system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

enumerating fuzzy switching functions and free kleene algebras abstract we investigate the free kleene algebra on n free generators fk n this algebra is essentially the same as the set of n variable fuzzy switching functions the cardinality of fk n has been previously determined only for n 1 2 and 3 we present a general method for enumeration problems of this kind we show fk 4 has 160 297 985 276 elements the final section contains new upper and lower bounds for the cardinality of fk n we also consider the number of regular functions and show that there are 11 197 129 615 and 430 904 428 717 such functions in 1 2 3 and 4 variables

gap metrics for time varying linear systems in a continuous time setting a recently proposed time varying generalisation of vinnicombe s v gap is shown to have the properties of a metric on a class of causal linear continuous time systems that admit normalised strong graph representations an alternative time varying gap metric is then defined in more general terms and a corresponding robust stability result is derived for linear feedback interconnections finally it is shown that the time varying gap and v gap metrics induce the same topology this being the coarsest topology under which feedback stability and closed loop sensitivity are both robust properties c 2014 elsevier b v all rights reserved

well posedness of feedback systems insights into exact robustness analysis and approximate computations this paper establishes a framework for robust stability analysis of linear time invariant uncertain systems the uncertainty is assumed to belong to an arbitrary subset of complex matrices the concept used here is well posedness of feedback systems leading to necessary and sufficient conditions for robust stability based on this concept some insights into exact robust stability conditions are given in particular frequency domain and state space conditions for well posedness are provided in terms of hermitian form inequalities it is shown that these inequalities can be interpreted as small gain conditions with a generalized class of scalings given by linear fractional transformations lft using the lft scaled small gain condition in the state space setting the duality is established between the h sub spl infin norm condition with frequency dependent scalings and the parameter dependent lyapunov condition connections to the existing results including the structured singular value and the integral quadratic constraints are also discussed finally we show that our well posedness conditions can be used to give a less conservative yet computable bound on the real structured singular value this result is illustrated by numerical examples

statistical inference for self designing clinical trials with a one sided hypothesis in the process of monitoring clinical trials it seems appealing to use the interim findings to determine whether the sample size originally planned will provide adequate power when the alternative hypothesis is true and to adjust the sample size if necessary in the present paper we propose a flexible sequential monitoring method following the work of fisher 1998 in which the maximum sample size does not have to be specified in advance the final test statistic is constructed based on a weighted average of the sequentially collected data where the weight function at each stage is determined by the observed data prior to that stage such a weight function is used to maintain the integrity of the variance of the final test statistic so that the overall type i error rate is preserved moreover the weight function plays an implicit role in termination of a trial when a treatment difference exists finally the design allows the trial to be stopped early when the efficacy result is sufficiently negative simulation studies confirm the performance of the method

parameter estimation following group sequential hypothesis testing summary parameter estimation techniques which fail to adjust for the interim analyses of group sequential test designs will introduce bias in much the same way that the repeated use of single sample hypothesis testing causes inflation of the type one statistical error rate methods based on the duality of hypothesis testing and interval estimation require definition of an ordering for the outcome space for the test statistic in this paper estimation following a group sequential hypothesis test for the mean of a normal distribution with known variance is investigated a proposed ordering of the sample space based on the maximum likelihood estimate of the mean is found to result in estimates which compare favourably with estimates computed from orderings investigated by tsiatis rosner mehta 1984 and chang o brien 1986 for a variety of group sequential designs the proposed ordering is then adapted for use when the sizes of groups accrued between analyses is random

improved approximation for estimation following closed sequential tests summary siegmund 1978 has developed a procedure for estimation following a closed sequential test known as the repeated significance tests here a more elaborate approximation is proposed for a wider class of closed sequential tests the major goal is to account for the excess over boundary in addition to the stopping time when the sequential test is terminated some hypothetical examples are used for comparing the two approximations to investigate the properties of these procedures further simulation results are presented in terms of estimated mean squared error bias and variance for point estimators and in terms of coverage probability and width for confidence intervals

confidence intervals for a binomial parameter following a multistage test with application to mil std 105d and medical trials this paper describes a method for constructing confidence intervals for a binomial parameter upon termination of a sequential or multistage test tables are presented for use with the mil std 105d multiple sampling plans for acceptance sampling also given are tables for use with some three stage schemes that have been proposed in connection with biomedical trials the results are compared with confidence intervals calculated as if the sampling plan had been one with a fixed sample size

use of logrank tests and group sequential methods at fixed calendar times this paper presents simulations to determine the operating characteristics of several group sequential boundaries when applied to the repeated analysis of survival data at equal intervals of calendar time because the group sequential boundaries of pocock 1977 biometrika 64 191 199 and o brien and fleming 1979 biometrics 35 549 556 were constructed on the assumption that each interim analysis provides an equal increment of statistical information these boundaries are not theoretically appropriate for interim analysis at prescheduled calendar times nonetheless our simulations show that these boundaries yield size and power near nominal levels for repeated logrank analyses at equal intervals of calendar time

sequential linear rank tests for two sample censored survival data under extremely general patterns of patient arrival allocation to treatment and loss to follow up in randomized clinical trial settings the sequentially computed logrank statistic mantel 1966 is shown under the null hypothesis of identically distributed lifetimes to have exactly uncorrelated increments and is shown via rebolledo s 1980 martingale invariance principle to satisfy a functional central limit theorem justifying sequential logrank tests of jones and whitehead 1979 generalizations are made to other twosample rank tests for censored survival data and practical applicability to real randomized clinical trials is discussed

a unifying framework for robust synchronization of heterogeneous networks via integral quadratic constraints a general framework for analysing robust synchronization in large scale heterogenous networks is proposed based on the theory of integral quadratic constraints iqcs dynamic agents are represented as linear time invariant single input single output systems the agents exchange information according to a sparse dynamical interconnection operator in order to achieve synchronization where their outputs are steered to the same possibly time varying signal the main technical hindrance to applying iqcs in this context lies with the presence of the marginally stable dynamics which define the trajectory to which the agents outputs synchronize it is shown that by working with conditions defined on modified signal spaces of interest and exploiting the graph structure underlying the connections between the dynamic systems iqc methods can be applied directly to synchronization analysis without recourse to loop transformations which may obscure the inherent structural properties of the multi agent networked systems decentralized and scalable conditions for synchronization are proposed within this setting the iqc framework is demonstrated to unify and generalize some of the existing results in the literature including certain nyquist type consensus certificates for time delay systems moreover it allows the role of feedback in robustness against uncertainty to be better manifested within the context of synchronization

regular ternary logic functions ternary logic functions suitable for treating ambiguity a special group of ternary functions called regular ternary logic functions are defined these functions are useful in switching theory programming languages algorithm theory and many other fields if we are concerned with the indefinite state in such fields this correspondence describes the fundamental properties and representations of the regular ternary logic functions

robustness analysis for feedback interconnections of distributed systems via integral quadratic constraints a framework is established for directly accommodating feedback interconnections of unstable distributed parameter transfer functions in robust stability analysis via integral quadratic constraints iqcs this involves transfer function homotopies that are continuous in a gap metric sense as such the development includes the extension of gap metric concepts to an irrational setting and the study of uncertainty set connectedness in these terms the main iqc based robust stability result is established for constantly proper transfer functions in the callier desoer algebra i e finitely many unstable poles and a constant limit at infinity problems of structured robust stability analysis and robust performance analysis are considered to illustrate use of the main result several numerical examples are also presented these include stability analysis of an autonomous system with uncertain time delay and a closed loop control system accounting for both the gain and phase characteristics of the distributed parameter uncertainty associated with the nominal rational plant model used for controller synthesis

asymmetric group sequential boundaries for monitoring clinical trials a group sequential method is proposed for the periodic analysis of accumulating data in clinical trials which are essentially one sided as in treatment placebo studies the proposed boundaries retain the early stopping properties of the procedure described previously by demets ware 1980 when treatment appears to be ineffective or harmful but incorporate the more stringent requirements of the model proposed by o brien fleming 1979 for early rejection of the null hypothesis in favour of treatment

the bias of the sample proportion following a group sequential phase ii clinical trial group sequential testing procedures have seen wide use in phase ii clinical trials the sample proportion p of responders is the commonly used estimator for the binomial response probability p it can be shown that p is the maximum likelihood estimator mle of p it is well known that mle can be in general whitehead and p in particular dupont biased estimators if ther computation follows a group sequential procedure in this paper we numerically investigate the bias of p we find that the magnitude of the bias of p is less than 0 025 in all cases we investigated we apply the idea in whitehead to propose a bias adjusted estimator that reduces the bias substantially and reduces the mean square error as well in a certain range of p we also evaluate the uniformly minimum variance unbiased umvu estimator if one does not mind a bias of 0 025 one may find the sample proportion a suitable estimator for p because of its simplicity and easy explanation if one is concerned with bias the bias adjusted estimator may be a good choice

spectral factorization and lq optimal regulation for multivariable distributed systems a necessary and sufficient condition is proved for the existence of a bistable spectral factor with entries in the distributed proper stable transfer function algebra x1d49c in the context of distributed multivariable convolution systems with no delays a by product is the existence of a normalized coprime fraction of the transfer function of such a possibly unstable system with entries in the algebra of fractions over x1d49c we next study semigroup state space systems sgb with bounded sensing and control having a transfer function with entries in and consider its standard lq optimal regulation problem having an optimal state feedback operator k0 for a system sgb a formula is given relating any spectral factor of a transfer function coprime fraction power spectral density to k0 a by product is the description of any normalized coprime fraction of the transfer function in terms of k0 finally we describe an alternative way of finding the solution operator k0 of the lq problem using spectral factor

duality in multiplier based robustness analysis frequency domain conditions involving multipliers are useful for robustness analysis the resulting analysis problem is generally convex but infinite dimensional and numerical solutions restricted to finite dimensional subspaces need to be considered the resulting finite dimensional problem can be transformed into a linear matrix inequality which can be solved using efficient algorithms the paper presents a format for the dual of the infinite dimensional problem the dual problem can be used to estimate the conservatism of particular finite dimensional subspaces

an evaluation of inferential procedures for adaptive clinical trial designs with pre specified rules for modifying the sample size summary many papers have introduced adaptive clinical trial methods that allow modifications to the sample size based on interim estimates of treatment effect there has been extensive commentary on type i error control and efficiency considerations but little research on estimation after an adaptive hypothesis test we evaluate the reliability and precision of different inferential procedures in the presence of an adaptive design with pre specified rules for modifying the sampling plan we extend group sequential orderings of the outcome space based on the stage at stopping likelihood ratio statistic and sample mean to the adaptive setting in order to compute median unbiased point estimates exact confidence intervals and p values uniformly distributed under the null hypothesis the likelihood ratio ordering is found to average shorter confidence intervals and produce higher probabilities of p values below important thresholds than alternative approaches the bias adjusted mean demonstrates the lowest mean squared error among candidate point estimates a conditional error based approach in the literature has the benefit of being the only method that accommodates unplanned adaptations we compare the performance of this and other methods in order to quantify the cost of failing to plan ahead in settings where adaptations could realistically be pre specified at the design stage we find the cost to be meaningful for all designs and treatment effects considered and to be substantial for designs frequently proposed in the literature

exact statistical inference for group sequential trials summary this paper considers clinical trials comparing two treatments with dichotomous responses where the data are examined periodically for early evidence of treatment difference the existing group sequential methods for such trials are based on the large sample normal approximation to the joint distribution of the estimators of treatment difference over interim analyses we demonstrate through extensive numerical studies that for small and even moderate sized trials these approximate procedures may lead to tests with supranominal size mainly when unpooled estimators of variance are used and confidence intervals with under nominal coverage probability we then study exact methods for group sequential testing repeated interval estimation and interval estimation following sequential testing the new procedures can accommodate any treatment allocation rules an example using real data is provided in clinical trials with sequential patient entry fixed sample size designs are sometimes unjustified on ethical grounds and fully sequential procedures are often impractical one compromise is the group sequential design under which multiple looks are performed during the trial it is widely recognized that the use of the standard single stage f1xedsample size test or confidence interval at each look would lead to an overall type i error or noncoverage rate significantly higher than the nominal level a number of methods have been proposed to determine proper critical values for maintaining a prespecified overall significance level in sequential testing to construct repeated confidence intervals with a given overall coverage probability and to obtain a valid confidence interval following a sequential trial an excellent review on the existing methodology was given by jennison and turnbull 1989 in this paper we are mainly concerned with the commonly used group sequential trials comparing two treatments a and b with relatively quick dichotomous responses success and failure the data are reviewed periodically with cumulative mk subjects receiving treatment a and nk subjects receiving treatment b by the kth look let random variables xk and yk denote the numbers of successes out of the msk and nk subjects respectively we assume that the maximum number of looks k is fixed in advance though some of the procedures developed here can be extended to the case where k is not fixed in advance

comparison of conditional bias adjusted estimators for interim analysis in clinical trials with survival data group sequential designs are widely used in clinical trials to determine whether a trial should be terminated early in such trials maximum likelihood estimates are often used to describe the difference in efficacy between the experimental and reference treatments however these are well known for displaying conditional and unconditional biases established bias adjusted estimators include the conditional mean adjusted estimator cmae conditional median unbiased estimator conditional uniformly minimum variance unbiased estimator cumvue and weighted estimator however their performances have been inadequately investigated in this study we review the characteristics of these bias adjusted estimators and compare their conditional bias overall bias and conditional mean squared errors in clinical trials with survival endpoints through simulation studies the coverage probabilities of the confidence intervals for the four estimators are also evaluated we find that the cmae reduced conditional bias and showed relatively small conditional mean squared errors when the trials terminated at the interim analysis the conditional coverage probability of the conditional median unbiased estimator was well below the nominal value in trials that did not terminate early the cumvue performed with less bias and an acceptable conditional coverage probability than was observed for the other estimators in conclusion when planning an interim analysis we recommend using the cumvue for trials that do not terminate early and the cmae for those that terminate early copyright 2017 john wiley sons ltd

tests and confidence intervals for secondary endpoints in sequential clinical trials in a sequential clinical trial whose stopping rule depends on the primary endpoint inference on secondary endpoints is an important long standing problem ignoring the possibility of early stopping based on the primary endpoint may result in substantial bias to address this problem a commonly used approach is to develop bias correction by estimating the bias in the case of bivariate normal outcomes and appealing to joint asymptotic normality of the statistics associated with the primary and secondary endpoints we propose herein a new approach that uses resampling and a novel ordering scheme in the sample space of sequential statistics observed up to a stopping time this approach is shown to provide accurate inference in complex clinical trials including time sequential trials with survival endpoints and covariates

estimation of a secondary parameter in a group sequential clinical trial summary in this article we investigate estimation of a secondary parameter in group sequential tests we study the model in which the secondary parameter is the mean of the normal distribution in a subgroup of the subjects the bias of the naive secondary parameter estimator is studied it is shown that the sampling proportions of the subgroup have a crucial effect on the bias as the sampling proportion of the subgroup at or just before the stopping time increases the bias of the naive subgroup parameter estimator increases as well an unbiased estimator for the subgroup parameter and an unbiased estimator for its variance are derived using simulations we compare the mean squared error of the unbiased estimator to that of the naive estimator and we show that the differences are negligible as an example the methods of estimation are applied to an actual group sequential clinical trial the beta blocker heart attack trial

confidence interval calculation for a sequential clinical trial of binary responses consider a sequential test applied to two streams of binary responses in a comparative clinical trial after the completion of such a trial woodroofe s 1992 technique for accurate confidence interval estimation for a treatment difference can be applied in this paper the technique is extended to provide confidence intervals for the individual absolute success probabilities of the two treatments in the trial for which no current methodology exists accuracy is explored by simulation of coverage probabilities and individual confidence limit probabilities the simulations concern a triangular test and an o brien fleming test

on the estimation of the binomial probability in multistage clinical trials due to the optional sampling effect in a sequential design the maximum likelihood estimator mle following sequential tests is generally biased in a typical two stage design employed in a phase ii clinical trial in cancer drug screening a fixed number of patients are enrolled initially the trial may be terminated for lack of clinical efficacy of treatment if the observed number of treatment responses after the first stage is too small otherwise an additional fixed number of patients are enrolled to accumulate additional information on efficacy as well as on safety there have been numerous suggestions for design of such two stage studies here we establish that under the two stage design the sufficient statistic i e stopping stage and the number of treatment responses for the parameter of the binomial distribution is also complete then based on the rao blackwell theorem we derive the uniformly minimum variance unbiased estimator umvue as the conditional expectation of an unbiased estimator which in this case is simply the maximum likelihood estimator based only on the first stage data given the complete sufficient statistic our results generalize to a multistage design we will illustrate features of the umvue based on two stage phase ii clinical trial design examples and present results of numerical studies on the properties of the umvue in comparison to the usual mle

confidence intervals following group sequential tests in clinical trials tsiatis rosner and mehta 1984 biometrics 40 797 803 proposed a procedure for constructing confidence intervals following group sequential tests of a normal mean this method is first extended for group sequential tests for which the sample sizes between interim analyses are not identical or the times are not equally spaced then properties of this confidence interval estimation procedure are studied by simulation the extension accommodates the flexible procedure by lan and demets 1983 biometrika 70 659 663 for constructing discrete group sequential boundaries to form a structure for monitoring and estimation following a class of group sequential tests finally it is demonstrated how to combine the procedures by lan and demets and by tsiatis rosner and mehta using a fortran program

estimating and reducing bias in group sequential designs with gaussian independent increment structure summary a concern in the use of sequential testing procedures is the bias associated with the estimates of treatment differences clinical trials that stop early because of evidence of therapeutic benefit are prone to exaggerate the magnitude of the treatment effect we consider methods for estimating and reducing the bias of treatment differences estimators in group sequential designs with gaussian independent increment structure we derive an analytical expression for the bias and give an easy to calculate approximate bound for its variation a simulation estimate of the bias based on a gaussian independent increment structure is also described and a related bias reduced estimator is considered

discrete interval truth values logic and its application in this paper we focus on functions defined on a special subset of the power set of 0 1 r 1 the elements in the subset will be called discrete interval truth values and operations on the truth values the operations discussed in this paper will be called regular because they are one of the extensions of the regularity which was first introduced by kleene in his ternary logic mukaidono investigated some properties of ternary functions which can be represented by the regular operations he called such ternary functions regular ternary logic functions regular ternary logic functions are useful for representing and analyzing ambiguities such as transient states and or initial states in binary logic circuits that boolean functions cannot cope with furthermore they are also applied to studies of fail safe systems for binary logic circuits in this paper we will discuss an extension of regular ternary logic functions to functions on the discrete interval truth values first we will suggest an extension of the regularity in the sense of kleene into operations on the discrete interval truth values we will then present some mathematical properties of functions on the discrete interval truth values consisting of regular operations and one application of these functions

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

testing secondary hypotheses following sequential clinical trials summary sequential monitoring in a clinical trial poses difficulty in hypotheses testing on secondary endpoints after the trial is terminated the conventional likelihood based testing procedure that ignores the sequential monitoring inflates type i error and undermines power in this article we show that the power of the conventional testing procedure can be substantially improved while the type i error is controlled the method is illustrated with a real clinical trial

group sequential tests for delayed responses with discussion summary group sequential methods are used routinely to monitor clinical trials and to provide early stopping when there is evidence of a treatment effect a lack of an effect or concerns about patient safety in many studies the response of clinical interest is measured some time after the start of treatment and there are subjects at each interim analysis who have been treated but are yet to respond we formulate a new form of group sequential test which gives a proper treatment of these pipeline subjects these tests can be applied even when the continued accrual of data after the decision to stop the trial is unexpected we illustrate our methods through a series of examples we define error spending versions of these new designs which handle unpredictable group sizes and provide an information monitoring framework that can accommodate nuisance parameters such as an unknown response variance by studying optimal versions of our new designs we show how the benefits of lower expected sample size that are normally achieved by a group sequential test are reduced when there is a delay in response the loss of efficiency for larger delays can be ameliorated by incorporating data on a correlated short term end point fitting a joint model for the two end points but still making inferences on the original longer term end point we derive p values and confidence intervals on termination of our new tests

point estimation in adaptive enrichment designs adaptive enrichment designs are an attractive option for clinical trials that aim at demonstrating efficacy of therapies which may show different benefit for the full patient population and a prespecified subgroup in these designs based on interim data either the subgroup or the full population is selected for further exploration when selection is based on efficacy data this introduces bias to the commonly used maximum likelihood estimator for the situation of two stage designs with a single prespecified subgroup we present six alternative estimators and investigate their performance in a simulation study the most consistent reduction of bias over the range of scenarios considered was achieved by a method combining the uniformly minimum variance conditionally unbiased estimator with a conditional moment estimator application of the methods is illustrated by a clinical trial example

resampling methods for confidence intervals in group sequential trials summary resampling methods are introduced for the construction of confidence intervals for treatment effects associated with the primary and secondary endpoints of a clinical trial whose stopping rule is specified by a group sequential test these methods are nonparametric and compare favourably with the exact methods that assume the responses to be normally distributed

repeated significance testing for a general class of statistics used in censored survival analysis abstract the asymptotic joint distribution of a class of sequentially computed statistics used in survival analysis is derived when the individuals under study enter serially and are subject to random loss to follow up these results are used for constructing repeated significance tests over real time

frequency domain uncertainty and the graph topology a new metric on linear time invariant systems is defined this metric is no greater than the gap metric and is in fact the smallest metric for which a certain robust stabilization result holds unlike other known metrics which induce the graph topology it has a clear frequency response interpretation this allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric

supplementary analysis of probabilities at the termination of a group sequential phase ii trial we consider estimation of various probabilities after termination of a group sequential phase ii trial a motivating example is that the stopping rule of a phase ii oncologic trial is determined solely based on response to a drug treatment and at the end of the trial estimating the rate of toxicity and response is desirable the conventional maximum likelihood estimator sample proportion of a probability is shown to be biased and two alternative estimators are proposed to correct for bias a bias reduced estimator obtained by using whitehead s bias adjusted approach and an unbiased estimator from the rao blackwell method of conditioning all three estimation procedures are shown to have certain invariance property in bias moreover estimators of a probability and their bias and precision can be evaluated through the observed response rate and the stage at which the trial stops thus avoiding extensive computation

unbiased estimation of secondary parameters following a sequential test we consider within a brownian motion framework estimation of secondary parameters following a sequential test whitehead 1986b proposed a method for reducing the bias of maximum likelihood estimators in this paper we present unbiased estimators to eliminate the bias

a review of methods for futility stopping based on conditional power conditional power cp is the probability that the final study result will be statistically significant given the data observed thus far and a specific assumption about the pattern of the data to be observed in the remainder of the study such as assuming the original design effect or the effect estimated from the current data or under the null hypothesis in many clinical trials a cp computation at a pre specified point in the study such as mid way is used as the basis for early termination for futility when there is little evidence of a beneficial effect brownian motion can be used to describe the distribution of the interim z test value the corresponding b value and the cp values under a specific assumption about the future data a stopping boundary on the cp value specifies an equivalent boundary on the b value from which the probability of stopping for futility can then be computed based on the planned study design sample size and duration and the assumed true effect size this yields expressions for the total type i and ii error probabilities as the probability of stopping increases the probability of a type i error alpha decreases from the nominal desired level e g 0 05 while the probability of a type ii error beta increases from the level specified in the study design thus a stopping boundary on the b value can be determined such that the inflation in type ii error probability is controlled at a desired level an iterative procedure is also described that determines a stopping boundary on the b value and a final test critical z value with specified type i and ii error probabilities the implementation in conjunction with a group sequential analysis for effectiveness is also described

p values after repeated significance testing a simple approximation method the p value after a repeated significance test is a useful measure of the strength of evidence against the null hypothesis its computation however requires a computer intensive numerical integration method the p value is not conceptually straightforward because it depends on how the sample space is ordered which can be arbitrary we look at two orderings of the sample space one proposed by tsiatis et al and the other by rosner and tsiatis and chang although studies have shown that the latter ordering gives more reasonable confidence intervals than the former the former gives a conservative and therefore more reasonable p value both however should yield an identical p value in most applications in this paper we present a simple method of approximating p values we provide tables to implement the method for two to ten stages with alpha 0 1 0 05 and 0 01 for the pocock and o brien fleming procedures the proposed method can be applied to both orderings

robust stability conditions for feedback interconnections of distributed parameter negative imaginary systems sufficient and necessary conditions for the stability of positive feedback interconnections of negative imaginary systems are derived via an integral quadratic constraint iqc approach the iqc framework accommodates distributed parameter systems with irrational transfer function representations while generalising existing results in the literature and allowing exploitation of flexibility at zero and infinite frequencies to reduce conservatism in the analysis the main results manifest the important property that the negative imaginariness of systems gives rise to a certain form of iqcs on positive frequencies that are bounded away from zero and infinity two additional sets of iqcs on the dc and instantaneous gains of the systems are shown to be sufficient and necessary for closed loop stability along a homotopy of systems

sequential analysis of the proportional hazards model abstract for the proportional hazards model of survival analysis an appropriate large sample theory is developed for cases of staggered entry and sequential analysis the principal techniques involve an approximation of the score process by a suitable martingale and a random rescaling of time based on the observed fisher information as a result we show that the maximum partial likelihood estimator behaves asymptotically like brownian motion author

some fundamental properties of multiple valued kleenean functions and determination of their logic formulas multiple valued kleenean functions that are models of a kleene algebra and are logic functions expressed by logic formulas composed of variables constants and logic operations and or and not are discussed the set of kleenean functions is a model with the largest number of logic functions among existing models of a kleene algebra such as fuzzy logic functions regular ternary logic functions and b ternary logic functions mainly it is shown that any p valued kleenean function is derived from a monotonic ternary input functions and any p valued unate function is derived from a unate binary input function the mapping relations between them and the method to determine the logic formula of the kleenean function and unate function from that of the monotonic ternary input function and unate binary input function respectively are classified 7 or less valued kleenean functions and unate functions of 3 or fewer variables are enumerated it is known that the number of p valued kleenean functions increases stepwise and that of unate functions increases smoothly as p becomes larger

estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials pour des essais de confirmation destines a prendre une decision correcte il est important que des experiences adaptees fournissent une inference avec un niveau nominal correct des estimateurs sans biais et des intervalles de confiance pour comparer les traitements dans les essais en cours mais une estimation ponctuelle simple et son intervalle de confiance sont souvent biaises dans des experiences sequentielles adaptatives nous developpons une nouvelle procedure d estimation suivant un test a partir d une taille d echantillon d une experience de re estimation pour obtenir un intervalle de confiance exact et une estimation ponctuelle la methode et basee sur la propriete generale de la distribution d une fonction pivot d un essai clinique sequentiel de groupe de shen et fisher 1999 on propose un estimateur modifie tenant explicitement compte de la limite d arret avec biais reduits quand les tailles de blocs sont petites nous montrons que les estimateurs sont convergents le calcul des estimations est elementaire nous fournissons aussi une fonction de ponderation modifiee pour ameliorer la puissance du test de nombreuses etudes de simulation montrent que les intervalles de confiance exacts ont une probabilite nominale de recouvrement precise et les estimateurs ponctuels sont pratiquement sans biais pour les tailles d echantillon habituels

confidence intervals for secondary parameters following a sequential test in sequential studies formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study however the completed data set may warrant a more detailed analysis involving the estimation of parameters corresponding to effects that were not used to determine when to stop and yet correlated with those that were this paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used the method assumes that information accumulates on the primary and secondary parameters at proportional rates this requirement will be valid in many potential applications but only in limited situations in survival analysis

exact confidence intervals following a group sequential trial a comparison of methods summary with interest in designing group sequential clinical trials increasing methodology for analysing the data arising in such trials is needed we discuss several approaches to constructing confidence intervals for the parameter of interest in a group sequential trial after the trial has ended we compare these met1fods to one already in the literature and point out interesting differences

p value adjustment in sequential clinical trials many randomized clinical trials utilize a group sequential monitoring procedure for assessing treatment efficacy during the course of the trial for a group sequential trial in which the null hypothesis is ultimately rejected the nominal p value generally overstates the statistical significance of the result and some adjustment is required several orderings of the sample space have been proposed for performing this adjustment each with unique operating characteristics in this article we compare four proposed methods with respect to the degree to which each captures the strength of evidence against the null hypothesis implied by the data we conclude that the ordering of the sample space induced by the observed z score has the most desirable operating characteristics

meaningful special classes of ternary logic functions regular ternary logic functions and ternary majority functions special ternary logic functions namely regular functions suitable for treating ambiguity and majority functions based on an extended majority principle are considered the one to one correspondence between n ary monotone regular ternary logic functions and n 1 ary monotone boolean functions is investigated as well as the one to one correspondence between n ary monotone ternary majority functions and n 1 ary monotone binary threshold functions the ternary majority functions are enumerated

approximately optimal one parameter boundaries for group sequential trials we present a class of group sequential tests indexed by a single parameter that yields approximately optimal results we also provide tables of key values to help in the design of group sequential tests that meet selected specifications

robustness analysis of nonlinear feedback systems an input output approach this paper presents an approach to robustness analysis for nonlinear feedback systems we pursue a notion of model uncertainty based on the closeness of input output trajectories which is not tied to a particular uncertainty representation such as additive parametric structured etc the basic viewpoint is to regard systems as operators on signal spaces we present two versions of a global theory where stability is captured by induced norms or by gain functions we also develop local approaches over bounded signal sets and give a treatment for systems with potential for finite time escape we compute the relevant stability margin for several examples and demonstrate robustness of stability for some specific perturbations e g small time delays we also present examples of nonlinear control systems which have zero robustness margin and are destabilized by arbitrarily small gap perturbations the paper considers the case where uncertainty is present in the controller as well as the plant and the generalization of the approach to the case where uncertainty occurs in several subsystems in an arbitrary interconnection

confidence intervals for a normal mean following a group sequential test this paper presents methodology based on the likelihood ratio test developed for construction of confidence intervals for a normal mean following a group sequential test examples show that the confidence intervals produced by this method have accurate nominal probability of coverage and have generally shorter average length than that produced by tsiatis rosner and mehta 1984 biometrics 40 797 803

exact confidence intervals following a group sequential test a numerical method is used to compute confidence intervals which have exact coverage probabilities for the mean of a normal distribution following a group sequential test this method which uses an ordering of the sample space similar to that employed by siegmund 1978 biometrika 65 341 349 is contrasted with the usual confidence interval for the mean

p values for group sequential testing summary group sequential testing procedures are widely employed in long term clinical trials we discuss several methods for computing the p value following a group sequential trial based on different outcome space orderings we use a criterion defined in this paper to compare these methods and point out interesting differences

a note on p values under group sequential testing and nonproportional hazards summary group sequential designs are often used for periodically assessing treatment efficacy during the course of a clinical trial following a group sequential test p values computed under the assumption that the data were gathered according to a fixed sample design are no longer uniformly distributed under the null hypothesis of no treatment effect various sample space orderings have been proposed for computing proper p values following a group sequential test although many of the proposed orderings have been compared in the setting of time invariant treatment effects little attention has been given to their performance when the effect of treatment within an individual varies over time our interest here is to compare two of the most commonly used methods for computing proper p values following a group sequential test based upon the analysis time at and z statistic orderings with respect to resulting power functions when treatment effects on survival are delayed power under the at ordering is shown to be heavily influenced by the presence of a delayed treatment effect while power functions corresponding to the z statistic ordering remain robust under time varying treatment effects

stability and l 1 l1 to l 1 l1 performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays abstractsolutions to the interval observation problem for delayed impulsive and switched systems with l1 performance are provided the approach is based on first obtaining stability and l1 l1 to l1

lpv control and full block multipliers a novel construction of linear parameter varying controllers is presented for arbitrarily fast parameter variations it is shown how full block multipliers allow to considerably reduce conservatism various auxiliary results of independent interest are included

operating characteristics of sample size re estimation with futility stopping based on conditional power various methods have been described for re estimating the final sample size in a clinical trial based on an interim assessment of the treatment effect many re weight the observations after re sizing so as to control the pursuant inflation in the type i error probability alpha lan and trost estimation of parameters and sample size re estimation proceedings of the american statistical association biopharmaceutical section 1997 48 51 proposed a simple procedure based on conditional power calculated under the current trend in the data cpt the study is terminated for futility if cpt or cu or re sized by a factor m to yield cpt cu if cl cpt cu where cl and cu are pre specified probability levels the overall level alpha can be preserved since the reduction due to stopping for futility can balance the inflation due to sample size re estimation thus permitting any form of final analysis with no re weighting herein the statistical properties of this approach are described including an evaluation of the probabilities of stopping for futility or re sizing the distribution of the re sizing factor m and the unconditional type i and ii error probabilities alpha and beta since futility stopping does not allow a type i error but commits a type ii error then as the probability of stopping for futility increases alpha decreases and beta increases an iterative procedure is described for choice of the critical test value and the futility stopping boundary so as to ensure that specified alpha and beta are obtained however inflation in beta is controlled by reducing the probability of futility stopping that in turn dramatically increases the possible re sizing factor m the procedure is also generalized to limit the maximum sample size inflation factor such as at m max 4 however doing so then allows for a non trivial fraction of studies to be re sized at this level that still have low conditional power these properties also apply to other methods for sample size re estimation with a provision for stopping for futility sample size re estimation procedures should be used with caution and the impact on the overall type ii error probability should be assessed

reconciling nu gap metric and iqc based robust stability analysis this technical note elaborates on the flexibility of an approach that combines gap metric and integral quadratic constraint iqc based analysis in the study of uncertain feedback interconnections of distributed parameter transfer functions it is established that a standard gap ball robust stability result can be recovered within the blended iqc gap framework which only requires the existence of gap continuous paths within the uncertainty set of interest this is achieved in part by showing that sufficiently small gap balls are pathwise connected in the graph topology a linear fractional characterisation of the gap is a key ingredient this characterisation is underpinned by a certain j spectral factorisation also shown to exist herein

on the bias of maximum likelihood estimation following a sequential test summary tlhe bias of maximum likelihood estimates calculated at the end of a sequential procedure is investigated for the two sequential designs considered in detail the sequential probability ratio test and the triangular test this bias is appreciable a method of calculating an adjusted estimate with reduced bias is described and an approximation to the standard error of the new estimate is provided examples of the implementation of the method are given and its advantages and disadvantages relative to alternative approaches are discussed

a computationally simpler algorithm for the umvue of a normal mean following a group sequential trial when using data collected in a group sequential clinical trial the sample mean is no longer the uniform minimum variance unbiased estimator umvue of the mean of a normal distribution emerson 1993 computers and biomedical research 26 68 73 described an algorithm for computing the umvue in this setting this algorithm although computationally expensive used only the basic software necessary for deriving group sequential boundaries in this paper we present an improved algorithm that results in greatly decreased computation times