RESUMEN
Various decision contexts require the calculation of smaller recurring changes accumulated over time and their comparison to larger one-time changes (e.g., $100 periodic increase in monthly rent every year vs. a $1000 increase in rent at the end of 5 years). In both hypothetical and incentivized studies, we demonstrate an inaccuracy of estimations involving total cumulations of smaller recurring changes and single lump sums. We document this effect when individuals process increasing or decreasing changes in gains or losses (e.g., raises in wages or rent, discounts in membership fees). Importantly, these biases occur even when the changes are provided to the consumers as clear absolute dollar values as opposed to complex percentages. We discuss the theoretical contributions of our study as well as its implications for consumers, managers, and policy makers.
RESUMEN
Tissue analysis is commonly used in ecology and agronomy to portray plant nutrient signatures. Nutrient concentration data, or ionomes, belongs to the compositional data class, i.e., multivariate data that are proportions of some whole, hence carrying important numerical properties. Statistics computed across raw or ordinary log-transformed nutrient data are intrinsically biased, hence possibly leading to wrong inferences. Our objective was to present a sound and robust approach based on a novel nutrient balance concept to classify plant ionomes. We analyzed leaf N, P, K, Ca, and Mg of two wild and six domesticated fruit species from Canada, Brazil, and New Zealand sampled during reproductive stages. Nutrient concentrations were (1) analyzed without transformation, (2) ordinary log-transformed as commonly but incorrectly applied in practice, (3) additive log-ratio (alr) transformed as surrogate to stoichiometric rules, and (4) converted to isometric log-ratios (ilr) arranged as sound nutrient balance variables. Raw concentration and ordinary log transformation both led to biased multivariate analysis due to redundancy between interacting nutrients. The alr- and ilr-transformed data provided unbiased discriminant analyses of plant ionomes, where wild and domesticated species formed distinct groups and the ionomes of species and cultivars were differentiated without numerical bias. The ilr nutrient balance concept is preferable to alr, because the ilr technique projects the most important interactions between nutrients into a convenient Euclidean space. This novel numerical approach allows rectifying historical biases and supervising phenotypic plasticity in plant nutrition studies.