Chap 10: Diagnostics, p384

Transcription

1 Chap 10: Dagnostcs, p384 Multcollnearty 10.5 p406 Defnton Multcollnearty exsts when two or more ndependent varables used n regresson are moderately or hghly correlated. - when multcollnearty exsts, regresson results can be confusng and msleadng. For example n a multple regresson model all partal slopes wll be sgnfcant wth a sgnfcant global F-test. Sgns of the regresson coeffcents mght not make sense. - Varance nflaton factors Tolerance : T = 1 R 2 VIF = 1 = 1 T 1 R 2 VIF = r 1 XX 1

2 /*Multcollnearty*/ optons ls=75; data cgar; nfle 'cgar.txt' frstobs=2; nput Row co tar ncotne weght; proc reg; model co = tar ncotne weght/ tol vf; model tar = ncotne weght; model ncotne = tar weght; model weght = tar ncotne; proc corr; var tar ncotne weght; run; Row co tar ncotne weght

3 Example (Cgarette) The REG Procedure Model: MODEL1 Dependent Varable: co Analyss of Varance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Parameter Estmates Parameter Standard Varable DF Estmate Error t Value Pr > t Tolerance Intercept tar ncotne weght Parameter Estmates Varance Varable DF Inflaton Intercept 1 0 tar ncotne weght

4 Model: MODEL2 Dependent Varable: tar Analyss of Varance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq The REG Procedure Model: MODEL3 Dependent Varable: ncotne Analyss of Varance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Model: MODEL4 Dependent Varable: weght Analyss of Varance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq

5 Pearson Correlaton Coeffcents, N = 25 Prob > r under H0: Rho=0 tar ncotne weght tar < ncotne < weght r XX = r 1 = XX

6 Outlers 10.2 p390 Deleted resduals d = y yˆ( ) y = th observed response y = the predcted value of the th repose ˆ() when the data for the th observaton s deleted from the analyss. Studentzed deleted resduals p396 s 2 d MSE = 1 h and d t = s d d t = sd n p 1 = e MSE 2 (1 h ) e 1/2 6

7 The studentzed deleted resdual t has a dstrbuton that s approxmated by a t- dstrbuton wth (n-1)-p d.f. The approprate Bonferron crtcal value therefore s t(1 α / 2 n, n 1 p) (p396) (n-1)-p= (24-1)-(4+1) = 18 and t = ( /24,18) Example Row cty traffc sales cty1 cty2 cty

8 /*outlers*/ optons ls=75; data nflu; nfle 'nflu.txt' frstobs=2; nput Row cty traffc sales cty1 cty2 cty3; proc reg; model sales = cty1 cty2 cty3 traffc / nfluence; output out=a cookd=cook h=h rstudent=tres; proc prnt data=a; var TRes h cook; run; 8

9 The REG Procedure Model: MODEL1 Dependent Varable: sales Analyss of Varance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Model: MODEL1 Dependent Varable: sales Output Statstcs Hat Dag Cov Obs Resdual RStudent H Rato DFFITS

10 Obs TRes h cook

11 Leverage values 10.3 p398 - h = Leverage of the th observaton. The leverage values are the dagonal elements of the hat matrx H= X( XX ) 1X Observatons wth 2p n h >. Are consdered by ths rule to ndcate outlyng cases wth regard to ther X values.. Note p n s the average leverage. 11

12 Cook s dstance, p402 - A measure of the overall nfluence of an observaton on the estmated β coeffcents. Cook s dstance: ( ˆ ) 2 y y h D = pmse (1 h )2 -Note that D depends on both the resdual ( y yˆ ) and the leverage h. - A large value of D ndcates that the th observaton has a strong nfluence on the estmated β coeffcents. - Values of D can be compared to the values of the F(p, n-p) Usually an observaton that falls above the 50 th percentle of the F dstrbuton s consdered to be an nfluental observaton. In fast-food sales example n = 24, p = 5 numerator d. f. = 5 and denomnator d. f = 24-5 = 19 and F =