The Weibull Function as a Dose‐Response Model to Describe Ozone Effects on Crop Yields1

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Abstract
Field experiments have demonstrated that ozone (O3) pollution can reduce yields of important crop species. The assessment of the impact of O3 is facilitated by the use of a family of dose response models that covers the range of responses observed in field O3 dose response studies. The Weibull dose response model is a flexible, biologically realistic, model that has been found useful for this purpose. The purposes of this study are i) to discuss analyses of dose response data using the Weibull model and its extensions, ii) to identify some of the problems inherent in the use of a nonlinear dose response model, and iii) to show application of the model to two sets of experimental data. The basic Weibuil model is a modification of the Weibull probability distribution function. Extensions of the basic model allow for class variables and other continuous variables in the experimental design. A bivariate form of the Weibull allows for independent effects of two pollutants. The use of the Weibull model with class variables (cultivars) is demonstrated using a 1977 winter wheat (Triticum aestivum L.) experiment grown at Raleigh, NC involving four cultivars exposed to four levels of O3 pollution. Tests of homogeneity over cultivars of the proportional yield responses show differential cultivar response to O3. The bivariate Weibull model is demonstrated using a 1981 soybean [Glycine max (L.) Merr.] experiment involving one cultivar exposed to the factorial set of treatments consisting of six levels of O3 and four levels of SO2 pollution. The test of lack of fit of the bivariate Weibuil model was nonsignificant, supporting the hypothesis that O3 and SO2 were acting independently. The percent predicted relative yield losses and their standard errors are given for two of the wheat cultivars and for the soybean cultivar.

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