arXiv subjects

David Azriel

Publications and source records attributed to David Azriel.

24 records · Page 2Linked to original sources

arXiv2020

Improving the Power of the Randomization Test

We consider the problem of evaluating designs for a two-arm randomized experiment with the criterion being the power of the randomization test for the one-sided null hypothesis. Our evaluation assumes a response that is linear in one observed covariate, an unobserved component and an additive treatment effect where the only randomness comes from the treatment allocations. It is well-known that the power depends on the allocations' imbalance in the observed covariate and this is the reason for the classic restricted designs such as rerandomization. We show that power is also affected by two other design choices: the number of allocations in the design and the degree of linear dependence among the allocations. We prove that the more allocations, the higher the power and the lower the variability in the power. Designs that feature greater independence of allocations are also shown to have higher performance. Our theoretical findings and extensive simulation studies imply that the designs with the highest power provide thousands of highly independent allocations that each provide nominal imbalance in the observed covariates. These high powered designs exhibit less randomization than complete randomization and more randomization than recently proposed designs based on numerical optimization. Model choices for a practicing experimenter are rerandomization and greedy pair switching, where both outperform complete randomization and numerical optimization. The tradeoff we find also provides a means to specify the imbalance threshold parameter when rerandomizing.

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arXiv2019

The conditionality principle in high-dimensional regression

Consider a high-dimensional linear regression problem, where the number of covariates is larger than the number of observations and the interest is in estimating the conditional variance of the response variable given the covariates. A conditional and unconditioned framework are considered, where conditioning is with respect to the covariates, which are ancillary to the parameter of interest. In recent papers, a consistent estimator was developed in the unconditional framework when the marginal distribution of the covariates is normal with known mean and variance. In the present work, a certain Bayesian hypothesis test is formulated under the conditional framework, and it is shown that the Bayes risk is a constant. This implies that no consistent estimator exists in the conditional framework. However, when the marginal distribution of the covariates is normal, the conditional error of the above consistent estimator converges to zero, with probability converging to one. It follows that even in the conditional setting, information about the marginal distribution of an ancillary statistic may have a significant impact on statistical inference. The practical implication in the context of high-dimensional regression models is that additional observations, where only the covariates are given, are potentially very useful and should not be ignored. This finding is most relevant to semi-supervised learning problems where covariate information is easy to obtain.

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arXiv2018

Harmonizing Fully Optimal Designs with Classic Randomization in Fixed Trial Experiments

There is a movement in design of experiments away from the classic randomization put forward by Fisher, Cochran and others to one based on optimization. In fixed-sample trials comparing two groups, measurements of subjects are known in advance and subjects can be divided optimally into two groups based on a criterion of homogeneity or "imbalance" between the two groups. These designs are far from random. This paper seeks to understand the benefits and the costs over classic randomization in the context of different performance criterions such as Efron's worst-case analysis. In the criterion that we motivate, randomization beats optimization. However, the optimal design is shown to lie between these two extremes. Much-needed further work will provide a procedure to find this optimal designs in different scenarios in practice. Until then, it is best to randomize.

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arXiv2016

Nearly Random Designs with Greatly Improved Balance

We present a new experimental design procedure that divides a set of experimental units into two groups so that the two groups are balanced on a prespecified set of covariates and being almost as random as complete randomization. Under complete randomization, the difference in covariate balance as measured by the standardized average between treatment and control will be $O_p(n^{-1/2})$. If the sample size is not too large this may be material. In this article, we present an algorithm which greedily switches assignment pairs. Resultant designs produce balance of the much lower order $O_p(n^{-3})$ for one covariate. However, our algorithm creates assignments which are, strictly speaking, non-random. We introduce two metrics which capture departures from randomization: one in the style of entropy and one in the style of standard error and demonstrate our assignments are nearly as random as complete randomization in terms of both measures. The results are extended to more than one covariate, simulations are provided to illustrate the results and statistical inference under our design is discussed. We provide an open source R package available on CRAN called GreedyExperimentalDesign which generates designs according to our algorithm.

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arXiv2011

Are adaptive allocation designs beneficial for improving power in binary response trials?

We consider the classical problem of selecting the best of two treatments in clinical trials with binary response. The target is to find the design that maximizes the power of the relevant test. Many papers use a normal approximation to the power function and claim that Neyman allocation that assigns subjects to treatment groups according to the ratio of the responses' standard deviations, should be used. As the standard deviations are unknown, an adaptive design is often recommended. The asymptotic justification of this approach is arguable, since it uses the normal approximation in tails where the error in the approximation is larger than the estimated quantity. We consider two different approaches for optimality of designs that are related to Pitman and Bahadur definitions of relative efficiency of tests. We prove that the optimal allocation according to the Pitman criterion is the balanced allocation and that the optimal allocation according to the Bahadur approach depends on the unknown parameters. Exact calculations reveal that the optimal allocation according to Bahadur is often close to the balanced design, and the powers of both are comparable to the Neyman allocation for small sample sizes and are generally better for large experiments. Our findings have important implications to the design of experiments, as the balanced design is proved to be optimal or close to optimal and the need for the complications involved in following an adaptive design for the purpose of increasing the power of tests is therefore questionable.

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arXiv2010

Convergence of Nonparametric Long-Memory Phase I Designs

We examine nonparametric dose-finding designs that use toxicity estimates based on all available data at each dose allocation decision. We prove that one such design family, called here "interval design", converges almost surely to the maximum tolerated dose (MTD), if the MTD is the only dose level whose toxicity rate falls within the pre-specified interval around the desired target rate. Another nonparametric family, called "point design", has a positive probability of not converging. In a numerical sensitivity study, a diverse sample of dose-toxicity scenarios was randomly generated. On this sample, the "interval design" convergence conditions are met far more often than the conditions for one-parameter design convergence (the Shen-O'Quigley conditions), suggesting that the interval-design conditions are less restrictive. Implications of these theoretical and numerical results for small-sample behavior of the designs, and for future research, are discussed.

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