Relative growth rate (RGR) and other confounded variables: mathematical problems and biological solutions

Relative growth rate (RGR) has a long history of use in biology. In its logged form, RGR = ln[(M + ΔM)/M], where M is size of the organism at the commencement of the study, and ΔM is new growth over time interval Δt. It illustrates the general problem of comparing non-independent (confounded) variables, e.g. (X + Y) vs. X. Thus, RGR depends on what starting M(X) is used even within the same growth phase. Equally, RGR lacks independence from its derived components, net assimilation rate (NAR) and leaf mass ratio (LMR), as RGR = NAR × LMR, so that they cannot legitimately be compared by standard regression or correlation analysis.

The mathematical properties of RGR exemplify the general problem of 'spurious' correlations that compare expressions derived from various combinations of the same component terms X and Y. This is particularly acute when X >> Y, the variance of X or Y is large, or there is little range overlap of X and Y values among datasets being compared. Relationships (direction, curvilinearity) between such confounded variables are essentially predetermined and so should not be reported as if they are a finding of the study. Standardizing by M rather than time does not solve the problem. We propose the inherent growth rate (IGR), lnΔM/lnM, as a simple, robust alternative to RGR that is independent of M within the same growth phase.

Although the preferred alternative is to avoid the practice altogether, we discuss cases where comparing expressions with components in common may still have utility. These may provide insights if (1) the regression slope between pairs yields a new variable of biological interest, (2) the statistical significance of the relationship remains supported using suitable methods, such as our specially devised randomization test, or (3) multiple datasets are compared and found to be statistically different. Distinguishing true biological relationships from spurious ones, which arise from comparing non-independent expressions, is essential when dealing with derived variables associated with plant growth analyses.