Analytical and monte carlo comparisons of six different linear least squares fits

Abstract

For many applications, particularly in allometry and astronomy, only a set of correlated data points (x_i:, y_i:) is available to fit a line. The underlying joint distribution is unknown, and it is not clear which variable is 'dependent' and which is 'independent'. In such cases, the goal is an intrinsic functional relationship between the variables rather than E(Y|X), and the choice of leastsquares line is ambiguous. Astronomers and biometricians have used as many as six different linear regression methods for this situation:the two ordinary least-squares (OLS) lines, Pearson's orthogonal regression, the OLS-bisector, the reduced major axis and the OLS-mean. The latter four methods treat the X and Y variables symmetrically. Series of simulations are described which compared the accuracy of regression estimators and their asymptotic variances for all six procedures. General relations between the regression slopes are also obtained. Among the symmetrical methods, the angular bisector of the OLS lines demonstrates the best performance. This line is used by astronomers and might be adopted for similar problems in biometry.