Normal Linear Model Notes

Assume p q so model under h0 is reduced compared to

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Unformatted text preview: Model with q − 1 covariates. H1 : β = β0 = (β1 , β2 , . . . , βp ). Model with p − 1 covariates. Assume p > q so model under H0 is reduced compared to model for H1 . Fit model under H0 (H1 ) and compute sum of squares S0 (S1 ) Since model under H1 has more parameters S0 > S1 , Test H0 with F= (S0 − S1 ) S1 (D0 − D1 ) D1 / = / ∼ F (p−q , N −p) (p − q ) (N − p) (p − q ) (N − p) . Reject H0 if Fobs > F(1−α) (p − q , N − p) UNM Residuals ˆ ˆ Based on ei = Yi − Yi = Yi − xiT b = obs − fitted ˆ Standardized residuals: ri = σ(1−ei )1/2 ; i = 1, 2, . . . , N ˆ hii where hi i is i − th element on the diagonal of H = X (X T X )−1 X T . Plots: Normal probability plots, independence, homoscedasticity, constant variance, etc. High leverage or influential: hii > 2(p/N ), DFITSi = ri (hii /(1 − hii )1/2 . Cook’s distance: Di = (1/p)(DFITS )2 . Large values means ”influential”. 1 Di = (b − b(i ) )T X T X (b − b(i ) ) p and b(i ) is the estimate of β without observation i . UNM...
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