be increased by changing the dimension 

 statements. Nineteen values of k from to 1.0 

 are automatically supplied by the program. 

 Other values of k may be designated by the 

 user. 



The means and variances of the variables are 

 printed by program RIDGE. The X'X and X'Y 

 matrices are transformed into the* correlation* 

 form and printed. 



The eigenvalues and corresponding matrix of 

 eigenvectors for the X'X matrix are calculated. 

 The presence of one or more zero eigenvalues in- 

 dicates linear dependencies between the in- 

 dependent variables. If this condition exists, 

 X'X is singular for k=0, and the program ter- 

 minates with an error message. If no linear 

 dependencies are present, an analysis of 

 variance table is printed. 



Standardized and actual regression coef- 

 ficients are printed for the different values of k. 

 The ridge trace can be plotted by the user from 

 the standardized coefficients. However, we 

 found that in most cases the tabled values of 

 standardized coefficients provide sufficient in- 

 formation for selecting the appropriate ridge 

 solution. 



The computer program is available from the 

 Biometrics Group, Northeastern Forest Experi- 

 ment Station. 



An Example of Ridge Regression 



Suppose we have 10 sample observations for 3 

 independent variables and 1 dependent variable 

 (table 1). Computer output from program 

 RIDGE for this example is given in the appen- 

 dix. Investigation of the correlation matrix 

 reveals high correlations between the predictor 

 variables; and one of the eigenvalues, 0.0138, is 

 small. These conditions suggest that ridge 

 regression be used to estimate the regression 

 coefficients. Since the F ratio for the least- 

 squares solution is highly significant, ridge- 

 regression coefficients and the residual sums of 

 squares were calculated for 19 values of k. 



The ridge trace was constructed by plotting 



Table 1.— Dafa for sample problem 



Y Xi X2 X:. 



223 



11 



11 



11 



223 



14 



15 



11 



292 



17 



18 



20 



270 



17 



17 



18 



285 



18 



19 



18 



304 



18 



18 



19 



311 



19 



18 



20 



314 



20 



21 



21 



328 



23 



24 



25 



340 



25 



25 



24 



Figure 1.— Ridge trace for 19 values of k. 



