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BACKGROUND: The proportion of nitrogen (N) derived from the atmosphere (Ndfa) is a fundamental component of the plant N demand in legume species. To estimate the N benefit of grain legumes for the subsequent crop in the rotation, a simplified N balance is frequently used. This balance is calculated as the difference between fixed N and removed N by grains. The Ndfa needed to achieve a neutral N balance (hereafter θ ) is usually estimated through a simple linear regression model between Ndfa and N balance. This quantity is routinely estimated without accounting for the uncertainty in the estimate, which is needed to perform formal statistical inference about θ . In this article, we utilized a global database to describe the development of a novel Bayesian framework to quantify the uncertainty of θ . This study aimed to (i) develop a Bayesian framework to quantify the uncertainty of θ , and (ii) contrast the use of this Bayesian framework with the widely used delta and bootstrapping methods under different data availability scenarios. RESULTS: The delta method, bootstrapping, and Bayesian inference provided nearly equivalent numerical values when the range of values for Ndfa was thoroughly explored during data collection (e.g., 6-91%), and the number of observations was relatively high (e.g., ≥ 100 ). When the Ndfa tested was narrow and/or sample size was small, the delta method and bootstrapping provided confidence intervals containing biologically non-meaningful values (i.e. < 0% or > 100%). However, under a narrow Ndfa range and small sample size, the developed Bayesian inference framework obtained biologically meaningful values in the uncertainty estimation. CONCLUSION: In this study, we showed that the developed Bayesian framework was preferable under limited data conditions ─by using informative priors─ and when uncertainty estimation had to be constrained (regularized) to obtain meaningful inference. The presented Bayesian framework lays the foundation not only to conduct formal comparisons or hypothesis testing involving θ , but also to learn about its expected value, variance, and higher moments such as skewness and kurtosis under different agroecological and crop management conditions. This framework can also be transferred to estimate balances for other nutrients and/or field crops to gain knowledge on global crop nutrient balances.

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Precise management of crop nitrogen nutrition is essential to maximize yields while limiting pollution risks. For several decades, the critical nitrogen (N) dilution curve - relating plant biomass (W) to N concentration (%N) - has become a key tool for diagnosing plant nutritional status. Increasing number of studies are being conducted to parameterize critical N dilution curves of a wide range of crop species in different environments and N-fertilized conditions. A global synthesis of the resulting data is lacking on this topic. Here, we conduct a systematic review of the experimental data collected worldwide to parametrize critical N dilution curves. The dataset consists of 36 papers containing a total of 4454 observations for 19 major crop species distributed in 16 countries. The key variables of this dataset are the W and %N collected at three or more sampling times, containing three or more fertilizer N rate levels. This dataset can guide the development of generic critical N dilution curves, helps scientists to identify factors influencing plant N status, and leads to the formulation of more robust N recommendations for a broad range of environmental conditions.

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The light attenuation process within a plant canopy defines energy capture and vertical distribution of light and nitrogen (N). The vertical light distribution can be quantitatively described with the extinction coefficient (k), which associates the fraction of intercepted photosynthetically active radiation (fPARi) with the leaf area index (LAI). Lower values of k correspond to upright leaves and homogeneous vertical light distribution, increasing radiation use efficiency (RUE). Yield gains in maize (Zea mays L.) were accompanied by increases in optimum plant density and leaf erectness. Thus, the yield-driven breeding programs and management changes, such as reduced row spacing, selected a more erect leaf habit under different maize production systems (e.g., China and the USA). In this study, data from Argentina revealed that k decreased at a rate of 1.1% year-1 since 1989, regardless of plant density and in agreement with Chinese reports (1.0% year-1 since 1981). A reliable assessment of changes in k over time is critical for predicting (i) modifications in resource use efficiency (e.g. radiation, water, and N), improving estimations derived from crop simulation models; (ii) differences in productivity caused by management practices; and (iii) limitations to further exploit this trait with breeding.

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BACKGROUND: The fraction of intercepted photosynthetically active radiation (fPARi) is typically described with a non-linear function of leaf area index (LAI) and k, the light extinction coefficient. The parameter k is used to make statistical inference, as an input into crop models, and for phenotyping. It may be estimated using a variety of statistical techniques that differ in assumptions, which ultimately influences the numerical value k and associated uncertainty estimates. A systematic search of peer-reviewed publications for maize (Zea Mays L.) revealed: (i) incompleteness in reported estimation techniques; and (ii) that most studies relied on dated techniques with unrealistic assumptions, such as log-transformed linear models (LogTLM) or normally distributed data. These findings suggest that knowledge of the variety and trade-offs among statistical estimation techniques is lacking, which hinders the use of modern approaches such as Bayesian estimation (BE) and techniques with appropriate assumptions, e.g. assuming beta-distributed data. RESULTS: The parameter k was estimated for seven maize genotypes with five different methods: least squares estimation (LSE), LogTLM, maximum likelihood estimation (MLE) assuming normal distribution, MLE assuming beta distribution, and BE assuming beta distribution. Methods were compared according to the appropriateness for statistical inference, point estimates' properties, and predictive performance. LogTLM produced the worst predictions for fPARi, whereas both LSE and MLE with normal distribution yielded unrealistic predictions (i.e. fPARi < 0 or > 1) and the greatest coefficients for k. Models with beta-distributed fPARi (either MLE or Bayesian) were recommended to obtain point estimates. CONCLUSION: Each estimation technique has underlying assumptions which may yield different estimates of k and change inference, like the magnitude and rankings among genotypes. Thus, for reproducibility, researchers must fully report the statistical model, assumptions, and estimation technique. LogTLMs are most frequently implemented, but should be avoided to estimate k. Modeling fPARi with a beta distribution was an absent practice in the literature but is recommended, applying either MLE or BE. This workflow and technique comparison can be applied to other plant canopy models, such as the vertical distribution of nitrogen, carbohydrates, photosynthesis, etc. Users should select the method balancing benefits and tradeoffs matching the purpose of the study.

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Targeting the right agronomic optimum plant density (AOPD) for maize (Zea mays L.) is a critical management decision, but even more when the seed cost and grain selling price are accounted for, i.e. economic OPD (EOPD). From the perspective of improving those estimates, past studies have focused on utilizing a Frequentist (classical) approach for obtaining single-point estimates for the yield-density models. Alternative analysis models such as Bayesian computational methods can provide more reliable estimation for AOPD, EOPD and yield at those optimal densities and better quantify the scope of uncertainty and variability that may be in the data. Thus, the aims of this research were to (i) quantify AOPD, EOPD and yield at those plant densities, (ii) obtain and compare clusters of yield-density for different attainable yields and latitudes, and (iii) characterize their influence on EOPD variability under different economic scenarios, i.e. seed cost to corn price ratios. Maize hybrid by seeding rate trials were conducted in 24 US states from 2010 to 2019, in at least one county per state. This study identified common yield-density response curves as well as plant density and yield optimums for 460 site-years. Locations below 40.5 N latitude showed a positive relationship between AOPD and maximum yield, in parallel to the high potential level of productivity. At these latitudes, EOPD depended mostly on the maximum attainable yield. For the northern latitudes, EOPD was not only dependent on the attainable yield but on the cost:price ratio, with high ratios favoring reductions in EOPD at similar yields. A significant contribution from the Bayesian method was realizing that the variability of the estimators for AOPD is sometimes greater than the adjustment accounting for seed cost. Our results point at the differential response across latitudes and commercial relative maturity, as well as the significant uncertainty in the prediction of AOPD, relative to the economic value of the crop and the seed cost adjustments.

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