Treatment allocation for linear models

This paper analyzes the benefits and the limits of a systematic allocation of treatments within a linear model framework. Linear models do not necessarily require the treatment allocation to be random. Since the variance of the treatment estimator within linear models does not depend on the realization of the dependent variable, whenever the covariate information is available prior to allocating treatments it is possible to allocate treatments in a way that minimizes the variance of the treatment estimator. I show that in each experiment satisfying the linear model assumptions, there exists at least one deterministic optimal design, i.e., a deterministic way of allocating treatments that minimizes the variance of the treatment estimator over all alternative ways of allocating treatments. Infinite samples, optimal design reduces the variance of the treatment estimator and increases statistical power compared to random allocation. For a given linear model and a given effect size, optimal design decreases the sample size necessary to detect a significant treatment effect on average by the number of covariates in the model. However, asymptotically, as the sample size goes to infinity, neither optimal design nor any alternative design yields any benefit over random allocation.