Ridge Regression Procedure

To employ the ridge regression, first (Step 1), the values are transformed to correlation form. Correlation for the y, value is 

The Xi values for each predictor variable are next transformed to correlation form using 

For example, referring to data from Table 6.1 A, the first x, in the data for the Xi variable is 

The entire 15 x 6X matrix is presented in Table 6.6. The transpose of X as X is presented in Table 6.7, a 6 x 15 matrix. The actual correlation matrix is presented in Table 6.8. 

Note This correlation form and the correlation form in Table 6.2 should be identical. They differ here because Table 6.2 was done in an autoselection of MiniTab and Table 6.8 was done via manual matrix manipulation using MiniTab. 

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The ridge regression procedure modifies the least-squares regression equation by introducing a constant (c) where c > 0. Generally, c is 0 < c < 1. The population ridge regression equation, then, is in correlation form... 

When collinearity is severe, regression procedures must be modified. Two ways to do this are (1) rescaling the data (2) using ridge regression. 

Chapter 6 discusses common problems encountered in multiple linear regression and the ways to deal with them. One problem is multiple collin-earity, in which some of the x, variables are correlated with other x, variables and the regression equation becomes unstable in applied work. A number of procedures are explained to deal with such problems and a biasing method called ridge regression is also discussed. 

Setting the c value is a trial-and-error process. Although c is generally a value between 0 and 1, many statisticians urge the researcher to assess 20 to 25 values of c. As c increases from 0 to 1, its effect on the regression parameters can vary dramatically. The selection procedure for a c value requires a ridge trace to find a c value where the fe values are stabilized. 

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