SUMMARY In situations such as allometry where a line is to be fitted to a bivariate sample but where an asymmetric choice of one or other variable as regressor cannot be made, the reduced major axis is often used. Existing tests of the slope of this line, particularly between samples, are not sufficiently accurate in view of the scarcity of the material to which such methods are often applied. Alternative test statistics are suggested and some of their properties derived from a computer implementation of k statistics. One often wishes to describe the relationship between two observed random variables without, in the usual regression terminology, having to specify one as dependent on the other. A typical case, in fact the one which led to this paper, is in allometry where the variables are anatomical measurements, the relationship between which determines shape and may be used as a basis for comparison between species. After suitable transformations, usually logarithmic, have been applied, some measure of the slope of the bivariate scatter plot is required that treats both variables symmetrically. Unless there are sufficient grounds for specifying an underlying model with estimable parameters a possible choice is the line whose sum of squared perpendicular distances from the sample points is a minimum, and it is well known that this is given by the eigenvector corresponding to the larger eigenvalue of the sample dispersion matrix, the smaller eigenvalue in this two variable case being the minimized sum of squares. For the bivariate normal distribution this line is the major axis of the ellipses of constant probability, and so has come to be called the major axis of the bivariate sample. Although invariant under rotation the major axis is altered in a complicated way by changes of scale and in practice preference in the specialist literature on allometry has been given to the line obtained by normalizing the variables to unit standard deviations, finding the major axis, and transforming back to the original scales of measurement. This has come to be called the reduced major axis. The purpose of this paper is to suggest some more precise methods of testing simple hypotheses about the reduced major axis than have hitherto been available.