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This article proposes a graphical model that handles mixed-type, multi-group data. The motivation for such a model originates from real-world observational data, which often contain groups of samples obtained under heterogeneous conditions in space and time, potentially resulting in differences in network structure among groups. Therefore, the iid assumption is unrealistic, and fitting a single graphical model on all data results in a network that does not accurately represent the between group differences. In addition, real-world observational data is typically of mixed discrete-and-continuous type, violating the Gaussian assumption that is typical of graphical models, which leads to the model being unable to adequately recover the underlying graph structure. Both these problems are solved by fitting a different graph for each group, applying the fused group penalty to fuse similar graphs together and by treating the observed data as transformed latent Gaussian data, respectively. The proposed model outperforms related models on learning partial correlations in a simulation study. Finally, the proposed model is applied on real on-farm maize yield data, showcasing the added value of the proposed method in generating new production-ecological hypotheses. An R package containing the proposed methodology can be found on https://CRAN.R-project.org/package=heteromixgm. Supplementary materials for this article are available online.

Improving crop yields is one of the main goals of agronomy. However, yield is determined by a complex interplay between Genotypic, Environmental and Management factors (G×E×M), which varies across time and space. Therefore, identifying the fundamental relations underlying yield variation is a principal aim of agricultural research. A narrow, and not necessarily appropriate, set of statistical methods tends to be used in the study of such relations, which is why we aim to introduce a diverse audience of agronomists, production ecologists, plant breeders and others interested in explaining yield variation to the use of graphical models. More specifically, we wish to demonstrate the usefulness of copula graphical models for heterogeneous mixed data. This new statistical learning technique provides a graphical representation of conditional independence relationships within data that is not necessarily normally distributed and consists of multiple groups for environments, management decisions, genotypes or abiotic stresses such as drought. This article introduces some basic graphical model terminology and theory, followed by an application on Ethiopian maize and wheat yield undergoing drought stress. The proposed method is accompanied with the R package heteromixgm https://CRAN.R-project.org/package=heteromixgm.

Context: Collection and analysis of large volumes of on-farm production data are widely seen as key to understanding yield variability among farmers and improving resource-use efficiency. Objective: The aim of this study was to assess the performance of statistical and machine learning methods to explain and predict crop yield across thousands of farmers’ fields in contrasting farming systems worldwide. Methods: A large database of 10,940 field-year combinations from three countries in different stages of agricultural intensification was analyzed. Random effects models were used to partition crop yield variability and random forest models were used to explain and predict crop yield within a cross-validation scheme with data re-sampling over space and time. Results: Yield variability in relative terms was smallest for wheat and barley in the Netherlands and for wheat in Ethiopia, intermediate for rice in the Philippines, and greatest for maize in Ethiopia. Random forest models comprising a total of 87 variables explained a maximum of 65 % of cereal yield variability in the Netherlands and less than 45 % of cereal yield variability in Ethiopia and in the Philippines. Crop management related variables were important to explain and predict cereal yields in Ethiopia, while predictive (i.e., known before the growing season) climatic variables and explanatory (i.e., known during or after the growing season) climatic variables were most important to explain and predict cereal yield variability in the Philippines and in the Netherlands, respectively. Finally, model cross-validation for regions or years not seen during model training reduced the R2 considerably for most crop x country combinations, while for wheat in the Netherlands this was model dependent. Conclusion: Big data from farmers’ fields is useful to explain on-farm yield variability to some extent, but not to predict it across time and space. Significance: The results call for moderate expectations towards big da

Grain legumes are key components of sustainable production systems in sub-Saharan Africa, but wide-spread nutrient deficiencies severely restrict yields. Whereas legumes can meet a large part of their nitrogen (N) requirement through symbiosis with N2-fixing bacteria, elements such as phosphorus (P), potassium (K) and secondary and micronutrients may still be limiting and require supplementation. Responses to P are generally strong but variable, while evidence for other nutrients tends to show weak or highly localised effects. Here we present the results of a joint statistical analysis of a series of on-farm nutrient addition trials, implemented across four legumes in four countries over two years. Linear mixed models were used to quantify both mean nutrient responses and their variability, followed by a random forest analysis to determine the extent to which such variability can be explained or predicted by geographic, environmental or farm survey data. Legume response to P was indeed variable, but consistently positive and we predicted application to be profitable for 67% of farms in any given year, based on prevailing input costs and grain prices. Other nutrients did not show significant mean effects, but considerable response variation was found. This response heterogeneity was mostly associated with local or temporary factors and could not be explained or predicted by spatial, biophysical or management factors. An exception was K response, which displayed appreciable spatial variation that could be partly accounted for by spatial and environmental covariables. While of apparent relevance for targeted recommendations, the minor amplitude of expected response, the large proportion of unexplained variation and the unreliability of the predicted spatial patterns suggests that such data-driven targeting is unlikely to be effective with current data.

This perspective describes the opportunities and challenges of data-driven approaches for crop diversity management (genebanks and breeding) in the context of agricultural research for sustainable development in the Global South. Data-driven approaches build on larger volumes of data and flexible analyses that link different datasets across domains and disciplines. This can lead to more information-rich management of crop diversity, which can address the complex interactions between crop diversity, production environments, and socioeconomic heterogeneity and help to deliver more suitable portfolios of crop diversity to users with highly diverse demands. We describe recent efforts that illustrate the potential of data-driven approaches for crop diversity management. A continued investment in this area should fill remaining gaps and seize opportunities, including i) supporting genebanks to play a more active role in linking with farmers using data-driven approaches; ii) designing low-cost, appropriate technologies for phenotyping; iii) generating more and better gender and socioeconomic data; iv) designing information products to facilitate decision-making; and v) building more capacity in data science. Broad, well-coordinated policies and investments are needed to avoid fragmentation of such capacities and achieve coherence between domains and disciplines so that crop diversity management systems can become more effective in delivering benefits to farmers, consumers, and other users of crop diversity.

Adjusting crop nutrient rates to local differences in soil fertility and yield potential is considered a promising way of improving fertilizer use efficiency in smallholder agriculture. Despite its obvious appeal, the benefits of this approach have not been evaluated empirically. In the absence of appropriate data, the theoretical potential of nutrient tailoring needs to be evaluated, and suitable models exist to do this. Such analysis may provide a benchmark for expectations of success and would form a valuable starting point for reflection and debate on the limitations of current data, models, and assumptions. This study presents a theoretical ex-ante assessment of the short-term economic benefits of fertilizer adjustments to differences in nutrient and water constraints, under the assumption that these are the only determinants of nutrient response. Combining two calibrations of a mathematical production function with digital maps of predicted nutrient and water availability, economically optimum macro-nutrient rates and resulting maize yields were calculated across a theoretical representation of soils across Sub-Saharan Africa. Consistent economic benefits from fertilizer application were predicted, raising theoretical yields from 19 to as much as 61 or 88% of water limited levels depending on the model used. Under the magnitude of soil fertility variation considered here, matching nutrient rates to indigenous levels was predicted to result in marginal added benefits at best, particularly at higher, more profitable, investment levels. Strong spatial heterogeneity in water limitation translated into more pronounced effects of rate adjustments to water limited yields, raising average predicted profit by up to 127 USD/ha compared to a fixed-rate recommendation. This suggests that tailoring to different yield levels can offer non-trivial benefits if good estimates of yield potential are available. Adding realistic levels of uncertainty around soil fertility and use

Leguminous plants are known to require phosphorus fertilizers and inoculation with nitrogen fixing rhizobia for optimum yield but other nutrients may also be lacking. In this study, the most limiting nutrients for legume growth were determined in soils where the crops had not responded to P and rhizobial inoculation in field trials, using the double pot technique. Soils were collected from 17 farmers' fields in West Kenya, Northern Nigeria, Eastern and Southern Rwanda, South-west and North-west Sierra Leone. Plant growth and mean biomass were measured on soils to which a full nutrient solution, containing phosphorus (P), potassium (K), magnesium (Mg), sulfur (S) and micronutrients (MN) were added, and which were compared to a control (no nutrient added), and individual omissions of each nutrient. The relationship between soil properties and nutrient deficiencies was explored. Nutrient limitations were found to differ between soils, both within and across countries. Generally, each soil was potentially deficient in at least one nutrient, with K, P, Mg, MN and S emerging as most limiting in 88, 65, 59, 18, and 12% of tested soils, respectively. While K was the most limiting nutrient in soils from Kenya and Rwanda, P was most limiting in soils from Nigeria. P and K were equally limiting in soils from Sierra Leone. Mg was found limiting in two soils from Kenya and three soils from Rwanda and one soil each in Nigeria and Sierra Leone. Micronutrients were found to be limiting in one soil from Nigeria and one soil from Rwanda. Estimates of nutrient deficiency using growth and mean biomass were found to be correlated with each other although the latter proved to be a more sensitive measure of deficiency. With few exceptions, the relation between soil parameters and nutrient deficiencies was weak and there were no significant relations between deficiency of specific nutrients and the soil content of these elements. Although our results cannot be translated directly to the fi

Rhizobia are soilborne bacteria that form symbiotic relations with legumes and fix atmospheric nitrogen. The nitrogen fixation potential depends on several factors such as the type of host and symbionts and on environmental factors that affect the distribution of rhizobia. We isolated bacteria nodulating common bean in Southern Ethiopia to evaluate their genetic diversity and phylogeography at nucleotide, locus (gene/haplotype) and species levels of genetic hierarchy. Phylogenetically, eight rhizobial genospecies (including previous collections) were determined that had less genetic diversity than found among reference strains. The limited genetic diversity of the Ethiopian collections was due to absence of many of the Rhizobium lineages known to nodulate beans. Rhizobium etli and Rhizobiumphaseoli were predominant strains of bean-nodulating rhizobia in Ethiopia. We found no evidence for a phylogeographic pattern in strain distribution. However, joint analysis of the current and previous collections revealed differences between the two collections at nucleotide level of genetic hierarchy. The differences were due to genospecies Rhizobium aethiopicum that was only isolated in the earlier collection.

Many interventions are assuming that introduced seeds diffuse. However, the details of this diffusion among farmers are poorly understood. This article presents data from eight sites in four on the diffusion of seed and associated information given to farmers involved in N2Africa’s demonstration trials. The study showed that 2–3 years after the trials had been organised, more than 90% of the farmers who had participating in the trial activities and were given a seed-input package with 1–5 kg of legume seed had shared this seed, on average with four other farmers. The farmers who received this seed from these directly involved farmers shared their seed less frequently. Eighty per cent of all the seed sharings were of 1–2 kg of seed given as a gift. Only 5% of the sharings involved a cash transaction. More than half of the seed sharings were with family members and around a third were between friends. Men shared at least as often as women and both men and women shared most with persons of their own sex. Information about rhizobium as an associated input for soya was shared by more than one-third of farmers, almost exclusively by farmers who had participated in the demonstration trials themselves. Extrapolation of data suggest that in addition to the 250,000 farmers who participated directly in the N2Africa demonstration trials, another 1,400,000 farmers may have received seed of a new legume crop or variety. The results show that knowing about the character of the seed sharing mechanisms may offer opportunities to influence the diffusion of seeds. Providing farmers with somewhat larger amounts of seeds, emphasise the importance of sharing seeds and information with relatives and friends could be an important factor in achieving a high multiplier effect.

Improving bacterial nitrogen fixation in grain legumes is central to sustainable intensification of agriculture in sub-Saharan Africa. In the case of soyabean, two main approaches have been pursued: first, promiscuous varieties were developed to form effective symbiosis with locally abundant nitrogen fixing bacteria. Second, inoculation with elite bacterial strains is being promoted. Analyses of the success of these approaches in tropical smallholder systems are scarce. It is unclear how current promiscuous and non-promiscuous soyabean varieties perform in inoculated and uninoculated fields, and the extent of variation in inoculation response across regions and environmental conditions remains to be determined.We present an analysis of on-farm yields and inoculation responses across ten countries in Sub Saharan Africa, including both promiscuous and non-promiscuous varieties. By combining data from a core set of replicated on-farm trials with that from a large number of farmer-managed try-outs, we study the potential for inoculation to increase yields in both variety types and evaluate the magnitude and variability of response.Average yields were estimated to be 1343 and 1227. kg/ha with and without inoculation respectively. Inoculation response varied widely between trials and locations, with no clear spatial patterns at larger scales and without evidence that this variation could be explained by yield constraints or environmental conditions. On average, specific varieties had similar uninoculated yields, while responding more strongly to inoculation. Side-by side comparisons revealed that stronger responses were observed at sites where promiscuous varieties had superior uninoculated yields, suggesting the availability of compatible, effective bacteria as a yield limiting factor and as a determinant of the magnitude of inoculation response.

To target food security interventions for smallholder households, decision makers need large-scale information, such as maps on poverty, food security and key livelihood activities. Such information is often based on expert knowledge or aggregated data, despite the fact that food security and poverty are driven largely by processes at the household level. At present, it is unclear if and how household level information can contribute to the spatial prediction of such welfare indicators or to what extent local variability is ignored by current mapping efforts. A combination of geo-referenced household level information with spatially continuous information is an underused approach to quantify local and large-scale variation, while it can provide a direct estimate of the variability of welfare indicators at the most relevant scale. We applied a stepwise regression kriging procedure to translate point information to spatially explicit patterns and create country-wide predictions with associated uncertainty estimates for indicators on food availability and related livelihood activities using household survey data from Uganda. With few exceptions, predictions of the indicators were weak, highlighting the difficulty in capturing variability at larger scale. Household explanatory variables identified little additional variation compared to environmental explanatory variables alone. Spatial predictability was strongest for indicators whose distribution was determined by environmental gradients. In contrast, indicators of crops that were more ubiquitously present across agroecological zones showed large local variation, which often overruled large-scale patterns. Our procedure adds to existing approaches that often only show large-scale patterns by revealing that local variation in welfare is large. Interventions that aim to target the poor must recognise that diversity in livelihood activities for income generation within any given area often overrides the variability of li

The impacts of rhizobium inoculation on growth and yield of chickpea have mainly been tested in experiments conducted in greenhouses or on research stations. We report the response of the crop to inoculation (I) and phosphorus fertilizer (P) application across a large number of smallholder's farms over four regions of Ethiopia, covering diverse soil fertility and agro-ecological conditions. Increased grain yields due to the soil fertility treatments was evident for 99% target farmers. On average, I and P increased grain yield by 21% and 25% respectively, while the combined application of I and P resulted in a 38% increase. However, observed grain yields on control plots and responses to the treatments on individual farms varied greatly, and relative yield responses (%; yield of P and/I minus control yield, divided by control yield) ranged from 3% to 138%. With the exception of a few extremely poorly yielding locations, average responses to P and I were high across a wide range of control yields, indicating the possibility of boosting chickpea productivity for smallholders with P fertilizer and inoculant technology. Variation in response to rhizobium inoculation was mostly independent of agro-ecology and soil type although it was found to be low on a number of farms with extremely high N contents (%). Assuming that a relative yield increase of 10% due to treatment effects is required to be visible, 71%, 73% and 92% of the farmers observed a yield benefit by applying P, I, and P + I, respectively. The results are discussed with respect to the additive benefits of P fertilizers and rhizobial inoculation and their implications for wide scale promotion of inoculant technology to smallholders.

Plants are sessile organisms and, consequently, are exposed to a plethora of stresses in their local habitat. As a result, different populations of a species are subject to different selection pressures leading to adaptation to local conditions and intraspecific divergence. The annual brassicaceous plant Arabidopsis thaliana is an attractive model for ecologists and evolutionary biologists due to the availability of a large collection of resequenced natural accessions. Accessions of A. thaliana display one of two different life cycle strategies: summer and winter annuals. We exposed a collection of 308 European Arabidopsis accessions, that have been genotyped for 250K SNPs, to a range of stresses: one abiotic stress (drought), four biotic stresses (Pieris rapae caterpillars, Plutella xylostella caterpillars, Frankliniella occidentalis thrips and Myzus persicae aphids) and two combined stresses (drought plus P. rapae and Botrytis cinerea fungus plus P. rapae). We identified heritable genetic variation for responses to the different stresses, estimated by narrow-sense heritability. We found that accessions displaying different life cycle strategies differ in their response to stresses. Winter annuals are more resistant to drought, aphids and thrips and summer annuals are more resistant to P. rapae and P. xylostella caterpillars. Summer annuals are also more resistant to the combined stresses of drought plus P. rapae and infection by the fungus Botryris cinerea plus herbivory by P. rapae. Adaptation to drought displayed a longitudinal gradient. Finally, trade-offs were recorded between the response to drought and responses to herbivory by caterpillars of the specialist herbivore P. rapae.

Background: Homoplasy affects demographic inference estimates. This effect has been recognized and corrective methods have been developed. However, no studies so far have defined what homoplasy metrics best describe the effects on demographic inference, or have attempted to estimate such metrics in real data. Here we study how homoplasy in chloroplast microsatellites (cpSSR) affects inference of population expansion time. cpSSRs are popular markers for inferring historical demography in plants due to their high mutation rate and limited recombination. Results: In cpSSRs, homoplasy is usually quantified as the probability that two markers or haplotypes that are identical by state are not identical by descent (Homoplasy index, P). Here we propose a new measure of multi-locus homoplasy in linked SSR called Distance Homoplasy (DH), which measures the proportion of pairwise differences not observed due to homoplasy, and we compare it to P and its per cpSSR locus average, which we call Mean Size Homoplasy (MSH). We use simulations and analytical derivations to show that, out of the three homoplasy metrics analyzed, MSH and DH are more correlated to changes in the population expansion time and to the underestimation of that demographic parameter using cpSSR. We perform simulations to show that Approximate Bayesian Computation (ABC) can be used to obtain reasonable estimates of MSH and DH. Finally, we use ABC to estimate the expansion time, MSH and DH from a chloroplast SSR dataset in Pinus caribaea. To our knowledge, this is the first time that homoplasy has been estimated in population genetic data. Conclusions: We show that MSH and DH should be used to quantify how homoplasy affects estimates of population expansion time. We also demonstrate how ABC provides a methodology to estimate homoplasy in population genetic data.