Choosing between the BP and BN sequential strategies

Cost and burden of diagnostic testing may be reduced if fewer tests can be applied. Sequential testing involves selecting a sequence of tests, but only administering subsequent tests dependent on results of previous tests. This research provides guidance to choosing between single tests or the believe the positive (BP) and believe the negative (BN) sequential testing strategies, using accuracy (as measured by the Youden Index) as the primary determinant. Approximately 75% of the parameter combinations examined resulted in either BP or BN being recommended based on a higher accuracy at the optimal point. In about half of the scenarios BP was preferred, and the other half, BN, with the choice often a function of the value of the ratio of standard deviations of those without and with disease (b). Large values of b for the first test of the sequence tended to be associated with preference for BN as opposed to BP, while small values of b appear to favor BP. When there was no preference between sequences and/or single tests based on the Youden Index, cost of the sequence was considered. In this case, disease prevalence plays a large role in the selection of strategies, with lower values favoring BN and sometimes higher values favoring BP. The cost threshold for the sequential strategy to be preferred over a single, more accurate test, was often quite high. It appears that while sequential strategies most often increase diagnostic accuracy over a single test, sequential strategies are not always preferred.     

A nonparametric repeated significance test with adaptive target sample size

In this article a general result is derived that, along with a functional central limit theorem for a sequence of statistics, can be employed in developing a nonparametric repeated significance test with adaptive target sample size. This method is used in deriving a repeated significance test with adaptive target sample size for the shift model. The repeated significance test is based on a functional central limit theorem for a sequence of partial sums of truncated observations. Based on numerical results presented in this article one can conclude that this nonparametric sequential test performs quite well.     

A rank test for stochastically ordered alternatives

A rank test based on the number of ‘near-matches’ among within-block rankings is proposed for stochastically ordered alternatives in a randomized block design with t treatments and b blocks. The asymptotic relative efficiency of this test with respect to the Page test is computed as number of blocks increases to infinity. A sequential analog of the above test procedure is also considered. A repeated significance test procedure is developed and average sample number is computed asymptotically under the null hypothesis as well as under a sequence of contiguous alternatives.     

DETERMINATION OF DOMAINS OF ATTRACTION BASED ON A SEQUENCE OF MAXIMA

Suppose that the maximum of a random sample from a distribution F(x) may be obtained in each of k equally spaced observation periods. This paper proposes a test to determine the domain of attraction of F(x), and investigates the properties when the sample size is very large and perhaps unknown and k is fixed and small. The test statistic is a function of the spacings between the order statistics based on the sequence of maxima and is suggested by reference to one studied previously when inference was based on the largest k observations of a random sample. A Monte Carlo study shows that the proposed test is more powerful than its main competitor. The test is illustrated by two examples.     

Detecting structural change in linear models

Consider a sequence of independent observations which change their marginal distribution at most once somewhere in the sequence and one is not certain where the change has occurred. One would be interested in detecting the change and determining the two distributions which would describe the sequence. On the other hand if no change had occurred, one would want to know the common distribution of the observations. This study develops a Bayesian test for detecting a switch from one linear model to another. The test is based on the marginal posterior mass function of the switch point and the posterior probability of a stable model. This test and an informal sequential procedure of Smith are illustrated with data generated from an unstable linear regression model, which changes the linear relationship between the dependent and independent variables     

A class of k-sample distribution-free tests for location against ordered alternatives

This paper introduces a new class of distribution-free tests for testing the homogeneity of several location parameters against ordered alternatives. The proposed class of test statistics is based on a linear combination of two-sample U-statistics based on subsample extremes. The mean and variance of the test statistic are obtained under the null hypothesis as well as under the sequence of local alternatives. The optimal weights are also determined. It is shown via Pitman ARE comparisons that the proposed class of test statistics performs better than its competitor tests in case of heavy-tailed and long-tailed distributions     

Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach

We consider a nonparametric autoregression model under conditional heteroscedasticity with the aim to test whether the innovation distribution changes in time. To this end, we develop an asymptotic expansion for the sequential empirical process of nonparametrically estimated innovations (residuals). We suggest a Kolmogorov–Smirnov statistic based on the difference of the estimated innovation distributions built from the first ?ns?and the last n ? ?ns? residuals, respectively (0 ≤ s ≤ 1). Weak convergence of the underlying stochastic process to a Gaussian process is proved under the null hypothesis of no change point. The result implies that the test is asymptotically distribution‐free. Consistency against fixed alternatives is shown. The small sample performance of the proposed test is investigated in a simulation study and the test is applied to a data example.     

Robust inference for group sequential trials

For ethical reasons, group sequential trials were introduced to allow trials to stop early in the event of extreme results. Endpoints in such trials are usually mortality or irreversible morbidity. For a given endpoint, the norm is to use a single test statistic and to use that same statistic for each analysis. This approach is risky because the test statistic has to be specified before the study is unblinded, and there is loss in power if the assumptions that ensure optimality for each analysis are not met. To minimize the risk of moderate to substantial loss in power due to a suboptimal choice of a statistic, a robust method was developed for nonsequential trials. The concept is analogous to diversification of financial investments to minimize risk. The method is based on combining P values from multiple test statistics for formal inference while controlling the type I error rate at its designated value.This article evaluates the performance of 2 P value combining methods for group sequential trials. The emphasis is on time to event trials although results from less complex trials are also included. The gain or loss in power with the combination method relative to a single statistic is asymmetric in its favor. Depending on the power of each individual test, the combination method can give more power than any single test or give power that is closer to the test with the most power. The versatility of the method is that it can combine P values from different test statistics for analysis at different times. The robustness of results suggests that inference from group sequential trials can be strengthened with the use of combined tests.     

A model for k-out-of-m load-sharing systems

ABSTRACT

In a load-sharing system, the failure of a component affects the residual lifetime of the surviving components. We propose a model for the load-sharing phenomenon in k-out-of-m systems. The model is based on exponentiated conditional distributions of the order statistics formed by the failure times of the components. For an illustration, we consider two component parallel systems with the initial lifetimes of the components having Weibull and linear failure rate distributions. We analyze one data set to show that the proposed model may be a better fit than the model based on sequential order statistics.     

Interim analysis of clinical trials based on urn models

Clinical trials usually involve efficient and ethical objectives such as maximizing the power and minimizing the total failure number. Interim analysis is now a standard technique in practice to achieve these objectives. Randomized urn models have been extensively studied in the literature. In this paper, we propose to perform interim analysis on clinical trials based on urn models and study its properties. We show that the urn composition, allocation of patients and parameter estimators can be approximated by a joint Gaussian process. Consequently, sequential test statistics of the proposed procedure converge to a Brownian motion in distribution and the sequential test statistics asymptotically satisfy the canonical joint distribution defined in Jennison & Turnbull (Jennison & Turnbull 2000. Group Sequential Methods with Applications to Clinical Trials, Chapman and Hall/CRC). These results provide a solid foundation and open a door to perform the interim analysis on randomized clinical trials with urn models in practice. Furthermore, we demonstrate our proposal through examples and simulations by applying sequential monitoring and stochastic curtailment techniques. The Canadian Journal of Statistics 40: 550–568; 2012 © 2012 Statistical Society of Canada     

Niveaudurchagangszeiten zur charakterisierung sequentieller schätzverfahren

Let θ be a parameter of a homogenous additive stochastic process. In order to get an unbiased and efficient estimator for a function h(v) one has often to use sequential procedures. In this paper we consider processes of the socalled exponential class. We study level crossing times, which characterize certain sequential estimations. It is shown that the family of level crossing times for an increasing sequence of levels is also a process of the exponential class. The density function of the one-dimensional probability distributions of this new process is given Examples and applications conclude the paper.     

A new extension strategy on three-level factorials under wrap-around L2-discrepancy

Sequential experiment is an indispensable strategy and is applied immensely to various fields of science and engineering. In such experiments, it is desirable that a given design should retain the properties as much as possible when few runs are added to it. The designs based on sequential experiment strategy are called extended designs. In this paper, we have studied theoretical properties of such experimental strategies using uniformity measure. We have also derived a lower bound of extended designs under wrap-around L₂-discrepancy measure. Moreover, we have provided an algorithm to construct uniform (or nearly uniform) extended designs. For ease of understanding, some examples are also presented and a lot of sequential strategies for a 27-run original design are tabulated for practice.     

Group sequential x2 statistic for testing the difference of two mean vectors in clinical trials

In this study we discuss the group sequential procedures for comparing two treatments based on multivariate observations in clinical trials. Also we suppose that a response vector on each of two treatments has a multivariate normal distribution with unknown covariance matrix. Then we propose a group sequential x² statistic in order to carry out repeated significance test for hypothesis of no difference between two population mean vectors. In order to realize the group sequential test where average sample number is reduced, we propose another modified group sequential x² statistic by extension of Jennison and Turnbull ( 1991 ). After construction of repeated confidence boundaries for making the repeated significance test, we compare two group sequential procedures based on two statistics regarding the average sample number and the power of the test in the simulations.     

A global stopping rule for recursive density estimators

In this paper asymptotic sequential fixed-width confidence bounds for an unknown density on the real line, based on integrated squared error, are studied. Using a sequence of Wolverton-Wagner kernel estimators, two classes of stopping rules are established.By the same approach, analogous results can be provided for other types of recursive density estimators.     

A note on sequential ML estimates and their asymptotic covariances

A marginal and sequential maximum likelihood estimation method is described which can be used instead of full information maximum likelihood estimation if the latter method is unfeasible. It is shown that the sequential procedure yields strongly consistent and asymptotically normal estimates under relatively general regularity conditions. It is shown that the covariance matrix of the sequential ML estimator does not coincide with the inverse of the Fisher information matrix. Hence, the corrected covariance matrix is derived. The application of the sequential procedure to the multivariate probit model with dichotomous, ordered categorical, single-sided censored and double-sided censored endogenous variables is included. This research was partially supported by a dissertation grant of theStudienstiftung des Deutschen Volkes. Comments and suggestions on earlier drafts by Gerhard Arminger, Giorgio Calzolari, Bernd Kortzen and an anonymous referee are gratefully acknowledged.     

Generalized run tests for statistical process control

ABSTRACT

In a sequence of elements, a run is defined as a maximal subsequence of like elements. The number of runs or the length of the longest run has been widely used to test the randomness of an ordered sequence. Based on two different sampling methods and two types of test statistics used, run tests can be classified into one of four cases. Numerous researchers have derived the probability distributions in many different ways, treating each case separately. In the paper, we propose a unified approach which is based on recurrence arguments of two mutually exclusive sub-sequences. We also consider the sequence of nominal data that has more than two classes. Thus, the traditional run tests for a binary sequence are special cases of our generalized run tests. We finally show that the generalized run tests can be applied to many quality management areas, such as testing changes in process variation, developing non-parametric multivariate control charts, and comparing the shapes and locations of more than two process distributions.     

Sequential empirical process in autoregressive models with measurement errors

In this paper, we study the weak convergence of the sequential empirical process based on the residuals from autoregressive models with measurement errors. It is shown that the sequential empirical process converges weakly to the sum of a Gaussian process which is the limit of a sequential empirical process of certain p-dependent random variables and an additional term depending on the parameter estimators of the model. As an application, we discuss the change point problem in the distribution of the error process in the autoregressive model. We present the numerical result of a simulation study for an asymptotically distribution-free test.     

A sequential testing procedure for outliers and structural change

In this paper a test for outliers based on externally studentized residuals is shown to be related to a test for predictive failure. The relationships between a test for outliers, a test for a correlated mean shift and a test for an intercept shift are developed. A sequential testing procedure for outliers and structural change is shown to be independent, so that the overall size of the joint test can be determined exactly. It is established that a joint test for outliers and constancy of variances cannot be performed.     

Estimating Process Capability Index C PM Using a Bootstrap Sequential Sampling Procedure

Construction of a confidence interval for process capability index C _PM is often based on a normal approximation with fixed sample size. In this article, we describe a different approach in constructing a fixed-width confidence interval for process capability index C _PM with a preassigned accuracy by using a combination of bootstrap and sequential sampling schemes. The optimal sample size required to achieve a preassigned confidence level is obtained using both two-stage and modified two-stage sequential procedures. The procedure developed is also validated using an extensive simulation study.     

ASYMPTOTIC EXPANSIONS APPLIED TO F-TEST CRITERIA

Using asymptotic expansions of the Kummer hypergeometric function, the sequential. F-test criterion is evaluated asymptotically in terms of the sample size. The continuation region inequalities for the test are inverted and expressed in terms of known test criteria. A rapidly converging algorithm for carrying out the sequential procedure is provided. This makes the F-test easier for the practitioner to use. Almost sure finite termination of the sequential. F-test is asserted by appealing to the continuation inequalities and a heuristic asymptotic expansion of the test criterion. Average stopping times of the sequential procedure for a variety of population means and population number configurations are tabulated. The computer symbolic manipulation program MAPLE was used to derive some formulae.