Liebig's Barrel, Dual Ratios and Pan Balance: Paradigm Change in Nutrient Diagnosis.

Liebig’s barrel where the shortest stave indicates the most limiting nutrient is a metaphor for the Law of minimum first stated by Carl Sprengel in 1828. Later, a critical value or range (CNR) approach was derived for use in plant and soil science. However, despite myriads of nutrient interactions in plants, no interaction was taken into account. Dual ratios were then proposed to represent nutrient interactions for diagnostic purposes. DRIS is an empirical tool to integrate dual ratios into nutrient indexes. The basic cation saturation ratio (BCSR) approach informs on relationships among cations and exchangeable acidity within the closed space of the cation exchange capacity (CEC). However, recent research in compositional data analysis (CoDa) showed that compositional data such as proportions and concentrations of components in soils and plants convey numerical biases, have inherent structure and have matrix of rank D-1, where D is the number of components of the whole. Log ratios avoid numerical biases. The additive log ratio could be used to avoid biases in stoichiometric rules such as the N:P:K:Ca:Mg proportions relative to N in plant dry matter or the C:S:N:P rule in humus formation. The centered log ratio was used to rectify DRIS and provide the required geometry in the Euclidean space. The isometric log ratio computes D-1 orthogonally arranged ad hoc balances based on principles of plant physiology and soil biogeochemistry or on specific crop management devices to facilitate interpreting the results of statistical analysis. CoDa thus provides robust mathematical ground for soil and plant nutrient diagnosis. We show that CNR, DRIS and BCSR are numerically biased. This requires a change of paradigm based on CoDa techniques and the clinical test concept to classify plant and soil health. We use the pan balance design to compare plant compositions, present reports and solve complex problems.