The Orthogonal Interactions Model for Unreplicated Factorial Experiments

Abstract

Unreplicated factorial experiments arise frequently in practice because of limited resources. When fitting the standard ANOVA model to data from unreplicated experiments, the data are not sufficient to estimate the interaction terms and error variance, thus limiting the possible inferences. In experiments such as crop yield trials, investigators are interested in estimating interactions but are unable to form replications. Existing methods, such as Tukey's One-Degree-of-Freedom model and the AMMI model, fit constrained interactions which allow for the error variance estimation.

We present the orthogonal interactions (OI) model for unreplicated factorial experiments. The model frees degrees of freedom for error by assuming that the main effects are orthogonal to the interactions. Through simulation we find that approximate F-statistics are appropriate in testing for main effects and interactions. The likelihood ratio test to compare the OI model to the ANOVA model leads to high type-I error rates, so we use a simulation corrected likelihood ratio test. Two real data sets (one with replication, one without) suggest that the OI model is appropriate for real data.

In working with the OI model and the existing models, we found a need for reliable degrees of freedom for complex statistical models. The resampling method for estimating degrees of freedom is motivated by the linear model, where there is a linear relationship between expected sums of squares and the variance of errors added to the response. Through simulation, we find that the resampling method provides reliable degrees of freedom for the OI model and for existing models as well.