the standardized regression coefficients against 

 values of k (fig. 1). The trace suggests that the 

 least-squares coefficients are too large in ab- 

 solute value, fh even having the wrong sign. At 

 k==0.2 the coefficients have stabilized, and the 

 residual sums of squares (SSE) has not substan- 

 tially increased. s- 

 The ridge regression 



Y* = 132.5+2. 870(X1)+1.650(X2)+3.934(X3) 



should be a better predicting equation than the 

 least-squares equation even though the coef- 

 ficients are biased. 



Summary 



Ridge regression is a statistical technique that 

 foresters should find useful. It is used to es- 

 timate coefficients for multiple-regression 

 models when the independent variables are 

 highly correlated. 



Considerable research has been done on ridge 

 regression. The paper by Hoerl and Kennard 

 (1970(0 introduced ridge-regression theory. 

 Although there is considerable matrix algebra 

 in this paper, it provides a sound backgi-ound for 

 the understanding and application of ridge 

 regi'ession. The subsequent paper by Hoerl and 

 Kennard (1970b) illustrates the applications of 

 ridge regi'ession, including its use as a guide to 

 variable selection. The article by Marquardt and 

 Snee (1975) is perhaps the most readable paper 



on ridge regression. All aspects of ridge regres- 

 sion are discussed at length, and many e.xamples 

 are included. Some of the other articles listed 

 are more mathematically sophisticated. 



Literature References 



Brown, P., and C. Payne. 



1975. Election night forecasting. J. Rov. statist. Soc. 



Ser. A. 38:463-498. 

 Conniffe, D., and Joan Stone. 



1974. A critical review of RIDGE regression. Statisti- 

 cian 22:181-187. 



Draper, N. R., and H. Smith. 



1966. Applied regression analysis. Wiley, New York. 407 

 P- 



Farebrother, R. W. 



1975. The minimum mean square error linear es- 

 timator and RIDGE regression. Technometrics 17:127- 

 128. 



Giulkey, David K., and James L. Murphy. 

 1975. Directed RIDGE regression techniques in cases 

 of multicollinearity. J. Am, Statist. Assoc. 70:769-775. 



Hemmerle, William J. 



1975. An explicit solution for generalized RIDGE 

 regression. Technometrics 17:309-314. 



Hoerl, Arthur E., and Robert W. Kennard. 



1970a. RIDGE regression: applications to non- 

 orthogonal problems. Technometrics 12:55-68. 



Hoerl, Arthur E., and Robert W. Kennard. 



1970b. RIDGE regression: biased estimation for non- 

 orthogonal problems. Technometrics 12:69-83. 



Marquardt, Donald W. 



1970. Generalized inverses, RIDGE regression, biased 

 linear estimation, and nonlinear estimation. 

 Technometrics 12:591-612. 



Marquardt, Donald W., and Ronald D. Snee. 



1975. RIDGE regression in practice. Am. Statist. 29:3- 

 19. 



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