Regression.pdf

# So the residuals unlike the errors do not have

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So the residuals (unlike the errors) do not have constant variance and are slightly correlated! Observations where X i is far away from X will have large values of h ii . However, not all observations with large leverages are necessarily outliers. Leverage is an index of the importance of an observation to a regression analysis: large deviations from mean are influential. STUDENTIZED RESIDUALS: To account for the different variances among residuals, we consider “studentizing” the residuals (i.e., dividing by an estimate of their standard devia- tion). The (internally) studentized residuals are defined as: r i = e i radicalbig MSE(1 h ii ) have E ( r i ) = 0, Var( r i ) = 1 so that the studentized residuals have a constant variance regardless of the location of the X ’s. Values of | r i | larger than 3 or so should cause concern. TRANSFORMATIONS: Transforming the data is a way of handling violations of the usual assumptions. In the regression context, this may be done in a number of ways. One way is to invoke an appropriate transformation, and then postulate a regression model on the PAGE 18

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1.4 DIAGNOSTICS and MODEL EVALUATION c circlecopyrt HYON-JUNG KIM, 2017 transformed scale. Sometimes it may be that, although the data do exhibit constant variance on the original scale, they may be analyzed better on some transformed scale. It is also important to remember that if a transformation is used, the resulting inferences apply to this transformed scale (and no longer to the original scale). One should remember that transforming the data may fix one problem, but it may create other violations of the model. Another approach is to proceed with a regression method known as weighted-least squares. In a weighted regression analysis, different responses are given different weights depending on their variances (to be discussed later). In general, transforming your data almost always involves lots of trial and error. BOX-COX TRANSFORMATION: The power transformation g ( Y ) = log( Y ) , λ = 0 Y λ , λ > 0 was suggested by Box and Cox (1964). The log and square root transformations are special cases with λ = 0 and λ = 1 / 2, respectively. Approximate 100(1 α ) percent confidence intervals for λ are available. OTHER TRANSFORMATIONS (mainly for simple linear regression models) Some guidelines: You may want to try transformations of the Y -variable when there is evidence of non-normality and/or non-constant variance problems in one or more residual plots. Try transformations of the X -variable(s) (e.g., X 1 , X 2 , ln ( X )) when there are strong nonlinear trends in one or more residual plots. If either of the above trials does not work, consider transforming both the response Y and the predictor X (e.g. ln ( Y ) , ln ( X )). If the error variances are unequal, try ‘stabilizing the variance’ by transforming Y : - When the response is a Poisson count, take the Y transformation.
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