lecture6 - ISYE6414 Summer 2010 Lecture 6 The Linear Model:...

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ISYE6414 Summer 2010 Lecture 6 The Linear Model: Multiple Regression Inference and Diagnostics Dr. Kobi Abayomi July 6, 2010 1 Introduction Inference on the parameters of multiple regression follow arguments similar to those in simple regression. Diagnostics, however, can be a bit more involved. 2 Inference The least squares and MLE of ˆ β are unbiased: E ( ˆ β ) = β ...just like in simple regression. This is the variance-covariance matrix Σ β = σ 2 ( X T X ) - 1 ; the full distribution for the estimators ˆ β is: ˆ β N ( β, Σ β ) and if we replace Σ β = σ 2 ( X T X ) - 1 with an estimate ˆ Σ β = ˆ σ 2 ( X T X ) - 1 which is ˆ Σ β = MSE · ( X T X ) - 1 , then full joint distribution for the estimators is the multivariate t-distribution. 2.1 Joint and Univariate inference on β coefficients Each individual univariate hypothesis test of H 0 : β j = 0 vs. H a : β j 6 = 0 uses a t - statistic: 1
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t = ˆ β j q ˆ V ar ( β j ) If we desire inference on g parameters from β 1 ,...,β k , we can appeal to univariate tests with a Bonferroni correction. We compute each CI or hypothesis test separately, using the adjusted Type I error level α 2 · g at each univariate. This yields overall Type I error level α . Or use the joint procedure outlined in section 6.1 of Lecture 4. 2.2 Inference on Mean Response and Prediction Call X h = (1 ,X h, 1 ,...,X h,k ), a vector of observed predictors. Under the model E ( Y h ) = X T h β and the fitted values ˆ Y h = X T h ˆ β . Of course, this fitted value the estimate of the expected value, or: ˆ Y h = ˆ E ( Y h ). We know this estimator is unbiased ( E ( ˆ Y h ) = X h T β ) and we can compute its variance ( V ar ( ˆ Y h ) = V ar ( X h ˆ β ) = σ · X T h ( X T X ) - 1 X h = X T h Σ ˆ β X h ); its distribution is also the distribution of the residuals: ˆ Y h = N ( X T h β, X T h Σ ˆ β X h ) If we have to use an estimate of the variance, i.e. we replace σ 2 with ˆ σ 2 = MSE then ˆ V ar ( ˆ Y h ) = MSE · X T h ( X T X ) - 1 X h = X T h ˆ Σ ˆ
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lecture6 - ISYE6414 Summer 2010 Lecture 6 The Linear Model:...

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