Figure S1, Assume that the research impact of individual researchers (I), measured in this case as number of publications, varies as an exponential function of grant size (F): I  = aFb, where a and b are empirical constants. Assume further that any researcher with $10,000 of grant funding produces a single publication. If 01, then impact is an accelerating function of funding (panel a) and researchers with large grants produce more publications per dollar than researchers with small grants (panel b). Consequently, if a granting agency has a fixed amount of money to invest (say, one million dollars), then the total impact of all researchers will be greater by spreading the money thinly if 01. In this study, we find that, for four different measures of scientific impact, the observed value of b is 0≤b≤1. Figure S2, Example of improvement of assumptions typically observed among tested models when untransformed data (panel A) were log transformed (panel B). Residuals vs. fitted and Scale-Location plots both support an improvement on homogeneity of variance between raw and transformed data. Normal Q-Q plots support also that the log transformation improves normality of residual. Residuals vs. Leverage plots support that there are no outliers. This example was made using the data from the Ecology and Evolution committee (n = 139), relating the number of articles published to the amount of NSERC funding received. (DOCX)