It is demonstrated in the appendix that if X and Y are two variables that are the cumulates of increments x and y respectively, X and Y are necessarily Linked to each other by a statistical linear relationship, providing that (1) the means mu(x) and mu(y) of X and y are different from zero and (2) the number of increments tends to infinity. Moreover, the least squares estimator of slope (a) in the relation Y = aX + b converges towards the ratio between the means of the elementary increments (mu(y)/mu(x)). This theoretical result is verified for the relationship between accumulated net photosynthesis of greenhouse-grown tomatoes and photosynthetically-active radiation intercepted by plants and accumulated on the same timesteps. In the appendix, simulations show that the smaller the coefficient of variation (standard deviation/mean) of x and y, the more rapidly the R-square of the X-to-Y relationship converges towards 1. For instance, when coefficients of variation equal 30%, R-square is often greater than 0.95 when there are more than 15 observations. Consequently, a Linear relationship between two cumulated variables is virtually always the result of a statistical artefact. Crop biomass, which is a cumulated growth variable, may thus exhibit Linear statistical relationships with any cumulated variable that may come to mind. The slopes of these relations, which only represent the ratio between the means of the elementary increments, are often biologically meaningless. As a conclusion, cumulated variables should be avoided in bio-environmental relationships: for instance, growth or photosynthesis rates should be used instead of their cumulated values. It is probably also preferable to use development rates (and not status). (C) 1997 Elsevier Science B.V.
