increasing the sample size during clinical trials with t distributed test statistics without inflating the type i error rate in clinical trials with t distributed test statistics the required sample size depends on the unknown variance taking estimates from previous studies often leads to a misspecification of the true value of the variance hence re estimation of the variance based on the collected data and re calculation of the required sample size is attractive we present a flexible method for extensions of fixed sample or group sequential trials with t distributed test statistics the method can be applied at any time during the course of the trial and does not require the necessity to pre specify a sample size re calculation rule all available information can be used to determine the new sample size the advantage of our method when compared with other adaptive methods is maintenance of the efficient t test design when no extensions are actually made we show that the type i error rate is preserved

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

density scaling a new monte carlo technique in statistical mechanics abstract we demonstrate the feasibility of using umbrella sampling to do monte carlo markov sampling runs each covering a substantial range of density density scaling monte carlo or dsmc one can obtain in this way not only the usual canonical averages but also the relative free energy as a function of density to test this it has been applied to systems for which there are some previous reliable results the hard sphere system and the restricted primitive model of 1 1 and 2 2 electrolytes the method proves to be startlingly powerful in that very extensive results can be obtained with very few dsmc runs an important further motivation is the prospect of using the technique to study phase transition regions

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

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

design of group sequential clinical trials with multiple endpoints abstract group sequential testing for randomized clinical trials designed with multiple endpoints is considered previously computed tables for single endpoints are still useful with a change in interpretation of certain parameters the advantage in setting sample size based on multiple endpoints is that the sample size required when a trial is designed using more than one endpoint is smaller than the sample size based on any single endpoint if the two calculations are undertaken using matching significance levels and powers an example is provided and possible extensions of the work are discussed

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

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 modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

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

approximation of discrete sequential boundaries on obtient des limites conservatives critiques pour des bornes sequentielles discretes lorsque le nombre d instants de decision et de tailles de groupes n est pas fixe d avance

maximum type i error rate inflation from sample size reassessment when investigators are blind to treatment labels consider a parallel group trial for the comparison of an experimental treatment to a control where the second stage sample size may depend on the blinded primary endpoint data as well as on additional blinded data from a secondary endpoint for the setting of normally distributed endpoints we demonstrate that this may lead to an inflation of the type i error rate if the null hypothesis holds for the primary but not the secondary endpoint we derive upper bounds for the inflation of the type i error rate both for trials that employ random allocation and for those that use block randomization we illustrate the worst case sample size reassessment rule in a case study for both randomization strategies the maximum type i error rate increases with the effect size in the secondary endpoint and the correlation between endpoints the maximum inflation increases with smaller block sizes if information on the block size is used in the reassessment rule based on our findings we do not question the well established use of blinded sample size reassessment methods with nuisance parameter estimates computed from the blinded interim data of the primary endpoint however we demonstrate that the type i error rate control of these methods relies on the application of specific binding pre planned and fully algorithmic sample size reassessment rules and does not extend to general or unplanned sample size adjustments based on blinded data 2015 the authors statistics in medicine published by john wiley sons ltd

an approach to the conditional error rate principle with nuisance parameters summary in the presence of nuisance parameters the conditional error rate principle is difficult to apply because of the dependency of the conditional error function of the preplanned test on nuisance parameters to use the conditional error rate principle with nuisance parameters we propose to search among tests that guarantee overall error control for the test that maximizes a weighted combination of the conditional error rates over possible values of the nuisance parameters we show that the optimization problem that defines such a test can be solved efficiently by existing algorithms

virial coefficients for hard discs and hard spheres the sixth seventh and eighth virial coefficients of hard discs and hard spheres are evaluated numerically monte carlo integration i improve on the best previous estimates for the seventh virial coefficients and the integration of the eighth virial coefficient is new the best estimates for these coefficients for hard discs are b7 b6 0 114 86 7 and b8 b7 0 065 14 8 and for hard spheres b7 b6 0 01307 7 and b8 b7 0 00432 10 b is the second virial coefficient in each case pade approximations to the excess pressure and the excess free energy are computed from these results and compared to data otherwise obtained

on special zeros of p adic l functions of hilbert modular forms let e be a modular elliptic curve over a totally real number field f we prove the weak exceptional zero conjecture which links a higher derivative of the p adic l function attached to e to certain p adic periods attached to the corresponding hilbert modular form at the places above p where e has split multiplicative reduction under some mild restrictions on p and the conductor of e we deduce the exceptional zero conjecture in the strong form i e where the automorphic p adic periods are replaced by the mathcal l invariants of e defined in terms of tate periods from a special case proved earlier by mok crucial for our method is a new construction of the p adic l function of e in terms of local data

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

adaptive designs for the one sample log rank test summary traditional designs in phase iia cancer trials are single arm designs with a binary outcome for example tumor response in some settings however a time to event endpoint might appear more appropriate particularly in the presence of loss to follow up then the one sample log rank test might be the method of choice it allows to compare the survival curve of the patients under treatment to a prespecified reference survival curve the reference curve usually represents the expected survival under standard of the care in this work convergence of the one sample log rank statistic to brownian motion is proven using rebolledo s martingale central limit theorem while accounting for staggered entry times of the patients on this basis a confirmatory adaptive one sample log rank test is proposed where provision is made for data dependent sample size reassessment the focus is to apply the inverse normal method this is done in two different directions the first strategy exploits the independent increments property of the one sample log rank statistic the second strategy is based on the patient wise separation principle it is shown by simulation that the proposed adaptive test might help to rescue an underpowered trial and at the same time lowers the average sample number asn under the null hypothesis as compared to a single stage fixed sample design

some approximate evidence factors in observational studies an observational study or nonrandomized experiment has exact evidence factors if it permits several statistically independent tests of the same null hypothesis h0 of no treatment effect where these several tests depend upon different assumptions about bias from nonrandom treatment assignment in an observational study we are typically uncertain about what assumptions truly describe treatment assignment if independent tests each reject h0 and if each of these tests is valid under assumptions about treatment assignment that would invalidate the others then rejection of h0 does not depend upon the truth of any one of these assumptions although exact evidence factors do exist the requirement of exact independence may limit the class of test statistics in ways that reduce pitman efficiency and design sensitivity so one might wish to be free of these limitations a much larger class of statistics permits approximate evidence factors which are not independent but which preserve some of the important con

an adaptive confirmatory trial with interim treatment selection practical experiences and unbalanced randomization this article has been motivated by an ongoing international adaptive confirmatory trial at the interim analysis of this two stage trial none one or two active treatment regimens are selected for further study in the second stage a combination test approach is used in this practical setting with an extension of the theory to unbalanced randomization we show that a combination test with suitable weights can still preserve the overall type i error rate provided that the test statistic is chosen appropriately and the unpooled z test for proportions is not used the accuracy of stagewise p values is also discussed in a more general framework monte carlo simulations confirm the validity of the approach retained and evaluate the necessary sample size additional issues addressed during the design of the trial are also examined such as multiplicity due to testing hypotheses on key secondary endpoints a non inferiority comparison to an active treatment and covariate adjusted analyses for various types of outcome

monitoring clinical trials with a conditional probability stopping rule conditional probability procedures offer a flexible means of performing sequential analysis of clinical trials since these procedures are not based on repeated significance tests the number and schedule of the interim analyses is less important than with group sequential procedures their main disadvantage is that the magnitude of their effect on the significance level is difficult to assess this paper describes a conditional probability procedure which attempts to maintain the overall significance level by balancing the probabilities of false early rejection and false early acceptance monte carlo sampling results suggest that this procedure can achieve a large reduction in expected sample size without greatly affecting either the significance level or power of the trial

shintani cocycles and the order of vanishing of p adic hecke l series at s 0 let chi be a hecke character of finite order of a totally real number field f by using hill s shintani cocycle we provide a cohomological construction of the p adic l series l p chi s associated to chi this is used to show that l p chi s has a trivial zero at s 0 of order at least equal to the number of places of f above p where the local component of chi is trivial

shortcuts for locally consonant closed test procedures the closed testing principle provides a general and simple framework to construct powerful multiple test procedures for k elementary null hypotheses while controlling the familywise error rate in the strong sense however the closed testing principle has the disadvantage of leading to the evaluation of o 2k intersection hypotheses multiple test procedures based on the closed testing principle may thus require substantial computational efforts consonant closed test procedures for unrestricted hypotheses have the advantage of rejecting at least one elementary null hypothesis whenever the global null hypothesis is rejected and thus admit shortcuts of size k if the elementary null hypotheses are restricted by logical constraints the closure of common tests such as max t or min p tests may not be consonant in the usual sense in this article we introduce a weaker consonance property denoted as local consonance and show that many closed test procedures with restricted hypotheses satisfy this condition

summation of series in statistical mechanics by continued exponentials abstract continued exponentials ces are used to extend a series when a finite number of coefficients are known since the ce does not have singularities built in as pade approximants and the maximum entropy me method do it is a useful technique to estimate thermodynamic functions from partial series right up to the transition point radius of convergence for first order phase transitions the method is tested on fisher s cluster model an ideal mixture of clusters with no excluded volume that contains a first order gas liquid phase transition with a critical point the one dimensional ising model an example of a fluid above the critical point and the two dimensional ising model an example of a fluid below the critical point where the effect of excluded volume is included in addition we compare the use of the ce to estimate the equation of state for hard spheres from the known partial virial series through b 8 with the results from pade approximants and the me method the results from the three methods are comparable with the results from the ce approximation agreeing best with simulation data ces can also be used to approximate series that are asymptotic to a power law in the case where the exponent is known we use the case of diffusion in one dimension as an example with power law behavior and compare the use of ces and the me method to treat this system

multiple testing in a two stage adaptive design with combination tests controlling fdr testing multiple null hypotheses in two stages to decide which of these can be rejected or accepted at the first stage and which should be followed up for further testing having had additional observations is of importance in many scientific studies we develop two procedures each with two different combination functions fisher s and simes to combine p values from two stages given prespecified boundaries on the first stage p values in terms of the false discovery rate fdr and controlling the overall fdr at a desired level the fdr control is proved when the pairs of first and second stage p values are independent and those corresponding to the null hypotheses are identically distributed as a pair p 1 p 2 satisfying the p clud property we did simulations to show that 1 our two stage procedures can have significant power improvements over the first stage benjamini hochberg bh procedure compared to the improvement offered by the ideal bh procedure that one would have used had the second stage

simultaneous confidence intervals that are compatible with closed testing in adaptive designs we describe a general method for finding a confidence region for a parameter vector that is compatible with the decisions of a two stage closed test procedure in an adaptive experiment the closed test procedure is characterized by the fact that rejection or nonrejection of a null hypothesis may depend on the decisions for other hypotheses and the compatible confidence region will in general have a complex nonrectangular shape we find the smallest cross product of simultaneous confidence intervals containing the region and provide computational shortcuts for calculating the lower bounds on parameters corresponding to the rejected null hypotheses we illustrate the method with an adaptive phase ii iii clinical trial

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

two hard spheres in a spherical pore exact analytic results in two and three dimensions the partition function and the one and two body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore the equivalent problem for hard disks is addressed too we establish a relation valid for any dimension between these partition functions second virial coefficient for inhomogeneous systems in a spherical pore and third virial coefficients for polydisperse hard spheres mixtures using the established relation we were able to evaluate the cluster integral b2 v related with the second virial coefficient for the hard disc system into a circular pore finally we analyse the behaviour of the obtained expressions near the maximum density

a simple multivariate test for one sided alternatives abstract a simple test for multivariate normal data is proposed that has good power for alternatives where the mean vector tends to be positive the test rejects if the quadratic form of the sample mean vector exceeds its 2 critical value and the sum of the elements of the mean vector exceeds zero the proposed test is shown to have type i error rate equal to whether or not the covariance matrix is known tight bounds on the power of the proposed test are provided and the test is compared to three likelihood ratio tests

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

a hybrid procedure for detecting global treatment effects in multivariate clinical trials theory and applications to fmri studies in multivariate clinical trials a key research endpoint is ascertaining whether a candidate treatment is more efficacious than an established alternative this global endpoint is clearly of high practical value for studies such as those arising from neuroimaging where the outcome dimensions are not only numerous but they are also highly correlated and the available sample sizes are typically small in this paper we develop a two stage procedure testing the null hypothesis of global equivalence between treatments effects and demonstrate its application to analysing phase ii neuroimaging trials prior information such as suitable statistics of historical data or suitably elicited expert clinical opinions are combined with data collected from the first stage of the trial to learn a set of optimal weights we apply these weights to the outcome dimensions of the second stage responses to form the linear combination z and t tests statistics while controlling the test s false positive rate we show that the proposed tests hold desirable asymptotic properties and characterise their power functions under wide conditions in particular by comparing the power of the proposed tests with that of hotelling s t2 we demonstrate their advantages when sample sizes are close to the dimension of the multivariate outcome we apply our methods to fmri studies where we find that for sufficiently precise first stage estimates of the treatment effect standard single stage testing procedures are outperformed copyright 2011 john wiley sons ltd

average case complexity of multivariate integration we study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical wiener sheet measure to derive the average case complexity one needs to obtain optimal sample points we prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years this relation enables us to show that optimal sample points can be derived from hammersley points extending the result of roth and using the recent result of wasilkowski we conclude that the average case complexity is e 1 lne 1 d 1 2

estimation of multidimensional integrals is monte carlo the best method the effectiveness of a deterministic algorithm recently proposed by wozniakowski 1991 for the numerical estimation of high dimensional integrals based on so called optimal sampling points is studied by using it to compute virial coefficients for the hard sphere fluid although the algorithm is inherently more efficient than the familiar monte carlo method for sufficiently large samples in practice this advantage diminishes rapidly with increasing dimensionality for the viral coefficient problem the authors find the new algorithm to be noticeably more efficient than crude monte carlo methods for dimensions less than 10 and of comparable efficiency for larger dimensionality up to 14

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

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

accurate p values for adaptive designs with binary endpoints adaptive designs encompass all trials allowing various types of design modifications over the course of the trial a key requirement for confirmatory adaptive designs to be accepted by regulators is the strong control of the family wise error rate this can be achieved by combining the p values for each arm and stage to account for adaptations including but not limited to treatment selection sample size adaptation and multiple stages while the theory for this is novel and well established in practice these methods can perform poorly especially for unbalanced designs and for small to moderate sample sizes the problem is that standard stagewise tests have inflated type i error rate especially but not only when the baseline success rate is close to the boundary and this is carried over to the adaptive tests seriously inflating the family wise error rate we propose to fix this problem by feeding the adaptive test with second order accurate p values in particular bootstrap p values secondly an adjusted version of the simes procedure for testing intersection hypotheses that reduces the built in conservatism is suggested numerical work and simulations show that unlike their standard counterparts the new approach preserves the overall error rate at or below the nominal level across the board irrespective of the baseline rate stagewise sample sizes or allocation ratio copyright 2017 john wiley sons ltd

the analysis of multiple endpoints in clinical trials treatment comparisons in randomized clinical trials usually involve several endpoints such that conventional significance testing can seriously inflate the overall type i error rate one option is to select a single primary endpoint for formal statistical inference but this is not always feasible another approach is to apply bonferroni correction i e multiply each p value by the total number of endpoints its conservatism for correlated endpoints is examined for multivariate normal data a third approach is to derive an appropriate global test statistic and this paper explores one such test applicable to any set of asymptotically normal test statistics quantitative binary and survival endpoints are all considered within this general framework two examples are presented and the relative merits of the proposed strategies are discussed

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 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

multivariate tests based on left spherically distributed linear scores in this paper a method for multivariate testing based on low dimensional data dependent linear scores is proposed the new approach reduces the dimensionality of observations and increases the stability of the solutions the method is reliable even if there are many redundant variables as a key feature the score coefficients are chosen such that a left spherical distribution of the scores is reached under the null hypothesis therefore well known tests become applicable in high dimensional situations too the presented strategy is an alternative to least squares and maximum likelihood approaches in a natural way standard problems of multivariate analysis thus induce the occurrence of left spherical nonnormal distributions hence new fields of application are opened up to the generalized multivariate analysis the proposed methodology is not restricted to normally distributed data but can also be extended to any left spherically distributed observations

optimized multi stage designs controlling the false discovery or the family wise error rate when a large number of hypotheses are investigated we propose multi stage designs where in each interim analysis promising hypotheses are screened which are investigated in further stages given a fixed overall number of observations this allows one to spend more observations for promising hypotheses than with single stage designs where the observations are equally distributed among all considered hypotheses we propose multi stage procedures controlling either the family wise error rate fwer or the false discovery rate fdr and derive asymptotically optimal stopping boundaries and sample size allocations across stages to maximize the power of the procedure optimized two stage designs lead to a considerable increase in power compared with the classical single stage design going from two to three stages additionally leads to a distinctive increase in power adding a fourth stage leads to a further improvement which is however less pronounced surprisingly we found only small differences in power between optimized integrated designs where the data of all stages are used in the final test statistics and optimized pilot designs where only the data from the final stage are used for testing however the integrated design controlling the fdr appeared to be more robust against misspecifications in the planning phase additionally we found that with increasing number of stages the drop in power when controlling the fwer instead of the fdr becomes negligible our investigations show that the crucial point is not the choice of the error rate or the type of design but the sequential nature of the trial where non promising hypotheses are dropped in the early phases of the experiment

analytical expressions for the fourth virial coefficient of a hard sphere mixture a method of numerical calculation of the fourth virial coefficients of the mixture of additive hard spheres is proposed the results are compared with an exact analytical formula for the fourth partial virial coefficient b 4 1 i e three spheres of diameters ensuremath sigma 1 and one sphere of diameter ensuremath sigma 2 and a semiempirical expression for b 4 2 i e two spheres of each kind it is shown that the first formula is nonanalytic and the implication to the equations of state for hard sphere mixtures is discussed

sequential monitoring of response adaptive randomized clinical trials clinical trials are complex and usually involve multiple objectives such as controlling type i error rate increasing power to detect treatment difference assigning more patients to better treatment and more in literature both response adaptive randomization rar procedures by changing randomization procedure sequentially and sequential monitoring by changing analysis procedure sequentially have been proposed to achieve these objectives to some degree in this paper we propose to sequentially monitor response adaptive randomized clinical trial and study it s properties we prove that the sequential test statistics of the new procedure converge to a brownian motion in distribution further we show that the sequential test statistics asymptotically satisfy the canonical joint distribution defined in jennison and turnbull 2000 therefore type i error and other objectives can be achieved theoretically by selecting appropriate boundaries these results open a door to sequentially monitor response adaptive randomized clinical trials in practice we can also observe from the simulation studies that the proposed procedure brings together the advantages of both techniques in dealing with power total sample size and total failure numbers while keeps the type i error in addition we illustrate the characteristics of the proposed procedure by redesigning a well known clinical trial of maternal infant hiv transmission

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

maximum entropy approach to classical hard sphere and hard disk equations of state the maximum entropy procedure for extrapolating series is applied to sum the known virial series for the hard disk and hard sphere fluids this procedure is found to produce numerical results more accurate than pade approximants and it agrees with monte carlo simulations of these systems the maximum entropy results extrapolate the series smoothly to densities arbitrarily near to maximum close packing and predict no singularities whatsoever

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

explicit and implicit finiteness corrections of virial coefficients the equation of state of finite systems deviates from thermodynamic limit the corresponding finiteness corrections of virial coefficients are studied most calculations are based on the canonical ensemble with periodic boundary conditions explicit andimplicit finiteness corrections occur they are displayed up to the eighth virial coefficient these results are applied to hard spheres in one two and three dimensions

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

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

an adaptive method for establishing a dose response relationship a multistage sampling method is proposed in dose response trials where dose adaptions can be performed in the preplanned adaptive interim analyses the overall test for proving a dose response relationship is performed by fisher s product criterion for the p values from the separate tests of a dose effect in the disjoint samples at the different stages based on these p values decision boundaries for early stopping with the rejection of the global null hypothesis of no existing dose response relationship are introduced the power of the adaptive two stage procedure using a particular adaptation rule is compared with the power of the test for a linear trend under analysis of variance assumptions in extensive simulations the bias in estimation is also quantified this procedure could be used for establishing a dose response relationship without including a placebo treatment

intersecting disks and spheres and statistical mechanics i mathematical basis the calculations of a previous paper on intersecting disks are completed further quantities of interest in connection with intersecting disks and spheres are defined the above considerations are extended to spherical boundary conditions then two applications are stated the penetrable sphere model of widom and rowlinson and the hard sphere system finally the generalization tod dimensional spheres is outlined

an approximate likelihood ratio test for a normal mean vector with nonnegative components with application to clinical trials summary a new multivariate statistic designed to be sensitive to the alternative hypothesis that a mean vector lies in the positive orthant rather than to all alternatives or to alternatives on a line o brien 1984 is proposed the statistic is shown to approximate the likelihood ratio test statistic for such alternatives kud5 1963 perlman 1969 its null distribution is a special case of the chi bar squared distribution which allows critical values to be obtained such a statistic may be used in two armed clinical trials where we are interested in showing that one treatment is better than another with respect to several responses as an example a clinical trial is reanalyzed with stronger conclusions than had been drawn originally

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

overlap graph representation of b6 and b7 the virial coefficients bn of the pressure of a thermodynamic system can be represented in terms of graphs the recently defined overlap graphs are studied in detail furthermore the overlap graph representation of the sixth and seventh virial coefficientsb 6 andb 7 is determined

procedures for comparing samples with multiple endpoints summary five procedures are considered for the comparison of two or more multivariate samples these procedures include a newly proposed nonparametric rank sum test and a generalized least squares test also considered are the following tests ordinary least squares hotelling s p and a bonferroni per experiment error rate approach applications are envisaged in which each variable represents a qualitatively different measure of response to treatment the null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others in all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test and is recommended for general use alternatively the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples a convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated clinical trials are often conducted for the purpose of evaluating the relative efficacy of two or more modes of therapy when the therapies are administered to independent groups and efficacy is measured on the basis of a single response variable appropriate parametric and nonparametric one way analysis of variance anova procedures and their properties are well known often however efficacy is measured by more than one variable although univariate methods for assessing each characteristic individually are useful in this setting there is often the additional need for a single overall objective probability statement that addresses the question of whether or not the experimental therapy is efficacious this need is particularly acute when the medical question is controversial and the sample size is small although a large literature exists on the comparison of multivariate samples the procedures tend to be infrequently used in practice as pointed out by meier 1975 p 523 the standard analysis for the comparison of two multivariate samples which is based on hotelling s t2 statistic addresses somewhat the wrong question and consequently has very poor power for the alternatives of primary interest specifically the t2 procedure addresses the question are one or more of the treatments different making no distinction between variables that change favorably and variables that change unfavorably a second approach would be to assign per experiment error rates to each of the univariate tests by using bonferroni s inequality i e by multiplying each univariate p value by the number of variables studied the desired overall probability statement is then given by the minimum of the per experiment univariate p values although this approach may be useful in some settings it may lack power for alternatives in which most or all measures of efficacy are improved this will be of particular concern when the number k of endpoints studied is large relative to sample size in fact statistical significance will not be possible in some instances this problem will be exacerbated when the measures of efficacy are highly

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

statistical mechanics of hard spheres a theoretical equation of state for a system of hard spheres is proposed it is based on the ideas of the scaled particle theory that has been developed by reiss et al j chem phys 31 369 1959 in this paper their method is generalized by expanding the function g y r in a set of functions ti r so that the low density behavior is described exactly not only for asymptotically large values of r but for the full range of r values as a consequence the virial coefficients b3 and b4 come out exactly in the approach to the scaled particle theory for the compressibility factor a pade like expression is obtained that gives exact values for b3 and b4 and reproduces b5 b6 b7 and b8 to within 1 3 1 4 0 74 and 5 5 respectively furthermore the equation of state agrees with the simulation data up to high densities where the fluid is metastable an expression for the surface tension is also found

a two stage procedure for comparing hazard rate functions summary comparison of two hazard rates is important in applications that are related to times to occurrence of a specific event conventional comparison procedures such as the log rank gehan wilcoxon and peto peto tests are powerful only when the two hazard rates do not cross each other because crossing hazard rates are common in practice several procedures have been proposed in the literature for comparing such rates however most of these procedures consider only the alternative hypothesis with crossing hazard rates many other realistic cases including those when the two hazard rates run parallel to each other are excluded from consideration we propose a two stage procedure that considers all possible alternatives including ones with crossing or running parallel hazard rates to define its significance level and p value properly a new procedure for handling the crossing hazard rates problem is suggested which has the property that its test statistic is asymptotically independent of the test statistic of the log rank test we show that the two stage procedure with the log rank test and the suggested procedure for handling the crossing hazard rates problem used in its two stages performs well in applications in comparing two hazard rates

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

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

sequential monitoring of clinical trials the role of information and brownian motion sequential monitoring has been a topic of major interest in clinical trials methodology over the past two decades this paper presents a unified conceptual framework for sequential monitoring that covers a wide variety of monitoring procedures in a wide variety of clinical trial settings the central elements of this framework consist of a suitable concept of statistical information and a scheme for using this concept as a basis for summarizing the accumulating results of a trial in a standardized form through a stochastic process that can be shown to approximate classical brownian motion the ideas are developed in a simple step by step fashion and illustrated by several practical examples

hard sphere gas state equation abstract the dynamic evolution of a gas of n hard spheres determines the equation of state once the equilibrium is reached after a short transient the system is investigated in the thermodynamic limit the algorithms based on a tricky management of the collisions list allow to simulate up to n 10 6 spheres rendering the statistical error sufficiently small in simulations involving 10 8 collisions the effect of boundaries is discussed and to avoid any dependence periodic boundary conditions are chosen in a box whose edge is much larger than the spheres radius the initial state is the symmetric close packing and by reducing the hard spheres radius we follow the evolution of the mean free path as a function of the density y where y is the ratio between the volume of the spheres and the total volume we observe the solid fluid first order phase transition and follow the fluid branch until the hard spheres gas is very dilute the phase transition is well resolved due to the improved statistics and to the choice as order parameter of the mean free path rather than z pv nkt which has a singularity at zero mean free path the equations of state in the fluid and solid branches are compared with the taylor series for the mean free path obtained from the virial expansions of z the second order truncations p 2 y appear to provide the best fit to the mean free path this suggests an approximation to z as z 1 b 2 y p 2 y where b 2 is the second virial coefficient

effect modification and design sensitivity in observational studies in an observational study of treatment effects subjects are not randomly assigned to treatment or control so differing outcomes in treated and control groups may reflect a bias from nonrandom assignment rather than a treatment effect after adjusting for measured pretreatment covariates perhaps by matching a sensitivity analysis determines the magnitude of bias from an unmeasured covariate that would need to be present to alter the conclusions of the naive analysis that presumes adjustments eliminated all bias other things being equal larger effects tend to be less sensitive to bias than smaller effects effect modification is an interaction between a treatment and a pretreatment covariate controlled by matching so that the treatment effect is larger at some values of the covariate than at others in the presence of effect modification it is possible that results are less sensitive to bias in subgroups experiencing larger effects two cases are considered i an a priori grouping into a few categ

a new graph expansion of virial coefficients it is well known that the virial coefficients of the pressure of thermodynamic systems can be represented in terms of graphs the existing graph expansions are compared with a new one the overlap graph expansion the merits of overlap graphs in general and especially for hard disks and spheres are discussed

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 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