The orthogonal estimation line or major axis

As already mentioned earlier (sections 20.2, 20.7, and 21.5), ordinary estimation lines minimize the sums of squares of residual deviations with respect to only one of two variates: in a direction parallel to the ordinate in the case of the Y.Xestimation line of Y from X (figure 22.1.1, left) but parallel to the abscissa in the case of the X.Yestimation line of X from Y(figure 22.1.1, right). Ordinary estimation lines are therefore appropriate either (1) when random fluctuations affect only one or mostly one of the two variates (sections 20.6 and 20.8) or (2) when random fluctuations affect both variates but it is clearly the estimation of one of the two variates from the other that is required. But it occurs often in biology (see section 20.7, for instance), as well as in other fields, that variates XandYare both affected by random fluctuations and that what is needed is a symmetrical description of their relationship rather than a one-way estimation of one variate from the other. Ordinary estimation lines are then unsatisfactory because there are no more reasons to choose the Y.Xline than the X.Y line and they can yield contradictory answers (section 20.2). When the random fluctuations of both variates are thought to have approximately the same magnitude, the orthogonal estimation line (also known as the orthogonal regression line) may be used. Because that line minimizes the sum of squares of residual deviations perpendicularly to itself (figure 22.1.1, center), it is taking into account the fluctuations of both variates jointly. The orthogonal estimation line is also called the major axis because its parametric (population) version coincides with the major axis of the equal probability density ellipses of a bivariate normal distribution (section 19.2; see also sections 30.8 and 31.1).