optimal group sequential designs for simultaneous testing of superiority and non inferiority confirmatory clinical trials comparing the efficacy of a new treatment with an active control typically aim at demonstrating either superiority or non inferiority in the latter case the objective is to show that the experimental treatment is not worse than the active control by more than a pre specified non inferiority margin we consider two classes of group sequential designs that combine the superiority and non inferiority objectives non adaptive designs with fixed group sizes and adaptive designs where future group sizes may be based on the observed treatment effect for both classes we derive group sequential designs meeting error probability constraints that have the lowest possible expected sample size averaged over a set of values of the treatment effect these optimized designs provide an efficient means of reducing expected sample size under a range of treatment effects even when the separate objectives of proving superiority and non inferiority would require quite different fixed sample sizes we also present error spending versions of group sequential designs that are easily implementable and can handle unpredictable group sizes or information levels we find the adaptive choice of group sizes to yield some modest efficiency gains alternatively expected sample size may be reduced by adding another interim analysis to a non adaptive group sequential design

recursive combination tests we present a method that extends the flexibility of adaptive designs to the number of interim analyses and to the choice of decision boundaries at each stage of the trial the design of the next stage can be determined using all of the information gathered so far additionally one can specify the next stage as the final stage or can plan a further interim analysis the method is based on a recursive application of the two stage combination tests for p values the crucial point is the appropriate definition of a p value function that combines p values from two separate stages to a single p value formally we start with a two stage combination test however the p value of the second stage can be replaced by the p value of a further combination test this applies if the experimenter decides in the first interim analysis to perform another interim analysis thereby extending the design to at least three stages obviously this can be also done in a recursive way in the following interim analyses the test

adaptive designs with correlated test statistics in clinical trials the collected observations such as clustered data or repeated measurements are often correlated as a consequence test statistics in a multistage design are correlated adaptive designs were originally developed for independent test statistics we present a general framework for two stage adaptive designs with correlated test statistics we show that the significance level for the bauer kohne design is inflated for positively correlated test statistics from a bivariate normal distribution the decision boundary for the second stage can be modified so that type one error is controlled this general concept is expandable to other adaptive designs in order to use these designs the correlation between test statistics has to be estimated for a known covariance matrix we show how correlation can be determined within the framework of linear mixed models a sample size reassessment rule is proposed and evaluated for an unknown covariance matrix by simulation as wald test statistics in linear mixed models have independent increments we use this property to create valid test procedures we compare these procedures with the proposed design in our simulations

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

group sequential methods for clinical trials allowing early acceptance of ho and incorporating costs summary when interim accumulated results from a clinical trial clearly indicate that the null hypothesis should be accepted or rejected ethical and cost considerations may dictate terminating the trial and making the appropriate decision pocock 1977 has described a group sequential procedure for clinical trials that permits early termination only for rejecting the null hypothesis this paper extends pocock s procedure to allow early termination with acceptance of the null hypothesis theoretical and simulation results demonstrate that designs allowing early acceptance of the null hypothesis can have lower expected costs than designs that do not even when the alternative hypothesis is true

construction of a continuous stopping boundary from an alpha spending function lan and demets 1983 biometrika 70 659 663 proposed a flexible method for monitoring accumulating data that does not require the number and times of analyses to be specified in advance yet maintains an overall type i error alpha their method amounts to discretizing a preselected continuous boundary by clumping the density of the boundary crossing time at discrete analysis times and calculating the resultant discrete time boundary values in this framework the cumulative distribution function of the continuous time stopping rule is used as an alpha spending function a key assumption that underlies this method is that future analysis times are not chosen on the basis of the current value of the statistic however clinical trials may be monitored more frequently when they are close to crossing the boundary in this situation the corresponding continuous time boundary should be used here we demonstrate how to construct a continuous stopping boundary from an alpha spending function this capability is useful in the design of clinical trials we use the beta blocker heart attack trial bhat and aids clinical trials group protocol 021 for illustration

sequential medical trials involving paired data summary a general approach to sequential decision theoretic problems is illustrated with anscombe s formulation of the problem of comparing two treatments in a medical trial the bayes sequential procedure for a continuous time verion of the problem is explicitly determined suboptimal procedures are proposed and evaluated asymptotic results and numerical descriptions are provided the continuous time version is found to provide accurate approximations even for clinical trials involving relatively small horizon sizes a natural formulation for many statistical problems is one combining bayesian sequential and decision theoretic aspects for deciding the sign of a normal mean chernoff 1961 1965a 1965b breakwell chernoff 1964 and bather 1962 replaced sums of successive observations by a continuous time wiener process the heat equation plays a prominent role in the resulting analysis since optimal procedures may be characterized in terms of solutions of free boundary problems involving the heat equation it would seem that the formidable appearance of some of the analysis involved in this approach has distracted potential users from taking advantage of other aspects which are easy or routine and which contribute clarity asymptotic results with important theoretical implications and numerical descriptions of sequential procedures and the resulting risks recent work by lai levin robbins siegmund 1980 and by begg mehta 1979 on a model for sequential medical trials proposed by anscombe 1963 has prompted us to investigate the same problem using the above approach in this paper we give the optimal procedure and show how it compares with suboptimal procedures secondly we use anscombe s model to illustrate how such results may be obtained for similar problems the key feature is the asymptotic resemblance of the problem to an optimal stopping problem for a wiener process since many sequential decision theoretic problems can be framed in terms of efficient scores which are asymptotically normal the generality of the approach should be apparent the formulation we refer to as anscombe s model is the following there is a horizon of n patients to be treated by one of two available treatments in the initial experimental phase n pairs of patients are treated sequentially with different treatments randomly assigned to the patients in each pair the differences xi in the values of a continuous outcome variable for the ith pair are assumed to be independently and normally

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

on the inefficiency of the adaptive design for monitoring clinical trials summary adaptive designs which allow the sample size to be modified based on sequentially computed observed treatment differences have been advocated recently for monitoring clinical trials although such methods have a great deal of appeal on the surface we show that such methods are inefficient and that one can improve uniformly on such adaptive designs using standard group sequential tests based on the sequentially computed likelihood ratio test statistic

on sample size and inference for two stage adaptive designs proschan and hunsberger 1995 biometrics 51 1315 1324 proposed a two stage adaptive design that maintains the type i error rate for practical applications a two stage adaptive design is also required to achieve a desired statistical power while limiting the maximum overall sample size in our proposal a two stage adaptive design is comprised of a main stage and an extension stage where the main stage has sufficient power to reject the null under the anticipated effect size and the extension stage allows increasing the sample size in case the true effect size is smaller than anticipated for statistical inference methods for obtaining the overall adjusted p value point estimate and confidence intervals are developed an exact two stage test procedure is also outlined for robust inference

stieltjes transforms and the stokes phenomenon recently berry olver and jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated these uniform expansions describe the rapid change in the behaviour of the remainders as a stokes line is crossed we show how such uniform expansions may be found when a function can be expressed as a stieltjes transform such an approach has the following advantages the uniform expansion is calculated directly non uniform expansions away from the stokes line are readily found and explicit error bounds may be established we illustrate the method by application to the modified bessel function k v z

sequential testing in clinical trials with repeated measurements summary in some clinical trials the number of subjects on each treatment is fixed and at successive equally spaced time points each subject yields a measurement which reflects response to treatment a model is presented for such repeated measurements data a sequential procedure is proposed for testing the null hypothesis of no difference in mean effect between two treatments in this procedure the maximum number of measurements per subject and absorption probabilities for intermediate significance tests are preassigned to yield a required approximate overall significance level the jth significance test follows the jth measurement on each subject and the test depends on values of model parameters estimated from all the data available so far simulations show that the proposed procedure yields overall significance levels close to 5 when required for data from a variety of

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

optimal asymmetric one sided group sequential tests summary we extend the optimal symmetric group sequential tests of eales jennison 1992 to the broader class of asymmetric designs two forms of asymmetry are considered involving unequal type i and type ii error rates and different emphases on expected sample sizes at the null and alternative hypotheses we discuss the properties of our optimal designs and use them to assess the efficiency of the family of tests proposed by pampallona tsiatis 1994 and two families of one sided tests defined through error spending functions we show that the error spending designs are highly efficient while the easily implemented tests of pampallona tsiatis are a little less efficient but still not far from optimal our results demonstrate that asymmetric designs can decrease the expected sample size under one hypothesis but only at the expense of a significantly larger expected sample size under the other hypothesis

two stage selection and testing designs for comparative clinical trials summary a two stage design which selects the best of several experimental treatments and compares it to a standard control is proposed the design allows early termination with acceptance of the global null hypothesis optimal sample size and cut off parameters are obtained by minimizing expected total sample size for fixed significance level and power with many other factors moreover the usual error rate computations associated with comparative testing do not account for the preliminary selection process consequently the overall procedure may be neither effective nor efficient for identifying an experimental treatment which is an improvement over c in this paper we propose a new approach to the problem of identifying the best of k experimental treatments and determining whether it is superior to a control we deal with the binomial setting where patient response may be characterized as either success or failure with ok the success probability for ek k 0 corresponds to c for ease of notation assume 01 ok we propose a two stage procedure which allows early termination with acceptance of h0 o 01 ok with design parameters chosen to minimize expected total sample size the design and accompanying generalized definitions of size a and power 1 3 are given in 2 the algorithm used for optimization is described in 3 numerical results are presented in 4 followed by a discussion of the relative merits of the proposed design

stepwise likelihood ratio statistics in sequential studies versus the simple null hypothesis we show that under the null hypothesis the probability that this stepwise likelihood ratio will ever be greater than c will not exceed 1 c in contrast under the composite alternative hypothesis this ratio will generally converge in probability to oo these results suggest that a stepwise likelihood ratio statistic can be useful in a sequential study for testing a composite alternative versus a simple null hypothesis for illustration we conduct two simulation studies one for a normal response and one for an exponential response to compare the performance of a sequential test based on a stepwise likelihood ratio statistic with a constant boundary versus some existing approaches

optimal tests of treatment effects for the overall population and two subpopulations in randomized trials using sparse linear programming we propose new optimal methods for analyzing randomized trials when it is suspected that treatment effects may differ in two predefined subpopulations such subpopulations could be defined by a biomarker or risk factor measured at baseline the goal is to simultaneously learn which subpopulations benefit from an experimental treatment while providing strong control of the familywise type i error rate we formalize this as a multiple testing problem and show it is computationally infeasible to solve using existing techniques our solution involves a novel approach in which we first transform the original multiple testing problem into a large sparse linear program we then solve this problem using advanced optimization techniques this general method can solve a variety of multiple testing problems and decision theory problems related to optimal trial design for which no solution was previously available in particular we construct new multiple testing procedures that satisfy minimax and bayes optimality criteria for a given optimality criterion our new approach yields the optimal tradeoff between power to detect an effect in the overall population versus power to detect effects in subpopulations we demonstrate our approach in examples motivated by two randomized trials of new treatments for hiv

exact analytical solution for coupled time independent schrodinger equations with certain model potentials an exact analytical solution is obtained for coupled time independent schrodinger equations with model potentials and this is made possible by solving a single fourth order ordinary differential equation derived from the original coupled equations exact closed form solutions for the non adiabatic transition matrices or scattering matrices are found for scattering boundary conditions with three two and one open channels respectively how to apply the present results to deal with general potentials is also briefly analysed

the dual test safeguarding p value combination tests for adaptive designs many modern adaptive designs apply an analysis where p values from different stages are weighted together to an overall hypothesis test one merit of this combination approach is that the design can be made very flexible however combination tests violate the sufficiency and conditionality principles as a consequence combination tests may lead to absurd conclusions such as proving a positive effect while the average effect is negative we explore the possibility of modifying the test so that such illogical conclusions are no longer possible the dual test requires both the weighted combination test and a nave test ignoring the adaptations to be statistically significant the result is that the flexibility and type i error level control of the combination test are preserved while the nave test adds a safeguard against unconvincing results the dual test is by construction at least as conservative as the combination test however many design changes will not lead to any power loss a typical situation where the combination approach can be used is two stage sample size reestimation ssr for this case we give a complete specification of all sample size modifications for which the two tests are equally powerful we also study the overall power loss for some suggested ssr rules rules based on conditional power generally lead to ignorable power loss while a decision analytic approach exhibits clear discrepancies between the two tests

increasing the sample size when the unblinded interim result is promising increasing the sample size based on unblinded interim result may inflate the type i error rate and appropriate statistical adjustments may be needed to control the type i error rate at the nominal level we briefly review the existing approaches which allow early stopping due to futility or change the test statistic by using different weights or adjust the critical value for final test or enforce rules for sample size recalculation the implication of early stopping due to futility and a simple modification to the weighted z statistic approach are discussed in this paper we show that increasing the sample size when the unblinded interim result is promising will not inflate the type i error rate and therefore no statistical adjustment is necessary the unblinded interim result is considered promising if the conditional power is greater than 50 per cent or equivalently the sample size increment needed to achieve a desired power does not exceed an upper bound the actual sample size increment may be determined by important factors such as budget size of the eligible patient population and competition in the market the 50 per cent conditional power approach is extended to a group sequential trial with one interim analysis where a decision may be made at the interim analysis to stop the trial early due to a convincing treatment benefit or to increase the sample size if the interim result is not as good as expected the type i error rate will not be inflated if the sample size may be increased only when the conditional power is greater than 50 per cent if there are two or more interim analyses in a group sequential trial our simulation study shows that the type i error rate is also well controlled

adaptive and nonadaptive group sequential tests summary methods have been proposed for redesigning a clinical trial at an interim stage in order to increase power in order to preserve the type i error rate methods for unplanned design change have to be defined in terms of nonsufficient statistics and this calls into question their efficiency and the credibility of conclusions reached we evaluate schemes for adaptive redesign extending the theoretical arguments for use of sufficient statistics of tsiatis mehta 2003 and assessing the possible benefits of preplanned adaptive designs by numerical computation of optimal tests these optimal adaptive designs are concrete examples of optimal sequentially planned sequential tests proposed by schmitz 1993 we conclude that the flexibility of unplanned adaptive designs comes at a price and we recommend that the appropriate power for a study should be determined as thoroughly as possible at the outset then standard error spending tests possibly with unevenly spaced analyses provide efficient designs but it is still possible to fall back on flexible methods for redesign should study objective change unexpectedly once the trial is under way

asymptotically optimal multistage tests of simple hypotheses a family of variable stage size multistage tests of simple hypotheses is described based on efficient multistage sampling procedures using a loss function that is a linear combination of sampling costs and error probabilities these tests are shown to minimize the integrated risk to second order as the costs per stage and per observation approach zero a numerical study shows significant improvement over group sequential tests in a binomial testing problem

exact confidence bounds following adaptive group sequential tests summary we provide a method for obtaining confidence intervals point estimates and p values for the primary effect size parameter at the end of a two arm group sequential clinical trial in which adaptive changes have been implemented along the way the method is based on applying the adaptive hypothesis testing procedure of muller and schafer 2001 to a sequence of dual tests derived from the stage wise adjusted confidence interval of tsiatis rosner and mehta 1984 in the non adaptive setting this confidence interval is known to provide exact coverage in the adaptive setting exact coverage is guaranteed provided the adaptation takes place at the penultimate stage in general however all that can be claimed theoretically is that the coverage is guaranteed to be conservative nevertheless extensive simulation experiments supported by an empirical characterization of the conditional error function demonstrate convincingly that for all practical purposes the coverage is exact and the point estimate is median unbiased no procedure has previously been available for producing confidence intervals and point estimates with these desirable properties in an adaptive group sequential setting the methodology is illustrated by an application to a clinical trial of deep brain stimulation for parkinson s disease

adaptive group sequential designs for clinical trials combining the advantages of adaptive and of classical group sequential approaches a general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing this allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs the concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans

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

sequential changepoint detection in quality control and dynamical systems after a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection this theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on line implementation and yet are nearly optimal under several performance criteria

adaptive design and estimation in randomized clinical trials with correlated observations summary clinical trial designs involving correlated data often arise in biomedical research the intracluster correlation needs to be taken into account to ensure the validity of sample size and power calculations in contrast to the fixed sample designs we propose a flexible trial design with adaptive monitoring and inference procedures the total sample size is not predetermined but adaptively reestimated using observed data via a systematic mechanism the final inference is based on a weighted average of the block wise test statistics using generalized estimating equations where the weight for each block depends on cumulated data from the ongoing trial when there are no significant treatment effects the devised stopping rule allows for early termination of the trial and acceptance of the null hypothesis the proposed design updates information regarding both the effect size and within cluster correlation based on the cumulated data in order to achieve a desired power estimation of the parameter of interest and its confidence interval are proposed we conduct simulation studies to examine the operating characteristics and illustrate the proposed method with an example

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

existence of minimizers for nonlevel convex supremal functionals the paper is devoted to determining necessary and sufficient conditions for existence of solutions to the problem inf mathop rm ess sup x in omega f nabla u x u in u 0 w 1 infty 0 omega when the supremand f is not necessarily level convex these conditions are obtained through a comparison with the related level convex problem and are written in terms of a differential inclusion involving the boundary datum several conditions of convexity for the supremand f are also investigated

optimization of multistage testing times and critical values in clinical trials optimal critical values and times of intermediate testing points are determined to minimize expected sample size in a variety of situations two stage and three stage models are considered for a variety of values of power we investigate the robustness of the optimality parameters

bayesian adaptive designs for clinical trials summary a bayesian adaptive design is proposed for a comparative two armed clinical trial using decision theoretic approaches a loss function is specified based on the cost for each patient and the costs of making incorrect decisions at the end of a trial at each interim analysis the decision to terminate or to continue the trial is based on the expected loss function while concurrently incorporating efficacy futility and cost the maximum number of interim analyses is determined adaptively by the observed data we derive explicit connections between the loss function and the frequentist error rates so that the desired frequentist properties can be maintained for regulatory settings the operating character istics of the design can be evaluated on frequentist grounds extensive simulations are carried out to compare the proposed design with existing ones the design is general enough to accommodate both continuous and discrete types of data we illustrate the methods with an animal study evaluating a medical treatment for cardiac arrest

semiclassically weak reflections above analytic and non analytic potential barriers the coefficient r for reflection above a barrier v x is computed semiclassically i e as h cross to 0 employing an exact multiple reflection series whose mth term is a 2m 1 fold integral if v x is analytic all terms have the same semiclassical order exp h cross 1 the multiple integrals are evaluated exactly and the series summed if v x has a discontinuous nth derivative the term m 1 dominates semiclassically and gives r approximately h cross n if v x has all derivatives continuous but possesses an essential singularity on the real axis the term m 1 again dominates semiclassically and for v approximately exp mod x mod n gives r approximately exp h cross n n 1 with an oscillatory factor corresponding to transmission resonances the formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de broglie wavelength is less than 1 of the barrier width and mod r mod 2 approximately 10 1000

efficient adaptive designs with mid course sample size adjustment in clinical trials adaptive designs have been proposed for clinical trials in which the nuisance parameters or alternative of interest are unknown or likely to be misspecified before the trial although most previous works on adaptive designs and mid course sample size re estimation have focused on two stage or group sequential designs in the normal case we consider here a new approach that involves at most three stages and is developed in the general framework of multiparameter exponential families this approach not only maintains the prescribed type i error probability but also provides a simple but asymptotically efficient sequential test whose finite sample performance measured in terms of the expected sample size and power functions is shown to be comparable to the optimal sequential design determined by dynamic programming in the simplified normal mean case with known variance and prespecified alternative and superior to the existing two stage designs and also to adaptive group sequential designs when the alternative or nuisance parameters are unknown or misspecified

histories of adiabatic quantum transitions the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation for a 2 state quantum system by transforming to a series of superadiabatic bases clinging ever closer to the exact evolving state it is shown that transition histories renormalize onto a universal one in which the amplitude grows to its final value as an error function rather than via large oscillations as in the ordinary adiabatic basis the time for the universal transition is of order surd hslash delta where delta is the small adiabatic slowness parameter in perturbation theory the pre exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value textstyle frac 1 3 pi for the ordinary adiabatic basis to the correct value unity the various histories could be observed in spin experiments

issues in designing flexible trials we outline the general framework of adaptive combination tests and discuss their relationship to flexible group sequential designs an important field of applications is sample size reassessment we discuss reassessment rules based on conditional power arguments using either the observed or the prefixed effect size these rules tend to lead to large expected sample sizes for small actual effects however the application of a maximal bound for the second stage sample size leads to more favourable properties additionally we consider an optimized reassessment rule in terms of expected sample sizes since the adaptive design does not use the classical test statistics for some types of sample size reassessments the adaptive test may reject the null hypothesis while the classical one sample test does not we characterize sample size reassessment rules where such inconsistencies are avoided finally the extension of flexibility to the number of stages is explored in the first interim analysis a second interim analysis is only planned if the chance to achieve a decision there is high this leads to savings in the average number of interim analysis performed without paying a noticeable price in terms of expected sample size

estimation in flexible two stage designs adaptive test designs for clinical trials allow for a wide range of data driven design adaptations using all information gathered until an interim analysis the basic principle is to use a test statistics which is invariant with respect to the design adaptations under the null hypothesis this allows for a control of the type i error rate for the primary hypothesis even for adaptations not specified a priori in the study protocol estimation is usually another important part of a clinical trial however is more difficult in adaptive designs in this research paper we give an overview of point and interval estimates for flexible designs and compare methods for typical sample size rules we also make some proposals for confidence intervals which have nominal coverage probability also after an unforeseen design adaptation and which contain the maximum likelihood estimate and the usual unadjusted confidence interval

quasi exactly solvable lie superalgebras of differential operators in this paper we study lie superalgebras of matrix valued first order differential operators on the complex line we first completely classify all such superalgebras of finite dimension among the finite dimensional superalgebras whose odd subspace is non trivial we find those admitting a finite dimensional invariant module of smooth vector valued functions and classify all the resulting finite dimensional modules the latter lie superalgebras and their modules are the building blocks in the construction of quasi exactly solvable quantum mechanical models for spin particles in one dimension

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

stokes constants for a certain class of second order ordinary differential equations by generalizing the coupled wave integral equations method devised by hinton the stokes phenomena of the standard second order ordinary differential equations with the following coefficient functions q z are analyzed i q z a2nz2n j n 1ajzj ii q z a2n 1z2n 1 j n 2ajz j iii q z j 04ajz j with a3 0 and iv q z j 03ajz j with a2 0 first the relations are derived among the stokes constants which enable the expression of all stokes constants in terms of only one furthermore this one stokes constant is shown to be expressed in the analytical form of a convergent infinite series as a function of the coefficients of q z this was done for all four cases listed above

waves near stokes lines the large k asymptotics of d 2 u z d z 2 k 2 r 2 z u z are studied near a stokes line z z 0 r d z real where z 0 is a zero of r 2 z of any order on which there is greatest disparity between the dominant and subdominant exponential waves in the phase integral wkb approximations the aim is to establish precisely how the multiplier b of the subdominant wave varies across the stokes line although b always has a total change proportional to i times the multiplier of the dominant wave the stokes phenomenon the form of the change depends on the convention used to define the two waves the optimal convention for which the variation is maximally compact and smooth is to define them by the phase integral approximation truncated at its least term whose order is proportional to k and therefore large asymptotics of asymptotics then the variation of b is proportional to the error function of the natural stokes crossing variable im k re this result is obtained without resumming divergent series thereby avoiding asymptotics of asymptotics of asymptotics an application is given to the birth of exponentially weak reflected waves in media with smoothly varying refractive index

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

an improved method for deriving optimal one sided group sequential tests summary we derive one sided group sequential tests for normal responses which minimize expected sample size minimization is at a single value of the normal mean or integrated with respect to a normal density the methods employed are much faster and also numerically more stable than those of jennison 1987 they provide solutions for cases with as many as 200 groups and for small numbers of groups facilitate optimization over the choice of group sizes we present a new method for constructing near optimal tests when group sizes are unpredictable based on interpolating the error spending function of an optimal test with a large number of groups

homogenization in l abstract homogenization of deterministic control problems with l running cost is studied by viscosity solutions techniques it is proved that the value function of an l problem in a medium with a periodic micro structure converges uniformly on the compact sets to the value function of the homogenized problem as the period shrinks to 0 our main convergence result extends that of ishii stochastic analysis control optimization and applications pp 305 324 birkhauser boston boston ma 1999 to the case of a discontinuous hamiltonian the cell problem is solved but as non uniqueness occurs the effective hamiltonian must be selected in a careful way the paper also provides a representation formula for the effective hamiltonian and gives illustrations to calculus of variations averaging and one dimensional problems

efficient group sequential tests with unpredictable group sizes summary we derive group sequential tests to minimize expected sample size a parametric family of group sequential tests is proposed containing tests which are nearly optimal in terms of minimizing expected sample size amongst all group sequential tests these parametric tests can be implemented when group sizes are unequal and unpredictable they are shown to achieve error rates close to their nominal level in a wide range of situations and to be efficient when compared with other tests based on the same group sizes and achieving the same error rates we consider group sequential procedures for choosing between two hypotheses concerning the mean of a normal distribution with known variance the group sequential approach pocock 1977 offers a practicable alternative to fully sequential tests which require accumulating data to be examined as each new observation is recorded it is also desirable to ensure the maximum sample size needed in a long term experiment is not much greater than that of the corresponding fixed sample size test and we shall take the specification of the maximum allowable sample size and the number of groups of observations as our starting point in deriving sequential procedures in 2 we formulate the testing problem and discuss its relevance to clinical trials in 3 we describe the derivation of boundaries which are optimal in the sense of minimizing various expected sample sizes subject to a fixed number of equally sized groups and a fixed maximum sample size in 4 we introduce a parametric family of group sequential tests containing tests which are nearly optimal by the criteria of 3 these tests are easily adapted to allow unequal and unpredictable group sizes we show in 5 that they remain efficient when actual group sizes are quite different from those anticipated programs used for the calculations are available from the author

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

interim analysis and sample size reassessment this article deals with sample size reassessment for adaptive two stage designs based on conditional power arguments utilizing the variability observed at the first stage fisher s product test for the p values from the disjoint samples at the two stages is considered in detail for the comparison of the means of two normal populations we show that stopping rules allowing for the early acceptance of the null hypothesis that are optimal with respect to the average sample size may lead to a severe decrease of the overall power if the sample size is a priori underestimated this problem can be overcome by choosing designs with low probabilities of early acceptance or by midtrial adaptations of the early acceptance boundary using the variability observed in the first stage this modified procedure is negligibly anticonservative and preserves the power

optimal flexible sample size design with robust power it is well recognized that sample size determination is challenging because of the uncertainty on the treatment effect size several remedies are available in the literature group sequential designs start with a sample size based on a conservative smaller effect size and allow early stop at interim looks sample size re estimation designs start with a sample size based on an optimistic larger effect size and allow sample size increase if the observed effect size is smaller than planned different opinions favoring one type over the other exist we propose an optimal approach using an appropriate optimality criterion to select the best design among all the candidate designs our results show that 1 for the same type of designs for example group sequential designs there is room for significant improvement through our optimization approach 2 optimal promising zone designs appear to have no advantages over optimal group sequential designs and 3 optimal designs with sample size re estimation deliver the best adaptive performance we conclude that to deal with the challenge of sample size determination due to effect size uncertainty an optimal approach can help to select the best design that provides most robust power across the effect size range of interest copyright 2016 john wiley sons ltd

full asymptotic expansion of transition probabilities in the adiabatic limit the authors consider a two level quantum mechanical system driven by an analytic time dependent hamiltonian of the form h in t in the adiabatic limit in or 0

sample size reestimation for clinical trials with censored survival data a flexible design with updating of sample size in clinical trials is proposed for censored survival data patients enter trials serially and are subject to random loss to follow up the statistical inference is based on a weighted average of the linear rank statistics where the weight function at each look depends on prior observed data a stopping rule is devised to allow early termination and acceptance of the null hypothesis when the experimental treatment offers no advantage or is inferior to the control the null hypothesis that two survival distributions are equal may be rejected only at the last step when the weight function is used up the overall type i error rate is preserved independent increments for the sequential linear rank statistic is key to deriving asymptotic properties of the test statistic some extensive simulations have been carried out to compare the operating characteristics of the method under different scenarios and for a comparison with the usual log rank test with fixed sam

two sample repeated significance tests based on the modified wilcoxon statistic abstract the asymptotic distribution theory of sequentially computed modified wilcoxon scores is developed for two sample survival data with random staggered entry and random loss to follow up the asymptotic covariance indicates generally dependent modified wilcoxon increments contradicting the authors reading of jones and whitehead 1979 a repeated significance testing procedure is presented for testing the equality of two survival distributions based on the asymptotic theory the early stopping properties of this procedure are illustrated by a prostate cancer example

two stage clinical trial stopping rules two stage stopping rules for clinical trials are considered the nominal significance level needed for the second stage test for any choice of first stage significance level is derived for rules with overall significance levels of 01 and 05 and for studies with either half or two thirds of the patients analyzed in the first stage a graphical demonstration is given of the inherent tradeoff between power and expected sample size or probability of early termination a specific rule intermediate to those advocated by pocock 1977 biometrika 64 191 199 and o brien and fleming 1979 biometrics 5 549 556 is recommended

efficient group sequential designs when there are several effect sizes under consideration we consider the construction of efficient group sequential designs where the goal is a low expected sample size not only at the null hypothesis and the alternative taken to be the minimal clinically meaningful effect size but also at more optimistic anticipated effect sizes pre specified type i error rate and power requirements can be achieved both by standard group sequential tests and by more recently proposed adaptive procedures we investigate four nested classes of designs a group sequential tests with equal group sizes and stopping boundaries determined by a monomial error spending function the rho family b as a but the initial group size is allowed to be different from the others c group sequential tests with arbitrary group sizes and arbitrary boundaries fixed in advance d adaptive tests as c but at each analysis future group sizes and critical values are updated depending on the current value of the test statistic by examining the performance of optimal procedures within each class we conclude that class b provides simple and efficient designs with efficiency close to that of the more complex designs of classes c and d we provide tables and figures illustrating the performances of optimal designs within each class and defining the optimal procedures of classes a and b

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

are flexible designs sound flexible designs allow large modifications of a design during an experiment in particular the sample size can be modified in response to interim data or external information a standard flexible methodology combines such design modifications with a weighted test which guarantees the type i error level however this inference violates basic inference principles in an example with independent n mu 1 observations the test rejects the null hypothesis of mu or 0 while the average of the observations is negative we conclude that flexible design in its most general form with the corresponding weighted test is not valid several possible modifications of the flexible design methodology are discussed with a focus on alternative hypothesis tests

group sequential analysis incorporating covariate information abstract in this article we survey existing results concerning the joint distribution of the sequence of estimates of the parameter vector when a model is fitted to accumulating data and provide a unified theory that explains the independent increments structure commonly seen in group sequential test statistics our theory covers normal linear models including the case of correlated observations and asymptotic results extend to generalized linear models and the proportional hazards regression model for survival data the asymptotic results are derived using standard methods for the nonsequential case and they hold as long as these nonsequential techniques are applicable at each individual analysis in all cases the joint distribution of the sequence of parameter estimates has the same form exactly or asymptotically as that of the sequence of means of an increasing number of independent identically distributed normal variables thus our results provide the formal basis for extending the scope of s

relaxed minimax control the relaxation of the optimal control problem with cost functional which is the supremum in time of some function h t x z is determined the trajectory is convexified in the usual way but the cost functional is convexified in a nonobvious manner thus if the original value function is v t x inf zeta in z left h s xi s zeta s right l infty t t cr then the relaxed value function is hat v t x mathop inf limits mu in hat z left t t right left left h s hat xi s z right l infty rm z mu s right l infty left t t right cr where the inner norm is the essential sup of h over z in rm z cr with respect to the measure mu s cr we prove that v and hat v coincide

designed extension of studies based on conditional power we propose a flexible method of extending a study based on conditional power the possibility for extension when the p value at the planned end is small but not statistically significant is built in to the design of the study the significance of the treatment difference at the planned end is used to determine the number of additional observations needed and the critical value necessary for use after accruing those additional observations it may therefore be thought of as a two stage procedure even though the observed treatment difference at stage 1 is used to make decisions the type i error rate is protected