Empirical Likelihood for the Analysis of Experimental Designs

Empirical likelihood enables a nonparametric, likelihood-driven style of inference without restrictive assumptions routinely made in parametric models. We develop a framework for applying empirical likelihood to the analysis of experimental designs, addressing issues that arise from blocking and multiple hypothesis testing. In addition to popular designs such as balanced incomplete block designs, our approach allows for highly unbalanced, incomplete block designs. Based on all these designs, we derive an asymptotic multivariate chi-square distribution for a set of empirical likelihood test statistics. Further, we propose two single-step multiple testing procedures: asymptotic Monte Carlo and nonparametric bootstrap. Both procedures asymptotically control the generalized family-wise error rate and efficiently construct simultaneous confidence intervals for comparisons of interest without explicitly considering the underlying covariance structure. A simulation study demonstrates that the performance of the procedures is robust to violations of standard assumptions of linear mixed models. Significantly, considering the asymptotic nature of empirical likelihood, the nonparametric bootstrap procedure performs well even for small sample sizes. We also present an application to experiments on a pesticide. Supplementary materials for this article are available online.