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Experimental design
The trials were conducted in five regions (Boro, Ugunja, Ukwala, Wagai and Yala) of Siaya County in western Kenya. Siaya County is located at 00°08.468′ N, 34°25.378′ E, and at an altitude of 1,336 m above sea level. The experimental sites were in lower midland 1(LM1) and lower midlands 2 (LM2) agro-ecological zones, which experience bimodal rainfall with long rains (LR) starting in March to July and short Rains (SR) starting in Late August to December41 and receive average annual rainfall of 1,500 mm42. The soils are mainly Ferralsols and Acrisols in the higher areas and Vertisols in the low areas.
Trials were conducted in 48 randomly selected villages, using stratification at the sub-county level. Half of them (randomly selected) participated in the trials in the long and short rain of 2014 and the long rain of 2015. The other half started in the short rain of 2014 and continued throughout the long and short rain of 2015. In each village 10 farmers participated in the trial. Half of them were specifically selected for participation in a community meeting. In those meetings, the researchers explained the objectives of the trials and asked the community members to nominate 5 farmers (as well as 5 potential substitutes), including two women, thought to be good farmers and interested in participating in the trials. Such non-random selection of farmers is common practice in research trials (Supplementary Table S1 online). The other half was selected randomly from the list of all the farmers in the village. All selected farmers were visited to obtain consent for the trials and identify the potential trial parcel (chosen by the farmer, conditional on fitting with some criteria for suitability to the research trials). A small number of replacements was done (but always keeping 5 selected by the community and 5 random). In each village 4 random farmers (2 random and 2 selected) were assigned to participate in the maize trial, while 3 random farmers (at least 1 random and 1 selected) were assigned to participate in the soybean trial. The other 3 farmers participated in a maize-soybean intercrop trial. During implementation, the assignment of inputs for the intercrop trial was, however, contaminated. As a result none of the plots in the intercrop trial received a best-bet input package, making the agronomic findings from that trial hard to interpret, and therefore not necessarily of interest for the decomposition proposed in this paper. Nevertheless, for completeness, results for these intercrop trials are shown in Supplementary Table S14 online.
Researcher-designed and farmer-managed trials
The trials would qualify as researcher-designed and farmer-managed (under the supervision of the researchers). The research team had full control over the design of the trials, from the choice of inputs to spacing and other management practices. All inputs were provided by the research team, with the exception of the local maize seed tested in two out of the six plots. A researcher (local expert agronomist) was present and led planting, gapping and thinning, all fertilizer applications, and harvesting. In these activities, labor was typically provided by the farmer. Planting dates were mostly decided by the researchers to best target the onset of rains, also responding to the farmers’ feedback on beginning of rains and availability to schedule the visit for planting. The farmer was in charge of land preparation, weeding and other management, with the researchers providing guidelines on those practices. In each village, a contact person (typically one of the ten farmers) visited the trials weekly to verify that the farmers fulfilled their responsibilities. Farmers were also asked to inform the contact person in case of any pest or disease, in which case the researcher provided the required pesticide or fungicide.
Treatment structure and application
Supplementary Table S2 online presents the full factorial designs of the multi-locational trials for maize and soybean, including details on crop varieties and quantity of inputs. The plot sizes were 4.5 × 5 m and the treatments were completely randomized between the six plots on each parcel. Plot sizes are of a similar order of magnitude as those found in other recently published work. A 1 m inter-plot spacing was planted with sweet potatoes to act as a buffer between plots to prevent inter-plot contamination. The sweet potatoes were planted at 50 cm from each plot, and border rows of the maize and soya plots were excluded for yield estimations to limit any edge effect. Hence the area harvested was 12.9 m2 for maize and 13.5 m2 for soybean. The experiments were repeated for three seasons, and plot layout and treatments were maintained for three seasons.
For the soybean trials, a soybean rhizobia inoculant was tested alone, with Minjingu hyper phosphate (0-30-0 + 38CaO) or Sympal (0:23:15 + 10CaO + 4S + 1MgO + 0.1Zn) in a full factorial design. Phosphorus rate of 30 kg P ha−1 was used to determine the quantity of Sympal and Minjingu hyper phosphate to be applied. On each farm only one replicate was used; hence, 6 plots were installed on each farm. Inoculation was done at planting as a seed coating using the directions for use in the respective product labels. Each plot had 6 soybean lines of 5 m in length each spaced at 5 cm from plant to plant within row and 50 cm from row to row. Inoculation was done on all the rows. Soybean variety TGx1740-2F with medium maturity (95–100 days)43 was used as the test crop. The spatial variability of the soybean response is studied in44. The soybean trials demonstrated that the combination of rhizobia inoculant and P-source led to important yield gains44.
The choice of inputs resulted from prior research conducted as part of the Compro project. Soybean was chosen as test crop mainly because in the prior phase of the project it had shown good response to rhizobia inoculation45 and was agro-ecologically suitable to the region. Kenya is an importer of soybean and multiple efforts are geared towards raising local production. In Compro I, the two rhizobia inoculants were tested and shown to be effective in increasing nodulation, nitrogen fixation and yield when inoculated on the tested soybean variety. Minjingu and Sympal were chosen based on their formulation with respect to the chemical characteristics of the soils in the test sites and results of earlier research46. The soils generally lack phosphorus and are acidic. A mapping study47 specifically identified Western Kenya as a potential K deficient area, and soil acidity has long been identified as a constraining factor in Western Kenya48 hence the importance of CaO. Results from soil sampling of the trial plots confirmed that more than 56.87% of soils were acidic (pH  More

Spatial patterns of stability proxies and background variables
Figure 2 a, b show the spatial distribution of our two proxy variables, the average rank of Rs per position (rankRs) and of the range of the ranks per position (rangeRs) in kriged maps. The middle to southern areas were found to have the largest, whilst the north-eastern areas the smallest rankRs values, whereas a slightly different pattern was characteristic for rangeRs with some additional north-western large values. Similarly, larger average soil organic carbon content (meanSOC) and average soil water content (meanSWC) (Fig. 2 c, d) were detected at the western-middle-southern regions and smaller at the north-eastern part of the study site.
Figure 2

Kriged patterns of stability proxies, rankRs (a) and rangeRs (b), as well as of background factors, meanSOC (%, c) and meanSWC (%, d).

Full size image

Correlations between stability proxies and background variables along DEMs: entire dataset
We investigated the potential direct effects of the different terrain attributes (local mean elevation (mALT), standard deviation of elevation (SD), topographic position index (TPI), slope (Sl), Easterness and Northness (East, North)) on the spatial distributions of our proxy variables by using the terrain attributes originating from differently smoothed DEM rasters. DEM1 was the original, 0.2 m resolution model, while DEMs 2–6 were progressively smoothed by a factor of two resulting in different resolution DEM rasters (DEM2: 0.4 m, DEM3: 0.8 m, DEM4: 1.6 m, DEM5: 3.2 m, DEM6: 6.4 m, respectively), and finally DEM7 met the resolution of the field measuring campaigns (10 m). The terrain attributes were filtered out from the rasters for the 78 measuring positions of the sampling grid.
On the basis of the correlation analysis we found an important difference in terrain attribute features between DEM 5 and 6, especially in SD, Sl, North and East. All subsequent results are then based on DEMs 1–5, which were found to be more similar to each other and to the original DEM1. The maps of terrain attributes with the box blur kernel from DEM1-5 can be found in the Supplementary Information (SI) together with the descriptions and calculations. As we couldn’t find any of the blur kernels superior to the other when considering correlations, the results hereafter are only presented for the box blur kernel calculations for simplicity.
When we considered the entire dataset (named hereafter: “A” dataset), we could only find significant correlation between rangeRs and TPI at less smoothed DEMs but the correlation was very weak (black symbols and line in Fig. 3).
Figure 3

Direct correlation between TPI and stability proxy, rangeRs at less smoothed DEMs, DEM1-2 for datasets A (black symbols and line) and S (blue symbols and line, see the information later on). The correlations were significant at p = 0.0076 and p  = 0.0172 levels, although they were weak, r2 = 0.09, r2 = 0.42 for A and S (see the information later on), respectively.

Full size image

Any other correlation between the proxies and the terrain attributes could only be deduced indirectly from the positive correlations between rankRs and meanSOC, meanSWC (cf. Table 1b). These correlations were scale-independent, i.e., we detected them at every DEMs. In general, the larger the soil carbon content and soil moisture at a position (cf. Figure 2c,d, showing quite similar patterns to the proxy patterns in the figure upper row), the larger the Rs activity detected and the opposite was true for lower carbon content and soil moisture positions.
Table 1 (a) Statistically significant (p  More

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