BRIAND, CHRISTOPHER H.*, ELICHIA A. VENSO, AND BARBARA A. WAINWRIGHT. Department of Biological Sciences and Environmental Health Science, and Department of Mathematical and Computer Sciences, Henson School of Science and Technology, Salisbury State University, Salisbury MD 21801, USA. - Axis rotation in allometric analysis: Which regression should I use?
One of the major difficulties when comparing the allometric scaling
coefficient (b), which is the slope of the power function, Y =
aXb, or more conveniently, log Y = log a + b log X, from
different studies is axis rotation. Using the example of the
relationship between tree height and diameter, there is no standard
for assigning a variable to a particular axis, as neither height nor
diameter are independent variables. Both of these variables are
dependent on tree age, but are usually highly correlated with each
other. The slope of the relationship between the two variables, can
be calculated using ordinary least squares, reduced major axis or
geometric mean regression. However, with ordinary least squares
regression, when the axes are rotated, the new slope estimate is not
the inverse of the original slope estimate before rotation. The
resulting error can be large and is negatively correlated to the
absolute value of the correlation coefficient (r). Only with reduced
major axis or geometric mean regression is it possible to obtain the
inverse relationship with a high degree of accuracy by simply taking
the inverse of the slope. With reduced major axis regression, the
allometric scaling coefficient is determined by dividing the slope
obtained by ordinary least squares regression by r. Therefore, the
allometric scaling coefficient can be calculated if authors also
include along with the slope, either the correlation coefficient (r)
or the coefficient of determination (r2). The allometric
scaling coefficient can also be calculated using the geometric mean
technique as long as authors include the slopes of both X versus Y and
Y versus X. It is imperative that authors state the type of regression
used if meaningful comparisons among studies are to be made.
Key words: allometry, axis rotation, regression