Generalized Sequential Probability Ratio Test for Separate Families of Hypotheses

Abstract

In this article, we consider the problem of testing two separate families of hypotheses via a generalization of the sequential probability ratio test. In particular, the generalized likelihood ratio statistic is considered and the stopping rule is the first boundary crossing of the generalized likelihood ratio statistic. We show that this sequential test is asymptotically optimal in the sense that it achieves asymptotically the shortest expected sample size as the maximal type I and type II error probabilities tend to zero.     

Design of Group Sequential Clinical Trials with Ordinal Categorical Data Based on the Mann–Whitney–Wilcoxon Test

Abstract

In this paper, methodology necessary do design group sequential trials based on ordinal categorical data based on the Mann–Whitney–Wilcoxon test is presented and illustrated in two practical examples. Curtailment for futility is an important component in improving the performance of the methods in terms of expected sample sizes under the null hypothesis. Simulation can also be employed to verify the asymptotic properties claimed by the method, and if necessary be used to tweak the design to better approximate the desired operating characteristics.     

EWMA Charts for Monitoring the Mean and the Autocovariances of Stationary Gaussian Processes

Abstract

In this article simultaneous individual control charts for the mean and the autocovariances of a stationary process are introduced. All control schemes are EWMA (exponentially weighted moving average) charts. A multivariate quality characteristic is considered. It describes the behavior of the mean and the autocovariances. This quantity is transformed to a one-dimensional variable by using the Mahalanobis distance. The control statistic is obtained by exponentially smoothing these variables. Another control procedure is based on a multivariate EWMA recursion applied directly to our multivariate quality characteristic. After that the resulting statistic is transformed to a univariate random variable. Besides modified control charts we consider residual charts. For the residual charts the same procedure is used but the original observations are replaced by the residuals. In an extensive simulation study all control schemes are compared with each other. The target process is assumed to be an ARMA(1, 1) process.     

Detection of Stationary Errors in Multiple Regressions with Integrated Regressors and Cointegration

Abstract

This article studies the problem of detecting sequentially stationary error terms in a multiple regression model with a difference-stationary multivariate I(1)-regressor. The detection of cointegration is covered as a special case. We provide the asymptotic distribution theory for a monitoring procedure that is related to a well-known nonparametric unit root test statistic calculated from sequentially updated least squares residuals. Functional limit theorems for the corresponding sequential processes and central limit theorems for the detectors used to raise an alarm are established under the no-change null hypothesis as well as under change-point models covering a change to I(0)-errors and a change of the regression coefficients as well. We also discuss extensions to the case that continuous time processes are discretely sampled to obtain the data allowing to apply the procedures to high-frequency data as well. Our results show that we can handle the infill asymptotics assuming that nonstationary continuous-time processes such as semimartingales are discretely observed, by virtue of the general assumptions that we impose. The finite sample properties are investigated by a simulation study.     

Unbalanced and Partial Group Sequential Methods for Normal Responses in Clinical Trials

Abstract

Group sequential methods for a two-treatment clinical trial with normal responses are discussed. First we consider the case where the sample sizes for two treatments are possibly unequal between the treatments due to an unequal randomization. Then we discuss group sequential design in the context of a historical control study, that is, under the partial sequential sampling scheme, in which the samples on one treatment, say control, are available at the outset, and the samples on the other treatment, say experimental, are obtained in the group sequential way. We discuss the cases of known and unknown variance for both unbalanced and partial group sequential setups. All of the procedures are discussed with numerical studies.     

Bayesian sequential joint detection and estimation

Abstract

Joint detection and estimation refers to deciding between two or more hypotheses and, depending on the test outcome, simultaneously estimating the unknown parameters of the underlying distribution. This problem is investigated in a sequential framework under mild assumptions on the underlying random process. We formulate an unconstrained sequential decision problem, whose cost function is the weighted sum of the expected run-length and the detection/estimation errors. Then, a strong connection between the derivatives of the cost function with respect to the weights, which can be interpreted as Lagrange multipliers, and the detection/estimation errors of the underlying scheme is shown. This property is used to characterize the solution of a closely related sequential decision problem, whose objective function is the expected run-length under constraints on the average detection/estimation errors. We show that the solution of the constrained problem coincides with the solution of the unconstrained problem with suitably chosen weights. These weights are characterized as the solution of a linear program, which can be solved using efficient off-the-shelf solvers. The theoretical results are illustrated with two example problems, for which optimal sequential schemes are designed numerically and whose performance is validated via Monte Carlo simulations.     

The steady-state behavior of multivariate exponentially weighted moving average control charts

Abstract

Multivariate exponentially weighted moving average (MEWMA) charts are popular, handy, and effective procedures to detect distributional changes in a stream of multivariate data. For doing appropriate performance analysis, dealing with the steady-state behavior of the MEWMA statistic is essential. Going beyond early papers, we derive quite accurate approximations of the respective steady-state densities of the MEWMA statistic. It turns out that these densities could be rewritten as the product of two functions depending on one argument only that allows feasible calculation. For proving the related statements, the presentation of the noncentral chi-square density deploying the confluent hypergeometric limit function is applied. Using the new methods it was found that for large dimensions, the steady-state behavior becomes different from what one might expect from the univariate monitoring field. Based on the integral equation driven methods, steady-state and worst-case average run lengths are calculated with higher accuracy than before. Eventually, optimal MEWMA smoothing constants are derived for all considered measures.     

An ARL-unbiased thinning-based EWMA chart to monitor counts

Abstract

Shewhart control charts are known to be somewhat insensitive to shifts of small and moderate size. Expectedly, alternative control schemes such as the exponentially weighted moving average (EWMA) charts have been proposed to speed up the detection of such shifts. Unfortunately, applying the ordinary EWMA recursion to count data leads to a control statistic no longer with a fixed discrete range. Therefore, we propose a novel chart which relies on a EWMA control statistic where the usual scalar product is replaced by a thinning operation. Actually, we use the new fractional binomial thinning to avoid the typical over-smoothing ascribable to ceiling, rounding, and flooring operations. The properties of this discrete statistic are similar to the ones of its continuous EWMA counterpart and the run length (RL) performance of the associated chart can be computed exactly using the Markov chain approach for independent and identically distributed (i.i.d.) counts. Moreover, this chart is set in such way that: the average run length (ARL) curve attains a maximum in the in-control situation, i.e., the chart is ARL-unbiased; and the in-control ARL is equal to a pre-specified value. We use the R statistical software to provide compelling illustrations of this unconventional EWMA chart and to compare its RL performance with the ones of a few competing control charts for the mean of i.i.d. Poisson counts.     

Discussion on “Second-Guessing Clinical Trial Designs” by Jonathan J. Shuster and Myron N. Chang

Abstract

Redesigning previously conducted clinical trials is a useful tool for better determining the impact of an alternative design. Shuster and Chang suggest the redesign of clinical trials that do not use sequential monitoring by imposing a sequential monitoring structure and determining what would have happened had such a structure been employed. However, early stopping decisions are complex and involve are intimate understanding of ethical, logistical, and statistical issues. “Second-guessing” a data and safety monitoring board is dangerous without this intimate understanding.     

A test of sphericity for high-dimensional data and its application for detection of divergently spiked noise

Abstract

In this article, we consider a test of the sphericity for high-dimensional covariance matrices. We produce a test statistic by using the extended cross-data-matrix (ECDM) methodology. We show that the ECDM test statistic is based on an unbiased estimator of a sphericity measure. In addition, the ECDM test statistic enjoys consistency properties and the asymptotic normality in high-dimensional settings. We propose a new test procedure based on the ECDM test statistic and evaluate its asymptotic size and power theoretically and numerically. We give a two-stage sampling scheme so that the test procedure can ensure a prespecified level both for the size and power. We apply the test procedure to detect divergently spiked noise in high-dimensional statistical analysis. We analyze gene expression data by the proposed test procedure.     

Approximately Optimal Continuous Stopping Boundaries in a One-Sided Standard Sequential Test

Abstract

In clinical trials, continuous monitoring is required when a large but nonsignificant test statistic is observed at an interim analysis. We consider a power family of continuous stopping boundaries that are optimal in the sense that they minimize the average stopping time for the given overall Type I error rate and power. The problem is formulated as a constrained optimization problem and is solved numerically using the differential evolution algorithm. The main results are the ready-to-use approximately optimal continuous stopping boundaries for a wide range of values of power and the most commonly used overall levels of significance.     

Semiparametric Surveillance of Monotonic Changes

Abstract

The detection of a change from a constant level to a monotonically increasing (or decreasing) regression is of special interest for the detection of outbreaks of epidemics but is also of interest in other areas. A maximum likelihood ratio statistic for the sequential surveillance of an “outbreak” situation is derived. The method is semiparametric in the sense that the regression model is nonparametric whereas the distribution belongs to the regular exponential family. The method is evaluated with respect to timeliness and predicted value in a simulation study that imitates the influenza outbreaks in Sweden. To illustrate its performance, the method is applied to Swedish influenza data for 6 years. The advantage of this semiparametric surveillance method, which does not rely on an estimated baseline, is illustrated by a Monte Carlo study. The advantage of information accumulation is illustrated.     

A hybrid Bayesian-frequentist predictive design for monitoring multi-stage clinical trials

Abstract

In this article, we propose a hybrid-Bayesian frequentist approach using a Bayesian sequential prediction of the index of satisfaction. For interim analysis that addresses prediction hypothesis, such as futility monitoring with delayed outcomes, the prediction of satisfaction properly accounts for the amount of data remaining to be observed in a clinical trial and has the flexibility to incorporate additional information via auxiliary variables. The prediction of satisfaction design guarantees the type I error rate and does not require intensive computation or comprehensive simulation. The design is retrospectively applied to a lung cancer clinical trial.     

On the distribution of a concomitant statistic in a sequential trial

In this paper the distribution of a concomitant statistic in a Gaussian sequential setting is characterized. It is shown that the distribution of such a statistic, conditional on the value of the stopping time, can be represented as a convolution of a normal random variable with a stochastic integral of the monitoring process of the trial. For continuous sequential designs the characteristic function of this stochastic integral is shown to satisfy an integral equation. The tools developed here are relevant for secondary inference problems in sequential clinical trials.     

Comparison of Stopping Rules in Sequential Estimation of the Number of Classes in a Population

Abstract

In sequential analysis, investigation of stopping rules is important, as they govern the sampling cost and derivation and accuracy of frequentist inference. We study stopping rules in sampling from a population comprised of an unknown number of classes where all classes are equally likely to occur in each selection. We adopt Blackwell's criterion for a “more informative experiment” to compare stopping rules in our context and derive certain complete class results, which provide some guidance for selecting a stopping rule. We show that it suffices to let the stopping probability, at any time, depend only on the number of selections and the number of discovered classes up to that time. A more informative stopping rule costs a higher expected sample size, and conversely, any given stopping rule can be improved with an increment in expected sample size. Admissibility within all stopping rules with a uniform upper bound on average sample size is also discussed. Any fixed-sample-size rule is shown to be admissible within an appropriate class. Finally, we show that for the minimal sufficient statistic to be complete, which is useful for unbiased estimation, the stopping rule must be nonrandomized.     

Sequential Classification on Lattices with Experiment-Specific Response Distributions

Abstract

A Bayesian framework for sequential classification on finite lattice models is described in which response distributions are allowed to vary according to experiment. Optimal rates of convergence in classification are established. Intuitive and computationally simple experiment selection rules are proposed, and it is shown that this class of rules attains optimal rates almost surely under general conditions. A simulation study demonstrates that sequential classification can be conducted efficiently on lattices, with potentially great savings in experiment adminstration while maintaining high classification accuracy. This framework can be applied to adaptive testing for cognitive assessment and to other sequential classification problems such as group testing when experimental response distributions depend on pool composition.     

Estimation of common change point and isolation of changed panels after sequential detection

Abstract

Quick detection of common changes is critical in sequential monitoring of multistream data where a common change is a change that only occurs in a portion of panels. After a common change is detected by using a combined cumulative sum Shiryaev-Roberts (CUSUM-SR) procedure, we first study the joint distribution for values of the CUSUM process and the estimated delay detection time for the unchanged panels. A Benjamini-Hochberg (BH) method using the asymptotic exponential property for the CUSUM process is developed to isolate the changed panels with control on the false discovery rate (FDR). The common change point is then estimated based on the isolated changed panels. Simulation results show that the proposed method can also control the false non-discovery rate (FNR) by properly selecting the FDR.     

On an Inequality That Implies the Lower Bound Formula for the Probability of Correct Selection in the Levin-Robbins-Leu Family of Sequential Binomial Subset Selection Procedures

Abstract

We study a key inequality that implies the lower bound formula for the probability of correct selection and other selection-related events of interest in the Levin-Robbins-Leu family of sequential binomial subset selection procedures. We present a strategy for the proof of the key inequality, and a mostly complete general proof is given. The strategy provides an entirely complete and rigorous proof of the inequality for as many as seven competing populations using computer-assisted symbolic manipulation.     

Weighted Logrank Statistics in Sequential Tests

Abstract

Fully sequential generalization of some group sequential tests currently in use in clinical trials are considered. The model is censored data with random staggered entry, and it is used for the comparison of two treatments. Results show that the sequential tests can outperform group sequential tests and provide more information about the nature of relationship of the two treatments. Furthermore, they allow complete freedom in scheduling data evaluation.