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

the tanh and the sine cosine methods for the complex modified k dv and the generalized k dv equations the complex modified k dv cmk dv equation and the generalized k dv equation are investigated by using the tanh method and the sine cosine method a variety of exact travelling wave solutions with compact and noncompact structures are formally obtained for each equation the study reveals the power of the two schemes where each method complements the other

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

the attractor basin portrait of a cellular automaton local space time structures such as domains and the intervening dislocations dominate a wide class of cellular automaton ca behavior for such spatially extended dynamics regular domains vicinities and attractors are introduced as organizing principles to identify the discretized analogs of attractors basins and separatrices structures used in classifying dissipative continuous state dynamical systems we describe the attractor basin portrait of nonlinear elementary ca rule 18 whose global dynamics is largely determined by a single regular attracting domain the latter s basin is analyzed in terms of subbasin and portal structures associated with particle annihilation the conclusion is that the computational complexity of such ca is more apparent than real transducer machines are constructed that automatically identify domain and dislocation structures in space time count the number of dislocations in a spatial pattern and implement an isomorphism between rule 18 and rule 90 we use a transducer to trace dislocation trajectories and confirm that in rule 18 isolated dislocation trajectories as well as a dislocation gas agree extremely well with the classical model of annihilating diffusive particles the ca efficiently transforms randomness of an initial pattern ensemble into a random walk of dislocations in space time

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

coupled map lattices one step forward and two steps back abstract we discuss the general notion of lattice dynamical systems and some recent progress in their studies we suggest also the approach that allows to extract some information on the dynamics of finitely extended real systems from the results obtained on the infinitely extended artificial ones

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

effects of assumption violations on type i error rate in group sequential monitoring we compare the degree to which the type i error probability can be inflated by assumption violations under the different sequential monitoring schemes for methods that assume equally spaced time intervals between looks at the data we examine the effects of unequal spacings for methods that do not specify the look times in advance we examine the degree of type 1 error inflation that can result when future look times are chosen based on data trends we also evaluate a bayesian method of monitoring from this non bayesian perspective of type i error inflation

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

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

aperiodicity in one dimensional cellular automata abstract cellular automata are a class of mathematical systems characterized by discreteness in space time and state values determinism and local interaction a certain class of one dimensional binary site valued nearest neighbor automata is shown to generate infinitely many aperiodic temporal sequences from arbitrary finite initial conditions on an infinite lattice the class of automaton rules that generate aperiodic temporal sequences are characterized by a particular form of injectivity in their interaction rules included are the nontrivial linear automaton rules that is rules for which the superposition principle holds certain nonlinear automata with injectivity properties similar to those of linear automata and a wider subset of nonlinear automata whose interaction rules satisfy a weaker form of injectivity together with certain symmetry conditions the evolution of this last subset of automata can be viewed as consisting of multiple domains within which behavior exactly mimics that of a linear automaton a linearization technique is used to establish that these domains coalesce in finite time to produce at most two domains and thereby to assert the aperiodicity of their temporal sequences

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

criticality in cellular automata abstract using recent results obtained for the transition to turbulence via spatiotemporal intermittency in extended dynamical systems critical cellular automata rules are built thanks to a systematic procedure the continuous phase transition observed in a coupled map lattice is translated into a sequence of cellular automata rules for which the critical properties of the original system are reproduced as precisely as desired it is shown that criticality as understood from the point of view of statistical mechanics is intimately related to the various characteristics of wolfram s class iv rules this suggests in turn the picture of a critical surface in the space of rules and provides the basis for a discussion of the problem of classification schemes we review recent cellular automata studies where questions linked to criticality arise and argue that they are unified in light of our results leaving the relationship between computational and statistical characterizations of critical rules as the central problem for future studies on this subject

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

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

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

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 phase transitions in coupled map lattices we study finite coupled map lattices of size d 2 with individual maps 0 1 0 1 and constant diffuse coupling for x 2x mod 1 we give sufficient conditions that the coupled system has a continuum of ergodic components in the case d 2 we determine the number of ergodic components for all coupling strengths if is a mixing tent map close to the transition from mixing to a periodic interval with period 2 the uncoupled system is mixing whereas numerical studies suggest that coupling with a suitable strength breaks up the phase space into domains which are interchanged with period 2 in case d 2 we prove this rigorously

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

an evaluation of inferential procedures for adaptive clinical trial designs with pre specified rules for modifying the sample size summary many papers have introduced adaptive clinical trial methods that allow modifications to the sample size based on interim estimates of treatment effect there has been extensive commentary on type i error control and efficiency considerations but little research on estimation after an adaptive hypothesis test we evaluate the reliability and precision of different inferential procedures in the presence of an adaptive design with pre specified rules for modifying the sampling plan we extend group sequential orderings of the outcome space based on the stage at stopping likelihood ratio statistic and sample mean to the adaptive setting in order to compute median unbiased point estimates exact confidence intervals and p values uniformly distributed under the null hypothesis the likelihood ratio ordering is found to average shorter confidence intervals and produce higher probabilities of p values below important thresholds than alternative approaches the bias adjusted mean demonstrates the lowest mean squared error among candidate point estimates a conditional error based approach in the literature has the benefit of being the only method that accommodates unplanned adaptations we compare the performance of this and other methods in order to quantify the cost of failing to plan ahead in settings where adaptations could realistically be pre specified at the design stage we find the cost to be meaningful for all designs and treatment effects considered and to be substantial for designs frequently proposed in the literature

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

on the relationship between fuzzy and boolean cellular automata fuzzy cellular automata fca are continuous cellular automata where the local rule is defined as the fuzzification of the local rule of a corresponding boolean cellular automaton in disjunctive normal form in this paper we are interested in the relationship between boolean and fuzzy models and for the first time we analytically show the existence of a strong connection between them by focusing on two properties density conservation and additivity we begin by showing that the density conservation property extensively studied in the boolean domain is preserved in the fuzzy domain a boolean ca is density conserving if and only if the corresponding fca is sum preserving a similar result is established for another novel spatial density conservation property second we prove an interesting parallel between the additivity of boolean ca and oscillations of the corresponding fuzzy ca around its fixed point in fact we show that a boolean ca is additive if and only if the behaviour of the corresponding fuzzy ca around its fixed point coincides with the boolean behaviour finally we give a probabilistic interpretation of our fuzzification which establishes an equivalence between convergent fuzzy ca and the mean field approximation on boolean ca an estimation of their asymptotic density these connections between the boolean and the fuzzy models are the first formal proofs of a relationship between them

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

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

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

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

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

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 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

point estimation in adaptive enrichment designs adaptive enrichment designs are an attractive option for clinical trials that aim at demonstrating efficacy of therapies which may show different benefit for the full patient population and a prespecified subgroup in these designs based on interim data either the subgroup or the full population is selected for further exploration when selection is based on efficacy data this introduces bias to the commonly used maximum likelihood estimator for the situation of two stage designs with a single prespecified subgroup we present six alternative estimators and investigate their performance in a simulation study the most consistent reduction of bias over the range of scenarios considered was achieved by a method combining the uniformly minimum variance conditionally unbiased estimator with a conditional moment estimator application of the methods is illustrated by a clinical trial example

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

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

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

convergence and aperiodicity in fuzzy cellular automata revisiting rule 90 abstract in this paper we consider a continuous version of cellular automata fuzzy ca obtained by fuzzification of the disjunctive normal form which describes the corresponding boolean rule we concentrate on fuzzy rule 90 whose boolean version deserves some attention for the complex patterns it generates we show that the behavior of fuzzy rule 90 is very simple in that the system always converges to a fixed point in the case of finite support configurations we also show aperiodicity of every temporal sequences extending and complementing jen s result on aperiodicity of boolean rule 90 we finally show and analyze the remarkable fact that depending on the level of state discreteness used to visualize the dynamics of fuzzy rule 90 the display might show after a transient the well known complex boolean behavior instead of the correct convergence to a fixed point the results of the analysis lead not only to a caveat on the dangers of visualization but also an unexpected explanation of the dynamics of boolean rule 90

a conditional error function approach for subgroup selection in adaptive clinical trials growing interest in personalised medicine and targeted therapies is leading to an increase in the importance of subgroup analyses if it is planned to view treatment comparisons in both a predefined subgroup and the full population as co primary analyses it is important that the statistical analysis controls the familywise type i error rate spiessens and debois cont clin trials 2010 31 647 656 recently proposed an approach specific for this setting which incorporates an assumption about the correlation based on the known sizes of the different groups and showed that this is more powerful than generic multiple comparisons procedures such as the bonferroni correction if recruitment is slow relative to the length of time taken to observe the outcome it may be efficient to conduct an interim analysis in this paper we propose a new method for an adaptive clinical trial with co primary analyses in a predefined subgroup and the full population based on the conditional error function principle the methodology is generic in that we assume test statistics can be taken to be normally distributed rather than making any specific distributional assumptions about individual patient data in a simulation study we demonstrate that the new method is more powerful than previously suggested analysis strategies furthermore we show how the method can be extended to situations when the selection is not based on the final but on an early outcome we use a case study in a targeted therapy in oncology to illustrate the use of the proposed methodology with non normal outcomes copyright 2012 john wiley sons ltd

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

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

exact inference for adaptive group sequential designs methods for controlling the type 1 error of an adaptive group sequential trial were developed in seminal papers by cui hung and wang biometrics 1999 lehmacher and wassmer biometrics 1999 and muller and schafer biometrics 2001 however corresponding solutions for the equally important and related problem of parameter estimation at the end of the adaptive trial have not been completely satisfactory in this paper a method is provided for computing a two sided confidence interval having exact coverage along with a point estimate that is median unbiased for the primary efficacy parameter in a two arm adaptive group sequential design the possible adaptations are not only confined to sample size alterations but also include data dependent changes in the number and spacing of interim looks and changes in the error spending function the procedure is based on mapping the final test statistic obtained in the modified trial into a corresponding backward image in the original trial this is an advance on previously available methods which either produced conservative coverage and no point estimates or provided exact coverage for one sided intervals only

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

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

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

lie symmetry analysis and exact solutions of 3 1 dimensional yutodasasafukuyama equation in mathematical physics in this paper the lie symmetry analysis method has been proposed for finding similarity reduction and exact solutions of nonlinear evolution equation here for illustrating the effectiveness and accuracy of proposed method we have taken 3 1 dimensional yutodasasafukuyama ytfs equation also by using symmetry reduction method we have reduced nonlinear partial differential equation npde to nonlinear ordinary differential equation which has advantage to provide semi analytical solution we have obtained infinitesimal generators the entire geometric vector field commutator table of lie algebra and symmetry group by using lie symmetry analysis method then we have used tanh method for finding new analytical exact solutions of some reduced transform equations

conservation laws and solutions of a generalized coupled 2 1 dimensional burgers system abstract in this paper we study a generalized coupled 2 1 dimensional burgers system which is a nonlinear version of a bilinear system under some dependent variable transformations it was introduced recently in the literature and has attracted a fair amount of interest from physicists the lie symmetry analysis together with the kudryashov approach are utilized to obtain new travelling wave solutions of the system furthermore for the first time conservation laws of the system are derived using the multiplier method

approximately optimal one parameter boundaries for group sequential trials we present a class of group sequential tests indexed by a single parameter that yields approximately optimal results we also provide tables of key values to help in the design of group sequential tests that meet selected specifications

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

subgroup detection and sample size calculation with proportional hazards regression for survival data in this paper we propose a testing procedure for detecting and estimating the subgroup with an enhanced treatment effect in survival data analysis here we consider a new proportional hazard model that includes a nonparametric component for the covariate effect in the control group and a subgroup treatment interaction effect defined by a change plane we develop a score type test for detecting the existence of the subgroup which is doubly robust against misspecification of the baseline effect model or the propensity score but not both under mild assumptions for censoring when the null hypothesis of no subgroup is rejected the change plane parameters that define the subgroup can be estimated on the basis of supremum of the normalized score statistic the asymptotic distributions of the proposed test statistic under the null and local alternative hypotheses are established on the basis of established asymptotic distributions we further propose a sample size calculation formula for detecting a given subgroup effect and derive a numerical algorithm for implementing the sample size calculation in clinical trial designs the performance of the proposed approach is evaluated by simulation studies an application to an aids clinical trial data is also given for illustration

design and monitoring of multi arm multi stage clinical trials two arm group sequential designs have been widely used for over 40 years especially for studies with mortality endpoints the natural generalization of such designs to trials with multiple treatment arms and a common control mams designs has however been implemented rarely while the statistical methodology for this extension is clear the main limitation has been an efficient way to perform the computations past efforts were hampered by algorithms that were computationally explosive with the increasing interest in adaptive designs platform designs and other innovative designs that involve multiple comparisons over multiple stages the importance of mams designs is growing rapidly this article provides break through algorithms that can compute mams boundaries rapidly thereby making such designs practical for designs with efficacy only boundaries the computational effort increases linearly with number of arms and number of stages for designs with both efficacy and futility boundaries the computational effort doubles with successive increases in number of stages

cellular automata in fuzzy backgrounds abstract the main purpose of this work is to understand some limitations introduced by the classical definitions of cellular automata ca to this end we have defined a new model of cas fuzzy ca s which allows the observation of interesting chaotic properties of elementary cas to date neither a formal nor a precise definition of chaos in cas exists we believe that the proposed model provides a sharper tool to detect which properties can be associated to a chaotic behavior we also define a measure rule entropy which gives information about the ca s dynamics solely on the basis of the rule table and provides theoretical explanations to some of the empirical observations

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

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

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

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

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

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