We proposed a workflow for nonlinear modeling of data from multiple‐harvest crops.
We demonstrated why the nonlinearity measures should be used to select nonlinear models.
We demonstrated as the critical points describe the multiple‐harvest crops production.
Logistic model parameters determine the precocity and the concentration of production.
Growth models are alternative to ANOVA in analyzing data from multiple‐harvest crops.
Nonlinear growth models have been widely used for analyzing production curves with a sigmoidal pattern; however, all benefits that these models provide are not being fully exploited. Our aim here is to provide a step‐by‐step guide on how to choose a nonlinear model with parameters close to being unbiased, and to show how to estimate and interpret the critical points of a model aimed at determining the precocity and concentration of the production. Data on two uniformity trials conducted with eggplant (Solanum melongena L.) was used for this purpose. The Brody, Gompertz, logistic, and von Bertalanffy models were fitted to predict the number and fresh mass of fruits per plant. The model with lower nonlinearity measures and lower bias of the parameter estimates was selected. All the tested models presented satisfactory goodness‐of‐fit measures, but they differed regarding nonlinearity measures. The logistic model was selected because it had lower intrinsic and parametric nonlinearity and lower bias in parameter estimates. The inflection point and maximum acceleration/deceleration points of this model provide detailed pieces of information of the production through the productive cycle. Finally, using the logistic model as an example, we demonstrate that lower values of β2 are related to an earlier maximum production rate, and higher values of β3 are related to an earlier production that is concentrated in fewer days. The nonlinearity measures were important for the model selection. Thus, it is strongly recommended that nonlinearity is estimated and used to select nonlinear models in future studies.
Core Ideas
The confidence interval (CI) of Pearson’s correlation coefficient (r) was investigated.Confidence interval width is inversely proportional to r and sample size (n).It is recommended to use 1000 or more bootstrap replicates in order to not underestimate CI width (CIw).A model to estimate CIw as a function of n and r is proposed.
The nonparametric bootstrap percentile method has been widely used to estimate confidence intervals (CI) for Pearson’s product‐moment correlation coefficient (r). However, because most studies provide results for specific crops and pre‐stablished CIs, an innovative approach to CI estimation is needed. The aim of this study was to propose a model that predicts CI width (CIw) as a function of the sample size (n) and the strength of association among traits. Additionally, we also investigated the extent to which the number of bootstrap replicates (BRs) influences CI estimation. Seventy‐eight different r magnitudes from a maize field experiment were used. The 95% CI half‐width for each trait combination was estimated based on 991 different sample sizes and seven different numbers of BRs. A simple nonlinear model with n and r as predictors is proposed for estimating the CIw: , where δ, β0, and β1 are the model coefficients. Based on our data, the fitted model was: . This model exhibited excellent goodness of fit (R2 = 0.988; root mean square error [RMSE] = 0.011). Considering an assumed magnitude of association (r), the n for a desired CIw can then be calculated as: . We also recommend using ≥1000 BRs, to prevent underestimating CIw. Finally, we present an intuitive table that provides previously estimated n for 9 levels of half‐widths for 95% CIs (0.05, 0.1,... 0.45) and 19 magnitudes for r (0.05, 0.10,..., 0.95).
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