Chapter4ModelAdequacyChecking.pptVIP

Chapter 4 Model Adequacy Checking;4.1 Introduction;Gross violations of the assumptions may yield an unstable model with opposite conclusions. The standard summary statistics: t-; F- statistics and R2 can not detect the departures from the underlying assumptions. Based on the study of the model residuals.;4.2 Residual Analysis;Properties: ;4.2.2 Methods for Scaling Residuals Scaling Residuals is helpful in detecting the outliers or extreme values. Standardized Residuals:;Studentized Residuals: The residual vector: e=(I-H)y, where H=X(X’X)-1X’ is the hat matrix. Since H is symmetric and idempotent, (I-H) is also symmetric and idempotent. Then;Hence Var(ei) = ?2(1-hii) Cov(ei, ej) = -?2hij Studentized residuals: ;ri di Constant variance, Var(ri) = 1 ri and di may be little difference and often convey equivalent information. PRESS Residuals: The prediction error (PRESS residuals) is the fitted value of the ith response based on all observations except the ith one. From Appendix C7, ;The variance of the ith PRESS residual is A standardized PRESS residual: A studentized PRESS residual: ;R-Student: Estimate variance based on a data set with the ith observation removed. From Appendix C.8, R-student If the ith observation is influential, then can differ significantly from MSRes , and thus R-student statistic will be more sensitive to this point.;Example 4.1 The Delivery Time Data See Table 4.1 ;4.2.3 Residual Plot Graphical analysis is a very effective way to investigate the adequacy of the fit of a regression model and to check the underlying assumption. Normal Probability Plot: If the errors come from a distribution with thicker or heavier tails than the normal, LS fit may be sensitive to a small subset of the data. Heavy-tailed error distributions often generate outliers that “pull” LS fit too much in their direction.;Normal probability plot: a simple way to check the normal assumption. Ranked residuals: e[1] … e[n] Plot e[i] against Pi = (i-1/2