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Type-I errorIn the case of two identical unimodal von Mises distributions, seven tests did not maintain Type-I error near the nominal 5% level, at least when sample sizes were small. These tests were the Kuiper two-sample test, the non equal concentration parameters approach ANOVA, the P-test, the Watson’s large-sample nonparametric test, the Watson–Williams test and the Rao dispersion test (Fig. 2). The Type-I error results were similar for the unimodal wrapped skew-normal distribution, except that the Wallraff test and Fisher’s method also showed Type-I error inflation (Fig. S1). No other methods showed evidence of failure to control Type-I error rate across different testing situations (Figs. S2–S5), except for the Log-likelihood ratio ANOVA in the case of two identical asymmetrical bimodal distributions (Fig. S3). In summary, only eight out of 18 tests reliably controlled the Type-I error rate near the nominal 5% level across all the situations investigated. These included five tests for identical distribution, the Watson’s U2 test, the Large-sample Mardia–Watson–Wheeler test, the Watson-Wheeler test, the embedding approach ANOVA, the MANOVA approach, the Rao polar test for differences in mean direction, and two tests for differences in concentration, the Levene’s test and the concentration test. We focus only on these tests in our explorations of statistical power.Figure 2Type-I error of all tests using von Mises distributions for different sample sizes: 10 and 10 (A), 20 and 20 (B), 50 and 50 (C), 20 and 30 (D) and 10 and 50 (E). Concentration (κ, kappa) increases for both distributions from 0 to 8. Tests are grouped according to their null hypotheses.Full size imagePower to detect differences in concentrationThe most powerful test to detect concentration differences between two von Mises distributions was the MANOVA approach, which offered superior power especially at lower sample sizes (Fig. 3). The Watson’s U2 test was also very powerful, followed by the Watson–Wheeler and the Large-sample Mardia–Watson–Wheeler tests with only marginally lower power. The embedding approach ANOVA had lower power, but, notably, was still more powerful than the Concentration test and Levene’s test, both specifically designed to detect differences in concentration. As expected, the Rao polar test was not sensitive to differences in concentration. The general results for two unimodal wrapped skew-normal distributions were comparable to the results for unimodal von Mises distributions, with the only exception of superior performance of Levene’s test in situations with highly asymmetric samples sizes (Fig. S6).Figure 3Power of all included tests when comparing von Mises distributions of differing concentrations using different sample sizes: 10 and 10 (A), 20 and 20 (B), 50 and 50 (C), 20 and 30 (D) and 10 and 50 (E). The first distribution is fixed at κ = 0, the second increases from 0 towards 8.Full size imageWhen comparing axial von Mises distributions, only the Watson’s U2 test offered acceptable power (Fig. S7). For the symmetrical trimodal distributions, overall power was very low, and again, only the Watson’s U2 providing some power (Fig. S8). The asymmetrical bimodal (Fig. S9) situation showed acceptable power of the MANOVA approach and Watson’s U2, however, for the asymmetrical trimodal distribution power was low with the Watson’s U2 providing the best results (Fig. S10).Power to detect differences in the mean/medianThe power to detect angular differences between two von Mises distributions was highest for the MANOVA approach at small sample sizes (n = 10), followed by the Watson’s U2, Watson-Wheeler test and the Large-sample Mardia-Watson-Wheeler test (Fig. 4). Notably, the Levene’s test also showed acceptable power levels, clearly failing to detect specifically concentration differences (to which it was less sensitive, see Fig. 4). The concentration test was not sensitive to the differences in mean direction. Special cases were the embedding approach ANOVA and the Rao polar test. The ANOVA approach showed, with the exception of very unequal sample sizes (n = 10/50), a unimodal response, with increasing power levels from 0° to 90° difference, but then rapidly decreasing power towards 180° difference. The Rao polar test showed an even stranger pattern, with, at higher sample sizes, very good power when the difference was either around 45° or 135°, but with power levels dropping to 0.05 in between these two peaks (at 90°). The results were similar for the wrapped skew-normal distribution, with the exceptions that the Rao polar test showed strongly reduced power and switched from a bimodal to a unimodal power curve with a peak around 60°, and the Levene’s test completely lost its power (Fig. S11).Figure 4Power of all included tests when comparing von Mises distributions (kappa for both = 2) of differing directions using different sample sizes: 10 and 10 (A), 20 and 20 (B), 50 and 50 (C), 20 and 30 (D) and 10 and 50 (E). The first distribution is fixed at 0°, the second increases from 0° towards 180.Full size imageFor axial distributions, only the Watson’s U2 test offered acceptable power levels, although large sample sizes (~ n = 100) were required for the power to reach over 50% (Fig. S12). All other tests failed to detect the difference in mean direction between two axial distributions. For symmetric trimodal distributions none of the tests used was sensitive to differences in mean direction (Fig. S13).When comparing asymmetrical bimodal distributions, the general trends were similar to the unimodal case. However, over all sample sizes the MANOVA approach offered the best power. The Watson–Wheeler test was considerably less powerful in this situation, as were the Watson’s U2 test and the Large-sample Mardia–Watson–Wheeler test (Fig. S14). The Levene’s test showed a unimodal-shaped power curve. The asymmetrical trimodal situation was, again, similar to the asymmetrical bimodal situation (Fig. S15), with the exception of the Levene’s test, which showed steady power increase with angular difference (instead of the hump-shaped curve).Power to detect differences in distribution typeWhen comparing a unimodal and an axial bimodal distribution, which increased similarly in concentration, we found that the MANOVA approach again offered the best power in particular at low samples sizes, followed by the Watson’s U2 test, the Large-sample Mardia–Watson–Wheeler test and Watson–Wheeler test (Fig. 5). While the embedding approach ANOVA and the Levene’s test had varying but usable power levels, the concentration test was only sensitive to such differences at low concentration values. The Rao polar test was not sensitive to such differences.Figure 5Power of all included tests when comparing von Mises distributions of differing number of modes (unimodal and axially bimodal) using different sample sizes: 10 and 20 (A), 20 and 40 (B), 50 and 100 (C), 20 and 60 (D) and 10 and 100 (E). The concentration (κ) of both increases from 0 to 8.Full size imageThe picture was only marginally different when comparing a von Mises with a wrapped skew-normal distribution (Fig. S16). For low sample sizes (n = 10) the MANOVA approach offered great power, followed by the embedding approach ANOVA. The latter offered good power throughout the range of sample sizes tested, followed by the Watson’s U2 test, the Large-sample Mardia–Watson–Wheeler test and Levene’s test. Also, the Rao polar test showed lower, but acceptable sensitivity to distribution type. The concentration test only showed very low power, that (as expected) increased with increasing concentrations of the respective distributions.We summarize the results obtained in the power analysis in Table 2. In all situations, either the Watson’s U2 test or the MANOVA approach offered the best power.Table 2 Ranking of tests based on the power comparisons for the main scenarios encountered in potential data sets (using different distributions: unimodal, axial, asymmetrical bimodal, symmetrical trimodal, asymmetrical trimodal), in cases were only one test performed acceptable the others ranks were left blank (see Table 1 for abbreviations).Full size tableReal data examplesTesting the performance of the robust tests on real data sets revealed, predominantly, the expected test behavior. In the example of homing pigeons where a difference in concentration was expected, all tests, with the exception of the Rao polar test and, notably, the concentration test, showed a significant difference between the distributions (Fig. 6A). Therefore, we can conclude, in accordance with the respective publication19, that sectioning of the olfactory nerve disrupted the homing behavior of pigeons.Figure 6Results from example data. Shown are results of pigeon (A), ant (B) and bat orientations (C). Control groups are on the left panels and experimental groups on the right. The tests are abbreviated according to Table 1, significant test results are indicated in red with asterisk and non-significant in blue. For each circular plot directional data is shown as dots on the circle (each dot is one individual), the arrows represent the mean direction and the dashed line the 95% confidence interval.Full size imageIn the ant example, where no difference between the groups was expected, there was no significant difference between the distributions detected by most of the tests (Fig. 6B). Only the concentration test showed a significant difference. Based on the other tests we would conclude that there was no biological meaningful difference between the two distributions. Therefore, ants appear to be able to transfer visual information from one eye to the other.In the bat example, where a difference in mean direction was expected, the Watson’s U2, the Mardia–Watson–Wheeler, Watson-Wheeler test and the MANOVA approach showed a significant difference (Fig. 6C). Notably, the Rao polar, Levene’s, and concentration tests and the embedding approach ANOVA failed to show a significant difference. At least for the Rao polar test, one would have expected a significant difference, as the two distributions are clearly 180° apart. This outcome concurs with our simulation results where the Rao polar test failed to distinguish distributions on the same and orthogonal axes (Fig. 4). As the results of the tests where quite mixed this example highlights the need for choosing a test with appropriate power to detect the expected differences. Based on the results of the most powerful tests, we conclude that the bats showed a mirrored orientation, as expected in the experimental design. More

Data acquisitionOur analysis was based on open access crop production data from the United Nation’s Food and Agricultural Organization (FAO) spanning from 1961 to 201718. We extracted data on area harvested (in ha) for 339 FAO-defined crop groups being grown in all UN-recognized countries. Since our research centred on understanding, quantifying, and mapping changes in crop diversity in current agricultural lands, countries that cease to exist (e.g., Yugoslavia) were not included in our analysis, resulting in data for 201 countries (Table S1). Prior to analyses, we adjusted certain crop group listings following our previous analyses of global changes in crop diversity8. Specifically, “Cottonlint” and “Cottonseed” were duplicated in our dataset and were therefore compiled as “Seedcotton”, while “Palmkernels” were renamed as “Oilpalmfruit.” Additionally, “Fruitpomenes”, “Fruitstonenes”, and “Grainmixed” were removed from analysis since these crop groupings are not associated with any specific crop species in the FAO database18. Finally, “Mushroomsandtruffles” were removed since it relates to non-plant species, and “Coir” was removed because it is a plant by-product.Changes in crop richness over timeAll statistical analyses were performed using R version 3.3.3 statistical software (R Foundation for Statistical Computing, Vienna, Austria). The initial step in our analysis was to calculate both crop richness and evenness for each country, at each individual year, using the vegan R package38. Based on these datasets, we then used the analytical framework developed by8 to evaluate how crop species richness and evenness have changed in each individual country across its entire data range.Specifically, in their analysis Martin et al.8 found that piecewise linear regression models provided the strongest descriptions of crop species richness change over time, across 21 of 22 FAO-defined regions globally. We therefore followed this approach by fitting a piecewise linear regression model for each country individually, that predicts changes in species richness over time. Piecewise model fitting was a two-step process, whereby for each country we first fit a linear regression model of the form:$$S = a + left( {b times {text{year}}} right)$$
(1)
where a is the intercept and b represents the rate of change in crop group richness (S) through time. This linear model (Eq. 1) was then used as the basis of a piecewise linear regression model, which was fitted in order to estimate breakpoints in the relationship between S and year. Specifically, piecewise models were fit using the segmented function in the segmented R package39, and were of the form:$$S = a + bleft( {{text{year}}} right) + left( {left( {c({text{year}} -uppsi _{1} } right) times Ileft( {{text{year}} >uppsi _{1} } right)} right) + left( {dleft( {{text{year}} -uppsi _{2} } right) times Ileft( {{text{year}} >uppsi _{2} } right)} right)$$
(2)
where a is as in Eq. (1), and b represents the slope of the S-year relationship prior to the first breakpoint (ψ1). Here, c represents the difference in the slope of the S-year relationship between the first and second piecewise model segments; the c parameter therefore applies only when the first conditional indicator function (denoted by “I”) is true. Similarly, d represents the difference in slopes for the S-year relationship between the first, second, and third segments, which only applies when the second conditional indicator function is true. In sum, the slope of the relationship between S and year is equal to b prior to the ψ1, is equal to b + c between ψ1and ψ2, and is equal to b + c + d after ψ2. Piecewise models were fit with initial starting parameters of 1975 and 2000 for ψ1 and ψ2, respectively. The ψ1 and ψ2 parameters were tuned manually for 29 countries with a shortened data range, following visual inspection of data (see Tables S1 and S2).Based on this piecewise regression model procedure, we then used parameters from Eq. (2) to determine three key indicator points of crop diversity change through time for each country (displayed visually in Fig. 1). Indicator 1 reflects the onset of diversification in each country, and was calculated as Breakpoint 1 (ψ1) in Eq. (2); this indicator therefore corresponds to the year in which notable changes in species richness began. Indicator 2 reflects the duration of the crop diversification period in each country, and was calculated as the difference between breakpoints 2 and 1 (i.e., ψ2-ψ1 from Eq. 2); this indicator therefore represents the duration of the period when crop prominent changes in crop diversity occurred. Finally, Indicator 3 reflects the rate at which crop diversity changed throughout the diversification period in each country; this indicator was calculated as the rate of crop diversity change (between ψ1 and ψ2), which in our models corresponded to the sum of the slopes (1) prior to the first breakpoint, and (2) between the first and second breakpoints (i.e., corresponding to b + c in Eq. 2). For each indicator we then calculated summary statistics as either mean ± standard deviations or median ± median absolute deviations (m.a.d.), where data was normally or log-normally distributed, respectively. Country values for each indicator were mapped using the mapCountryData function in the rworldmap R package40.Changes in crop evenness over timeEvaluations of temporal changes in crop evenness at national scales followed this same analytical approach as above. First, for each country-by-year combination we calculated Pielou’s evenness index (J′)—which ranges from 0 to 1, with values closer to 0 indicating less evenness or greater abundance of a few dominant crop groups, and values closer to 1 representing more equitable abundances of crop groups—as:$$J^{prime} = frac{H^prime }{{ln left( S right)}}$$
(3)
where S is again crop richness, and H′ is the Shannon–Weiner diversity index calculated as:$$H^prime = – mathop sum limits_{i = 1}^{S} p_{i} ln p_{i}$$
(4)
where pi represents the relative proportion of the ith crop group for a given country-by-year combination. In these evenness calculations, all values of pi were estimated as the relative proportion of agricultural area (measured in ha) occupied by a given crop commodity group, within a country at a given year; this analytical approach was employed by Martin et al.8 when assessing crop group composition at supra-national scales. We then evaluated how J′ values changed in each country through time by replicating our stepwise modelling analyses above, substituting J′ for S in Eqs. (1) and (2), and extracting the same model indicators (Fig. 1). Finally, we calculated summary statistics and mapped each of these indicators, as described above.Changes in crop composition across countries and over timeWe used multivariate analyses to evaluate how temporal changes in S and J′ influenced crop composition across countries and over time. To do so, we created a community composition matrix whereby national-level crop assemblages were estimated for each of the country-by-year combinations. In this matrix, area harvested was taken as an approximation of the abundance of each crop group within each country-by-year combination (again following Martin et al.8). Since these abundances (or area harvested) across country-by-year combinations varied over orders of magnitude, we used non-metric multidimensional scaling (NMDS) to analyze and visualize spatial (country) and temporal (year) differences in crop diversity. Specifically, we used the vegan R package38 to calculate all 58,899,231 Bray–Curtis dissimilarities among all 10,854 data points (i.e., crop group composition in every country-by-year data point), as:$$BC_{jk} = frac{{sum i left| {x_{ij} – x_{ik} } right|}}{{sum i left( {x_{ij} + x_{ik} } right)}}$$
(5)
where BCjk represents the dissimilarity between the jth and kth community, xij represents the abundance (i.e., area harvested) of crop group i in sample j, and xik represents the abundance of crop group i in sample k. We then used a multivariate analysis of variance (i.e., an Adonis test), to test for significant differences in Bray–Curtis distances as a function of country, year, and a country-by-year interaction. Significance was assessed using a permutation test, with 99 permutations used.Latitudinal gradients in crop richnessTo test our hypotheses surrounding the presence of, and temporal changes in, latitudinal gradients in crop group diversity, we focused on 164 countries for which crop group diversity was available in both 1961 and 2017. For each of these two datasets, we fit a separate linear regression model that predicts crop group richness as a function of latitude (expressed as an absolute value) and a 2nd-order polynomial term for the ‘latitude2’ variable. From both of these models, we extracted and compared latitude value at which crop group richness was estimated/ modelled to peak.Predictors of change in crop diversity and compositionWe tested if Human Development Index (HDI) was correlated with patterns of change in crop diversity and composition. Briefly, the HDI is a composite index of four metrics related to socio-economic status, including life expectancy at birth, expected years of schooling for children at a school-centring age, mean years of schooling for adults ≥ 25 years of age, and log-transformed gross national income per capita. These values are then aggregated on a per country basis, into an HDI index that ranges from 0–1 with higher scores denoting higher performance in these indicators. We employed 2017 HDI values in our analysis here, in order to include the most countries possible in each analysis (since earlier HDI scores are less readily available)41.We then used linear mixed effects models to test if patterns of change in crop diversity and evenness varied systematically with HDI values. This entailed fitting six linear mixed models, where each of our six indicators (i.e., Indicators 1–3 for both S and J′) were predicted as a function of HDI; these models also accounted for potential spatial autocorrelation in Indicator values by including the FAO-defined continent identity and FAO-defined region identity of each country, as a nested random variable. Models were fit using the lme function in the nlme R package41. We then estimated the proportion of variation in each indicator that is explained by HDI, continent identity, and region identity, using the varcomp function in the ape R package42—which partitioned explained variation across continents and regions—as well as the sem.model.fits function in the piecewiseSEM R package43—which partitioned explained variation across the fixed (i.e., model intercept and HDI) vs. random (i.e., continent and region) effects. Due to differences in HDI data availability and in the number of piecewise models that converged, n = 152 countries for all models of S indicators and n = 139 countries for all models of J′ indicators. Log-transformed values of Indicators were used in these analyses where they better approximated a log-normal distribution, as determined using the fitdistrplus function in the fitdistrplus R package44. More

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AffiliationsEarth, Atmospheric, and Planetary Sciences Department, Massachusetts Institute of Technology, Cambridge, MA, USAGabriela Serrato MarksSchool of Science, Technology, Accessibility, Mathematics and Public Health, Gallaudet University, Washington DC, USACaroline SolomonScience, Technology & Society Department, Rochester Institute of Technology, Rochester, NY, USAKaitlin Stack WhitneyAuthorsGabriela Serrato MarksCaroline SolomonKaitlin Stack WhitneyCorresponding authorsCorrespondence to
Gabriela Serrato Marks, Caroline Solomon or Kaitlin Stack Whitney. More