all be expressed in matrix form as follows….
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STA302/1001 - week 10 2 Multiple Regression A multiple regression model is a model that has more than one explanatory variable in it. Some of the reasons to use multiple regression models are: Often multiple X ’s arise naturally from a study. We want to control for some X ’s Want to fit a polynomial Compare regression lines for two or more groups
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STA302/1001 - week 10 3 Multiple Linear Regression Model In a multiple linear regression model there are p predictor variables. The model is This model is linear in the β ’s. The variables may be non-linear, e.g., log( X 1 ), X 1 * X 2 etc. We need to estimate p +1 β ’s and σ 2 . There are p +2 parameters in this model and so we need at least that many observations to be able to estimate them, i.e., need n > p +2. n i X X X Y i p i p i i i ..., , 1 , 2 2 1 1 0 = + + + + + = ε β β β β
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STA302/1001 - week 10 4 Multiple Regression Model in Matrix Form In matrix notation the multiple regression model is: Y=Xβ + ε where Note, Y and ε are vectors, β is a vector and X is a matrix. The matrix X is called the ‘design matrix’. The Gauss-Markov assumptions are: E ( ε | X ) = 0 , Var( ε | X ) = σ 2 I . These result in E ( Y | X ) = 0 , Var( Y | X ) = σ 2 I . The Least-Square estimate of β is = = = = p n n p p p n n X X X X X X Y Y Y 1 2 21 1 11 1 0 2 1 2 1 1 1 1 , , , X Y β β β β ε ε ε ε ( 29 . ' ' 1 Y X X X b - = 1 × n ( 29 1 1 × + p ( 29 1 + × p n
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STA302/1001 - week 10 5 Estimate of σ 2 The estimate of σ 2 is: It has n-p -1 degrees of freedom because… Claim: s 2 is unbiased estimator of σ 2 . Proof: 1 ' 2 2 - - = = = p n error df e MSE s i e e
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STA302/1001 - week 10 6 General Comments about Multiple Regression The regression equation gives the mean response for each combination of explanatory variables.
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