Daniel and Wood

A complication arises. We learn from considerations of multiple regression analysis that when two (or more) variables are correlated, the standard error of both variables is increased over what would be obtained if equivalent but uncorrelated variables are used. This is discussed by Daniel and Wood (see p. 55 in [9]), who show that the variance of the estimates of coefficients (their standard errors) is increased by a factor of... 

Since, as we shall see, we do not need to deal with the general case, we can use a simpler method to orthogonalize the variables, based on Daniel and Wood, who showed how a variable can be transformed so that the square of that variable is uncorrelated with the variable. This is a matter of creating a new variable by simply calculating a quantity Z and subtracting that from each of the original values of X. A symmetric distribution of the data is not required since that is taken into account in the formula. Z is calculated using the expression (see p. 121 in [9]). In Appendix A, we present the derivation of this formula ... 

Equation 66-A6 appears to differ from the expression in Daniel and Wood [9], in that the denominator expressions differ. To show that they are equivalent, we start with the denominator term of the expression on p. 121 of [9] ... 

This example was first introduced by Gorman and Toman (1966) and since then was extensively used in the statistical literature. The data set contains 36 data points of 10 dependent variables and one independent variable, where each row in the data set represents one day of operation of a petroleum refining unit. The complete data set of this example is given in Daniel and Wood (1980). They have also carried out a stepwise regression analysis of the data set using a linear model that includes a free parameter and the transformation In y for the dependent variable. This corresponds to Step 1 of the proposed algorithm with A = 0. The optimal solution obtained for this case by SROV is shown in Table 1. Note that the range of the dependent variable, the parameter values... 

Daniel, C., and Wood, F., Fitting Equations to Data. 2nd ed. Wiley, New York, 1980. Deming, W., Out of the Crisis. Massachusetts Institute of Technology, Center for Advanced Engineering Study, Cambridge, MA, 1986. 

Andre s Explosive, patented in 1895, contained AN 85, K nitrate 3 and wood flour 12% cf Electronite No 2 Ref- Daniel(1902),28... 

Daniel, C and Wood, F.S. (1971). Fitting Equations to Data. Wiley, New York. 

Daniel, C. and Wood, F.S. "Fitting Equations to Data" Wiley Interscience New York, N.Y., 1971. 

Filonova, L Gunnarsson, LC Daniel, G Ohlin, M. Synthetic xylan-binding modules for mapping of pulp fibres and wood sections. BMC Plant Biology, 2007, 7(54). 

R.A. Blanchette, T. Nilsson, G. Daniel and A. Abad, in R.M. Rowell and R.J. Barbour, Archaeological Wood. Properties, Chemistry, and Preservation. Advances in Chemistry Series 225, American Chemical Society, Washington DC, 1990, p. 141. 

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