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Statistical Designs and Response Models for Mixtures of Cultivars¹

In spite of current interest in growing mixtures (composites) of cultivars and in multiple cropping systems, appropriate response models, definitions, treatment designs, and statistical analyses are not available when more than two cultivars are involved. Since the beneficial or detrimental effects of using mixtures of size k equal to or greater than two cultivars (lines, species) as compared to sole cropping (monoculture) needs to be assessed, the objective of this study is to present response models, treatment designs (the selection of entities or treatments for an experiment), definitions of effects, and statistical analyses for estimating and comparing the various effects involved in experiments on mixtures of k cultivars. The concepts of general mixing (blending), bi-specific mixing, tri-specific mixing, . . . , n^th specific mixing effects are formulated. Treatment designs are derived from weighing, balanced incomplete block, and supplemented block experiment designs. It is shown that solutions for parameters in the response models are dependent upon the number and types of mixtures used in the treatment design. A statistical analysis is presented for a particular treatment design involving mixtures of all possible pairs of eight dry bean cultivars plus the cultivars themselves. Finally, it is noted that the results may be useful for a number of other types of experiments involving mixtures, such as herbicide, nutrition, medicine, education, etc.

Key Words: Treatment design • Balanced incomplete block designs • General mixing or blending effects • Biblend or bispecific mixing effect • k-Blend or k-specific mixing effect

² Liberty Hyde Bailey Professor of biological statistics.

Received for publication May 22, 1978.