The concepts and quantification of yield gap using boundary lines. A review

Context: The potential yield of crops is not usually realised on farms creating yield gaps. Methods are needed to diagnose yield gaps and to select interventions. One method is the boundary line model in which the upper bound of a plot of yield against a potentially limiting factor is viewed as the most efficient response to that factor and anything below it has a yield gap caused by inefficiency of other factors. If many factors are studied, the cause of the yield gap can be identified (yield gap analysis, YGA). Though the boundary line is agronomically interpretable, its estimation and statistical inference are not straightforward and there is no standard method to fit it to data. Objective: We review the different methods used to fit the boundary line, their strengths and weaknesses, interpretation, factors influencing the choice of method and its impact on YGA. Methods: We searched for articles that used boundary lines for YGA, using the Boolean “Boundary*” AND “Yield gap*” in the Web of Science. Results: Methods used to fit boundary lines include heuristic methods (visual, Binning, BOLIDES and quantile regression) and statistical methods (Makowski quantile regression, censored bivariate model and stochastic frontier analysis). In contrast to heuristic methods, which in practice require ad hoc decisions such as the quantile value in the quantile regression method, statistical methods are typically objective, repeatable and offer a consistent basis to quantify parameter uncertainty. Nonetheless, most studies utilise heuristic methods (87% of the articles reviewed) which are easier to use. The boundary line is usually interpreted in terms of the Law of the Minimum or the Law of Optimum to explain yield gaps. Although these models are useful, their interpretation holds only if the modelled upper limit represents a boundary and not just a particular realization of the upper tail of the distribution of yield. Therefore, exploratory and inferential analysis tools that inform boundary characteristics in data are required if the boundary line is to be useful for YGA. Conclusions and implications: Statistical methods to fit boundary line models consistently and repeatably, with quantified uncertainty and evidence that there is a boundary limiting the observed yields, are required if boundary line methods are to be used for YGA. Practical and conceptual obstacles to the use of statistical methods are required. Bayesian methods should also be explored to extend further the capacity to interpret uncertainty of boundary line models.