Computation and uses of central trend lines

Abstract

A bivariate array of naturally variable observations can take many different forms, depending on the relative lengths of the measurement units used. Each of these has a different central trend or major axis. In a standard presentation the major axis has a slope of ± 1 obtained when 1 standard deviation (s) of each variate, Y and X, occupies the same distance on its coordinate axis. With any other presentation the position of the standard trend is indicated by a line whose slope is the ratio of the standard deviations; it is called the standard (or reduced) major axis, or geometric mean regression line (GMR). The GMR is symmetrical, invariant with change of scale, and "robust." Besides indicating the central trend, it is a suitable line for estimating Y from X, or X from Y, in two common situations where ordinary regressions fail: (i) when the sampling procedure was not random with respect to the entire population (but was random with respect to its standard trend); (ii) when the population sampled departs seriously from a bivariate normal configuration. In the latter case an alternative "Schnute" line is appropriate if components of the population may have different s Y /s X ratios.