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

centers and isochronous centers for generalized quintic systems in this paper we classify the centers and the isochronous centers of certain polynomial differential systems in r 2 of degree d 5 odd that in complex notation are z i z z z d 5 2 a z 5 b z 4 z c z 3 z 2 d z 2 z 3 e z z 4 f z 5 where z x i y r and a b c d e f c note that if d 5 we obtain the class of polynomial differential systems in the form of a linear system with homogeneous polynomial nonlinearities of degree 5 due to the huge computations required for computing the necessary and sufficient conditions for the characterization of the centers and isochronous centers our study uses algorithms of computational algebra based on the grobner basis theory and on modular arithmetics

the fisher neyman pearson theories of testing hypotheses one theory or two abstract the fisher and neyman pearson approaches to testing statistical hypotheses are compared with respect to their attitudes to the interpretation of the outcome to power to conditioning and to the use of fixed significance levels it is argued that despite basic philosophical differences in their main practical aspects the two theories are complementary rather than contradictory and that a unified approach is possible that combines the best features of both as applications the controversies about the behrens fisher problem and the comparison of two binomials 2 2 tables are considered from the present point of view

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

on the inappropriateness of an em algorithm based procedure for blinded sample size re estimation when planning a clinical trial the sample size calculation is commonly based on an a priori estimate of the variance of the outcome variable misspecification of the variance can have substantial impact on the power of the trial it is therefore attractive to update the planning assumptions during the ongoing trial using an internal estimate of the variance for this purpose an em algorithm based procedure for blinded variance estimation was proposed for normally distributed data various simulation studies suggest a number of appealing properties of this procedure in contrast we show that i the estimates provided by this procedure depend on the initialization ii the stopping rule used is inadequate to guarantee that the algorithm converges against the maximum likelihood estimator and iii the procedure corresponds to the special case of simple randomization which however in clinical trials is rarely applied further we show that maximum likelihood estimation leads to no reasonable results for blinded sample size re estimation due to bias and high variability the problem is illustrated by a clinical trial in asthma

a comparison of permutation and mixed model regression methods for the analysis of simulated data in the context of a group randomized trial our first purpose was to determine whether in the context of a group randomized trial grt with gaussian errors permutation or mixed model regression methods fare better in the presence of measurable confounding in terms of their monte carlo type i error rates and power our results indicate that given a proper randomization the type i error rate is similar for both methods whether unadjusted or adjusted even in small studies however our results also show that should the investigator face the unfortunate circumstance in which modest confounding exists in the only realization available the unadjusted analysis risks a type i error in this regard there was little to distinguish the two methods finally our results show that power is similar for the two methods and not surprisingly better for the adjusted tests our second purpose was to examine the relative performance of permutation and mixed model regression methods in the context of a grt when the normality assumptions underlying the mixed model are violated published studies have examined the impact of violation of this assumption at the member level only our findings indicate that both methods perform well when the assumption is violated so long as the icc is very small and the design is balanced at the group level however at icc 0 01 the permutation test carries the nominal type i error rate while the model based test is conservative and so less powerful binomial group and member level errors did not otherwise change the relative performance of the two methods with regard to confounding copyright 2005 john wiley sons ltd

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

re estimating the sample size of an on going blinded trial based on the method of randomization block sums we extend a method we had previously described statist med 2005 to estimate the within group variance of a continuous endpoint without breaking the blind in a randomized clinical trial specifically we a explain how the method may be used for a wider set of designs than we had previously indicated b obtain a within group covariate adjusted blinded variance estimator c illustrate use of the method for sample size re estimation and d describe a procedure to determine whether or not the blinded variance estimator works well not just on average but for the data set at hand the proposed method is simple to use and makes no additional assumptions than is made for unblinded analysis simulations show that for realistic sample sizes there is virtually no inflation in the type i error rate when weighing the burden imposed by interim unblinded re estimation with the loss in precision with blinded re estimation it may be advantageous for some trials to use the blinded method

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

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

controlling test size while gaining the benefits of an internal pilot design summary to compensate for a power analysis based on a poor estimate of variance internal pilot designs use some fraction of the planned observations to reestimate error variance and modify the final sample size ignoring the randomness of the final sample size may bias the final variance estimate and inflate test size we propose and evaluate three different tests that control test size for an internal pilot in a general linear univariate model with fixed predictors and gaussian errors test 1 uses the first sample plus those observations guaranteed to be collected in the second sample for the final variance estimate test 2 depends mostly on the second sample for the final variance estimate test 3 uses the unadjusted variance estimate and modifies the critical value to bound test size we also examine three sample size modification rules only test 2 can control conditional test size align with a modification rule and provide simple power calculations we recommend it if the minimum second incremental sample is at least moderate perhaps 20 otherwise the bounding test appears to have the highest power in small samples reanalyzing published data highlights some advantages and disadvantages of the various tests

the center problem for a 1 4 resonant quadratic system abstract the main objective of this paper is to find necessary and sufficient conditions for a 1 4 resonant system of the form x x a 10 x 2 a 01 x y a 12 y 2 y 4 y b 2 1 x 2 b 10 x y b 01 y 2 to have a center at the origin since applying a linear change of variables any system of this form can be transformed either to system with a 10 1 or a 10 0 these are the two cases considered here when a 10 1 there appear 46 resonant center conditions and for a 10 0 there are 9 center conditions to obtain necessary conditions for integrability the computation of the resonant saddle quantities focus quantities and the decomposition of the variety of the ideal generated by an initial string of them were used the theory of darboux first integrals and some other methods as the monodromy arguments for instance are used to show the sufficiency since decomposition of the variety mentioned above was performed using modular computations the obtained conditions of integrability represent the complete list of the integrability conditions only with very high probability and there remains an open problem to verify this statement

numerical boundaries for some classical banach spaces abstract globevnik gave the definition of boundary for a subspace a c b this is a subset of that is a norming set for a we introduce the concept of numerical boundary for a banach space x a subset b x is a numerical boundary for a subspace a c b b x x if the numerical radius of f is the supremum of the modulus of all the evaluations of f at b for every f in a we give examples of numerical boundaries for the complex spaces x c 0 c k and d w 1 the predual of the lorentz sequence space d w 1 in all these cases if k is infinite we show that there are closed and disjoint numerical boundaries for the space of the functions from b x to x which are uniformly continuous and holomorphic on the open unit ball and there is no minimal closed numerical boundary in the case of c 0 we characterize the numerical boundaries for that space of holomorphic functions

an improved double sampling procedure based on the variance summary sample size calculations for a continuous outcome require specification of the anticipated variance inaccurate specification can result in an underpowered or overpowered study for this reason adaptive methods whereby sample size is recalculated using the variance of a subsample have become increasingly popular the first proposal of this type stein 1945 annals of mathematical statistics16 243 258 used all of the data to estimate the mean difference but only the first stage data to estimate the variance stein s procedure is not commonly used because many people perceive it as ignoring relevant data this is especially problematic when the first stage sample size is small as would be the case if the anticipated total sample size were small a more naive approach uses in the denominator of the final test statistic the variance estimate based on all of the data applying the helmert transformation we show why this naive approach underestimates the true variance and how to construct an unbiased estimate that uses all of the data we prove that the type i error rate of our procedure cannot exceed

how conservative is fisher s exact test a quantitative evaluation of the two sample comparative binomial trial the debate as to which statistical methodology is most appropriate for the analysis of the two sample comparative binomial trial has persisted for decades practitioners who favor the conditional methods of fisher fisher s exact test fet claim that only experimental outcomes containing the same amount of information should be considered when performing analyses hence the total number of successes should be fixed at its observed level in hypothetical repetitions of the experiment using conditional methods in clinical settings can pose interpretation difficulties since results are derived using conditional sample spaces rather than the set of all possible outcomes perhaps more importantly from a clinical trial design perspective this test can be too conservative resulting in greater resource requirements and more subjects exposed to an experimental treatment the actual significance level attained by fet the size of the test has not been reported in the statistical literature berger j r statist soc d the statistician 2001 50 79 85 proposed assessing the conservativeness of conditional methods using p value confidence intervals in this paper we develop a numerical algorithm that calculates the size of fet for sample sizes n up to 125 per group at the two sided significance level alpha 0 05 additionally this numerical method is used to define new significance levels alpha alpha epsilon where epsilon is a small positive number for each n such that the size of the test is as close as possible to the pre specified alpha 0 05 for the current work without exceeding it lastly a sample size and power calculation example are presented which demonstrates the statistical advantages of implementing the adjustment to fet using alpha instead of alpha in the two sample comparative binomial trial

strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices abstract using the variational method it is shown that the set of all strong peak functions in a closed algebra a of c b k is dense if and only if the set of all strong peak points is a norming subset of a as a corollary we can induce the denseness of strong peak functions on other certain spaces in case that a set of uniformly strongly exposed points of a banach space x is a norming subset of p x n then the set of all strongly norm attaining elements in p x n is dense in particular the set of all points at which the norm of p x n is frechet differentiable is a dense g subset in the last part using reisner s graph theoretic approach we construct some strongly norm attaining polynomials on a cl space with an absolute norm then we show that for a finite dimensional complex banach space x with an absolute norm its polynomial numerical indices are one if and only if x is isometric to l n moreover we give a characterization of the set of all complex extreme points of the unit ball of a cl space with an absolute norm

proper metrics for clinical trials transformations and other procedures to remove non normality effects a simulation study was performed to study the effects of non normality and to examine procedures to ameliorate possible loss of power when data are incorrectly assumed to be normally distributed it was found that only distributions with high asymmetry or heavy tails seriously affect the t test the box cox likelihood ratio test appears to have some advantages over the others but this must be offset by the greater complexity in making the results understandable to non statisticians the variability in outcomes with the different procedures demonstrates the importance of specifying such procedures a priori

a method to estimate the variance of an endpoint from an on going blinded trial blinded estimation of variance allows for changing the sample size without compromising the integrity of the trial some of the methods that estimate the variance in a blinded manner either make untenable assumptions or are only applicable to two treatment trials we propose a new method for continuous endpoints that makes minimal assumptions the method uses the enrollment order of subjects and the randomization block size to estimate the variance it can be applied to normal or non normal data trials with two or more treatments equal or unequal allocation schemes fixed or random randomization block sizes and single or multi centre trials the variance estimator is unbiased and performs best when the randomization block size is the smallest simulation results suggest that for many commonly used randomization block sizes the proposed estimator is expected to perform well the proposed method is used to estimate the variance of the endpoint for two trials and is shown to perform well by comparison with its unblinded counterpart

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

sequential equivalence testing and repeated confidence intervals with applications to normal and binary responses summary we propose group sequential tests of the equivalence of two treatments based on ideas related to repeated confidence intervals these tests adapt readily to unpredictable group sizes to the possibility of continuing even though a boundary has been crossed and to nonnormal observations in comparing two binomial distributions the required sample size depends strongly on the average success probability and an adaptive choice of group size is needed to produce an efficient test meeting specified error probability constraints a special case is the experiment where interim analyses are performed not for the purpose of early termination but simply to adjust the sample size so that nominal error rates will be guaranteed despite the presence of a nuisance parameter

local bifurcations of critical periods in a generalized 2d lv system abstract in this paper we investigate a generalized two dimensional lotka volterra system which has a center we give an inductive algorithm to compute polynomials of periodic coefficients find structures of solutions for systems of algebraic equation corresponding to isochronous centers and weak centers of finite order and derive conditions on parameters under which the considered equilibrium is an isochronous center or a weak center of finite order we show that with appropriate perturbations at most two critical periods bifurcate from the center

the solution of the 1 3 resonant center problem in the quadratic case abstract the 1 3 resonant center problem in the quadratic case is to find necessary and sufficient conditions for the existence of a local analytic first integral for the differential system x x a 10 x 2 a 01 xy a 12 y 2 y 3 y b 21 x 2 b 10 xy b 01 y 2 there appear 25 center cases for a 10 1 and 11 cases for a 01 0 the necessity is obtained using modular arithmetics and with high probability we have all the resonant center cases we show that in each case there exists a local analytic first integral sufficient condition around the origin this sufficient condition is proved using different classical criteria and in one case monodromy arguments

internal pilots for a class of linear mixed models with gaussian and compound symmetric data an internal pilot design uses interim sample size analysis without interim data analysis to adjust the final number of observations the approach helps to choose a sample size sufficiently large to achieve the statistical power desired but not too large which would waste money and time we report on recent research in cerebral vascular tortuosity curvature in three dimensions which would benefit greatly from internal pilots due to uncertainty in the parameters of the covariance matrix used for study planning unfortunately observations correlated across the four regions of the brain and small sample sizes preclude using existing methods however as in a wide range of medical imaging studies tortuosity data have no missing or mistimed data a factorial within subject design the same between subject design for all responses and a gaussian distribution with compound symmetry for such restricted models we extend exact small sample univariate methods for internal pilots to linear mixed models with any between subject design not just two groups planning a new tortuosity study illustrates how the new methods help to avoid sample sizes that are too small or too large while still controlling the type i error rate

pros and cons of permutation tests in clinical trials hypothesis testing in which the null hypothesis specifies no difference between treatment groups is an important tool in the assessment of new medical interventions for randomized clinical trials permutation tests that reflect the actual randomization are design based analyses for such hypotheses this means that only such design based permutation tests can ensure internal validity without which external validity is irrelevant however because of the conservatism of permutation tests the virtues of permutation tests continue to be debated in the literature and conclusions are generally of the type that permutation tests should always be used or permutation tests should never be used a better conclusion might be that there are situations in which permutation tests should be used and other situations in which permutation tests should not be used this approach opens the door to broader agreement but begs the obvious question of when to use permutation tests we consider this issue from a variety of perspectives and conclude that permutation tests are ideal to study efficacy in a randomized clinical trial which compares in a heterogeneous patient population two or more treatments each of which may be most effective in some patients when the primary analysis does not adjust for covariates we propose the p value interval as a novel measure of the conservatism of a permutation test that can be defined independently of the significance level this p value interval can be used to ensure that the permutation test have both good global power and an acceptable degree of conservatism copyright 2000 john wiley sons ltd

variance estimation in clinical studies with interim sample size reestimation summary we consider clinical studies with a sample size reestimation based on the unblinded variance estimation at some interim point of the study because the sample size is determined in such a flexible way the usual variance estimator at the end of the trial is biased we derive sharp bounds for this bias these bounds have a quite simple form and can help for the decision if this bias is negligible for the actual study or if a correction should be done an exact formula for the bias is also provided we discuss possibilities to get rid of this bias or at least to reduce the bias substantially for this purpose we propose a certain additive correction of the bias we see in an example that the significance level of the test can be controlled when this additive correction is used

conditional monte carlo randomization tests for regression models we discuss the computation of randomization tests for clinical trials of two treatments when the primary outcome is based on a regression model we begin by revisiting the seminal paper of gail tan and piantadosi 1988 and then describe a method based on monte carlo generation of randomization sequences the tests based on this monte carlo procedure are design based in that they incorporate the particular randomization procedure used we discuss permuted block designs complete randomization and biased coin designs we also use a new technique by plamadeala and rosenberger 2012 for simple computation of conditional randomization tests like gail tan and piantadosi we focus on residuals from generalized linear models and martingale residuals from survival models such techniques do not apply to longitudinal data analysis and we introduce a method for computation of randomization tests based on the predicted rate of change from a generalized linear mixed model when outcomes are longitudinal we show by simulation that these randomization tests preserve the size and power well under model misspecification

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

numerical peak holomorphic functions on banach spaces abstract we introduce the notion of numerical strong peak function and investigate the denseness of the norm and numerical peak functions on complex banach spaces let a b b x x be the banach space of all bounded continuous functions f on the unit ball b x of a banach space x and their restrictions f b x to the open unit ball are holomorphic in finite dimensional spaces we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense g subset of a b b x x we also prove that if x is a smooth banach space with the radon nikod m property then the set of all numerical strong peak functions is dense in a b b x x in particular when x l p 1 p or x l 1 it is shown that the intersection of the set of all norm strong peak functions and the set of all numerical strong peak functions is a dense g subset of a b b x x as an application the existence and properties of numerical boundary of a b b x x are studied finally the numerical peak function in a b b x x is characterized when x c k and some negative results on the denseness of numerical strong peak holomorphic functions are given

planning and revising the sample size for a trial the sample size for a trial depends on the type i and type ii error rates and on the minimum relevant clinical difference all of which are known and on the anticipated but unknown value of a measure of variation for the key response this measure is the overall response rate when the key response is binomially distributed or the residual variance in each treatment group when the key response is continuous and normally distributed since the true value of the measure is unknown it must be guessed or estimated from previous trials we describe approaches to determine an appropriate value for it both before the trial begins and after it has begun for use in calculating the final sample size these approaches differ from previously described internal pilot methods in not requiring unblinding of the treatment assignments in the trial they preserve the power and do not affect the type i error rate materially the approaches can be applied to longitudinal studies where the rate of change over time is the response of interest and to group sequential trials

integrability of complex planar systems with homogeneous nonlinearities abstract in this paper we obtain sufficient conditions for the existence of a local analytic first integral for a family of quintic systems having homogeneous nonlinearities the family studied in this work is the largest one classified until now for systems with such nonlinearities we propose also an approach to find reversible systems within polynomial families of lotka volterra systems with homogeneous nonlinearities

decomposition of algebraic sets and applications to weak centers of cubic systems there are many methods such as grobner basis characteristic set and resultant in computing an algebraic set of a system of multivariate polynomials the common difficulties come from the complexity of computation singularity of the corresponding matrices and some unnecessary factors in successive computation in this paper we decompose algebraic sets stratum by stratum into a union of constructible sets with sylvester resultants so as to simplify the procedure of elimination applying this decomposition to systems of multivariate polynomials resulted from period constants of reversible cubic differential systems which possess a quadratic isochronous center we determine the order of weak centers and discuss the bifurcation of critical periods

on the use of a pilot sample for sample size determination to compute the sample size needed to achieve the planned power for a t test one needs an estimate of the population standard deviation sigma if one uses the sample standard deviation from a small pilot study as an estimate of sigma it is quite likely that the actual power for the planned study will be less than the planned power monte carlo simulations indicate that using a 100 1 gamma per cent upper one sided confidence limit on sigma will provide a sample size sufficient to achieve the planned power in at least 100 1 gamma per cent of such trials

isochronicity of a z2 equivariant quintic system abstract the main purpose of this paper is to study the problem of simultaneous existence of two isochronous centers in some families of planar polynomial differential systems with symmetry more precisely we present conditions for a family of z 2 equivariant systems of degree 5 to have an isochronous bi center we also study their global phase portraits in the poincare disk and present considerations about the number of simultaneous centers in z 2 equivariant systems of degree 3 and 5 this study provides examples of quintic systems possessing three and five isochronous centers simultaneously

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

linearizability conditions for lotka volterra planar complex cubic systems in this paper we investigate the linearizability problem for the two dimensional planar complex system the necessary and sufficient conditions for the linearizability of this system are found from them the conditions for isochronicity of the corresponding real system can be derived

darboux linearization and isochronous centers with a rational first integral abstract in this paper we study isochronous centers of polynomial systems it is known that a center is isochronous if and only if it is linearizable we introduce the notion of darboux linearizability of a center and give an effective criterion for verifying darboux linearizability if a center is darboux linearizable the method produces a linearizing change of coordinates most of the known polynomial isochronous centers are darboux linearizable moreover using this criterion we find a new two parameter family of cubic isochronous centers and give the linearizing changes of coordinates for centers belonging to that family we also determine all hamiltonian cubic systems which are darboux linearizable in the second part of this work we restrict to the study of isochronous centers having a rational first integral we prove that under certain conditions the cycle vanishing at the isochronous center is either zero homologous in the closure of a generic fiber or the function obtained from the first integral by eliminating the indeterminacy points has several critical points in the singular fiber passing through the isochronous center

blinded sample size reassessment in non inferiority and equivalence trials even in situations where the design and conduct of clinical trials is highly standardized there may be a considerable between study variation in the observed variability of the primary outcome variable as a consequence performing a study in a fixed sample size design implies a considerable risk of resulting in a too high or too low sample size this difficulty can be alleviated by applying a design with internal pilot study after a provisional sample size calculation in the planning stage a portion of the planned sample is recruited and the sample size is recalculated on the basis of the observed variability to comply with the requirement of some regulatory guidelines only blinded data should be used for the reassessment procedure furthermore the effect on the type i error rate should be quantified the current literature presents analytical results on the actual level in the t test situation only for superiority trials in these situations blinded sample size recalculation does not lead to an inflation of the type i error rate we extended the methodology to non inferiority and equivalence trials with normally distributed outcome variable and hypotheses formulated in terms of the ratio and difference of means surprisingly in contrast to the case of testing superiority we observed actual type i error rates above the nominal level the extent of inflation depends on the required sample size the sample size of the internal pilot study and the standardized equivalence or non inferiority margin it turned out that the elevation of the significance level is negligible for most practical situations nevertheless the consequences of sample size reassessment have to be discussed case by case and regulatory concerns with respect to the actual size of the procedure cannot generally be refuted by referring to the fact that only blinded data were used

linearizability of linear systems perturbed by fifth degree homogeneous polynomials we present the necessary and sufficient conditions for linearizability of the planar complex system where p and q are homogeneous polynomials of degree 5 using these conditions we also give the complete solution for the isochronicity of real systems in the form of linear oscillator perturbed by fifth degree homogeneous polynomials

a method for determining the size of internal pilot studies the assessment of sample size in clinical trials comparing population means requires a variance estimate of the main efficacy variable when this variance estimate has a low precision it may be appropriate to use the data from the first patients entered in the trial internal pilot study to estimate the sample size we suggest a method for determining the size of internal pilot studies which aims at ensuring that this size is as large as possible but not larger than the optimal size of the planned study advantages and limitations of the method are discussed

simple procedures for blinded sample size adjustment that do not affect the type i error rate for normally distributed data determination of the appropriate sample size requires a knowledge of the variance because of the uncertainty in the planning phase two stage procedures are attractive where the variance is reestimated from a subsample and the sample size is adjusted if necessary from a regulatory viewpoint preserving blindness and maintaining the ability to calculate or control the type i error rate are essential recently a number of proposals have been made for sample size adjustment procedures in the t test situation unfortunately none of these methods satisfy both these requirements we show through analytical computations that the type i error rate of the t test is not affected if simple blind variance estimators are used for sample size recalculation furthermore the results for the expected power of the procedures demonstrate that the methods are effective in ensuring the desired power even under initial misspecification of the variance a method is discussed that can be applied in a more general setting and that assumes analysis with a permutation test this procedure maintains the significance level for any design situation and arbitrary blind sample size recalculation strategy

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

nondegenerate centers for abel polynomial differential equations of second kind in this paper we study the center problem for abel polynomial differential equations of second kind computing the focal values and using modular arithmetics and grbner bases we find the center conditions for such systems for lower degrees

permutation tests for detecting and estimating mixtures in task performance within groups we propose a two sample permutation test incorporating mixture models as a general tool for detecting and quantifying effects on task performance we illustrate the proposed method with examples where the dependent measures under investigation are recorded for normal controls and relatives of patients with schizophrenia on a delayed response spatial and object working memory task our mixture modelling in relatives allows the component distributions to arise from different continuous parametric families we also investigate the effects of the within family correlation and the prior distribution of the mixing proportion on the test results the power of the test depends on sample sizes the mixing proportion the difference in component means and the ratio of component variances copyright 2002 john wiley sons ltd

1 3 resonant centers on c2 with homogeneous cubic nonlinearities we characterize the collection of systems of differential equations on c 2 of the form email protected x p x y email protected 3y q x y where p and q are homogeneous polynomials of degree three either of which may be zero that possess a first integral in a neighborhood of 0 0 of the form x 3y where omitted terms are of order at least five such systems are called 1 3 resonant centers

properties of internal pilots with the univariate approach to repeated measures uncertainty surrounding the error covariance matrix often presents the biggest barrier to achieving accurate power analysis in the univariate approach to repeated measures analysis of variance unirep a poor choice gives either an overpowered study which wastes resources or an underpowered study with little chance of success internal pilot designs were introduced to resolve such uncertainty about error variance for t tests in earlier papers we extended the use of internal pilots to any univariate linear model with fixed predictors and independent gaussian errors here we further extend our exact and approximate results to unirep analysis for a fixed treatment effect the inaccuracy in a power calculation depends only on the ratio of the true variance to the value used for planning the greater complexity of repeated measures requires generalizing misspecification of error variance to the misspecification of the eigenvalues of the error covariance we recommend approximating the misspecification in terms of the first and second moments of the eigenvalues for both fixed sample and internal pilot designs we also describe an unadjusted approach for internal pilots with repeated measures simulations illustrate the fact that both positive and negative properties in the univariate setting extend to repeated measures analysis in particular internal pilots allow maintaining power or reducing expected sample size when the covariance matrix used for planning differs from the true value however an unadjusted approach can inflate test size at least with small to moderate sample sizes hence new adjusted methods must be developed for small samples at this time we caution against using an internal pilot design with repeated measures without first conducting simulations to document the amount of test size inflation possible for the conditions of interest

on holomorphic functions attaining their norms abstract we show that on a complex banach space x the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that x has the radon nikodym property we also show that the same result holds for banach spaces having a strengthened version of the approximation property but considering just functions which are also weakly uniformly continuous on the unit ball we prove that there exists a polynomial such that for any fixed positive integer k it cannot be approximated by norm attaining polynomials with degree less than k for x d 1 a predual of a lorentz sequence space we prove that the product of two polynomials with degree less than or equal two attains its norm if and only if each polynomial attains its norm

isochronicity and linearizability of a planar cubic system abstract in this paper we investigate the problem of linearizability for a family of cubic complex planar systems of ordinary differential equations we give a classification of linearizable systems in the family obtaining conditions for linearizability in terms of parameters we also discuss coexistence of isochronous centers in the systems

bi center problem for some classes of z2 equivariant systems we investigate the simultaneous existence of two centers and their isochronicity for two families of planar z2 equivariant differential systems first for a family studied by liu and li 2011 we present the necessary and sufficient conditions for the existence of an isochronous bi center next we give conditions for the existence of a bi center and study its isochronicity for a planar z2 equivariant quintic system having two weak foci or centers we also give two examples of non hamiltonian cubic systems with three isochronous centers

connections between permutation and t tests relevance to adaptive methods a permutation test assigns a p value by conditioning on the data and treating the different possible treatment assignments as random the fact that the conditional type i error rate given the data is controlled at level ensures validity of the test even if certain adaptations are made we show the connection between permutation and t tests and use this connection to explain why certain adaptations are valid in a t test setting as well we illustrate this with an example of blinded sample size re calculation

maximum likelihood estimation for mixed continuous and categorical data with missing values summary maximum likelihood procedures for analysing mixed continuous and categorical data with missing values are presented the general location model of olkin tate 1961 and extensions introduced by krzanowski 1980 1982 form the basis for our methods maximum likelihood estimation with incomplete data is achieved by an application of the em algorithm dempster laird rubin 1977 special cases of the algorithm include orchard woodbury s 1972 algorithm for incomplete normal samples fuchs s 1982 algorithms for log linear modelling of partially classified contingency tables and day s 1969 algorithm for multivariate normal mixtures applications include a imputation of missing values b logistic regression and discriminant analysis with missing predictors and unclassified observations c linear regression with missing continuous and categorical predictors and d parametric cluster analysis with incomplete data methods are illustrated using data from the st louis risk research project some key word8 cluster analysis discriminant analysis em algorithm incomplete data linear regression logistic regression log linear model mixture model

isochronicity of analytic systems via urabe s criterion a method for studying isochronous oscillations in some systems of ode reducible to the equation is described it is applied to obtain the necessary and sufficient conditions for isochronicity of a cubic two dimensional autonomous system depending on six parameters for all isochronous systems in this family the urabe function is explicitly constructed

an approach to solving systems of polynomials via modular arithmetics with applications the objective of this paper is twofold first we describe a method to solve large systems of polynomial equations using modular arithmetics then we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations

first integrals of a three dimensional system in the case of one zero eigenvalue in this paper we propose an efficient computational approach to find systems with a first integral within some families of polynomial systems of ordinary differential equations in the case when the matrix of the linear approximation has one zero eigenvalue while the other eigenvalues have negative real parts we apply it to find conditions for the existence of first integrals for a three dimensional system involving cubic polynomials

estimating the sample size for a t test using an internal pilot if the sample size for a t test is calculated on the basis of a prior estimate of the variance then the power of the test at the treatment difference of interest is not robust to misspecification of the variance we propose a t test for a two treatment comparison based on stein s two stage test which involves the use of an internal pilot to estimate variance and thus the final sample size required we evaluate our procedure s performance and show that it controls the type i and ii error rates more closely than existing methods for the same problem we also propose a rule for choosing the size of the internal pilot and show that this is reasonable in terms of the efficiency of the procedure

a comparison of permutation and parametric testing for between group effective connectivity differences using dcm effective connectivity is becoming an increasingly popular technique for obtaining additional information from functional magnetic resonance imaging fmri cognitive activation studies it is potentially important for investigating psychiatric illnesses which are thought to depend on disrupted connections and in observing the action of psychoactive drugs used to treat these disorders if researchers are to apply these techniques confidently then it is important to establish the level of power that is available in an experiment this study compares the level of power available when applying effective connectivity to test for differences between groups using parametric tests and permutation testing permutation testing has previously been shown to have superior sensitivity to parametric tests in fmri studies as an illustrative example both the parametric t test and equivalent permutation test were applied to a comparison between healthy controls and remitted depressed volunteers performing an emotional face processing task permutation testing was found to provide superior power compared with the nonparametric equivalent

on integrability conditions and limit cycle bifurcations for polynomial systems we present an approach to finding general conditions for integrability of a given family of two dimensional polynomial systems using conditions computed when some parameters were fixed we apply it to obtain integrability conditions for a lotka volterra planar complex quartic system having homogeneous nonlinearities we also study bifurcations of limit cycles from each component of the center variety of the corresponding quartic real system

a comparison of adaptive allocation rules for group sequential binary response clinical trials in clinical trials to compare two or more treatments with dichotomous responses group sequential designs may reduce the total number of patients involved in the trial and response adaptive designs may result in fewer patients being assigned to the inferior treatments in this paper we combine group sequential and response adaptive designs extending recent work on sample size re estimation in trials to compare two treatments with normally distributed responses to analogous binary response trials we consider the use of two parameters of interest in the group sequential design the log odds ratio and the simple difference between the probabilities of success in terms of the adaptive sampling rules we study two urn models the drop the loser rule and the randomized polya urn rule and compare their properties with those of two sequential maximum likelihood estimation rules which minimize the expected number of treatment failures we investigate two ways in which adaptive urn designs can be used in conjunction with group sequential designs the first method updates the urn at each interim analysis and the second method continually updates the urn after each patient response assuming immediate patient responses our simulation results show that the group sequential design which uses the drop the loser rule applied fully sequentially is the most effective method for reducing the expected number of treatment failures and the average sample number whilst still maintaining the nominal error rates over a range of success probabilities

analysis of small sample size studies using nonparametric bootstrap test with pooled resampling method experimental studies in biomedical research frequently pose analytical problems related to small sample size in such studies there are conflicting findings regarding the choice of parametric and nonparametric analysis especially with non normal data in such instances some methodologists questioned the validity of parametric tests and suggested nonparametric tests in contrast other methodologists found nonparametric tests to be too conservative and less powerful and thus preferred using parametric tests some researchers have recommended using a bootstrap test however this method also has small sample size limitation we used a pooled method in nonparametric bootstrap test that may overcome the problem related with small samples in hypothesis testing the present study compared nonparametric bootstrap test with pooled resampling method corresponding to parametric nonparametric and permutation tests through extensive simulations under various conditions and using real data examples the nonparametric pooled bootstrap t test provided equal or greater power for comparing two means as compared with unpaired t test welch t test wilcoxon rank sum test and permutation test while maintaining type i error probability for any conditions except for cauchy and extreme variable lognormal distributions in such cases we suggest using an exact wilcoxon rank sum test nonparametric bootstrap paired t test also provided better performance than other alternatives nonparametric bootstrap test provided benefit over exact kruskal wallis test we suggest using nonparametric bootstrap test with pooled resampling method for comparing paired or unpaired means and for validating the one way analysis of variance test results for non normal data in small sample size studies copyright 2017 john wiley sons ltd

optimization and differentiation in banach spaces abstract we show that some old ideas of smulian can be used to give another proof of a theorem of bourgain we characterize subsets of banach spaces having the radon nikodym property by means of optimization results

isochronicity and linearizability of planar polynomial hamiltonian systems in this paper we study isochronicity and linearizability of planar polynomial hamiltonian systems first we prove a theorem which supports a negative answer to the following open question stated by jarque and villadelprat in 15 do there exist planar polynomial hamiltonian systems of even degree having an isochronous center additionally we obtain some conditions for linearizability of complex cubic hamiltonian systems

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

internal pilot studies ii comparison of various procedures the two stage design involves sample size recalculation using an interim variance estimate stein proposed the design in 1945 biostatisticians recently have shown renewed interest in it wittes and brittain proposed a modification aimed at greater efficiency gould and shih proposed a similar procedure but with a different interim variance estimate based on blinded data we compare the power of stein s original test an idealized version of the wittes brittain test and a theoretical optimal test which can be approximated in practice we also compare two procedures that control the conditional type i error rate given the actual final sample size gould and shih s procedure and a newly proposed second segment procedure the comparison among the first three procedures indicates that the stein test is unexpectedly the test of choice under the original design alternative whereas the approximate optimal and wittes brittain procedures appear to have superior power for detecting smaller treatment differences as between the latter two procedures the second segment procedure is more powerful when many observations are likely to be taken after the interim resizing whereas otherwise the gould shih procedure is superior

sample sizes for event rate equivalence trials using prior information trials for demonstrating the equivalence of active standard and test treatments generally require large sample sizes that depend on the definition of equivalence and the overall event rate when the outcome is incidence of an event such as morality the planning of sample sizes for such trials requires specification of a value for the overall event rate this value often will reflect the outcomes of previous trials of the standard treatment and is subject to uncertainty that needs some accommodation to protect against an inadequate sample bayes and empirical bayes methods can be used to incorporate information from one or more previous trials into the sample size calculation when equivalence means high confidence that the event rate ratio is less than some specified value

convex hull test for ordered categorical data when testing for stochastic order in ordered 2 x j contingency tables it is common to select the cutoff required to declare significance so as to ensure that the size of the test is exactly alpha conditionally on the margins it is valid however to use the margins to select not only the cutoff but also the form of the test linear rank tests which are locally most powerful and frequently used in practice suffer from the drawback that they may have power as low as zero to detect some alternatives of interest when the margins satisfy certain conditions the smirnov and convex hull tests are shown through exact conditional power calculations and simulations to avoid this drawback the convex hull test is also admissible and palindromic invariant and minimizes the required significance level to have limiting power of one as the alternative moves away from the null in any direction

linearizability conditions of time reversible quartic systems having homogeneous nonlinearities abstract in this paper we investigate the linearizability problem for the two dimensional time reversible complex system x x p x y y y q x y where p and q are homogeneous quartic polynomials the necessary and sufficient conditions for the linearizability of this system are found as a corollary the necessary and sufficient conditions for the origin to be an isochronous center of the planar time reversible real system u v f u v v u g u v where f and g are homogeneous quartic polynomials are obtained

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

the center problem for a 2 3 resonant cubic lotka volterra system in this paper we obtain conditions on the coefficients of a cubic lotka volterra system of the form 1 x x 2 a 2 0x 2 a 1 1xy a 0 2y 2 y y 3 b 2 0x 2 b 1 1xy b 0 2y 2 which fulfillment yields the existence in a neighborhood of the origin of a first integral of the form f x y x 3y 2 h o t in which case the origin is termed a 2 3 resonant center this system was studied in 13 where due to computational constrains the consideration was limited to the cases where either one or both coefficients a 1 1 b 1 1 in system 1 were equal to zero or both coefficients were equal to 1 here we are studying the case where the coefficient a 1 1 is equal to 1 and b 1 1 is arbitrary the obtained results represent the study of the center problem for general system 1 since by a linear substitution any system of the form 1 can be transformed either to system 1 with a 1 1 1 or to one of systems studied in 13 computation of the resonant saddle quantities focus quantities and the decomposition of the variety of the ideal generated by an initial string of them were used to obtain necessary conditions of integrability and the theory of darboux first integrals and some other methods to show the sufficiency since the decompositions of the variety mentioned above was performed using modular computations the obtained 19 conditions of integrability represent the complete list of the integrability conditions only with very high probability and there remains an open problem to verify this statement

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

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

on the numerical index of banach spaces we give a partial answer to the problem of computing the numerical index of lp space 1 p also we give an estimate of the numerical index of the two dimensional real space lp2 1 p for the lp space we show that its numerical index is greater than the lp space one

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

exact test size and power of a gaussian error linear model for an internal pilot study wittes and brittain recommended using an internal pilot study to adjust sample size the approach involves five steps in testing a general linear hypothesis for a general linear univariate model with gaussian errors first specify the design hypothesis desired test size power a smallest clinically meaningful effect and a speculated error variance second conduct a power analysis to choose provisionally a planned sample size third collect a specified proportion of the planned sample as the internal pilot sample and estimate the variance but do not test the hypothesis fourth update the power analysis with the variance estimate to adjust the total sample size fifth finish the study and test the hypothesis with all data we describe methods for computing exact test size and power under this scenario our analytic results agree with simulations of wittes and brittain furthermore our exact results apply to any general linear univariate model with fixed predictors which is much more general than the two sample t test considered by wittes and brittain in addition our results allow for examination of the impact on test size of internal pilot studies for more complicated designs in the framework of the general linear model we examine the impact of i small samples ii allowing the planned sample size to decrease iii the choice of internal pilot sample size and iv the maximum allowable size of the second sample all affect test size power and expected total sample size we present a number of examples including one that uses an internal pilot study in a three group analysis of variance

internal pilot studies i type i error rate of the naive t test when sample size is recalculated using unblinded interim data use of the usual t test at the end of a study may lead to an elevated type i error rate this paper describes a numerical quadrature investigation to calculate the true probability of rejection as a function of the time of the recalculation the magnitude of the detectable treatment effect and the ratio of the guessed to the true variance we consider both restricted designs those that require final sample size at least as large as the originally calculated size and unrestricted designs those that permit smaller final sample sizes than originally calculated our results indicate that the bias in the type i error rate is often negligible especially in restricted designs some sets of parameters however induce non trivial bias in the unrestricted design

centers for a class of generalized quintic polynomial differential systems abstract we characterize the centers of the planar polynomial differential systems of degree d 5 odd that in complex notation writes as z iz z z d 5 2 az 5 bz 4 z cz 3 z 2 dz 2 z 3 ez z 4 f z 5 where a b c d e f c and either a re e 0 or a im e 0 or re a e 0 or im a e 0