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

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

mastrovito multiplier for general irreducible polynomials we present a new formulation of the mastrovito multiplication matrix for the field gf 2 sup m generated by an arbitrary irreducible polynomial we study in detail several specific types of irreducible polynomials e g trinomials all one polynomials and equally spaced polynomials and obtain the time and space complexity of these designs particular examples illustrating the properties of the proposed architecture are also given the complexity results established in this paper match the best complexity results known to date the most important new result is the space complexity of the mastrovito multiplier for an equally spaced polynomial which is found as m sup 2 spl delta xor gates and m sup 2 and gates where spl delta is the spacing factor

polynomial factorization over f 2 we describe algorithms for polynomial factorization over the binary field f2 and their implementation they allow polynomials of degree up to 250 000 to be factored in about one day of cpu time distributing the work on two processors

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

degree reduction of bezier curves and filter banks abstract we consider the degree elevation and reduction of bezier curves as the filter bank process the process consists of the synthesis filters and the analysis filters using the relationship of basis changes we find what these filters are and how these filters are related explicit forms of each filter are given and the best degree reduced bezier curves in the l2 norm l norm and l1 norm are obtained easily from the analysis filters

a new approach to subquadratic space complexity parallel multipliers for extended binary fields based on toeplitz matrix vector products and coordinate transformation techniques we present a new scheme for subquadratic space complexity parallel multiplication in gf 2n using the shifted polynomial basis both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers for example with n being a power of 2 the space complexity is about 8 percent better while the asymptotic gate delay is about 33 percent better respectively another advantage of the proposed matrix vector product approach is that it can also be used to design subquadratic space complexity polynomial dual weakly dual and triangular basis parallel multipliers to the best of our knowledge this is the first time that subquadratic space complexity parallel multipliers are proposed for dual weakly dual and triangular bases a recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers this design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature

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

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

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

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

constrained degree reduction of polynomials in bernstein bezier form over simplex domain in this paper we show that the orthogonal complement of a subspace in the polynomial space of degree n over d dimensional simplex domain with respect to the l 2 inner product and the weighted euclidean inner product of bb bezier bernstein coefficients are equal using it we also prove that the best constrained degree reduction of polynomials over the simplex domain in bb form equals the best approximation of weighted euclidean norm of coefficients of given polynomial in bb form from the coefficients of polynomials of lower degree in bb form

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

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

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

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

good degree reduction of bezier curves using jacobi polynomials abstract the constrained chebyshev polynomial is the error function of the best degree reduction of polynomial with c1 continuity in this paper we propose the constrained jacobi polynomial as an alternative error function for good degree reduction although the degree reduction is not the best approximation it is more useful than the constrained chebyshev polynomial since its coefficients are represented explicitly but the coefficients of the constrained chebyshev polynomial are not we present the uniform error bounds of the constrained jacobi polynomials and subdivision scheme for degree reduction within given tolerance we also apply our method to an example and compare its result to that of the best degree reduction

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

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

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

arithmetic operations in finite fields of medium prime characteristic using the lagrange representation in this paper we propose a complete set of algorithms for the arithmetic operations in finite fields of prime medium characteristic the elements of the fields ifpk are represented using the newly defined lagrange representation where polynomials are expressed using their values at sufficiently many points our multiplication algorithm which uses a montgomery approach can be implemented in o k multiplications and o k2 log k additions in the base field ifp for the inversion we propose a variant of the extended euclidean gcd algorithm where the inputs are given in the lagrange representation the lagrange representation scheme and the arithmetic algorithms presented in the present work represent an interesting alternative for elliptic curve cryptography

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

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

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

subquadratic space complexity binary field multiplier using double polynomial representation this paper deals with binary field multiplication we use the bivariate representation of binary field called double polynomial system dps presented in this concept generalizes the composite field representation to every finite field as shown in the main interest of dps representation is that it enables to use lagrange approach for multiplication and in the best case fast fourier transform approach which optimizes lagrange approach we use here a different strategy from to perform reduction and we also propose in this paper some new approaches for constructing dps we focus on dps which provides a simpler and more efficient method for coefficient reduction this enables us to avoid a multiplication required in the montgomery reduction approach of and thus to improve the complexity of the dps multiplier the resulting algorithm proposed in the present paper is subquadratic in space o n1 31 and logarithmic in time the space complexity is 33 percent better than in and 18 percent faster it is asymptotically more efficient than the best known method specifiably more efficient than when n 3 000 furthermore our proposal is available for every n and not only for n a power of two or three

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

a note on constrained degree reduction of polynomials in bernstein bezier form over simplex domain in the paper h s kim y j ahn constrained degree reduction of polynomials in bernstein bezier form over simplex domain j comput appl math 216 2008 14 19 kim and ahn proved that the best constrained degree reduction of a polynomial over d dimensional simplex domain in l 2 norm equals the best approximation of weighted euclidean norm of the bernstein bezier coefficients of the given polynomial in this paper we presented a counterexample to show that the approximating polynomial of lower degree to a polynomial is virtually non existent when d 2 furthermore we provide an assumption to guarantee the existence of solution for the constrained degree reduction

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

gf 2 sup m multiplication and division over the dual basis in this paper an algorithm for gf 2 sup m multiplication division is presented and a new more generalized definition of duality is proposed from these the bit serial berlekamp multiplier is derived and shown to be a specific case of a more general class of multipliers furthermore it is shown that hardware efficient bit parallel dual basis multipliers can also be designed these multipliers have a regular structure are easily extended to different gf 2 sup m and hence suitable for vlsi implementations as in the bit serial case these bit parallel multipliers can also be hardwired to carry out constant multiplication these constant multipliers have reduced hardware requirements and are also simple to design in addition the multiplication division algorithm also allows a bit serial systolic finite field divider to be designed this divider is modular independent of the defining irreducible polynomial for the field easily expanded to different gf 2 sup m and its longest delay path is independent of m

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

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

a generalized method for constructing subquadratic complexity gf 2 sup k multipliers we introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions the construction is obtained by recursively extending short convolution algorithms and nesting them to obtain the short convolution algorithms the winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication we present a recursive construction technique that extends any d point multiplier into an n d sup k point multiplier with area that is subquadratic and delay that is logarithmic in the bit length n we present a thorough analysis that establishes the exact space and time complexities of these multipliers using the recursive construction method we obtain six new constructions among which one turns out to be identical to the karatsuba multiplier all six algorithms have subquadratic space complexities and two of the algorithms have significantly better time complexities than the karatsuba algorithm

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

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

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

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

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

constrained multi degree reduction of triangular bezier surfaces using dual bernstein polynomials we propose a novel approach to the problem of multi degree reduction of bezier triangular patches with prescribed boundary control points we observe that the solution can be given in terms of bivariate dual discrete bernstein polynomials the algorithm is very efficient thanks to using the recursive properties of these polynomials the complexity of the method is o n 2m 2 n and m being the degrees of the input and output bezier surfaces respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined triangular bezier surfaces the result is a composite surface of global c r continuity with a prescribed order r some illustrative examples are given

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

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

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

degree reduction of bezier curves by l1 approximation with endpoint interpolation abstract we consider the one degree reduction problem with endpoint interpolation in the l1 norm we obtain the best one degree reduction of bezier curve of the degree n 5 with endpoint interpolation by using perfect splines for the general degree n we propose a good one degree reduction by use of an appropriate transform of the tchebycheff polynomials un x of the second kind of degree n by use of the good one degree reduction subdivision algorithm is given to get one degree reduced bezier curve within a given tolerance some numerical experiments are also given

subquadratic computational complexity schemes for extended binary field multiplication using optimal normal bases based on a recently proposed toeplitz matrix vector product approach a subquadratic computational complexity scheme is presented for multiplications in binary extended finite fields using type i and ii optimal normal bases

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

multi degree reduction of tensor product bezier surfaces with general boundary constraints abstract we propose an efficient approach to the problem of multi degree reduction of rectangular bezier patches with prescribed boundary control points we observe that the solution can be given in terms of constrained bivariate dual bernstein polynomials the complexity of the method is o mn 1 n 2 with m min m1 m2 where n1 n2 and m1 m2 is the degree of the input and output bezier surface respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined rectangular bezier surfaces the result is a composite surface of global cr continuity with a prescribed r 0 in the detailed discussion we restrict ourselves to r 0 1 which is the most important case in practical application some illustrative examples are given

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

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

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

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

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

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

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

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

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

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

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

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures