Irreversible random sequential filling of lattices by monte carlo simulation

By means of Monte Carlo simulations we study the irreversible, random, sequential filling of small clusters (e.g., pairs, triples,...) on linear, square, and cubic lattices. In particular, we are interested in the fraction of sites filled at saturation (the point at which further filling is not possible without rearrangement of the filled and empty sites). The results obtained show good agreement with those of previously developed analytic techniques.

We present the first extensive results for filling linear strings of lattice sites by use of the end-on mechanism (where the ends of the string are chosen sequentially rather than simultaneously as in conventional filling). For end-on filling we find that the saturation coverage increases, relative to conventional filling, for short strings, but decreases as we go to the limit of infinitely long strings (the car-parking problem).

An examination of the Palasti conjecture (and its extension to discrete lattices) is also made.     

Approximations of distributions for some standardized partial sums in sequential analysis

In sequential analysis it is often necessary to determine the distributions of √t Y _t and/or √a Y _t , where t is a stopping time of the form t = inf{ n ≥ 1 : n+S_n+ξ_n> a }, Y _n is the sample mean of n independent and identically distributed random variables (iidrvs) Y_i with mean zero and variance one, S_n is the partial sum of iidrvs X_i with mean zero and a positive finite variance, and { ξ_n } is a sequence of random variables that converges in distribution to a random variable ξ as n →∞ and ξ_n is independent of ( X_n+1, Y_n+1), (X_n+2, Y_n+2), . . . for all n ≥ 1. Anscombe's (1952) central limit theorem asserts that both √t Y _t and √a Y _t are asymptotically normal for large a , but a normal approximation is not accurate enough for many applications. Refined approximations are available only for a few special cases of the general setting above and are often very complex. This paper provides some simple Edgeworth approximations that are numerically satisfactory for the problems it considers.     

A note regarding meta‐analysis of sequential trials with stopping for efficacy

It is shown that fixed‐effect meta‐analyses of naïve treatment estimates from sequentially run trials with the possibility of stopping for efficacy based on a single interim look are unbiassed (or at the very least consistent, depending on the point of view) provided that the trials are weighted by information provided. A simple proof of this is given. An argument is given suggesting that this also applies in the case of multiple looks. The implications for this are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Depth and outliers for samples of sets and random sets distributions

We suggest several constructions suitable to define the depth of set-valued observations with respect to a sample of convex sets or with respect to the distribution of a random closed convex set. With the concept of a depth, it is possible to determine if a given convex set should be regarded an outlier with respect to a sample of convex closed sets. Some of our constructions are motivated by the known concepts of half-space depth and band depth for function-valued data. A novel construction derives the depth from a family of non-linear expectations of random sets. Furthermore, we address the role of positions of sets for evaluation of their depth. Two case studies concern interval regression for Greek wine data and detection of outliers in a sample of particles.     

Informing the selection of futility stopping thresholds: case study from a late-phase clinical trial

In an environment where (i) potential risks to subjects participating in clinical studies need to be managed carefully, (ii) trial costs are increasing, and (iii) there are limited research resources available, it is necessary to prioritize research projects and sometimes re-prioritize if early indications suggest that a trial has low probability of success. Futility designs allow this re-prioritization to take place. This paper reviews a number of possible futility methods available and presents a case study from a late-phase study of an HIV therapeutic, which utilized conditional power-based stopping thresholds. The two most challenging aspects of incorporating a futility interim analysis into a trial design are the selection of optimal stopping thresholds and the timing of the analysis, both of which require the balancing of various risks. The paper outlines a number of graphical aids that proved useful in explaining the statistical risks involved to the study team. Further, the paper outlines a decision analysis undertaken which combined expectations of drug performance with conditional power calculations in order to produce probabilities of different interim and final outcomes, and which ultimately led to the selection of the final stopping thresholds.     

Maximizing survival time in a random walk on an interval

A gambler buys N tokens that enable him to play N rounds of the following game. A symmetric random walk on a discrete interval { ? r, …, r} starts from the point 0. The gambler knows only the number of steps made so far, but is unaware of the current position of the walk. Once the walk hits one of the barriers ? r or r for the first time in the current round, the round ends with no payoff. The gambler can start a new round by inserting a new token, if there are any tokens left. The gambler can end the game at any time getting the payoff equal to the number of steps made in the current round. We find the optimal stopping strategy for this game and calculate the expected payoff once the optimal strategy is applied.     

Point estimation after early stopping in a repeated measures trial

When the individual measurements are statistically independent, the maximum likelihood estimator calculated at the end of a sequential procedure overestimates the underlying effect. There are many clinical trials in which we are interested in comparing changes in responses between two treatment groups sequentially. Lee and DeMets (1991, JASA 86, 757–762) proposed a group sequential method for comparing rates of change when a response variable is measured for eaeh patient at successive follow-up visits. They assumed that the response follows the linear mixed effects model and derived the asymptotic joint distribution of the sequentially computed statistics. In this article, we consider the maximum likelihood estimator (MLE), the median unbiased estimator (MUE) and the midpoint of a 100(1-α)% confidence interval as point estimators for the rate of change in the linear mixed effects model, and investigate their properties by Monte Carlo simulation.     

Utilizing auxiliary variables in sequential ratio and regression estimation schemes

Ratio and regression estimators for a mean are considered in conjunction with certain sequential sampling schemes. An auxiliary variable is assumed present and both fixed-cost and fixed- width confidence interval stopping rules are investigated. The asymptotic distributions of the estimators are derived as well as optimal probabilities pertinent to the schemes. Comparisons are made with results of certain double sampling procedures. Estimation of the ratio of two means is also considered and the results of a Monte Carlo simulation are included.     

Optimal stopping for Brownian motion with applications to sequential analysis and option pricing

Herman Chernoff made fundamental contributions to analytical and computational methods for solving optimal stopping problems for Brownian motion. He also showed how these optimal stopping problems are closely related to some basic problems in sequential analysis and singular stochastic control. This paper gives a survey of these and related developments and describes some recent applications to option valuation in financial economics.     

Bayesian prediction for two-stage sequential analysis in clinical trials

This article deals with a Bayesian predictive approach for two-stage sequential analyses in clinical trials, applied to both frequentist and Bayesian tests. We propose to make a predictive inference based on the notion of satisfaction index and the data accrued so far together with future data. The computations and the simulation results concern an inferential problem, related to the binomial model.     

Fixed-width sequential confidence interval for the correlation coefficient of a bivariate normal distribution

A sequential confidence interval of fixed width 2d d > 0, is constructed for the correlation coefficient of a bivariate normal distribution. It is shown that the coverage probability is approximately equal to a preassigned number γ, 0 < γ < as d → 0.     

A two-stage sequential design and analysis for comparing binomial rates with possibly random dropouts

Many clinical trials involve two-stage sequential designs with one interim analysis (Elashoff and Reedy, 1984, Biometrics 41, 791-795.) In this paper we present a situation where events are counted only at two fixed calendar time points and some patients may dropout during the time intervals. In the two-stage case, naive application of Tsiatis’s (1984, JASA 77, 855-861) logrank and Wilcoxon tests, which are for continuous survival time, is shown to lead to conservative type-I error rates and lower power. The two-stage sequential boundaries can also be calculated directly, rather than by simulation as was done by DeMets and Gail (1985, Biometrics 41, 1039-1044) with the assumption of some survival models, and are shown to be more flexible than the Pocock (1977, Biometrika 64, 191-199) and O’Brien-Fleming (1983, Biometrics 35, 549-556) boundaries since the former do not require an assumption on the correlation of the test statistics for the two stages. Repeated confidence intervals are also discussed. The design and approach are motivated by clinical trials studying treatment effects on vertebral fracture rates in elderly osteoporotic women. An example (Tilyard, et al. New England Journal of Medicine, 1992) is given to illustrate the method.     

Incorporating scientific,ethical and economic considerations into the design of clinical trials in the pharmaceutical industry: a sequential approach

A flexible sequential approach to the design of clinical trials is discussed herein. This approach is based on a “confidence sequence” viewpoint instead of the rigid stopping and terminal decision rules in conventional sequential testing theory. By using an appropriate confidence sequence, one can always ensure a prescribed degree of scientific rigor (confidence) in establishing the drug to be effective. Moreover, one also has the option of terminating the trial early when there is already enough statistical evidence for concluding that the drug is effective, or when the drug shows uniorseen harmful effects, or when the data predict that there is little chance of arriving at a definitive conclusion in favor of the drug by the scheduled end of the trial. We discuss how these and other ethical and economic considerations can be readily incorporated into the stopping criteria of the trial.     

An approximation to cluster size distribution of two Gaussian random fields conjunction with application to FMRI data

In brain mapping, the regions of the brain that are ‘activated’ by a task or external stimulus are detected by thresholding an image of test statistics. Often the experiment is repeated on several different subjects or for several different stimuli on the same subject, and the researcher is interested in the common points in the brain where ‘activation’ occurs in all test statistic images. The conjunction is thus defined as those points in the brain that show ‘activation’ in all images. We are interested in which parts of the conjunction are noise, and which show true activation in all test statistic images. We would expect truly activated regions to be larger than usual, so our test statistic is based on the volume of clusters (connected components) of the conjunction. Our main result is an approximate P-value for this in the case of the conjunction of two Gaussian test statistic images. The results are applied to a functional magnetic resonance experiment in pain perception.     

On the extremal behavior of a nonstationary normal random field

Let X={_Xn}n?1$\mathbf{X}=\{X_{\mathbf{n}}{\}}_{\mathbf{n}?\mathbf{1}}$ be a nonstationary random field satisfying a long range weak dependence for each coordinate at a time and a local dependence condition that avoids clustering of exceedances of high values. For these random fields, the probability of no exceedances of high values can be approximated by exp(−τ)$\exp (-\tau )$, where τ$\tau$ is the limiting mean number of exceedances. We present a class of nonstationary normal random fields for which this result can be applied.     

A Bayesian approach to sequential analysis in post‐licensure vaccine safety surveillance

With rapid development of computing technology, Bayesian statistics have increasingly gained more attention in various areas of public health. However, the full potential of Bayesian sequential methods applied to vaccine safety surveillance has not yet been realized, despite acknowledged practical benefits and philosophical advantages of Bayesian statistics. In this paper, we describe how sequential analysis can be performed in a Bayesian paradigm in the field of vaccine safety. We compared the performance of the frequentist sequential method, specifically, Maximized Sequential Probability Ratio Test (MaxSPRT), and a Bayesian sequential method using simulations and a real world vaccine safety example. The performance is evaluated using three metrics: false positive rate, false negative rate, and average earliest time to signal. Depending on the background rate of adverse events, the Bayesian sequential method could significantly improve the false negative rate and decrease the earliest time to signal. We consider the proposed Bayesian sequential approach to be a promising alternative for vaccine safety surveillance.     

Convergence rates of sequential confidence intervals and tests for the mean of a u-statistic

Csenki(1980a) utilized the rate of convergence results of Landers and Rogge(1976b) for suitably normalized random means to obtain the rate of convergence of the coverage probability for Chow-Robbins'(1965) fixed-width confidence interval procedure. In this paper, we utilize the rate of convergence results of Ghosh and DasGupta(1980) for suitably normalized random U-statistics to derive the rate of convergence of the coverage probability for estimating the mean of a U-statistic through Sproule's(1969, 1974) procedure. We show that Csenki's(1980a) convergence rate can be achieved with a substantial economy on moment condition. We also propose a two-stage procedure for this pro-1970 AMS Classification: 60F05, 62L10, 62G10     

Early stopping by using stochastic curtailment in a three-arm sequential trial

Summary. Interim analysis is important in a large clinical trial for ethical and cost considerations. Sometimes, an interim analysis needs to be performed at an earlier than planned time point. In that case, methods using stochastic curtailment are useful in examining the data for early stopping while controlling the inflation of type I and type II errors. We consider a three-arm randomized study of treatments to reduce perioperative blood loss following major surgery. Owing to slow accrual, an unplanned interim analysis was required by the study team to determine whether the study should be continued. We distinguish two different cases: when all treatments are under direct comparison and when one of the treatments is a control. We used simulations to study the operating characteristics of five different stochastic curtailment methods. We also considered the influence of timing of the interim analyses on the type I error and power of the test. We found that the type I error and power between the different methods can be quite different. The analysis for the perioperative blood loss trial was carried out at approximately a quarter of the planned sample size. We found that there is little evidence that the active treatments are better than a placebo and recommended closure of the trial.     

A sequential procedure for testing the existence of a random walk model in finite samples

Given the random walk model, we show, for the traditional unrestricted regression used in testing stationarity, that no matter what the initial value of the random walk is or its drift or its error standard deviation, the sampling distributions of certain statistics remain unchanged. Using Monte Carlo simulations, we estimate, for different finite samples, the sampling distributions of these statistics. After smoothing the percentiles of the empirical sampling distributions, we come up with a new set of critical values for testing the existence of a random walk, if each statistic is being used on an individual base. Combining the new sets of critical values, we finally suggest a general methodology for testing for a random walk model.     

Basic concepts of group sequential and adaptive group sequential test procedures

Based on the concept of repeated significance tests, an empirical study may be planned in subsequent stages. Group sequential test procedures offer the possibility of performing the study with a fixed number of observations per stage. At least, the number of observations must be chosen independently of the observed data. In adaptive group sequential test procedures, the number of observations can be changed during the course of the study using all results observed so far. In this article, the basic concepts of these two designs are reviewed. Recent developments in adaptive designs are outlined and potential fields of application are given.