Regression Diagnostics

Outlying Y Observations – Standardized Residuals

Outlying Y Observations – Studentized Deleted Residuals

Outlying X-Cases – Hat Matrix Leverage Values Cases with X-levels close to the “center” of the sampled X-levels will have small leverages. Cases with “extreme” levels have large leverages, and have the potential to “pull” the regression equation toward their observed Y-values. Large leverage values are > 2p’/n (2 times larger than the mean) New cases with leverage values larger than those in original dataset are extrapolations

Identifying Influential Cases I – Fitted Values

Influential Cases II – Regression Coefficients

Standardized Regression Model - I Useful in removing round-off errors in computing (X’X)-1 Makes easier comparison of magnitude of effects of predictors measured on different measurement scales Coefficients represent changes in Y (in standard deviation units) as each predictor increases 1 SD (holding all others constant) Since all variables are centered, no intercept term

Standardized Regression Model - II

Standardized Regression Model - III

Multicollinearity Consider model with 2 Predictors (this generalizes to any number of predictors) Yi = b0+b1Xi1+b2Xi2+ei When X1 and X2 are uncorrelated, the regression coefficients b1 and b2 are the same whether we fit simple regressions or a multiple regression, and: SSR(X1) = SSR(X1|X2) SSR(X2) = SSR(X2|X1) When X1 and X2 are highly correlated, their regression coefficients become unstable, and their standard errors become larger (smaller t-statistics, wider CIs), leading to strange inferences when comparing simple and partial effects of each predictor Estimated means and Predicted values are not affected

Multicollinearity - Variance Inflation Factors Problems when predictor variables are correlated among themselves Regression Coefficients of predictors change, depending on what other predictors are included Extra Sums of Squares of predictors change, depending on what other predictors are included Standard Errors of Regression Coefficients increase when predictors are highly correlated Individual Regression Coefficients are not significant, although the overall model is Width of Confidence Intervals for Regression Coefficients increases when predictors are highly correlated Point Estimates of Regression Coefficients arewrong sign (+/-)

Variance Inflation Factor