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Unformatted text preview: PubH 7405: REGRESSION ANALYSIS DIAGNOSTICS IN MULTIPLE REGRESSION limitation of the usual residual plots (say, residuals against values of a predictor variable): they may not show the nature of the “ additional contribution ” of a predictor variable to those by other variables already in the model. For example, we consider a multiple regression model with 2 independent variables X 1 and X 2 ; is the relationship between Y and X 1 linear? “ Addedvariable plots ” (also called “partial regression plots” or “adjusted variable plots”) are refined residual plots that provide graphic information about the marginal importance of a predictor variable given the other variables already in the model. ADDEDVARIABLE PLOTS In an addedvariable plot, both the response variable Y and the predictor variable under investigation (say, X 1 ) are both regressed against the other predictor variables already in the regression model and the residuals are obtained for each . These two sets of residuals reflect the part of each (Y and X 1 ) that is not linearly associated with the other predictor variables . The plot of one set of residuals against the other set would show the marginal contribution of the candidate predictor in reducing the residual variability as well as the information about the nature of its marginal contribution. PARTIAL CORRELATION IN MLR These two sets of residuals reflect the part of each (Y and X 1 ) that is not linearly associated with the other predictor variables. Result: The (simple) correlation between the above two sets of residuals is equal to the Coefficient of Partial Correlation between Y and X1 (adjusted for all others). SIMPLE & SPECIFIC EXAMPLE Consider the case in which we already have a regression model of Y on predictor variable X 2 and is now considering if we should add X 1 into the model (if we do, we would have a multiple regression model of Y on (X 1 ,X 2 )). In order to decide, we investigate 2 simple linear regression models : (a) The (old) regression of Y on X 2 and obtained residuals, and (b) The (new) regression of X 1 on X 2 and obtain 2 sets of residuals : 2 X by explained not Y of part : #1 Regression the represent residuals These )  ( model) old or (first X on Y ^ 2 2 2 ^ 2 i i i i i Y Y X Y e X b b Y 2 1 X in contained not X of part : #2 Regression the represent residuals These )  ( model) new or (second X on X 1 ^ 1 2 1 2 * 2 * 1 ^ 2 1 i i i i i X X X X e X b b X We now do a “regression of e i (YX 2 ) – as new dependent variable on e i (X 1 X 2 ) – as independent variable: That is to see if “the part of X 1 not contained in X 2 ” can further explained “the part of Y not explained by X 2 ” ; (if it can, X 1 should be added to the model for Y which already has X 2 in it). There are three possibilities: The “horizontal band” shows that X 1 contains no “additional information” useful for the prediction of Y beyond that contained in and provided for by X 2 This “linear band with a nonzero...
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 Fall '08
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 Regression Analysis, residuals

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