a calculus for design of two stage adaptive procedures we propose calculus for designing two stage adaptive procedures we describe the design components that specify a two stage adaptive design define their interrelationships and demonstrate how changes in one design component effect changes in the other components our technique allows us to control many aspects of a two stage adaptive clinical trial including the type i and type ii error rates and the maximum total sample size we also conduct an anova type study to understand the effects of different components of the design specification on the performance characteristics of the design stage i components namely sample size type i error rate and type ii error rate are found to be the most influential

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 enrichment design incorporating subgroup selection an important component of clinical trials in drug development is the analysis of treatment efficacy in patient subgroups subpopulations because of concerns of multiplicity and of the small sample sizes often involved such analyses can present substantial statistical challenges and may lead to misleading conclusions as a confirmatory seamless phase ii iii design we propose an adaptive enrichment group sequential procedure whereby resources can be concentrated on subgroups most likely to respond to treatment stopping boundaries are defined through upper and lower spending functions the procedure is presented in terms of the efficient score enabling the analysis of normal binary or time to event data it addresses the dilution effect by eliminating populations at the first stage that appear likely to derive no therapeutic benefit it subsequently proceeds with the definitive assessment of treatment efficacy among the remaining pooled populations using a group sequential design the procedure provides strong protection of familywise type i error rate and we employ a bootstrap algorithm to obtain point and interval estimates that are adjusted for the selection bias we give examples to demonstrate how the design is used we make comparisons with adaptive two stage combination test procedures and with a group sequential test that does not account for the presence of subgroups numerical results show that the procedure has high power to detect subgroup specific effects and the use of multiple interim analysis points can lead to substantial sample size savings

a projection fem for variable density incompressible flows abstract this work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density one original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid it is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step in particular a second order accurate bdf projection method is presented and its numerical performance is illustrated by test computations and comparisons

blinded sample size re estimation in three arm trials with gold standard design in this article we study blinded sample size re estimation in the gold standard design with internal pilot study for normally distributed outcomes the gold standard design is a three arm clinical trial design that includes an active and a placebo control in addition to an experimental treatment we focus on the absolute margin approach to hypothesis testing in three arm trials at which the non inferiority of the experimental treatment and the assay sensitivity are assessed by pairwise comparisons we compare several blinded sample size re estimation procedures in a simulation study assessing operating characteristics including power and type i error we find that sample size re estimation based on the popular one sample variance estimator results in overpowered trials moreover sample size re estimation based on unbiased variance estimators such as the xing ganju variance estimator results in underpowered trials as it is expected because an overestimation of the variance and thus the sample size is in general required for the re estimation procedure to eventually meet the target power to overcome this problem we propose an inflation factor for the sample size re estimation with the xing ganju variance estimator and show that this approach results in adequately powered trials because of favorable features of the xing ganju variance estimator such as unbiasedness and a distribution independent of the group means the inflation factor does not depend on the nuisance parameter and therefore can be calculated prior to a trial moreover we prove that the sample size re estimation based on the xing ganju variance estimator does not bias the effect estimate copyright 2017 john wiley sons ltd

optimal conditional error functions for the control of conditional power ethical considerations and the competitive environment of clinical trials usually require that any given trial have sufficient power to detect a treatment advance if at an interim analysis the available data are used to decide whether the trial is promising enough to be continued investigators and sponsors often wish to have a high conditional power which is the probability to reject the null hypothesis given the interim data and the alternative of interest under this requirement a design with interim sample size recalculation which keeps the overall and conditional power at a prespecified value and preserves the overall type i error rate is a reasonable alternative to a classical group sequential design in which the conditional power is often too small in this article two stage designs with control of overall and conditional power are constructed that minimize the expected sample size either for a simple point alternative or for a random mixture of alternatives given by a prior density for the efficacy parameter the presented optimality result applies to trials with and without an interim hypothesis test in addition one can account for constraints such as a minimal sample size for the second stage the optimal designs will be illustrated with an example and will be compared to the frequently considered method of using the conditional type i error level of a group sequential design

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

controlling the type i error rate in two stage sequential adaptive designs when testing for average bioequivalence in a 2 2 crossover trial for establishing average bioequivalence abe of a generic agent and a currently marketed drug the recommended approach to hypothesis testing is the two one sided test tost procedure which depends among other things on the estimated within subject variability the power of this procedure and therefore the sample size required to achieve a minimum power depends on having a good estimate of this variability when there is uncertainty it is advisable to plan the design in two stages with an interim sample size reestimation after the first stage using an interim estimate of the within subject variability one method and 3 variations of doing this were proposed by potvin et al using simulation the operating characteristics including the empirical type i error rate of the 4 variations called methods a b c and d were assessed by potvin et al and methods b and c were recommended however none of these 4 variations formally controls the type i error rate of falsely claiming abe even though the amount of inflation produced by method c was considered acceptable a major disadvantage of assessing type i error rate inflation using simulation is that unless all possible scenarios for the intended design and analysis are investigated it is impossible to be sure that the type i error rate is controlled here we propose an alternative principled method of sample size reestimation that is guaranteed to control the type i error rate at any given significance level this method uses a new version of the inverse normal combination of p values test in conjunction with standard group sequential techniques that is more robust to large deviations in initial assumptions regarding the variability of the pharmacokinetic endpoints the sample size reestimation step is based on significance levels and power requirements that are conditional on the first stage results this necessitates a discussion and exploitation of the peculiar properties of the power curve of the tost testing procedure we illustrate our approach with an example based on a real abe study and compare the operating characteristics of our proposed method with those of method b of povin et al

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

empirical bayes estimation of the selected treatment mean for two stage drop the loser trials a meta analytic approach point estimation for the selected treatment in a two stage drop the loser trial is not straightforward because a substantial bias can be induced in the standard maximum likelihood estimate mle through the first stage selection process research has generally focused on alternative estimation strategies that apply a bias correction to the mle however such estimators can have a large mean squared error carreras and brannath stat med 32 1677 90 have recently proposed using a special form of shrinkage estimation in this context given certain assumptions their estimator is shown to dominate the mle in terms of mean squared error loss which provides a very powerful argument for its use in practice in this paper we suggest the use of a more general form of shrinkage estimation in drop the loser trials that has parallels with model fitting in the area of meta analysis several estimators are identified and are shown to perform favourably to carreras and brannath s original estimator and the mle however they necessitate either explicit estimation of an additional parameter measuring the heterogeneity between treatment effects or a quite unnatural prior distribution for the treatment effects that can only be specified after the first stage data has been observed shrinkage methods are a powerful tool for accurately quantifying treatment effects in multi arm clinical trials and further research is needed to understand how to maximise their utility

adaptive randomized trial designs that cannot be dominated by any standard design at the same total sample size summary prior work has shown that certain types of adaptive designs can always be dominated by a suitably chosen standard group sequential design this applies to adaptive designs with rules 10 for modifying the total sample size a natural question is whether analogous results hold for other types of adaptive designs we focus on adaptive enrichment designs which involve preplanned rules for modifying enrollment criteria based on accrued data in a randomized trial such designs often involve multiple hypotheses e g one for the total population and one for a predefined subpopulation such as those with high disease se verity at baseline we fix the total 15 sample size and consider overall power defined as the proba bility of rejecting at least one false null hypothesis we present adaptive enrichment designs whose overall power at two alternatives cannot simultaneously be matched by any standard design in some scenarios there is a substantial gap between the overall power achieved by these adaptive designs and that of any standard design we also prove that such gains in overall power come at a cost to attain overall power 20 above what is achievable by certain standard designs it is necessary to increase power to reject some hypotheses and reduce power to reject others we conclude by showing the class of adaptive enrichment designs allows certain power tradeoffs that are not available when restricting to standard designs we illustrate our results in the context o f planning a hypothetical randomized trial of a new antidepressant using data distributions fro m kirsch et al 2008 25

modification of sample size in group sequential clinical trials in group sequential clinical trials sample size reestimation can be a complicated issue when it allows for change of sample size to be influenced by an observed sample path our simulation studies show that increasing sample size based on an interim estimate of the treatment difference can substantially inflate the probability of type i error in most practical situations a new group sequential test procedure is developed by modifying the weights used in the traditional repeated significance two sample mean test the new test has the type i error probability preserved at the target level and can provide a substantial gain in power with the increase of sample size generalization of the new procedure is discussed

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

adaptive sample size modification in clinical trials start small then ask for more we consider sample size re estimation in a clinical trial in particular when there is a significant delay before the measurement of patient response mehta and pocock have proposed methods in which sample size is increased when interim results fall in a promising zone where it is deemed worthwhile to increase conditional power by adding more subjects our analysis reveals potential pitfalls in applying this approach mehta and pocock use results of chen demets and lan to identify when increasing sample size but applying a conventional level significance test at the end of the trial does not inflate the type i error rate we have found the greatest gains in power per additional observation are liable to lie outside the region defined by this method mehta and pocock increase sample size to achieve a particular conditional power calculated under the current estimate of treatment effect this leads to high increases in sample size for a small range of interim outcomes whereas we have found it more efficient to make moderate increases in sample size over a wider range of cases if the aforementioned pitfalls are avoided we believe the broad framework proposed by mehta and pocock is valuable for clinical trial design working in this framework we propose sample size rules that apply explicitly the principle of adding observations when they are most beneficial the resulting trial designs are closely related to efficient group sequential tests for a delayed response proposed by hampson and jennison copyright 2015 john wiley sons ltd

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

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

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

adaptive two stage test procedures to find the best treatment in clinical trials a main objective in clinical trials is to find the best treatment in a given finite class of competing treatments and then to show superiority of this treatment against a control treatment the traditional procedure estimates the best treatment in a first trial then in an independent second trial superiority of this treatment estimated as best in the first trial is to be shown against the control treatment by a size alpha test par we investigate these two trials of this traditional procedure as a two stage test procedure additionally we introduce competing two stage group sequential test procedures then we derive formulae for the expected number of patients these formulae depend on unknown parameters when we have a prior for the unknown parameters we can determine the two stage test procedure of size alpha and power beta that is optimal in that it needs a minimal number of observations the results are illustrated by a numerical example which indicates the superiority of the group sequential procedures

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

mixed stabilized finite element methods based on backward difference adams bashforth scheme for the time dependent variable density incompressible flows in this paper we consider a second order mixed stabilized finite element method based on pressure projection method for variable density incompressible flows the originality of the proposed approach is to use a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for variable density incompressible flows approximated by the lowest equal order finite element pairs a second order backward difference bdf for the temporal term and a second order adams bashforth ab for the explicit treatment of the nonlinear term lead to the presented second order bdf ab scheme the stability of the method was proved and the performance of the method is numerically illustrated finally comparison with some established methods a series of numerical experiments are given to show that this method has better stability and accuracy

nonparametric estimation of a delay distribution based on left censored and right truncated data this paper proposes a nonparametric method for estimating a delay distribution based on left censored and right truncated data a variance covariance estimator is provided the method is applied to the australian aids data in which some data are left censored and some data are not left censored this situation arises with aids case reporting data in australia because reporting delays were recorded only from november 1990 rather than from the beginning of the epidemic there it is shown that inclusion of the left censored data as opposed to analyzing only the uncensored data improves the precision of the estimate

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

a new stabilized finite element method for the transient navier stokes equations abstract this paper is concerned with the development and analysis of a new stabilized finite element method based on two local gauss integrations for the two dimensional transient navier stokes equations by using the lowest equal order pair of finite elements this new stabilized finite element method has some prominent features parameter free avoiding higher order derivatives or edge based data structures and stabilization being completely local at the element level an optimal error estimate for approximate velocity and pressure is obtained by applying the technique of the galerkin finite element method under certain regularity assumptions on the solution compared with other stabilized methods using the same pair of mixed finite elements for the two dimensional transient navier stokes equations through a series of numerical experiments it is shown that this new stabilized method has better stability and accuracy results

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

selection and bias two hostile brothers we consider the situation where in a first stage of a clinical trial several treatments are compared with a single control and the best treatment s are selected in an interim analysis to be carried on to the second stage we quantify the mean bias and mean square error of the conventional estimates after selection depending on the number of treatments and the selection time during the trial the cases without or with reshuffling the planned sample size of the dropped treatments to the selected ones are investigated the mean bias shows very different patterns depending on the selection rule and the unknown parameter values we stress the fact that the quantification of the bias is possible only in designs with planned adaptivity where the design allows reacting to new evidence but the decision rules are laid down in advance finally we calculate the mean bias which arises in a simple but influential regulatory selection rule to register a new medical therapy only when two pivotal trials have both proven an effect by a statistical test

high dimensional ordinary least squares projection for screening variables variable selection is a challenging issue in statistical applications when the number of predictors p far exceeds the number of observations n in this ultra high dimensional setting the sure independence screening sis procedure was introduced to significantly reduce the dimensionality by preserving the true model with overwhelming probability before a refined second stage analysis however the aforementioned sure screening property strongly relies on the assumption that the important variables in the model have large marginal correlations with the response which rarely holds in reality to overcome this we propose a novel and simple screening technique called the high dimensional ordinary least squares projection holp we show that holp possesses the sure screening property and gives consistent variable selection without the strong correlation assumption and has a low computational complexity a ridge type holp procedure is also discussed simulation study shows that holp performs competitively compared to many other marginal correlation based methods an application to a mammalian eye disease data illustrates the attractiveness of holp

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

probabilistic foundation of confirmatory adaptive designs adaptive designs allow the investigator of a confirmatory trial to react to unforeseen developments by changing the design this broad flexibility comes at the price of a complex statistical model where important components such as the adaptation rule remain unspecified it has thus been doubted whether type i error control can be guaranteed in general adaptive designs this criticism is fully justified as long as the probabilistic framework on which an adaptive design is based remains vague and implicit therefore an indispensable step lies in the clarification of the probabilistic fundamentals of adaptive testing we demonstrate that the two main principles of adaptive designs namely the conditional type i error rate and the conditional invariance principle will provide type i error rate control if the conditional distribution of the second stage data given the first stage data can be described in terms of a regression model a similar assumption is required for regression analysis where the dis

evaluation of experiments with adaptive interim analyses summary a general method for statistical testing in experiments with an adaptive interim analysis is proposed the method is based on the observed error probabilities from the disjoint subsamples before and after the interim analysis formally an intersection of individual null hypotheses is tested by combining the twop values into a global test statistic stopping rules for fisher s product criterion in terms of critical limits for the p value in the first subsample are introduced including early stopping in the case of missing effects the control of qualitative treatment stage interactions is considered a generalization to three stages is outlined the loss of power when using the product criterion instead of the optimal classical test on the whole sample is calculated for the test of the mean of a normal distribution depending on increasing proportions of the first subsample in relation to the total sample size an upper bound on the loss of power due to early stopping is derived a general example is presented and rules for assessing the sample size in the second stage of the trial are given the problems of interpretation and precautions to be taken for applications are discussed finally the sources of bias for estimation in such designs are described

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

adaptive extensions of a two stage group sequential procedure for testing primary and secondary endpoints ii sample size re estimation in this part ii of the paper on adaptive extensions of a two stage group sequential procedure gsp proposed by tamhane mehta and liu 1 referred to as tml hereafter for testing a primary and a secondary endpoint we focus on the second stage sample size re estimation based on thefirst stage data first we show that if we use the cui huang and wang 2 referred to as chw hereafter statistics at the second stage then we can use the same primary and the secondary boundaries as for the original procedure without sample size re estimation and still control the type i familywise error rate fwer this extends their result for the single endpoint case we further show that the secondary boundary can be sharpened in this case by taking the unknown correlation coefficient between the primary and secondary endpoints into account through the use of the confidence limit method proposed in part i of this paper 3 if we use the sufficient statistics instead of the chw statistics then we need to modify both the primary and secondary boundaries otherwise the error rate can get inflated we show how to modify the boundaries of the original gsp to control the fwer power comparisons between competing procedures are provided the procedures are illustrated with a clinical trial example copyright c 0000 john wiley sons ltd

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

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

adaptive multivariate global testing we present a methodology for dealing with recent challenges in testing global hypotheses using multivariate observations the proposed tests target situations often arising in emerging applications of neuroimaging where the sample size n is relatively small compared with the observations dimension k we employ adaptive designs allowing for sequential modifications of the test statistics adapting to accumulated data the adaptations are optimal in the sense of maximizing the predictive power of the test at each interim analysis while still controlling the type i error optimality is obtained by a general result applicable to typical adaptive design settings further we prove that the potentially high dimensional design space of the tests can be reduced to a low dimensional projection space enabling us to perform simpler power analysis studies including comparisons to alternative tests we illustrate the substantial improvement in efficiency that the proposed tests can make over standard tests especia

optimizing randomized trial designs to distinguish which subpopulations benefit from treatment it is a challenge to evaluate experimental treatments where it is suspected that the treatment effect may only be strong for certain subpopulations such as those having a high initial severity of disease or those having a particular gene variant standard randomized controlled trials can have low power in such situations they also are not optimized to distinguish which subpopulations benefit from a treatment with the goal of overcoming these limitations we consider randomized trial designs in which the criteria for patient enrollment may be changed in a preplanned manner based on interim analyses since such designs allow data dependent changes to the population enrolled care must be taken to ensure strong control of the familywise type i error rate our main contribution is a general method for constructing randomized trial designs that allow changes to the population enrolled based on interim data using a prespecified decision rule for which the asymptotic familywise type i error rate is strongly controlled at a specified level as a demonstration of our method we prove new sharp results for a simple two stage enrichment design we then compare this design to fixed designs focusing on each design s ability to determine the overall and subpopulation specific treatment effects

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

sample size recalculation using conditional power the sample size required to achieve a given power at a prespecified absolute difference in mean response may depend on one or more nuisance parameters which are usually unknown proposed methods for using an internal pilot to recalculate the sample size using estimates of these parameters have been well studied most of these methods ignore the fact that data on the parameter of interest from within this internal pilot will contribute towards the value of the final test statistic we propose a method which involves recalculating the target sample size by computing the number of further observations required to maintain the probability of rejecting the null hypothesis at the end of the study under the prespecified absolute difference in mean response conditional on the data observed so far we do this within the framework of a two group error spending sequential test modified so as to prevent inflation of the type i error rate

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

neighborhoods of algebraic sets in differential topology a smooth submanifold in a manifold has a tubular neighborhood and in piecewise linear topology a subcomplex of a simplicial complex has a regular neighborhood the purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets the neighbor hoods will be defined as level sets of polynomial or semialgebraic functions introduction let m be an algebraic set in real n space r and let x be a compact algebraic subset of m containing its singular locus if any an algebraic neighborhood of a in m is defined to be a 1 0 8 where o 0 is sufficiently small and a m r is a proper polynomial function for which a 0 and a 1 0 x such an a will be called a rug function occasionally we will need rational or analytic a but this is not a significant generalization algebraic neighborhoods always exist the curve selec tion lemma is used to prove uniqueness anyone familiar with milnor 2 will recognize the technique since uniqueness is a crucial result and since there are a few trouble some small points this proof is given in detail the uniqueness theorem shows that the link of a singularity of an algebraic set m is independent of the embedding of m in its ambient space i have been unable to find a proof of this result in the litera ture in addition an algebraic neighborhood of a nonsingular x in m is shown to be a tubular neighborhood in the sense of differential topology this material is in 1 in 2 the theory is rapidly developed for real and complex projective space by embedding these spaces in real affine space as an application it is shown that when m is an affine algebraic set with projective completion m then the complement in m of a large ball centered at the origin of affine space is an alge braic neighborhood of the intersection of m with the hyperplane at infinity in 3 the theory is generalized to the case where m is a semialgebraic subset of r and x is a compact semialgebraic subset of m for example x could be a non isolated singular point of m a semialgebraic neighborhood of x in m is defined to be a 1 0 o where o 0 is sufficiently small and a m r is a proper semialgebraic function for which a 0 and a 1 0 x again these neighborhoods exist and are unique this is shown by mimicking the proof in the algebraic case using semialgebraic

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

extension of finite volume compressible flow solvers to multi dimensional variable density zero mach number flows when attempting to compute unsteady variable density flows at very small or zero mach number using a standard finite volume compressible flow solver one faces at least the following difficulties i spatial pressure variations vanish as the mach numberm

a splitting method for incompressible flows with variable density based on a pressure poisson equation a new fractional time stepping technique for solving incompressible flows with variable density is proposed the main feature of this method is that as opposed to other known algorithms the pressure is determined by just solving one poisson equation per time step which greatly reduces the computational cost the stability of the method is proved and the performance of the method is numerically illustrated

adaptive group sequential and decision theoretic approaches to sample size determination this paper presents two adaptive methods for sample size re estimation within a unified group sequential framework the conceptual and practical distinction between these adaptive modifications and more traditional sample size changes due to revised estimates of nuisance parameters is highlighted the motivation for the adaptive designs is discussed having established that adaptive sample size modifications can be made without inflating the type 1 error the paper concludes with a novel decision theoretic approach for determining the magnitude of the sample size modification

reduced order modeling of time dependent incompressible navier stokes equation with variable density based on a local radial basis functions finite difference lrbf fd technique and the pod deim method abstract the main propose of this investigation is to introduce a rapid and impressive numerical procedure to simulate the time dependent incompressible navier stokes equation with variable density the developed formulation is constructed by using the meshfree rbf fd technique also to improve the numerical results an extra diffusion term has been added to the density equation with a small coefficient on the other hand a reduced order technique e g proper orthogonal decomposition pod has been employed to get a fast numerical method and to decrease the elapsed cpu time on the other hand since the considered problem is fully nonlinear thus in the reduced order model based on the pod idea there are some nonlinear terms that they take more computational time hence we employ the discrete empirical interpolation method deim to overcome the nonlinear terms four test problems have been proposed to show the ability of the numerical simulations

an extended residual based variational multiscale method for two phase flow including surface tension abstract in this study an extended residual based variational multiscale method is proposed for two phase flow including surface tension the extended residual based variational multiscale method combines a residual based form of the variational multiscale method and the extended finite element method xfem by extending the solution spaces it is possible to reproduce discontinuities of the solution fields inside elements intersected by the interface in particular we propose a quasi static enrichment to reproduce time dependent discontinuities kink enrichments of both velocity and pressure as well as kink enrichments of velocity combined with jump enrichments of pressure are considered here to capture the interface between the phases on a fixed grid a level set approach is used a residual based variational multiscale method is employed for computing both flow and interface motion the presented method is tested for various two phase flow examples exhibiting small and large density and viscosity ratios with and without surface tension a two phase couette flow a rayleigh taylor instability a sloshing tank and a three dimensional rising bubble to the best of our knowledge these are the first simulation results for representative time dependent three dimensional two phase flow problems using an extended finite element method stable and accurate results are obtained for all test examples

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

gauge uzawa methods for incompressible flows with variable density two new gauge uzawa schemes are constructed for incompressible flows with variable density one is in the conserved form while the other is in the convective form it is shown that the first order versions of both schemes in their semi discretized form are unconditionally stable numerical experiments indicate that the first order resp second order versions of the two schemes lead to first order resp second order convergence rate for all variables and that these schemes are suitable for handling problems with large density ratios such as in the situation of air bubble rising in water

a calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves any graph manifold can be obtained by plumbing according to some plumbing graph i a calculus for plumbing which includes normal forms for such graphs is developed this is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves for instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity in this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3 manifolds m tx and a ftj ob tained by plumbing according to two graphs tx and t2 are homeomorphic 3 mani folds are oriented 3 manifolds throughout this paper and homeomorphisms of 3 manifolds are orientation preserving we then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one parameter families of complex curves these results are described below since the class of 3 manifolds obtainable by plumbing is precisely the class of graph manifolds which were classified with minor exceptions by waldhausen 24 a calculus for plumbing is in some sense implicit in waldhausen s work moreover it has been known for some time that the calculus can be put in a form like the one given here but the details have never appeared in the literature a related calculus for plumbing trees has been worked by bonahon and siebenmann 1 in order to classify their algebraic knots we describe the calculus in greater generality than is needed for the present applications since this involves minimal extra work and the calculus is needed elsewhere 4 14 and 15 in particular in an appendix we describe two generalizations of it the calculus consists of a collection of moves one can do to a plumbing graph t without altering the plumbed manifold m t to see these moves are sufficient we describe how they can be used to reduce any graph to a normal form which is

interval estimation for drop the losers designs summary in the first stage of a two stage drop the losers design a candidate for the best treatment is selected at the second stage additional observations are collected to decide whether the candidate is actually better than the control the design also allows the investigator to stop the trial for ethical reasons at the end of the first stage if there is already strong evidence of futility or superiority two types of tests have recently been developed one based on the combined means and the other based on the combined p values but corresponding interval estimators are unavailable except in special cases the problem is that in most cases the interval estimators depend on the mean configuration of all treatments in the first stage which is unknown in this paper we prove a basic stochastic ordering lemma that enables us to bridge the gap between hypothesis testing and interval estimation the proposed confidence intervals achieve the nominal confidence level in certain special cases simulations show that decisions based on our intervals are usually more powerful than those based on existing methods

error analysis of a fractional time stepping technique for incompressible flows with variable density in this paper we analyze the convergence properties of a new fractional time stepping technique for the solution of the variable density incompressible navier stokes equations the main feature of this method is that contrary to other existing algorithms the pressure is determined by just solving one poisson equation per time step first order error estimates are proved and stability of a formally second order variant of the method is established

α investing a procedure for sequential control of expected false discoveries summary investing is an adaptive sequential methodology that encompasses a large family of procedures for testing multiple hypotheses all control mfdr which is the ratio of the expected number of false rejections to the expected number of rejections mfdr is a weaker criterion than the false discovery rate which is the expected value of the ratio we compensate for this weakness by showing that investing controls mfdr at every rejected hypothesis investing resembles spending that is used in sequential trials but it has a key difference when a test rejects a null hypothesis investing earns additional probability towards subsequent tests investing hence allows us to incorporate domain knowledge into the testing procedure and to improve the power of the tests in this way investing enables the statistician to design a testing procedure for a specific problem while guaranteeing control of mfdr

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

a stabilized finite element method based on two local gauss integrations for the stokes equations this paper considers a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for the stokes equations approximated by the lowest equal order finite element pairs i e the p 1 p 1 and q 1 q 1 pairs this method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf sup condition optimal error estimates are obtained by applying the standard galerkin technique finally the numerical illustrations agree completely with the theoretical expectations

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

a group sequential t test with updating of sample size if the maximum sample size for a group sequential t test is calculated on the basis of a prior estimate of the variance power at a fixed distance from the null hypothesis is not robust to misspecification of the variance we propose a new group sequential t test based on stein s 1945 two stage test in which we use internal estimates of the variance to update the maximum sample size and allow early stopping to reject the null hypothesis through a lan demets 1983 error spending function we evaluate our procedure s performance and show that it closely controls the type i and ii error rates

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

precision of maximum likelihood estimation in adaptive designs there has been increasing interest in trials that allow for design adaptations like sample size reassessment or treatment selection at an interim analysis ignoring the adaptive and multiplicity issues in such designs leads to an inflation of the type 1 error rate and treatment effect estimates based on the maximum likelihood principle become biased whereas the methodological issues concerning hypothesis testing are well understood it is not clear how to deal with parameter estimation in designs were adaptation rules are not fixed in advanced so that in practice the maximum likelihood estimate mle is used it is therefore important to understand the behavior of the mle in such designs the investigation of bias and mean squared error mse is complicated by the fact that the adaptation rules need not be fully specified in advance and hence are usually unknown to investigate bias and mse under such circumstances we search for the sample size reassessment and selection rules that lead to the maximum bias or maximum mse generally this leads to an overestimation of bias and mse which can be reduced by imposing realistic constraints on the rules like for example a maximum sample size we consider designs that start with k treatment groups and a common control and where selection of a single treatment and control is performed at the interim analysis with the possibility to reassess each of the sample sizes we consider the case of unlimited sample size reassessments as well as several realistically restricted sample size reassessment rules

on adaptive extensions of group sequential trials for clinical investigations in group sequential trials it is important to obtain adequate data to assess overall benefits and risks of an experimental treatment for patients to achieve this goal we provide a general formal framework for adaptively extending a group sequential trial to stop at any interim analysis time often after a significance boundary for a clinical endpoint is crossed for statistical inference we propose to order the sample space by a class of well ordered group sequential tests on this basis we develop a unified sequential statistical inference approach that is applicable to both interim monitoring and final analysis we also show that the new ordering provides the foundation for the repeated confidence intervals procedure of jennison and turnbull

an hybrid finite volume finite element method for variable density incompressible flows this paper is devoted to the numerical simulation of variable density incompressible flows modeled by the navier stokes system we introduce an hybrid scheme which combines a finite volume approach for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint the breakthrough relies on the definition of a suitable footbridge between the two methods through the design of compatibility condition in turn the method is very flexible and allows to deal with unstructured meshes several numerical tests are performed to show the scheme capabilities in particular the viscous rayleigh taylor instability evolution is carefully investigated

maximum inflation of the type 1 error rate when sample size and allocation rate are adapted in a pre planned interim look we calculate the maximum type 1 error rate of the pre planned conventional fixed sample size test for comparing the means of independent normal distributions with common known variance which can be yielded when sample size and allocation rate to the treatment arms can be modified in an interim analysis thereby it is assumed that the experimenter fully exploits knowledge of the unblinded interim estimates of the treatment effects in order to maximize the conditional type 1 error rate the worst case strategies require knowledge of the unknown common treatment effect under the null hypothesis although this is a rather hypothetical scenario it may be approached in practice when using a standard control treatment for which precise estimates are available from historical data the maximum inflation of the type 1 error rate is substantially larger than derived by proschan and hunsberger biometrics 1995 51 1315 1324 for design modifications applying balanced samples before and after the interim analysis corresponding upper limits for the maximum type 1 error rate are calculated for a number of situations arising from practical considerations e g restricting the maximum sample size not allowing sample size to decrease allowing only increase in the sample size in the experimental treatment the application is discussed for a motivating example

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

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

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

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

regular article extension of finite volume compressible flow solvers to multi dimensional variable density zero mach number flows when attempting to compute unsteady variable density flows at very small or zero mach number using a standard finite volume compressible flow solver one faces at least the following difficulties i spatial pressure variations vanish as the mach number m 0 but they do affect the velocity field at leading order ii the resulting spatial homogeneity of the leading order pressure implies an elliptic divergence constraint for the energy flux iii violations of this constraint crucially affect the transport of mass preventing a code to properly advect even a constant density distribution we overcome these difficulties through a new algorithm for constructing numerical fluxes in the context of multi dimensional finite volume methods in conservation form the construction of numerical fluxes involves 1 an explicit upwind step yielding predictions for the nonlinear convective flux components 2 a first correction step that introduces pressure gradients which guarantee compliance of the convective fluxes with a divergence constraint this step requires the solution of a first poisson type equation 3 a second projection step which provides the yet unknown non convective pressure contribution to the total flux of momentum this second projection requires the solution of another poisson type equation and yields the cell centered velocity field at the new time this velocity field exactly satisfies a divergence constraint consistent with the asymptotic limit step 1 can be done by any standard finite volume compressible flow solver the input to steps 2 and 3 involves solely the fluxes from step 1 and is independent of how these were obtained thus our approach allows any such solver to be extended to compute variable density incompressible flows

the succession procedure for interim analysis extensions for early acceptance of h0 and for flexible times of analysis the succession procedure is a new method for group sequential analysis based on a succession of significant tests in this paper the procedure is extended by the addition of a lower boundary which allows the early acceptance of the null hypothesis ho according to the stochastic curtailing approach the succession procedure is also adapted to the type i error spending rate function alpha t leading to the method of semisequential analysis for which the times of analysis need not be specified in advance

conditional estimation in two stage adaptive designs we consider conditional estimation in two stage sample size adjustable designs and the consequent bias more specifically we consider a design which permits raising the sample size when interim results look rather promising and which retains the originally planned sample size when results look very promising the estimation procedures reported comprise the unconditional maximum likelihood the conditionally unbiased rao blackwell estimator the conditional median unbiased estimator and the conditional maximum likelihood with and without bias correction we compare these estimators based on analytical results and a simulation study we show how they can be applied in a real clinical trial setting

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

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

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

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

adaptive sample size calculations in group sequential trials a method for group sequential trials that is based on the inverse normal method for combining the results of the separate stages is proposed without exaggerating the type i error rate this method enables data driven sample size reassessments during the course of the study it uses the stopping boundaries of the classical group sequential tests furthermore exact test procedures may be derived for a wide range of applications the procedure is compared with the classical designs in terms of power and expected sample size

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

the reassessment of trial perspectives from interim data a critical view if an interim analysis is performed during a trial it is tempting to determine the conditional power to reach a rejection in the trial given the observed results in the interim analysis since the true effect size is unknown the conditional power may be calculated by using the effect size which the study has been powered for in the planning phase or by using an interim estimate of the true size or a combination of both in either case the conditional power is a random variable and its density is investigated depending on the analysis time and the true effect size under the null hypothesis in early interim analyses after a small proportion of sample units the conditional power typically will be close to the overall power when the effect size from the planning stage is used for calculation in this case the majority of observations must still be made and the small first stage sample in general will be dominated by the hypothetical second stage chance based on the wrong parameter value it is shown that the conditional power in moderately underpowered studies can have a distribution symmetric around 0 5 when using the interim estimate for calculating the conditional power the density in general will be u shaped the impact of using conditional power to reassess the sample size using flexible two stage combination tests is shown for a specific example in terms of overall power and average sample size as compared to the corresponding group sequential design for small true effect sizes mid trial sample size recalculation based on an interim estimate may lead to an overly large price to be paid in average sample size in relation to the gain in overall power finally the problem is discussed in terms of estimating the true conditional power

a unified theory of two stage adaptive designs adaptation of a clinical trial occurs when there is a change from the planned course of action in design sample size or method of analysis there are many proposed adaptations in the literature but no comprehensive mathematical theory to support them the standard neyman pearson theory is not suitable for the setting and its use may result in invalid inference this article provides a general formulation of adaptation under which its validity can be broadly and rigorously established the basic theory on distribution stochastic independence point estimation confidence intervals hypothesis testing and asymptotics is developed various examples are given to illustrate applications of the theory the goal is to provide a solid theoretical foundation to aid the development of adaptive methods for clinical investigations