Randomization tests for small samples: an application for genetic expression data

Summary. An advantage of randomization tests for small samples is that an exact P -value can be computed under an additive model. A disadvantage with very small sample sizes is that the resulting discrete distribution for P -values can make it mathematically impossible for a P -value to attain a particular degree of significance. We investigate a distribution of P -values that arises when several thousand randomization tests are conducted simultaneously using small samples, a situation that arises with microarray gene expression data. We show that the distribution yields valuable information regarding groups of genes that are differentially expressed between two groups: a treatment group and a control group. This distribution helps to categorize genes with varying degrees of overlap of genetic expression values between the two groups, and it helps to quantify the degree of overlap by using the P -value from a randomization test. Moreover, a statistical test is available that compares the actual distribution of P -values with an expected distribution if there are no genes that are differentially expressed. We demonstrate the method and illustrate the results by using a microarray data set involving a cell line for rheumatoid arthritis. A small simulation study evaluates the effect that correlated gene expression levels could have on results from the analysis.     

Estimating standard error of intra-class correlation coefficients up to three level unbalanced nested clinical trials

ABSTRACT

Very often researchers plan a balanced design for cluster randomization clinical trials in conducting medical research, but unavoidable circumstances lead to unbalanced data. By adopting three or more levels of nested designs, they usually ignore the higher level of nesting and consider only two levels, this situation leads to underestimation of variance at higher levels. While calculating the sample size for three-level nested designs, in order to achieve desired power, intra-class correlation coefficients (ICCs) at individual level as well as higher levels need to be considered and must be provided along with respective standard errors. In the present paper, the standard errors of analysis of variance (ANOVA) estimates of ICCs for three-level unbalanced nested design are derived. To conquer the strong appeal of distributional assumptions, balanced design, equality of variances between clusters and large sample, general expressions for standard errors of ICCs which can be deployed in unbalanced cluster randomization trials are postulated. The expressions are evaluated on real data as well as highly unbalanced simulated data.     

Optimal Bayesian Randomization

Randomization is a puzzle for Bayesians. The intuitive need for randomization is clear, but there is a standard result that Bayesians need not randomize. In this paper we propose a model in which randomization is a strictly optimal procedure. The most important aspect of our model is that there are several parties who make different decisions and observe different data. The result also sheds light on the ethical considerations involving randomization in a clinical trial.     

Randomization tests in recursive response-adaptive randomization procedures

In a randomized clinical trial, response-adaptive randomization procedures use the information gathered, including the previous patients' responses, to allocate the next patient. In this setting, we consider randomization-based inference. We provide an algorithm to obtain exact p-values for statistical tests that compare two treatments with dichotomous responses. This algorithm can be applied to a family of response adaptive randomization procedures which share the following property: the distribution of the allocation rule depends only on the imbalance between treatments and on the imbalance between successes for treatments 1 and 2 in the previous step. This family includes some outstanding response adaptive randomization procedures. We study a randomization test to contrast the null hypothesis of equivalence of treatments and we show that this test has a similar performance to that of its parametric counterpart. Besides, we study the effect of a covariate in the inferential process. First, we obtain a parametric test, constructed assuming a logit model which relates responses to treatments and covariate levels, and we give conditions that guarantee its asymptotic normality. Finally, we show that the randomization test, which is free of model specification, performs as well as the parametric test that takes the covariate into account.     

The Performance of Randomization Tests that Use Permutations of Independent Variables

ABSTRACT

In this article we evaluate the performance of a randomization test for a subset of regression coefficients in a linear model. This randomization test is based on random permutations of the independent variables. It is shown that the method maintains its level of significance, except for extreme situations, and has power that approximates the power of another randomization test, which is based on the permutation of residuals from the reduced model. We also show, via an example, that the method of permuting independent variables is more valuable than other randomization methods because it can be used in connection with the downweighting of outliers.     

Adjusting for bias in randomized cluster trials

The randomized cluster design is typical in studies where the unit of randomization is a cluster of individuals rather than the individual. Evaluating various intervention strategies across medical care providers at either an institutional level or at a physician group practice level fits the randomized cluster model. Clearly, the analytical approach to such studies must take the unit of randomization and accompanying intraclass correlation into consideration. We review alternative methods to the typical Pearson's chi-square analysis and illustrate these alternatives. We have written and tested a Fortran program that produces the statistics outlined in this paper. The program, in an executable format is available from the author on request.     

Randomization Inference for Outcomes with Clumping at Zero

While randomization inference is well developed for continuous and binary outcomes, there has been comparatively little work for outcomes with nonnegative support and clumping at zero. Typically, outcomes of this type have been modeled using parametric models that impose strong distributional assumptions. This article proposes new randomization inference procedures for nonnegative outcomes with clumping at zero. Instead of making distributional assumptions, we propose various assumptions about the nature of the response to treatment and use permutation inference for both testing and estimation. This approach allows for some natural goodness-of-fit tests for model assessment, as well as flexibility in selecting test statistics sensitive to different potential alternatives. We illustrate our approach using two randomized trials, where job training interventions were designed to increase earnings of participants.     

Randomization with non-uniform allocations: fulfillment time distributions and bias properties

This article compares the properties of two balanced randomization schemes with several treatments under non-uniform allocation probabilities. According to the first procedure, the so-called truncated multinomial randomization design, the process employs a given allocation distribution, until a treatment receives its quota of subjects, after which this distribution switches to the conditional distribution for the remaining treatments, and so on. The second scheme, the random allocation rule, selects at random any legitimate assignment of the given number of subjects per treatment. The behavior of these two schemes is shown to be quite different: the truncated multinomial randomization design's assignment probabilities to a treatment turn out to vary over the recruitment period, and its accidental bias can be large, whereas the random allocation rule's this bias is bounded. The limiting distributions of the instants at which a treatment receives the given number of subjects is shown to be that of weighted spacings for normal order statistics with different variances. Formulas for the selection bias of both procedures are also derived.     

Randomization of neighbour balanced generalized Youden designs

If a crossover design with more than two treatments is carryover balanced, then the usual randomization of experimental units and periods would destroy the neighbour structure of the design. As an alternative, Bailey [1985. Restricted randomization for neighbour-balanced designs. Statist. Decisions Suppl. 2, 237–248] considered randomization of experimental units and of treatment labels, which leaves the neighbour structure intact. She has shown that, if there are no carryover effects, this randomization validates the row–column model, provided the starting design is a generalized Latin square. We extend this result to generalized Youden designs where either the number of experimental units is a multiple of the number of treatments or the number of periods is equal to the number of treatments. For the situation when there are carryover effects we show for so-called totally balanced designs that the variance of the estimates of treatment differences does not change in the presence of carryover effects, while the estimated variance of this estimate becomes conservative.     

An analysis of variance type test for comparing clusters of DNA sequences based on randomization test methodologies

SUMMARY A method for comparing groupings of DNA sequences is presented, which utilizes randomization test methods to assign significance levels to a test statistic defined in terms of the Hamming distance between two sequences. The method, which is intuitively motivated by the analysis of variance procedure, partitions the variation caused by differences between clusters from the variation attributable to differences at random base pair locations within clusers. Implementation issues are discussed, and an example of the application of the method is provided.     

Randomization and blocking in the design of experiments

It is suggested that, whenever possible, an experiment be run in a completely randomized fashion. One reason for randomizing Is to protect against violations in the usual linear model assump¬tions. The protection has always been argued on qualitative grounds. This paper quantitatively demonstrates the protection by hypothesizing models in violation of the usual assumptions, mathe¬matically representing the physical act of randomization, and algebraically deriving expected mean squares, EMS, and F tests. It is shown that randomization offers considerable but not com¬plete protection against model violations.

The same methodology is also applied to blocked experiments, i.e. to experiments performed under a specific type of incomplete randomization commonly referred to as blocking. It is shown that blocking offers little protection against certain model viola¬tions. The common practice of representing blocks as a treatment factor applied to the experimental units approximates the form of the EMS derived under the violated assumptions model.     

基于白化权函数分类区分度的变权灰色聚类

针对灰色聚类指标权重确定的问题,通过定义白化权函数的分类区分度来度量各指标对聚类对象的分类所作的贡献,并据此确定分类指标的权重。在此基础上,提出了变权灰色聚类方法。结果表明,该方法能够融合聚类对象的样本信息和专家的经验,有效确定不同聚类对象的各指标权重,且适用于聚类指标的量纲不同、数量级悬殊较大的情形。最后通过一个实例说明了变权灰色聚类的实用性和有效性。     

Approximating Matrix-Exponential Distributions by Global Randomization

Abstract

Based on the general concept of randomization, we develop linear-algebraic approximations for continuous probability distributions that involve the exponential of a matrix in their definitions, such as phase types and matrix-exponential distributions. The approximations themselves result in proper probability distributions. For such a global randomization with the Erlang-k distribution, we show that the sequences of true and consistent distribution and density functions converge uniformly on [0, ∞). Furthermore, we study the approximation errors in terms of the power moments and the coefficients of the Taylor series, from which the accuracy of the approximations can be determined apriori. Numerical experiments demonstrate the feasibility of the presented randomization technique – also in comparison with uniformization.     

Sample size determination for cluster randomization phase II trials of dichotomous data

One of the main goals for a phase II trial is to screen and select the best treatment to proceed onto further studies in a phase III trial. Under the flexible design proposed elsewhere, we discuss for cluster randomization trials sample size calculation with a given desired probability of correct selection to choose the best treatment when one treatment is better than all the others. We develop exact procedures for calculating the minimum required number of clusters with a given cluster size (or the minimum number of patients with a given number of repeated measurements) per treatment. An approximate sample size and the evaluation of its performance for two arms are also given. To help readers employ the results presented here, tables are provided to summarize the resulting minimum required sample sizes for cluster randomization trials with two arms and three arms in a variety of situations. Finally, to illustrate the sample size calculation procedures developed here, we use the data taken from a cluster randomization trial to study the association between the dietary sodium and the blood pressure.     

Transient Analysis of Large Markov Models with Absorbing States Using Regenerative Randomization

ABSTRACT

In this article, we develop a new method, called regenerative randomization, for the transient analysis of continuous time Markov models with absorbing states. The method has the same good properties as standard randomization: numerical stability, well-controlled computation error, and ability to specify the computation error in advance. The method has a benign behavior for large t and is significantly less costly than standard randomization for large enough models and large enough t. For a class of models, class C, including typical failure/repair reliability models with exponential failure and repair time distributions and repair in every state with failed components, stronger theoretical results are available assessing the efficiency of the method in terms of “visible” model characteristics. A large example belonging to that class is used to illustrate the performance of the method and to show that it can indeed be much faster than standard randomization.     

Statistical methods for continuous outcomes in partially clustered designs

We address statistical issues involved in the partially clustered design where clusters are only employed in the intervention arm, but not in the control arm. We develop a cluster adjusted t-test to compare group treatment effects with individual treatment effects for continuous outcomes in which the individual level data are used as the unit of the analysis in both arms, we develop an approach for determining sample sizes using this cluster adjusted t-test, and use simulation to demonstrate the consistent accuracy of the proposed cluster adjusted t-test and power estimation procedures. Two real examples illustrate how to use the proposed methods.     

Valid Randomization for Models with Neighbour Effects From Treatments

Valid methods of randomization have been proposed for several classes of neighbour-balanced designs, but the assumed models did not include the neighbour effects from treatments. We present sufficient conditions for such randomizations to be also valid for direct and neighbour effects simultaneously. It is shown through several examples that these sufficient conditions can be satisfied for uni- or bi-directional neighbour effects, provided a particular block structure is used. The covariance between estimators of direct and neighbour effects over the randomization is also studied.     

Subject‐level matching for imbalance in cluster randomized trials with a small number of clusters

In a cluster randomized controlled trial (RCT), the number of randomized units is typically considerably smaller than in trials where the unit of randomization is the patient. If the number of randomized clusters is small, there is a reasonable chance of baseline imbalance between the experimental and control groups. This imbalance threatens the validity of inferences regarding post‐treatment intervention effects unless an appropriate statistical adjustment is used. Here, we consider application of the propensity score adjustment for cluster RCTs. For the purpose of illustration, we apply the propensity adjustment to a cluster RCT that evaluated an intervention to reduce suicidal ideation and depression. This approach to adjusting imbalance had considerable bearing on the interpretation of results. A simulation study demonstrates that the propensity adjustment reduced well over 90% of the bias seen in unadjusted models for the specifications examined. Copyright © 2013 John Wiley & Sons, Ltd.     

The Role of Randomization in Model-Based Inference

A common justification for randomization in sampling and experimental design is that it creates certain desirable invariance properties, e.g., exchangeability, in the distribution of the sample responses, and this is so regardless of the distribution of the population responses. This paper argues that for model-based inference randomization's role is much more limited, viz., to preserve invariance properties already embodied in the model for the population responses.     

Randomization tests for multiple regression

Four generic means of conducting randomization tests in the context of multiple regression are analysed. Based on their performance in traditional repeated samples, three of these are shown to be inappropriate or applicable only in special circumstances; their shortcomings are illustrated via Monte Carlo studies