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Author(s): Peng, Hua | Advisor(s): Jeske, Daniel R | Abstract: Group sequential designs play an important role in monitoring clinical trials. In this dissertation, we consider the construction of one-sided group sequential designs where the stopping rule includes boundaries for early stopping to accept for futility and to reject for efficacy. The traditional assumption that all patients have the same likelihood of benefiting from the treatment is sometimes unrealistic and it can underestimate the required sample size. This motivates us to power the design for an alternative where the treatment group observations come from a mixture of normal distributions. For the proposed setting, we use standardized test statistics based on sample means. Stopping boundaries and arm size for the design are determined by Type I and Type II error spending equations. The unknown variance case is discussed, and we introduce the group sequential t-test for the mixture setting. With the mixture model, we discuss a more general definition of treatment effect. The maximum likelihood estimator and method of moment estimator for the treatment effect are discussed.

The problem of asymptotic efficiency of adaptive one-step predictors for a stable multivariate first-order autoregressive process (AR(1)) with unknown parameters is considered. The predictors are based on the truncated estimators of the dynamic matrix parameter. The truncated estimation method is a modification of the truncated sequential estimation method that makes it possible to obtain estimators of ratio-type functionals with a given accuracy by samples of fixed size. The criterion of optimality is based on the loss function, defined as a sum of sample size and squared prediction error's sample mean. The cases of known and unknown variance of the noise model are studied. In the latter case the optimal sample size is a special stopping time. The simulation results are given.

In this paper, 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.

In this article, we respond to the comments made by the 10 discussants on “Two-Stage Procedures for High-Dimensional Data.” We also give some new results along with their brief explanations.

In this article, we consider a variety of inference problems for high-dimensional data. The purpose of this article is to suggest directions for future research and possible solutions about p n problems by using new types of two-stage estimation methodologies. This is the first attempt to apply sequential analysis to high-dimensional statistical inference ensuring prespecified accuracy. We offer the sample size determination for inference problems by creating new types of multivariate two-stage procedures. To develop theory and methodologies, the most important and basic idea is the asymptotic normality when p → ∞. By developing asymptotic normality when p → ∞, we first give (a) a given-bandwidth confidence region for the square loss. In addition, we give (b) a two-sample test to assure prespecified size and power simultaneously together with (c) an equality-test procedure for two covariance matrices. We also give (d) a two-stage discriminant procedure that controls misclassification rates being no more than a prespecified value. Moreover, we propose (e) a two-stage variable selection procedure that provides screening of variables in the first stage and selects a significant set of associated variables from among a set of candidate variables in the second stage. Following the variable selection procedure, we consider (f) variable selection for high-dimensional regression to compare favorably with the lasso in terms of the assurance of accuracy and the computational cost. Further, we consider variable selection for classification and propose (g) a two-stage discriminant procedure after screening some variables. Finally, we consider (h) pathway analysis for high-dimensional data by constructing a multiple test of correlation coefficients., Editor's Special Invited Paper。 この招待論文は、Abraham Wald Prize in Sequential Analysis 2012の受賞論文となっております。

Because of rare but serious adverse events, pharmaceutical drugs and vaccines are sometimes withdrawn from the market, either by a government agency such as the Food and Drug Administration (FDA) in the United States or by the manufacturing pharmaceutical company. In other cases, a drug may be generally safe but increase the risk for serious adverse events for certain subpopulations such as pregnant women or people with heart problems. Due to limited sample size and selected study populations, rare adverse events are often impossible to detect during phase 3 trials conducted before the drug is approved for general use. It is then important to conduct post-approval drug safety surveillance, using, for example, health insurance claims data. In such surveillance, the goal should be to detect serious adverse events as early as possible without too many false alarms, and it is then natural to use a continuous or near-continuous sequential test procedure that reevaluates the data on a daily or weekly b...

The paper deals with two classes of unbiased nonparametric estimators of survival and cumulative hazard functions in a population subject to right-censoring. Both classes of estimators are based on a sequential sampling scheme, and are similar to the well-known Kaplan–Meier and Nelson–Aalen estimators.

In this paper, we consider sequential estimation of the location parameter based on the midrange in the presence of an unknown scale parameter when the underlying distribution has a bounded support. The estimation is done under squared loss plus cost of sampling. Stopping rules based on the range are proposed and are shown to be asymptotically efficient. The risks of the sequential procedures are compared with the Robbins sequential estimation procedure based on the sample mean. The former are shown to be asymptotically more efficient than the latter in the sense of the sample size when the density function changes sharply at the end points of the support. Koike (2007) observed a similar asymptotic superiority of the sequential estimation procedure based on the midrange in the sequential interval estimation procedure under the same condition.

Motivated by applications to confirmatory clinical trials for testing a new treatment against a placebo or active control when the new treatment has k possible treatment strategies (arms)-for example, k possible doses for a new drug-we develop an asymptotic theory for efficient outcome-adaptive randomization schemes and optimal stopping rules. Our approach consists of developing asymptotic lower bounds for the expected sample sizes from the k treatment arms and the control arm and using generalized sequential likelihood ratio procedures to achieve these bounds. Implementation details of our design and analysis and comparative simulation studies are also provided.

Repeated significance tests are frequently used in areas of science and technology in which the data is accumulated sequentially over time. In this article we develop a repeated significance test for independent and identically distributed observations from a continuous symmetric distribution with heavy tails, infinite variance, and possibly no mean. This repeated significance test is nonparametric in nature, as it is applicable to a class of distributions with a specified tail behavior. We present an algorithm for selecting the continuation region associated with the repeated significance test that achieves specified significance level and power at a given alternative. Moreover, we derive approximations for the power function and expected sample size. Numerical results are presented to evaluate the performance of these approximations.

We consider group sequential comparisons of two treatments in which subjects' treatment assignments may be based on previously observed responses. We present theory for the joint distribution of the sequence of test statistics in such a study and show that this has a standard form when treatment assignment is based only on the current estimate of the treatment difference. This theory supports the inclusion of adaptive sampling rules in standard group designs, maintaining the usual Type I and II error probabilities; results extend to the case where a linear model is fitted at each analysis to adjust for covariates. Simulation results show the expected number of subjects receiving the inferior treatment can be reduced by up to 10 or 15% with increases in total sample size of around 1 to 3%. More aggressive strategies can reduce the number on the inferior treatment by up to 30 or 40%, on average, but at the price of substantial increases in total sample size.

Buzaianu and Chen apply strong curtailment to modify the two-stage select-and-test clinical trial design proposed by Thall et al. (1988). The modification reduces the expected sample size while maintaining overall power but requires continuous monitoring in stage 1. I will review the history of this type of design and discuss practical issues related to the use of strong curtailment that arise in trial conduct.

Bechhofer and Kulkarni (1982) proposed procedures for selecting that one of k Bernoulli populations with the largest single trial success probability. They showed that their procedure for k = 2 minimizes the expected total sample size amongst a class of procedures, all of which attain the same probability of correct selection. Kulkarni and Jennison (1986) generalized this result to the case k ≥ 3. In this article we prove the stronger result that the Bechhofer-Kulkarni procedure for each k≥2 stochastically minimizes the distribution of sample size amongst procedures in the same class. That is, the distribution of sample size for the Bechhofer-Kulkarni procedure is the same as or stochastically smaller than that for any other procedure in the class.

Bechhofer and Kulkarni (1982a) proposed a sequential procedure for selecting the best of k≥2 Bernoulli populations, and in a subsequent paper (1982b) gave an upper bound for the expected number of observations taken from each population by this procedure. In this note we present an asymptotically correct approximation to the expected sample size taken from each population and a slightly improved upper bound on these expected sample sizes.

Bounded risk estimation of the mean of a finite population is considered under three simple random sampling strategies: (i) with replacement, mean per unit estimation, (ii) with replacement, mean per distinct unit estimation, and (iii) without replacement, mean per unit estimation. It is well known that in the fixed sample size scheme, (iii) fares better than (ii) and (ii) better than (i). However, in the current context, the sample sizes are dictated by (possibly, degenerate) stopping times, and visualizing the cost (due to measurements/recording, etc.) as a function of the number of distinct units in the sample, it is shown that the second strategy still fares better than the first, although the third strategy may not perform better than the second one. Actually, in the case of known population variance, it is shown that in the light of the number of distinct units, the difference of ASN for the second and third strategies can never be greater than two or less than minus one. A similar relationship also...

In this article, we consider a geometric classifier that is applicable to multiclass classification for high-dimensional data. We show the consistency property and the asymptotic normality of the geometric classifier under certain mild conditions. We discuss sample size determination so that the geometric classifier can ensure that its misclassification rates are less than prespecified thresholds. We give a two-stage procedure to estimate the sample sizes required in such a geometric classifier and propose a misclassification rate–adjusted classifier (MRAC) based on the geometric classifier. We evaluate the performance of the MRAC theoretically and numerically. Finally, we demonstrate the MRAC in actual data analyses by using a microarray data set.