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# 469 Class Note for STAT 462 at PSU

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This 10 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 28 views.

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Date Created: 02/06/15

ltCOOzmgtNlt gt20 ltgtNgtZOm zugtOz ugtOONm n 03583033 Pool of available predictorsterms from them in the data set Related to model selection are the questions What is the relative importance of different terms what is the sign and magnitude of their effect on Y what is their contributions to explaining the variability of Y Can a term be dropped should a term be added to the model because its contribution is small large When the terms are not correlated with one another answers are straightforward eg yz39 o lxu ZxZJ 8139 yz39 o lxu 5i if corx1 x2 0 then the LS coefficient and the 0 bl 2 b1 response variability explained by x1 alone 39 SSROQ I x2 SSEx2 SSEx12 x2 SSRx1 and together with x2 are the same F Chiaromonte 2 However when terms are correlated with one another things become complicated 7 LS coefficient and contribution to explaining y variability can change depending on what other terms are considered with xj SSROCJ I xha h 75 J in the model they are context dependent bj can change in magnitude and even sign depending on the presence of terms correlated to xj interpretation of bj as the average change in y when xj increases by one and all other terms are held constant becomes ambiguous in practice can you hold the other terms constant while changing xj the SSR attributable to xj can decrease but also increase depending on the presence of terms correlated to xj eg decrease if both xj and other terms are correlated with y and with one another increase if xj is not correlated with y per se but is correlated with other terms which in turn are correlated with y F Chiaromonte 3 In addition when terms are correlated with one another the sampling variance of the LS regression coefficients and therefore their standard errors 02bJ 02X 39X 1JJ S6bj S X 39XYILJ Our estimates become less accurate When correlations among terms are ven strong the inversion of X X becomes numerically unstable det close to 0 so our estimates b X X71X IY are not just ven variable they are poorly determined increase Many different b vectors provide ven similar LS fit to the data F Chiaromonte Because both the elements in b and their se are affected if terms are correlated it is hard to use the t ratios 5 J l J sebj as indicative of importance many pvalues for individual t tests can be nonsignificant although some terms are obviously relevant and the regression is significant as a whole overall F test pvalues can change dramatically when droppingadding The regression equation is Systol 225 131 Years 450 Weight 00381 YearsA2 00504 WeightA2 00503 YearsWeight Predictor Coef SE Coef T P Constant 2251 1155 195 0060 Years 1308 1919 068 0500 Weight 4499 3906 115 0258 YearsA2 003809 001388 274 0010 WeightA2 005042 003325 152 0139 YearsWeight 005032 003247 155 0131 S 946320 R Sg 548 R Sgadj 479 Analysis of Variance Source DF SS MS F P Regression 5 357621 71524 799 0000 Residual Error 33 295522 8955 Total 38 653144 F Chiaromonte 5 Diagnose pairwise correlations among terms scatterplot matrix and correlation coefficients matrix Matrix Plot of x1 x2 x3 x1 Correlations xl X2 X2 0674 x3 0830 0853 x2 x3 F Chiaromonte However these diagnostics are incomplete because the real issue is the presence of linear interdependencies among the terms These can be strong even when pairwise correlations are relatively week Given the model y o 1x12 jxjg p1xp1 8 Consider each term as a linear function of all others fitting regressions of the type 2 2 x a0 Zazxw err gt R R x xw 2 j 1p 1 1 share of the variability of xj explained by a linear form in the other terms variance of regr coeff 1 2 1 R 02 9 72 X 39 X 1jj 0C 2 Vle variance in ation factor Rules of thumb serious multicollinearity if max VIF 210 andor VT 2 j1p 1 1 P4 ZVIFJ 210 p1j1 F Chiaromonte 7 Simulated example yl O x11 x21 x31 81 81 iid N O l D x1 x2 15 10 F Chiaromonte The regression equation is y 0115 131 x1 0109 x2 138 x3 Predictor Coef SE Coef T P VIF Constant 01148 03523 033 0745 x1 1312 1031 127 0204 224 145 x2 01093 09971 011 0913 207526 x3 13798 07323 188 0061 454378 5 101361 R Sq 944 R Sqadj 943 Analysis of Variance Source DF 55 MS F P Regression 3 34111 11370 110670 0000 Residual Error 196 2014 10 Total 199 36124 Correlations I x1 X2 x2 0995 X3 0998 0998 839 x11 2 x small gaussian noise x11 2 x xii small gaussian noise 8 One remedy dropping terms as needed eg in the simulated example we have The regression equation is y 0051 275 X1 123 X2 Predictor Coef SE Coef T P VIE Constant 00507 03529 014 0886 X1 27457 07001 392 0000 102060 X2 12299 07037 175 0082 102060 S 102015 R Sq 943 R Sqadj 943 Analysis of Variance Source DE SS MS F P Regression 2 34074 17037 163708 0000 Residual Error 197 2050 10 Total 199 36124 The regression equation is y 0107 396 X1 Predictor Coef SE Coef T P VIE Constant 01073 03532 030 0762 X1 396326 006965 5690 0000 1000 S 102543 R Sq 942 R Sqadj 942 Analysis of Variance Source DE SS MS F P Regression 1 34042 34042 323751 0000 Residual Error 198 2082 11 Total 199 36124 F Chiaromonte 9 More sophisticated remedies orthogonalize terms at the outset but new terms are linear combs of original ones harder to interpret fit a ridge regression Important multicollinearity does not affect prediction fitted values their sampling variability and stability are not affected it s just that very similar fitted values can be produced by very different models F Chiaromonte 1O

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