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Solutions to assignment 6 of statistics 333, focusing on polynomial regression and model diagnostics. Topics covered include least squares fitting, stepwise backward elimination, f-statistic testing, standardized residuals, cook's distance, and leverage values.

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Download Statistical Analysis: Assignment 6 - Polynomial Regression & Model Diagnostics and more Assignments Statistics in PDF only on Docsity! Statistics 333 Solutions of Selected Questions from Assignment 6 Nov. 26, 2003 A6-1. ii) In LS fitting of the full second-order polynomial model Y = β 0 + β 1 X1 + β 2 X2 + β 3 X3 + β 11 X21 + β 22 X22 + β 33 X23 + β 12 X1 X2 + β 13 X1 X3 + β 23 X2 X3 + ε to the data, we obtain results with SSRm = 20. 7771, SSE = 2. 4926, S2 = 2. 4926/(30 − 10) = 0. 1246, and R2 = 0. 893. Using a stepwise backward elimination procedure, based on removing a predictor variable whose t-statistic was not significant at α = 0. 05 and was the smallest such non-significant term, the predictors X1 X2, X1 X3, X 2 3 , and X2 X3 were removed in that order, one at a time. Thus, we arrive at and fitted a reduced model with following form and results, Y = − 0. 4738 + 0. 4859X1 + 0. 5023X2 + 0. 007148X3 − 0. 02799X21 − 0. 026497X22 + ε with all the remaining coefficient estimates highly significant. This reduced fitted model gives the results with SSRm = 20. 4627, SSE = 2. 8070, S2 = 2. 8070/(30 − 6) = 0. 1170, and R2 = 0. 879. (The reduced model does satisfy the origin shift criterion.) We obtain the F-statistic for (simultaneously) testing whether the β coefficients of all the omitted predictor variables are zero, that is, H0 : β 33 = β 12 = β 13 = β 23 = 0 as F = [(2. 8070 − 2. 4926)/4]/[2. 4926/(30 − 10)] = 0. 631 << F (.05)4,20 = 2. 87. So we do not have any evidence to reject H0. The model with additional terms does not give any improvement to fit beyond what is already provided by terms in the ‘reduced’ model. iii) Various plots of standardized residuals si are considered, as are Cook statistic values Di and leverage values hii. There are no unusual/extreme values among the si, Di, or hii, so no suggestion of outliers or evidence of extremely influential data cases. However, plot of Yi versus Ŷi does show an ‘interesting flattening out feature’ for cases Ŷi > 4. 25 roughly. A6-2. i) For the simple regression model in ‘centered’ form Yi = β *0 + β 1 (Xi − X) + ε i, X′ = 1 X1 − X 1 X2 − X . . . . . . 1 Xi − X . . . . . . 1 Xn − X so that, since Σni=1 (Xi − X) = 0, we have X′X = n 0 0 S xx and then (X′X)−1 = 1/n 0 0 1/S xx , where S xx = Σni=1 (Xi − X)2. Thus, hii = X i′(X′X)−1 X i = [ 1, Xi − X ] Diag[ 1/n, 1/S xx ] [ 1, Xi − X ]′ = (1/n) + (Xi − X)2/S xx . ii) Write model in form as Yi = β *0 + β 1 (Xi1 − X1) + β 2 (Xi2 − X2) + ε i, and so we have X′X = n 0 0 0 S x1 x1 S x1 x2 0 S x1 x2 S x2 x2 . But then if S x1 x2 = Σni=1 (Xi1 − X1)(Xi2 − X2) = 0, that is, the vectors of values {Xi1 − X1} and {Xi2 − X2} are orthogonal, we get X′X = Diag[ n, S x1 x1 , S x2 x2 ],