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Multicollinearity arises for several reasons. It will occur when the analyst collects data such that a linear constraint holds approximately among the columns of the X matrix. For example, if four regressor variables are the components of a mixture, such a constraint will always exist because the sum of the components is always constant. Usually, these constraints do not hold exactly, and the analyst might not know that they exist. The presence of multicollinearity can be detected in several ways. Two of the more easily understood of these will be discussed briefly. 1. The variance inflation factors, defined in equation 12-50, are very useful measures of multicollinearity. The larger the variance inflation factor, the more severe the multicollinearity. Some authors have suggested that if any variance inflation factor exceeds 10, multicollinearity is a problem. Other authors consider this value too liberal and suggest that the variance inflation factors should not exceed 4 or 5. Minitab will calculate the variance inflation factors. Table 12-4 presents the Minitab multiple regression output for the wire bond pull strength data. Since both VIF1 and VIF2 are small, there is no problem with multicollinearity. 2. If the F-test for significance of regression is significant, but tests on the individual regression coefficients are not significant, multicollinearity may be present.

Several remedial measures have been proposed for solving the problem of multicollinearity. Augmenting the data with new observations specifically designed to break up the approximate linear dependencies that currently exist is often suggested. However, this is sometimes impossible because of economic reasons or because of the physical constraints that relate the xj. Another possibility is to delete certain variables from the model, but this approach has the disadvantage of discarding the information contained in the deleted variables. Since multicollinearity primarily affects the stability of the regression coefficients, it would seem that estimating these parameters by some method that is less sensitive to multicollinearity than ordinary least squares would be helpful. Several methods have been suggested. One alternative to ordinary least squares, ridge regression, can be useful in combating multicollinearity. For more details on ridge regression, see Section 12-6.5 on the CD material or the more extensive presentations in Montgomery, Peck, and Vining (2001) and Myers (1990).

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The event= ProcName attribute defines the name of a procedure that will be executed with the event is raised. The attribute creates a link, or Event Handler, to connect the control to the procedure that will be executed. In the following example, the lblDateTime label control is populated with the current date and time when btnSelect is clicked.

12-6.5 12-6.6

Ridge Regression (CD Only) Nonlinear Regression (CD Only)

<HTML> <HEAD> <title>Hello World Web Page</title> <script runat= server > sub ShowDateTime(sender as object, e as System.EventArgs) lblDateTime.Text = DateTime.Now end sub </script> </HEAD> <body> <form id= Form1 method= post runat= server > <asp:label id= lblDateTime runat= server > </asp:label>

EXERCISES FOR SECTION 12-6

12-46. An article entitled A Method for Improving the Accuracy of Polynomial Regression Analysis in the Journal of Quality Technology (1971, pp. 149 155) reported the following data on y ultimate shear strength of a rubber compound (psi) and x cure temperature ( F). y x y x 770 280 735 298 800 284 640 305 840 292 590 308 810 295 560 315 (d) Compute the residuals from part (a) and use them to evaluate model adequacy. 12-47. Consider the following data, which result from an experiment to determine the effect of x test time in hours at a particular temperature on y change in oil viscosity: y x y x 1.42 .25 3.15 1.50 1.39 .50 4.05 1.75 1.55 .75 5.15 2.00 1.89 1.00 6.43 2.25 2.43 1.25 7.89 2.50

(a) Fit a second-order polynomial to these data. (b) Test for significance of regression using 0.05. (c) Test the hypothesis that 11 0 using 0.05.

<asp:button id= btnSelect Text= Select Runat= server OnClick= ShowDateTime > </asp:button> </form> </body> </HTML>

(a) Fit a second-order polynomial to the data. (b) Test for significance of regression using 0.05. (c) Test the hypothesis that 11 0 using 0.05.

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