Analysis of Trials with Complex Treatment Structure Using Multiple Contrast Tests

Experiments with complex treatment structures are not uncommon in horticultural research. For example, in augmented factorial designs, one control treatment is added to a full factorial arrangement, or an experiment might be arranged as a two-factorial design with some groups omitted because they are practically not of interest. Several statistical procedures have been proposed to analyze such designs. Suitable linear models followed by F-tests provide only global inference for main effects and their interactions. Orthogonal contrasts are demanding to formulate and cannot always reflect all experimental questions underlying the design. Finally, simple mean comparisons following global F-tests do not control the overall error rate of the experiment in the strong sense. In this article, we show how multiple contrast tests can be used as a tool to address the experimental questions underlying complex designs while preserving the overall error rate of the conclusions. Using simultaneous confidence intervals allows for displaying the direction, magnitude, and relevance of the mean comparisons of interest. Along with application in statistical software, shown by two examples, we discuss the possibilities and limitations of the proposed approach. In agricultural and horticultural research, controlled experiments are set up to evaluate the effect of several treatments and their interactions on physiological or developmen- tal variables. If the levels of a first factor of potential influence are combined with all levels of a second factor, the resulting exper- iment has a two-factorial, completely cross- classified treatment structure. However, often the experimental questions are manifold and, together with the background knowledge on the practical problem, lead to complex treat- ment structures. Augmented factorial designs