Predictors of treatment means for a one factor completely randomized design
A one factor experimental design is developed based on the potential observable outcome framework, sampling from finite population of units and random allocation of treatments. No assumptions are made about the distribution of units and distribution of treatments. We introduce sampling and treatment assignment random variables to represent the joint permutation of the potentially observable population. The joint roles of sampling and treatment assignments are considered. The predictors for treatment means, presented by the observed values in the sample and unobserved values of the remainder, are obtained by Royall’s (1976) prediction theory. We take three cases into account: the latent values correspond to a no interaction model and there are no response errors; the latent values correspond to a model with interaction and there are no response errors; the latent values correspond to a model with interaction and response errors. The predictors with the property of being “shrunk” towards the overall mean are similar to realized random effects in the one way random effect model. If the treatment is not selected in the sample, the predictors correspond to the overall mean. The model is based solely on the population sampling and treatment assignments, and provides a design based framework for inference of linear combinations of the treatment means. The population of treatments can be of small size and up to all of the treatments can be assigned to the samples. This model extended the randomization model developed by Kempthorne (1952) via introducing random allocation of the treatments. Theoretically the predictors provide smaller MSEs than using the linear combination of sample means. When the variance components are unknown, the empirical predictors are considered. The confidence intervals for the empirical predictors are calculated using bootstrapping methods. Several bootstrap methods, such as bootstrapping with replacement (BWR) or bootstrapping without replacement (BWO) are introduced. Each bootstrap method is developed to account for the sampling from the finite population of units and random allocation of treatments. Comparisons of the different bootstrapping methods are made methodologically, and via simulation. We discuss these comparisons, and recommend an appropriate bootstrapping method for statistical inference in the one factor experimental design.