nonparametric tests under restricted treatment assignment rules abstract this article considers comparative clinical trials with r r 2 treatments and allocation designs that aim to balance the assignment of treatments while controlling selection bias a general form of these restricted designs was presented by smith 1984a his class of designs encompasses most of the other designs including efron s 1971 biased coin design bcd widespread use of these designs is impeded by the lack of knowledge concerning the proper statistical analyses of the resulting data in particular little is known about the small sample randomization distributions of test statistics thus current approaches rely on large sample approximations simulation techniques and distributional results derived under classical schemes such as the complete randomization design clearly none of these approaches is exact a recursion procedure is presented for computing the exact conditional randomization distribution of a wide class of test statistics under these restricted designs this proced

importance sampling for estimating exact probabilities in permutational inference abstract this article discusses importance sampling as an alternative to conventional monte carlo sampling for estimating exact significance levels in a broad class of two sample tests including all of the linear rank tests with or without censoring homogeneity tests based on the chi squared hypergeometric and likelihood ratio statistics the mantel haenszel trend test and zelen s test for a common odds ratio in several 2 2 contingency tables inference is based on randomly selecting 2 k contingency tables from a reference set of all such tables with fixed marginals through a network algorithm the tables are selected in proportion to their importance for reducing the variance of the estimated monte carlo p value spectacular gains up to four orders of magnitude are achieved relative to conventional monte carlo sampling the technique is illustrated on four real data sets

sequential monitoring of randomization tests stratified randomization in many clinical trials it is desirable to establish a sequential monitoring plan whereby the test statistic is computed at an interim point or points in the trial and a decision is made whether to stop early due to evidence of treatment efficacy in this article we will set up a sequential monitoring plan for randomization based inference under the permuted block design stratified block design and stratified urn design we will also propose a definition of information fraction in these settings and discuss its calculation under these different designs

asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials a general doubly adaptive biased coin design is proposed for the allocation of subjects to k treatments in a clinical trial this design follows the same spirit as efron s biased coin design and applies to the cases where the desired allocation proportions are unknown but estimated sequentially strong consistency a law of the iterated logarithm and asymptotic normality of this design are obtained under some widely satisfied conditions for two treatments a new family of designs is proposed and shown to be less variable than both the randomized play the winner rule and the adaptive randomized design also the proposed design tends toward a randomization scheme with a fixed target proportion as the size of the experiment increases

eigenvalue bifurcations in kac murdock szegő matrices with a complex parameter abstract for complex the spectral properties of the toeplitz matrix k n j k j k 1 n often called the kac murdock szeg matrix have been examined in detail in two recent papers the second paper in particular introduced the concept of borderline curves these are two closed curves in the complex plane that consist of all the for which k n possesses some eigenvalue whose magnitude equals the matrix dimension n the purpose of the present paper is to examine eigenvalue bifurcations in both a qualitative and a quantitative manner and to discuss connections between bifurcations and the borderline curves

a kawamata viehweg vanishing theorem on compact kahler manifolds we prove a kawamata viehweg vanishing theorem on a normal compact kahler space x if l is a nef line bundle with l 6 0 then h x kx l 0 for q dimx 1 as an application we complete a part of the abundance theorem for minimal kahler threefolds if x is a minimal kahler threefold then the kodaira dimension x is nonnegative 0 introduction in this paper we establish the following kawamata viehweg type vanishing theorem on a compact kahler manifold or more generally a normal compact kahler space 0 1 theorem let x be a normal compact kahler space of dimension n and l a nef line bundle on x assume that l 6 0 then h x kx l 0 for q n 1 in general one expects a vanishing h x kx l 0 for q n 1 l where l is the numerical kodaira dimension of the nef line bundle l i e l is the largest integer such that l 6 0 of course when x is projective theorem 0 1 is contained in the usual kawamataviehweg vanishing theorem but the methods of proof in the algebraic case clearly fail in the general kahler setting instead we proceed in the following way clearly we may assume that x is smooth and by serre duality only the cohomology group h is of interest take a singular metric h on l with positive curvature current t with local weight function h by si74 de93a there exists a decomposition t jdj g d where j 1 are irreducible divisors and g is a pseudo effective current such that g di is pseudo effective for all i consider the multiplier ideal sheaf i h we associate to h another upper regularized multiplier ideal sheaf i h by setting i h lim e 0 i h lim e 0 i

optimum biased coin designs for sequential treatment allocation with covariate information randomized optimum designs of biased coin type are compared with other strategies for the sequential allocation of two or more treatments in a clinical trial the emphasis is on the variance of estimated treatment contrasts this variance which depends on the design strategy employed may be interpreted as the number of patients on whom information is lost simulations provide clear plots of the evolution of this loss during the course of the clinical trial

spectral properties of kac murdock szego matrices with a complex parameter abstract when 0 1 the kac murdock szego matrix k n j k j k 1 n is a toeplitz correlation matrix with many applications and very well known spectral properties we study the eigenvalues and eigenvectors of k n for the general case where is complex pointing out similarities and differences to the case 0 1 we then specialize our results to real with 1 emphasizing the continuity of the eigenvalues as functions of for 1 we develop simple approximate formulas for the eigenvalues and pinpoint all eigenvalues locations our study starts from a certain polynomial whose zeros are connected to the eigenvalues by elementary formulas we discuss relations of our results to earlier results of w f trench

constructing the exact significance level for a person fit statistic this paper uses an extension of the network algorithm originally introduced by mehta and patel to construct exact tail probabilities for testing the general hypothesis that item responses are distributed according to the rasch model by assuming that item difficulties are known the algorithm is applicable to the statistical tests either given the maximum likelihood ability estimate or conditioned on the total score a simulation study indicates that the network algorithm is an efficient tool for computing the significance level of a person fit statistic based on test lengths of 30 items or less

adaptive biased coin designs for skewing the allocation proportion in clinical trials with normal responses response adaptive designs are used in phase iii clinical trial for skewing the allocation pattern towards the better treatments we use optimum design theory to derive adaptive designs when the responses are normally distributed the performance of the designs is studied with respect to the loss and the proportion of allocation to different treatments the adaptive design does not affect inference

constructing exact significance tests with restricted randomization rules summary in a clinical trial comparing two treatments suppose that subjects are assigned sequentially to the treatment groups by a restricted randomization rule under the randomization model an efficient recursive algorithm is provided to generate the exact permutational distribution for linear rank statistics given the final imbalance between the number of patients in the two groups if only the significance level of the test is required the efficiency of the algorithm can be improved significantly in general by the time the size of the trial reaches 30 the large sample approximation to the permutation distribution wei smythe smith 1986 is satisfactory the new procedure is illustrated by censored survival data from a prostatic cancer clinical trial we show that if the study subjects are not a random sample from a well defined population then the treatment allocation rule used at the design stage should not be ignored in the analysis

k treatment comparisons with restricted randomization rules in clinical trials in the course of conducting a clinical trial to compare k 2 treatments it is often desirable to balance the trial with respect to the assignments of patients to treatments on the other hand some form of randomization of treatment assignments is essential for reducing experimental bias in this article the large sample approximation to the null distribution of k sample randomization tests generated from a broad class of restricted randomization rules is derived the implication of this result for conditional inference is also discussed

nef dimension of minimal models we reduce the abundance conjecture in dimension 4 to the following numerical statement if the canonical divisor k is nef and has maximal nef dimension then k is big from this point of view we classify in dimension 2 nef divisors which have maximal nef dimension but which are not big

estimating exact p values by the method of control variates or monte carlo rescue abstract despite algorithmic advances in exact nonparametric inference problems often occur that are too large for exact p value computations but too sparse for reliable asymptotic results in these situations monte carlo methods are a good compromise they bound the true p value within a confidence interval but a factor discouraging the use of monte carlo p values is their sensitivity to the random number sequence one can overcome this drawback by computing a 99 c i confidence interval of width less than 0001 then the estimated p values become insensitive to the random number sequence up to three decimals for all practical purposes these estimates are invariant to random number sequences the usual monte carlo method requires millions of samples to yield such an invariant estimate the monte carlo scheme presented here decreases the sample size by two to three orders of magnitude we illustrate this method with tests for r c tables two sample survival data and stratified 2 2 tables

central limit theorems for doubly adaptive biased coin designs asymptotic normality of the difference between the number of subjects assigned to a treatment and the desired number to be assigned is established for allocation rules which use eisele s biased coin design subject responses are assumed to be independent random variables from standard exponential families in the proof it is shown that the difference may be magnified by appropriate constants so that the magnified difference is nearly a martingale an application to the behrens fisher problem in the normal case is described briefly

constrained randomization in a therapeutic efficacy trial the use of valid constrained randomization is presented for a therapeutic efficacy trial the construction and evaluation of randomization schemes are studied for two treatment replicates per block and for hypothesized linear gradients two different isomorphism classes of designs are compared using the variance of the mean squared error criterion the results indicate that valid constrained randomization schemes can be constructed that are superior to ordinary randomization a mastitis efficacy trial provided the specific problem to be solved

optimum biased coin designs for sequential clinical trials with prognostic factors patients in a clinical trial arrive sequentially and are assigned to one of t treatments this assignment should maintain a balance between the numbers receiving each treatment yet should be sufficiently random to avoid any suspicion of conscious or unconscious cheating to balance these requirements efron 1971 introduced biased coin designs for the comparison of two treatments in which allocation of the treatment is determined probabilistically but with a bias towards the underrepresented treatment one disadvantage of efron s scheme is that it does not include balance over covariates or prognostic factors which may affect the response of the patient to the treatment biased coin schemes which do force balance over both treatments and prognostic factors are given by pocock simon 1975 and efron 1980 the properties of the designs have been elucidated by numerical studies and they are now being increasingly used in clinical trials reviews of the literature and discussions of the practical implications of the designs are given by pocock 1979 and simon 1979 however the designs suffer from the disadvantage that they rely on arbitrary functions to achieve the desired balance the procedures thus lack a firm theoretical framework an alternative approach in the presence of prognostic factors begg iglewicz 1980 uses optimum design theory to suggest a deterministic design criterion which is then modified for computational convenience in this paper i use optimum design theory to provide a procedure of the biased coin type for an arbitrary number of treatments in the presence or absence of prognostic factors this has the theoretical advantage of obviating dependence on a series of arbitrary functions the necessary optimum design theory is presented in 2 and in 3 applied to biased coin experiments the simplest case that of two treatments in the absence of prognostic factors is studied in 4 the extension to three or more treatments is in 5 followed in 6 by the allowance for prognostic factors although in the biased coin designs allocation of treatments is made probabilistically the results of the paper can be used for the sequential construction of da optimum designs in which randomization is not introduced by the experimenter all the designs mentioned so far are sequential but not adaptive in the useful distinction made by pericchi 1981 they are data dependent but not outcomedependent each allocation depends on the previous allocations and prognostic factors

significance tests with restricted randomization design in an experiment comparing the effectiveness of two treatments a and b subjects arrive at an experimental site sequentially and must be assigned immediately to one of the two treatment groups the complete randomization scheme tossing a fair coin each time independently to decide which treatment will be used for the next subject has been widely employed because this scheme is free of different kinds of experimental bias efron 1971 and it also provides a basis for statistical inference lehmann 1975 chapter 1 but in a small experiment complete randomization may result in a severe imbalance between the numbers of subjects assigned to the different treatment groups a simple way to attain a perfectly balanced experiment is to use the systematic design abab however this plan can easily bias the results of the experiment in a number of ways for example if the experimenter is predisposed toward treatment a and if he knows or guesses that treatment a will be the next assignment then he may tend to choose a subject with a high expected response because of this bias at the end of the experiment treatment a might appear to be significantly better than treatment b even if there is in fact no difference between the two obviously the systematic design maximizes this kind of bias several compromises between a perfectly balanced assignment scheme and a completely randomized one have been proposed for example the permuted block design the biased coin design efron 1971 the urn design wei 1977 1978a the adaptive biased coin design wei 1978b and the random allocation design stigler 1969 wei 1978c simon 1979 has given an excellent review in particular the urn and adaptive biased coin designs force a small experiment to be balanced but they tend to the complete randomization scheme as the size of the experiment increases recently a specific urn design has been shown to be d optimal by atkinson 1982 sometimes it may be inappropriate to postulate a population from which the subjects in the experiment have been obtained by random sampling instead a randomization model lehmann 1975 chapter 1 is utilized when the data are analysed with this approach the significance tests used for testing the null hypothesis ho that there is no difference between a and b must be only those tests generated by the experimental randomization design actually employed for any given sequence of responses from the subjects one can tabulate all possible patterns of treatment assignments to subjects using the restricted randomization rule and calculate the corresponding treatment

sequential monitoring with conditional randomization tests sequential monitoring in clinical trials is often employed to allow for early stopping and other interim decisions while maintaining the type i error rate however sequential monitoring is typically described only in the context of a population model we describe a computational method to implement sequential monitoring in a randomization based context in particular we discuss a new technique for the computation of approximate conditional tests following restricted randomization procedures and then apply this technique to approximate the joint distribution of sequentially computed conditional randomization tests we also describe the computation of a randomization based analog of the information fraction we apply these techniques to a restricted randomization procedure efron s biometrika 58 1971 403 417 biased coin design these techniques require derivation of certain conditional probabilities and conditional covariances of the randomization procedure we employ combinatoric techniques to derive these for the biased coin design 1 introduction sequential monitoring refers to analyzing data periodically during the course of a clinical trial with the purpose of detecting early evidence in support of or against a hypothesis a desirable feature of such a monitoring plan would be flexible inspections of the data that can occur at arbitrary time points at the same time sequentially tested hypotheses must maintain the overall probability of type i error at the prespecified level since repeated testing is known to inflate it the lan and demets 1983 error spending approach for sequential monitoring allows this the approach makes use of a type i error spending function which depends on the amount of statistical information available at the time of the interim inspection in

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

power and sample size calculations for exact conditional tests with ordered categorical data we develop an algorithm for computing sample sizes equal or unequal for categorical data we illustrate its use in the two sample setting using the wilcoxon rank sum statistic but the algorithm accommodates the entire class of linear rank statistics and can be extended to include nonlinear rank statistics as well the sample size determinations can be based on either exact power or on a very precise monte carlo estimate of it to reduce the computations further power can be computed as a function of asymptotic critical values when the number of categories is not too small for the wilcoxon statistic we show that this approximation works well if there are more than five response categories

the generalized hodge conjecture for the quadratic complex of lines in projective four space we verify the generalized hodge conjecture ghc x 5 2 for the quadratic complex of lines in projective four space

allocation of effort in monte carlo simulation for power of permutation tests abstract consider a randomization test in which an observed statistic is compared to a null distribution constructed by first drawing a random sample from the set of all possible permutations of the data and then calculating the statistic anew for each member of the random sample if the power of such a test is investigated by simulation the computer code must have two main loops in the outer loop which has o iterations data are generated under a specific alternative hypothesis for each outer iteration there are i inner iterations in each of which the data are permuted and the statistic calculated assume that the costs of inner and outer iterations are known and the total cost of the study is fixed so that i and o are constrained by o io for fixed this article considers what the optimal choice for i is in a mean squared sense a general expression is derived for the mean squared error of the distribution of p values if the alternative hypothesis is close to the null the optimal c

exact significance testing to establish treatment equivalence with ordered categorical data this communication concerns the problem of establishing the therapeutic equivalence of two treatments that are being compared on the basis of ordered categorical data the problem is formulated as a significance test in which the null hypothesis specifies a treatment difference an efficient numerical algorithm for computing the exact significance level is provided along with a simple method for obtaining the asymptotic significance level both methods are applied to a clinical trial of a new agent versus an active control guidelines for when to use the exact procedure and when to rely on asymptotic theory are provided

double borderline and extraordinary eigenvalues of kac murdock szegő matrices with a complex parameter abstract for all sufficiently large complex and for arbitrary matrix dimension n it is shown that the kac murdock szeg matrix k n j k j k 1 n possesses exactly two eigenvalues whose magnitude is larger than n we discuss a number of properties of the two extraordinary eigenvalues conditions are developed that given n allow us without actually computing eigenvalues to find all values that give rise to eigenvalues of magnitude n termed borderline eigenvalues the aforementioned values of form two closed curves in the complex plane we describe these curves which are n dependent in detail an interesting borderline case arises when an eigenvalue of k n equals n apart from certain exceptional cases this occurs if and only if the eigenvalue is a double one and if and only if the point is a cusp like singularity of one of the two closed curves

bounding scalar curvature for global solutions of the kahler ricci flow we show that the scalar curvature is uniformly bounded for the normalized k ahler ricci flow on a k ahler manifold with semi ample canonical bundle in particular the normalized k ahler ricci flow has long time existence if and only if the scalar curvature is uniformly bounded for k ahler surfaces projective manifolds of complex dimension three and for projective manifolds of all dimensions if assuming the abundance conjecture

eigenvectors of hermitian toeplitz matrices with smooth simple loop symbols abstract the paper is devoted to the structure and the asymptotics of the eigenvector matrix of hermitian toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line the results extend existing results on banded toeplitz matrices to full toeplitz matrices with temperate decay of the entries in the first row and column we establish formulas for both the exact and the asymptotic computation of arbitrarily prescribed individual components of arbitrarily prescribed individual eigenvectors these formulas are in terms of the wiener hopf factorization of a function which depends solely on the symbol and the corresponding eigenvalue or even only on an approximation to this eigenvalue the paper does not aim at providing numerical algorithms its main purpose is rather to reveal the structure underneath the eigenvectors which are shown to be the sum of two harmonics and certain edge corrections

building blocks of etale endomorphisms of complex projective manifolds etale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true they are the endomorphisms of abelian varieties and the nearly etale rational endomorphisms of weak calabi yau varieties

properties of biased coin designs in sequential clinical trials martingale methods and the martingale invariance principle are used to derive central limit theorems and related results for biased coin designs of the kind previously studied by efron wei and many others the results are applied to the study of selection bias the method is developed for the simplest two treatment case and then extended first to the case of several treatments and secondly to the case of two treatments with prognostic factors

dynamic randomization and a randomization model for clinical trials data randomization models are useful in supporting the validity of linear model analyses applied to data from a clinical trial that employed randomization via permuted blocks here a randomization model for clinical trials data with arbitrary randomization methodology is developed with treatment effect estimators and standard error estimators valid from a randomization perspective a central limit theorem for the treatment effect estimator is also derived as with permuted blocks randomization a typical linear model analysis provides results similar to the randomization model results when roughly unit effects display no pattern over time a key requirement for the randomization inference is that the unconditional probability that any patient receives active treatment is constant across patients when this probability condition is violated the treatment effect estimator is biased from a randomization perspective most randomization methods for balanced 1 to 1 treatment allocation satisfy this condition however many dynamic randomization methods for planned unbalanced treatment allocation like 2 to 1 do not satisfy this constant probability condition and these methods should be avoided copyright 2012 john wiley sons ltd

a theorem on isotropic null vectors and its application to thermoelasticity let a be a symmetric n times n matrix over an arbitrary field we show that if a possesses a non zero null vector x which is also isotropic i e such that x cdot x 0 then both the trace of the adjugate and the determinant of a vanish for the complex field the converse is also true in purely mechanical elasticity it is known that an isotropic wave amplitude vector which corresponds to a circularly polarized wave is necessarily associated with a double root of the characteristic equation of wave propagation whose roots give essentially the squared wave speeds we apply the preceding results to obtain concise conditions for the occurrence both of isotropic wave amplitudes and of double roots in the theory of thermoelastic wave propagation in sharp contrast with the purely mechanical theory we find that these two phenomena generally occur separately in thermoelasticity

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

interval estimation with restricted randomization rules summary in comparing two treatments in a clinical trial suppose that patients are assigned sequentially to treatments by a restricted randomization rule exact and approximate confidence intervals for treatment difference are derived without assuming that the study subjects were obtained by random sampling from a well defined population we show that if there is a time trend among the responses of study subjects one should not ignore the design in the analysis all the procedures are illustrated with a real life example

coupling of eigenvalues of complex matrices at diabolic and exceptional points the paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters the cases of weak and strong coupling are distinguished and their geometric interpretation in two and three dimensional spaces is given general asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios two physical examples illustrate effectiveness and accuracy of the presented theory

the covariate adaptive biased coin design for balancing clinical trials in the presence of prognostic factors the present paper deals with sequential designs intended to balance the allocations of two competing treatments in the presence of prognostic factors after giving a theoretical framework on the optimality of balanced designs that can arise when covariates are taken into account we propose a new family of covariate adaptive randomized designs that represents higher order approximation to balance treatments both globally and also across covariates we derive the theoretical properties of the suggested designs in terms of loss of precision and predictability the performance of this proposal is illustrated through a simulation study and compared with those of other procedures suggested in the literature copyright 2011 oxford university press