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

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

singular p laplacian equations with superlinear perturbation abstract we consider a nonlinear dirichlet problem driven by the p laplace operator and with a right hand side which has a singular term and a parametric superlinear perturbation we are interested in positive solutions and prove a bifurcation type theorem describing the changes in the set of positive solutions as the parameter 0 varies in addition we show that for every admissible parameter 0 the problem has a smallest positive solution u and we establish the monotonicity and continuity properties of the map u

design and analysis of group sequential tests based on the type i error spending rate function summary lan demets 1983 devised a method of constructing discrete group sequential boundaries by using the type i error spending rate function it is extended so as to generate asymmetric as well as symmetric two sided boundaries for clinical trials the design aspect of this procedure is explored in terms of the maximum sample size needed to achieve a desired level of power and the expected stopping times under both null and alternative hypotheses finally these properties are employed in search of appropriate type i error spending rate functions for differing situations

solutions of perturbed p laplacian equations with critical nonlinearity in this paper we study a perturbed p laplacian equation under some given conditions on v x we prove that the equation has at least one positive solution provided that e e for any n n it has at least n pairs solutions if e en and suppose there exists an orthogonal involution t rn rn such that v x p x and k x are t invariant then it has at least one pair of solutions which change sign exactly once provided that e e where e and en are sufficiently small positive numbers moreover these solutions ue 0 in w1 p rn as e 0

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

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

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

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

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

non analytic bergman and szego kernels for weakly pseudoconvex tube domains in mathbb c 2 for any weakly pseudoconvex tube domain in mathbb c 2 with real analytic boundary there exist points on the boundary off the diagonal where the bergman kernel and the szego kernel fail to be real analytic

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

solutions and positive solutions for superlinear robin problems we consider nonlinear nonhomogeneous robin problems with a p 1 superlinear reaction term which need not satisfy the ambrosetti rabinowitz condition we look for positive solutions and prove existence and multiplicity theorems for the particular case of the p laplacian we prove existence results under a different geometry near the origin we consider nonlinear nonhomogeneous robin problems with a p 1 superlinear reaction term which need not satisfy the ambrosetti rabinowitz condition we look for positive solutions and prove existence and multiplicity theorems for the particular case of the p laplacian we prove existence results under a different geometry near the origin

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

solutions of nonlinear schrodinger equation with fractional laplacian without the ambrosetti rabinowitz condition this paper is concerned with the existence of two nonnegative radial solutions of following nonlinear schrodinger equation with fractional laplacian u u f u in r n u h r n where 0 1 under certain assumptions we obtain that the above problem has at least two nontrivial radial solutions without assuming the ambrosetti rabinowitz condition by variational methods and concentration compactness principle the result extends one of the main results of felmer et al 2012

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

on a robin problem with indefinite weight and asymmetric reaction superlinear at abstract the existence of at least three nontrivial smooth solutions to a semilinear robin problem with indefinite potential and asymmetric nonlinearity super linear at is established both crossing and resonance are allowed proofs exploit variational methods truncation techniques and morse theory

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

multiplicity of weak solutions to non local elliptic equations involving the fractional p x laplacian this paper is devoted to study the several existence results of a sequence of infinitely many solutions to the nonlocal elliptic problem involving the fractional p x laplacian without assuming the ambrosetti and rabinowitz type condition the strategy of the proof for these results is to approach the problem variationally by using the fountain theorem and the dual fountain theorem in addition we prove that the sequence of weak solutions becomes bounded solutions

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

on a nonlinear and non homogeneous problem without a r type condition in orlicz sobolev spaces using variational methods we prove some existence and multiplicity results for a class of nonlinear and non homogeneous problems without a r type condition in orlicz sobolev spaces

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

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

existence of strong solutions of a p x laplacian dirichlet problem without the ambrosetti rabinowitz condition abstract in this paper we consider the existence of strong solutions of the following p x laplacian dirichlet problem via critical point theory d i v u p x 2 u f x u in u 0 on we give a new growth condition under which we use a new method to check the cerami compactness condition hence we prove the existence of strong solutions of the problem as above without the growth condition of the well known ambrosetti rabinowitz type and also give some results about multiplicity of the solutions

anisotropic equations with indefinite potential and competing nonlinearities abstract we consider a nonlinear dirichlet problem driven by a variable exponent p laplacian plus an indefinite potential term the reaction has the competing effects of a parametric concave sublinear term and a convex superlinear perturbation the anisotropic concave convex problem we prove a bifurcation type theorem describing the changes in the set of positive solutions as the positive parameter varies also we prove the existence of minimal positive solutions

superlinear problems without ambrosetti and rabinowitz growth condition superlinear elliptic boundary value problems without ambrosetti and rabinowitz growth condition are considered existence of nontrivial solution result is established by combining some arguments used by struwe and tarantello and schechter and zou also by wang and wei firstly by using the mountain pass theorem due to ambrosetti and rabinowitz is constructed a solution for almost every parameter by varying the parameter then it is considered the continuation of the solutions

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

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

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

unbiased estimation following a group sequential test it is shown that in a group sequential test about the drift 0 of a brownian motion x t stopped at time t the sufficient statistic t x t is not complete for 0 there exist infinitely many unbiased estimators of 0 and none has uniformly minimum variance a truncation adaptable criterion is proposed and the uniformly minimum variance estimator among all truncation adaptable unbiased estimators is found this estimator is identical to estimators of ferebee 1983 and emerson fleming 1990

robin problems with a general potential and a superlinear reaction abstract we consider semilinear robin problems driven by the negative laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the ambrosetti rabinowitz condition we prove existence and multiplicity theorems producing also an infinity of smooth solutions using variational tools truncation and perturbation techniques and morse theory critical groups

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

counterexamples to analytic hypoellipticity for domains of finite type consider the hypersurface z1 z2 q z2 p zi cc c2 where p r is a subharmonic nonharmonic polynomial such a surface is pseudoconvex more precisely is the boundary of a pseudoconvex domain and of finite type it is the boundary of a model domain f z2 p zi so called because such domains provide good approximations to general pseudoconvex domains of finite type in c2 see 37 11 a nonvanishing antiholomorphic tangent vector field is a oaz 1 2i ap oaz i o a29z as coordinates for the surface we use c x jr 3 z t z t ip z the vector field pulls back to l o oz i op 9z a3 t equip c x jr with lebesgue measure and consider the orthogonal projection from l2 onto its intersection with the kernel of l the intersection of this kernel with the schwartz space may be shown to have infinite dimension and it is well known that l fails to be c hypoelliptic however c hypoellipticity does hold in a modified form see 30 if lu e c in an open set and if in addition u lv for some v e l2 where l is the formal adjoint of l then u e c there our main purpose is to point out that the analogous assertion for real analytic hypoellipticity is false if p z rz m m an even integer 4 specifically for p a general homogeneous subharmonic nonharmonic polynomial let s z t w s be the szeg6 kernel that is the distribution kernel associated to the operator defined by the orthogonal projection of l2 c x jr with respect to lebesgue measure onto the kernel of l we shall see in section 5 that this projection maps co to c further s is smooth off the diagonal by 37 define the distribution

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

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

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

wang s multiplicity result for superlinear equations without the ambrosetti rabinowitz condition we consider a nonlinear elliptic equation driven by the sum of a p laplacian and a q laplacian where 1 q 2 p with a p 1 superlinear caratheodory reaction term which doesn t satisfy the usual ambrosetti rabinowitz condition using variational methods based on critical point theory together with techniques from morse theory we show that the problem has at leat three nontrivial solutions among them one is positive and one is negative

semiparametric efficiency and its implication on the design and analysis of group sequential studies abstract authors have shown that the time sequential joint distributions of many statistics used to analyze data arising from group sequential time to event and longitudinal studies are multivariate normal with an independent increments covariance structure in theorem 1 of this article we demonstrate that this limiting distribution arises naturally when one uses an efficient test statistic to test a single parameter in a semiparametric or parametric model because we are able to think of many of the statistics in the literature in this fashion the limiting distribution under investigation is just a special case of theorem 1 using this general structure we then develop an information based design and monitoring procedure that can be applied to any type of model for any type of group sequential study provided that there is a unique parameter of interest that can be efficiently tested

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

the existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p laplacian type without the ambrosetti rabinowitz condition abstract in this paper we study the existence of a nontrivial solution to the following nonlinear elliptic boundary value problem of p laplacian type p p u f x u x u 0 x where p 1 r 1 r n is a bounded domain and p u d i v u p 2 u is the p laplacian of u f c 0 r 1 r 1 is p superlinear at t 0 and subcritical at t we prove that under suitable conditions for all 0 the problem p has at least one nontrivial solution without assuming the ambrosetti rabinowitz condition our main result extends a result for p for when p 2 given by miyagaki and souto 2008 in 8 to the general problem p where p 1 meanwhile our result is stronger than a similar result of li and zhou 2003 given in 15

asymptotic properties of the sequential empirical roc ppv and npv curves under case control sampling the receiver operating characteristic roc curve the positive predictive value ppv curve and the negative predictive value npv curve are three measures of performance for a continuous diagnostic biomarker the roc ppv and npv curves are often estimated empirically to avoid assumptions about the distributional form of the biomarkers recently there has been a push to incorporate group sequential methods into the design of diagnostic biomarker studies a thorough understanding of the asymptotic properties of the sequential empirical roc ppv and npv curves will provide more flexibility when designing group sequential diagnostic biomarker studies in this paper we derive asymptotic theory for the sequential empirical roc ppv and npv curves under case control sampling using sequential empirical process theory we show that the sequential empirical roc ppv and npv curves converge to the sum of independent kiefer processes and show how these results can be used to derive asymptotic results for summaries of the sequential empirical roc ppv and npv curves

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

existence of solutions to a class of semilinear elliptic equations involving general subcritical growth in this paper we consider the semilinear elliptic equation u f x u with the dirichlet boundary value and under suitable assumptions on the nonlinear term f with a more general growth condition some existence results of solutions are given for all 0 via the variational method and some analysis techniques

multiple solutions for a class of double phase problem without the ambrosetti rabinowitz conditions abstract in the present paper in view of the variational approach we consider the existence and multiplicity of weak solutions for a class of the double phase problem div u p 2 u a x u q 2 u f x u in u 0 on where n 2 and 1 p q n firstly by the fountain and dual theorem with cerami condition we obtain some existence of infinitely many solutions for the above problem under some weaker assumptions on f secondly we prove that this problem has at least one nontrivial solution for any parameter 0 small enough and also that the solution blows up in the sobolev norm as 0 finally by imposing additional assumptions on f we establish the existence of infinitely many solutions by using krasnoselskii s genus theory for the above equation

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

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

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

inference about a secondary process following a sequential trial summary we consider the following sequential testing problem a group sequential or fullysequential test is carried out for a primary parameter using a score process or an effective score process to eliminate nuisance parameters after stopping the possibility of additional parameters is considered and appropriate tests and estimators are desired that recognise the sequential stopping rule we formulate an asymptotic multi dimensional gaussian process form of such problems and then construct tests and confidence procedures optimality conditions are given and an example is summarised

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