highly significantP <0.01
The following criteria were used to identify the best-fit model for each crop–region yield trend: highly significant (Student’s t -test; P <0.01) model parameters, highest r 2 and lowest r.m.s.e. compared with other models, and independent, normally distributed yield residuals with homogenous variance and unbiased distribution when residuals were plotted against year ( Fig. 4 ).