Sequential Stopping Research Articles

Multivariate output analysis for Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem (CLT). The multivariate nature of this Monte Carlo error largely has been ignored in the MCMC literature. We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule; which we demonstrate is an asymptotically valid procedure. The finite sample properties of the proposed method are demonstrated in a variety of examples.

Speculative Trading, Prospect Theory and Transaction Costs

A speculative agent with Prospect Theory preference chooses the optimal time to purchase and then to sell an indivisible risky asset as to maximize the expected utility of the round-trip profit net of transaction costs. The optimization problem is formulated as a sequential optimal stopping problem and we provide a complete characterization of the solution. Depending on the preference and market parameters as well as the initial price of the asset, the optimal strategy can be buy and hold, buy low sell high, buy high sell higher or no Transaction costs do not necessarily curb speculative trading. For example, while a large proportional transaction cost on sale can unambiguously suppress trading participation, introducing a fixed market entry fee will indeed encourage trading when the asset price level is high.

Nonparametric group sequential methods for evaluating survival benefit from multiple short-term follow-up windows.

This article takes a fresh look at group sequential methods applied to two-sample tests of censored survival data and proposes an alternative method of defining and evaluating treatment benefit. Our method re-purposes traditional censored event time data into a sequence of short-term outcomes taken from (potentially overlapping) follow-up windows. A new two-sample restricted means test based on this restructured follow-up data is proposed along with group sequential methods for its use in the clinical trial setting. This method compares favorably with existing methods for group sequential monitoring of time-to-event outcomes, including methods for monitoring the restricted means test and the logrank test. Our method performs particularly well in cases where there is a delayed treatment effect and/or a subset of cured patients. As part of developing group sequential methods for these analyses, we consider asymmetric error spending approaches that differentially limit the chances of stopping incorrectly for perceived efficacy versus perceived harm attributed to the investigational arm of the trial. Recommendations for how to choose proper group sequential stopping boundaries are given, with supporting simulations and an example from the AIDS Clinical Trial Group.

A combination of max‐type and distance based schemes for simultaneous monitoring of time between events and event magnitudes

AbstractTraditionally, two isolated sequential stopping rules are employed for monitoring the time of occurrence of an event (T) and the magnitude of an event (X). Recently, several researchers recommend monitoring T and X together using some unified approach. A unified approach based on combinations of two statistics, one for monitoring T and the other for X, is often more efficient. Likewise, a new approach of simultaneous monitoring of location and scale parameters of a process, combining a max and a distance based statistics, is recently introduced in literature. Motivated by such emerging concepts, we design a new scheme combining a Max‐type and a Distance‐type schemes, referred to as the MT scheme, to monitor T and X simultaneously and efficiently. It retains the advantages of both the Max‐type and the Distance‐type schemes for joint inference. The proposed scheme is very competent in detecting a shift in the process distribution of T or X or both. Moreover, it is computationally simpler. It has nice exact expressions for design parameters. Therefore, it is easier to implement. It has a distinct advantage over its traditional counterparts in detecting moderate to large shifts. Finally, we illustrate the implementation of the proposed scheme with a real dataset of damage caused by outbreak of fire disaster.

EDA on the asymptotic normality of the standardized sequential stopping times, Part-I: Parametric models

ABSTRACTIn sequential analysis, an experimenter may gather information regarding an unknown parameter by observing random samples in successive steps. We emphasize a number of specific parametric models under a variety of loss functions. The total number of observations collected at termination is a positive integer-valued random variable, customarily denoted by . The exact probability distribution of N is often hard to obtain. However, under a set of regulatory conditions, our standardized version of the stopping variable from Definition 2.1 in Section 2.2 would follow a normal distribution in the asymptotic sense. In this article, we first show how these regulatory conditions build upon one another in order to conclude the asymptotic normality of such standardized stopping variable, . We provide exploratory data analysis (EDA) via a number of interesting illustrations obtained through large-scale simulation studies. We demonstrate the broad applicability of purely sequential methodologies included in this article along with the appropriateness of our conclusions regarding the asymptotic normality of the standardized stopping variable as a practical guideline.

A Tutorial on the Practical Use and Implication of Complete Sufficient Statistics

SummaryCompleteness means that any measurable function of a sufficient statistic that has zero expectation for every value of the parameter indexing the parametric model class is the zero function almost everywhere. The property is satisfied in many simple situations in view of parameters of direct scientific interest, such as in regression models fitted to data from a random sample with fixed size. A random sample is not always of a fixed, a priori determined size. Examples include sequential sampling and stopping rules, missing data and clusters with random size. Often, there then is no complete sufficient statistic. A simple characterisation of incompleteness is given for the exponential family in terms of the mapping between the sufficient statistic and the parameter, based upon the implicit function theorem. Essentially, this is a comparison of the dimension of the sufficient statistic with the length of the parameter vector. This results in an easy verifiable criterion for incompleteness, clear and simple to use, even for complex settings as is shown for missing data and clusters of random size.This tutorial exemplifies the (in)completeness property of a sufficient statistic, thereby illustrating our proposed characterisation. The examples are organised from more classical, simple examples to gradually more advanced settings.

Controlling alpha for mixed effects models for repeated measures

ABSTRACTMixed Effects Models for Repeated Measures (MMRM) is often used in clinical trials with longitudinal data. However, there has not been an in-depth examination available on how investigators can implement interim analysis while also controlling the overall alpha for clinical trials under an MMRM analysis framework. Statistical independence among measurements, which is often assumed in group sequential testing (GST), is not valid under an MMRM framework due to the correlations induced by longitudinal within-subject measurements. Therefore, methods associated with GST derived under independence need to be adjusted accordingly. While these correlations can be estimated from the study data, regulatory agencies may not accept results based on these estimated correlations since there is no guarantee that the overall alpha is strongly controlled. In this article, we propose a new AC-Hybrid-approach for controlling the overall alpha. The AC-Hybrid-approach has two key attributes. First, we apply the MMRM analysis framework on all available data at every analysis timepoint. Second, we use complete-case information fractions to derive the group sequential stopping boundaries. We prove that the overall alpha is controlled regardless of the correlations among within-subject measurements. We also show the impact of this approach on the alpha and the power through examples.

Early stopping for statistical inverse problems via truncated SVD estimation

We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension $D$. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level $\widehat{m}\in \{1,\ldots ,D\}$ only based on the knowledge of the first $\widehat{m}$ singular values and vectors. We analyse in detail whether sequential early stopping rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity.

Variable criteria sequential stopping rule: Validity and power with repeated measures ANOVA, multiple correlation, MANOVA and relation to Chi-square distribution.

The variable criteria sequential stopping rule (vcSSR) is an efficient way to add sample size to planned ANOVA tests while holding the observed rate of Type I errors, αo, constant. The only difference from regular null hypothesis testing is that criteria for stopping the experiment are obtained from a table based on the desired power, rate of Type I errors, and beginning sample size. The vcSSR was developed using between-subjects ANOVAs, but it should work with p values from any type of F test. In the present study, the αo remained constant at the nominal level when using the previously published table of criteria with repeated measures designs with various numbers of treatments per subject, Type I error rates, values of ρ, and four different sample size models. New power curves allow researchers to select the optimal sample size model for a repeated measures experiment. The criteria held αo constant either when used with a multiple correlation that varied the sample size model and the number of predictor variables, or when used with MANOVA with multiple groups and two levels of a within-subject variable at various levels of ρ. Although not recommended for use with χ2 tests such as the Friedman rank ANOVA test, the vcSSR produces predictable results based on the relation between F and χ2. Together, the data confirm the view that the vcSSR can be used to control Type I errors during sequential sampling with any t- or F-statistic rather than being restricted to certain ANOVA designs.

Timing options for a startup with early termination and competition risks

This paper analyzes the timing options embedded in a startup firm, and the associated market entry and exit timing decisions under the exogenous risks of early termination and competitor's entry. Our valuation approach leads to the analytical study of a non-standard perpetual American installment option nested with an optimal sequential stopping problem. Explicit formulas are derived for the firm's value functions. Analytically and numerically, we show that early termination risk leads to earlier voluntary entry or exit, and the threat of competition has a non-trivial effect on the firm's entry and abandonment strategies.

The application of group sequential stopping boundaries to evaluate the treatment effect of an experimental agent across a range of biomarker expression

Using stopping boundaries developed for group sequential trials, we control the overall type 1 error for a series of tests that evaluate the treatment effect of an experimental agent in each of several predefined marker subgroups. We then go on to present a procedure based on these group sequential stopping boundaries that provides strong control of type 1 error for testing a series of hypothesis regarding the treatment effect over a range of marker expression. Finally this use of group sequential procedures to control the type 1 error in biomarker subset testing will be compared with some other benchmark procedures in a simulation.

Coupled sequential detection and estimation procedure for track‐before‐detect

In this study, the authors consider the coupled sequential detection and estimation of weak targets for track-before-detect algorithm in long range early warning radar. They present a new strategy and develop a coupled cost function consisting of detection, position estimation and tracking estimation loss. In the proposed procedure, the sequential stopping and detection rules are simplified. Moreover, they derive the lower bounds of the detection and estimation probabilities, which can be used to design the suitable related parameter values. Finally, several numerical simulations of the new strategy and the comparisons are presented and discussed.

Motion Planning for Continuous-Time Stochastic Processes: A Dynamic Programming Approach

We study stochastic motion planning problems which involve a controlled process, with possibly discontinuous sample paths, visiting certain subsets of the state-space while avoiding others in a sequential fashion. For this purpose, we first introduce two basic notions of motion planning, and then establish a connection to a class of stochastic optimal control problems concerned with sequential stopping times. A weak dynamic programming principle (DPP) is then proposed, which characterizes the set of initial states that admit a control enabling the process to execute the desired maneuver with probability no less than some pre-specified value. The proposed DPP comprises auxiliary value functions defined in terms of discontinuous payoff functions. A concrete instance of the use of this novel DPP in the case of diffusion processes is also presented. In this case, we establish that the aforementioned set of initial states can be characterized as the level set of a discontinuous viscosity solution to a sequence of partial differential equations, for which the first one has a known boundary condition, while the boundary conditions of the subsequent ones are determined by the solutions to the preceding steps. Finally, the generality and flexibility of the theoretical results are illustrated on an example involving biological switches.

A Practical Sequential Stopping Rule for High-Dimensional Markov Chain Monte Carlo

A current challenge for many Bayesian analyses is determining when to terminate high-dimensional Markov chain Monte Carlo simulations. To this end, we propose using an automated sequential stopping procedure that terminates the simulation when the computational uncertainty is small relative to the posterior uncertainty. Further, we show this stopping rule is equivalent to stopping when the effective sample size is sufficiently large. Such a stopping rule has previously been shown to work well in settings with posteriors of moderate dimension. In this article, we illustrate its utility in high-dimensional simulations while overcoming some current computational issues. As examples, we consider two complex Bayesian analyses on spatially and temporally correlated datasets. The first involves a dynamic space-time model on weather station data and the second a spatial variable selection model on fMRI brain imaging data. Our results show the sequential stopping rule is easy to implement, provides uncertainty estimates, and performs well in high-dimensional settings. Supplementary materials for this article are available online.

Timing Options for a Startup with Early Termination and Competition Risks

This paper analyzes the timing options embedded in a startup firm, and the associated market entry and exit timing decisions under the exogenous risks of early termination and competitor's entry. Our valuation approach leads to the analytical study of a non-standard perpetual American installment option nested with an optimal sequential stopping problem. Explicit formulas are derived for the firm's value functions. Analytically and numerically, we show that early termination risk leads to earlier voluntary entry or exit, and the threat of competition has a non-trivial effect on the firm's entry and abandonment strategies.

An Incentive-Compatible Routing Protocol for Two-Hop Delay-Tolerant Networks

Delay-tolerant networks (DTNs) rely on the mobility of nodes and their contacts to make up with the lack of continuous connectivity and, thus, enable message delivery from source to destination in a “store-carry-forward” fashion. Since message delivery consumes resource such as storage and power, some nodes may choose not to forward or carry others' messages while relying on others to deliver their locally generated messages. These kinds of selfish behaviors may hinder effective communications over DTNs. In this paper, the authors present an efficient incentive-compatible (IC) routing protocol (ICRP) with multiple copies for two-hop DTNs based on the algorithmic game theory. It takes both the encounter probability and transmission cost into consideration to deal with the misbehaviors of selfish nodes. Moreover, the authors employ the optimal sequential stopping rule and Vickrey–Clarke–Groves (VCG) auction as a strategy to select optimal relay nodes to ensure that nodes that honestly report their encounter probability and transmission cost can maximize their rewards. The authors attempt to find the optimal stopping time threshold adaptively based on realistic probability model and propose an algorithm to calculate the threshold. Based on this threshold, the authors propose a new method to select relay nodes for multicopy transmissions. To ensure that the selected relay nodes can receive their rewards securely, the authors develop a signature scheme based on a bilinear map to prevent the malicious nodes from tampering. Through simulations, the authors demonstrate that ICRP can effectively stimulate nodes to forward/carry messages and achieve higher packet delivery ratio with lower transmission cost.

Relative fixed-width stopping rules for Markov chain Monte Carlo simulations

Markov chain Monte Carlo (MCMC) simulations are commonly employed for estimating features of a target distribution, particularly for Bayesian inference. A fundamental challenge is determining when these simulations should stop. We consider a sequential stopping rule that terminates the simulation when the width of a confidence interval is sufficiently small relative to the size of the target parameter. Specifically, we propose relative magnitude and relative standard deviation stopping rules in the context of MCMC. In each setting, we develop sufficient conditions for asymptotic validity, that is conditions to ensure the simulation will terminate with probability one and the resulting confidence intervals will have the proper coverage probability. Our results are applicable in a wide variety of MCMC estimation settings, such as expectation, quantile, or simultaneous multivariate estimation. Finally, we investigate the finite sample properties through a variety of examples and provide some recommendations to practitioners.

An Introduction to Practical Sequential Inferences via Single-Arm Binary Response Studies Using the binseqtest R Package

We review sequential designs, including group sequential and two-stage designs, for testing or estimating a single binary parameter. We use this simple case to introduce ideas common to many sequential designs, which in this case can be explained without explicitly using stochastic processes. We focus on methods provided by our newly developed R package, binseqtest, which exactly bound the Type I error rate of tests and exactly maintain proper coverage of confidence intervals. Within this framework, we review some allowable practical adaptations of the sequential design. We explore issues such as the following: How should the design be modified if no assessment was made at one of the planned sequential stopping times? How should the parameter be estimated if the study needs to be stopped early? What reasons for stopping early are allowed? How should inferences be made when the study is stopped for crossing the boundary, but later information is collected about responses of subjects that had enrolled before the decision to stop but had not responded by that time? Answers to these questions are demonstrated using basic methods that are available in our binseqtest R package. Supplementary materials for this article are available online.

A Bayesian Stopping Rule for Sequential Monitoring of Serious Adverse Events.

In an ongoing clinical trial, there will always be a risk for unanticipated critical safety problems, such as excessive occurrence of serious adverse events. When such a problem arises, the trial administrators must conduct an immediate evaluation to determine whether the trial should be terminated to protect patients. This decision is complicated but may be aided by statistical stopping rules. Sequential stopping rules are appropriate for immediate decisions, but frequentist approaches may not be useful because the unknown truncated end of the trial makes it impossible to define type I errors. Thus, a Bayesian stopping rule is proposed that is based on the posterior distribution with an informative prior distribution, and a guideline to construct this stopping rule is presented. Some operating characteristics are evaluated and compared with those of the modified sequential probability ratio test (SPRT), the maximized SPRT, and Pocock's test. The proposed method has flexibility for construction and could provide a more desirable performance than the other compared methods.

Flexible Stopping Boundaries When Changing Primary Endpoints After Unblinded Interim Analyses

It has been widely recognized that interim analyses of accumulating data in a clinical trial can inflate type I error. Different methods, from group sequential boundaries to flexible alpha spending functions, have been developed to control the overall type I error at prespecified level. These methods mainly apply to testing the same endpoint in multiple interim analyses. In this article, we consider a group sequential design with preplanned endpoint switching after unblinded interim analyses. We extend the alpha spending function method to group sequential stopping boundaries when the parameters can be different between interim, or between interim and final analyses.