On the Design of Experiments with Ordered Treatments

Abstract

Summary There are many situations where one expects an ordering among K⩾2 experimental groups or treatments. Although there is a large body of literature dealing with the analysis under order restrictions, surprisingly, very little work has been done in the context of the design of experiments. Here, a principled approach to the design of experiments with ordered treatments is provided. In particular we propose two classes of designs which are optimal for testing different types of hypotheses. The theoretical findings are supplemented with thorough numerical experimentation and a concrete data example. It is shown that there is a substantial gain in power, or alternatively a reduction in the required sample size, when an experiment is both designed and analysed by using methods which account for order restrictions.

Optimal designs for testing the efficacy of heterogeneous experimental groups

“…Hence, adoptingρ, the power of Wald's test can be approximated by Pr χ 2 1 (nφ(ρ)) > q 1,α . As correctly stated by Singh and Davidov [12], althoughρ is a degenerate target, it is still optimal since Pr χ 2 1 (nφ(ρ…”

“…This section is dedicated to the performance assessment of the newly introduced optimal designs. Starting with the normal model, ρ * andρ will be compared with the balanced allocation ρ B and the design ρ M = (e 1 + e K )/2 proposed by Baldi Antognini et al [3] and Singh and Davidov [12], which is the optimal design for normal homoscedastic data (i.e.,ρ = ρ M ) and it is also the target maximizing the minimum power for both restricted and unrestricted likelihood ratio tests under the simple order restriction θ 1 ≥ . .…”

“…The case of exponential responses with censoring tends to be similar to that of Normal heteroscedastic data with variances ordered as the treatment effects, and it is strongly affected by the chosen censoring scheme. Finally, inspired by [12], it is interesting to compare the above-considered designs when the values of the model parameters induce unfavorable configurations. From (2.1), due to the heteroscedasticity, the NCP tends to vanish as the variances of the treatment group grow, regardless of the chosen design; so, for any fixed maximum difference between the treatment effects, there is no unique least favorable configuration, but different scenarios that tend to degenerate as the signal to noise ratio vanishes.…”

“…Recently, Baldi Antognini et al [3] derived the design maximizing the power of the test of homogeneity for normal homoscedastic data, which is a balanced allocation involving only the best and the worst treatments; moreover, by imposing the ethical constraint that the treatment allocation proportions should reflect the a-priori unknown ordering among their effects, they also derived a non-degenerate optimal target implementable via response-adaptive randomization. Under the same framework, assuming that the treatment ordering is known, Singh and Davidov [12] discussed the optimal designs for restricted and unrestricted statistical inference (which are equivalent for large samples) by adopting a maxi-min approach, in order to overcome local optimality problems.…”

“…For the normal homoscedastic, exponential and Poisson models θ t = (θ 1 , 2, 1)and θ t = (θ 1 , 2.5, 2, 1.5, 1), while for binary data θ t = (θ 1 , 0.2, 0.1) and θ t = (θ 1 , 0.25, 0.2, 0.15, 0.1). For the normal heteroscedastic case, scenario (a) corresponds to v t = (12, 9, 2) and v t = (20, 16, 12, 9, 2) and scenario (b) to v t = (2, 12, 9) and v t =(2,12, 20,16,9).…”