Next Contents Previous

7.3 Linear Fits When Both Variables Have Errors

In the previous examples, it was assumed that the independent variables xi were completely free of errors. Strictly speaking, of course, this is never the case, although in many problems the errors on x are small with respect to those on y so that they may be neglected. In cases where the errors on both variables are comparable, however, ignoring the errors on x leads to incorrect parameters and an underestimation of their errors. For these problems the effective variance method may be used. Without deriving the result which is discussed by Lybanon [Ref. 7] and Orear [Ref. 8] for Gaussian distributed errors, the method consists of simply replacing the variance sigmai2 in (70) by

Equation 84 (84)

where sigmax and sigmay, are the errors on x and y respectively. Since the derivative is normally a function of the parameters aj, S is nonlinear and numerical methods must be used to minimize S.