independent elements integrally closed ideals and quasi unmixedness abstract let i a denote the integral closure of an ideal i in a noetherian ring a then it is shown that sup a i the maximum number of elements which are i n a independent as n exists and is equal to min height i a p z z p is a prime divisor of i n a for all large n and z is a minimal prime ideal in the completion a p of a p from this it follows that a is locally quasi unmixed if and only if sup a i height i for all ideals i in a

improved approximation for estimation following closed sequential tests summary siegmund 1978 has developed a procedure for estimation following a closed sequential test known as the repeated significance tests here a more elaborate approximation is proposed for a wider class of closed sequential tests the major goal is to account for the excess over boundary in addition to the stopping time when the sequential test is terminated some hypothetical examples are used for comparing the two approximations to investigate the properties of these procedures further simulation results are presented in terms of estimated mean squared error bias and variance for point estimators and in terms of coverage probability and width for confidence intervals

interim analyses in clinical trials classical vs bayesian approaches this paper concerns interim analysis in clinical trials involving two treatments from the points of view of both classical and bayesian inference i criticize classical hypothesis testing in this setting and describe and recommend a bayesian approach in which sampling stops when the probability that one treatment is the better exceeds a specified value i consider application to normal sampling analysed in stages and evaluate the gain in average sample number as a function of the number of interim analyses

point and interval estimation following a sequential clinical trial summary the usual methodology employed in analysis after a sequential clinical trial is based on orderings of the possible samples resulting from the design however this approach lacks flexibility for use in wider applications in this paper two estimation techniques not based on orderings are considered and modified to obtain improved accuracy a bias adjusted maximum likelihood estimate together with a new and general method for setting confidence limits are discussed the realistic scenario of group sequential monitoring is assumed and methods for exact estimation are given accuracy of the methodology after a triangular test and an o brien fleming test are demonstrated through simulation

confidence interval calculation for a sequential clinical trial of binary responses consider a sequential test applied to two streams of binary responses in a comparative clinical trial after the completion of such a trial woodroofe s 1992 technique for accurate confidence interval estimation for a treatment difference can be applied in this paper the technique is extended to provide confidence intervals for the individual absolute success probabilities of the two treatments in the trial for which no current methodology exists accuracy is explored by simulation of coverage probabilities and individual confidence limit probabilities the simulations concern a triangular test and an o brien fleming test

exact and explicit solutions to some nonlinear evolution equations by utilizing the g g expansion method in this paper we demonstrate the effectiveness of the so called g g expansion method by examining some nonlinear evolution equations with physical interest our work is motivated by the fact that the g g expansion method provides not only more general forms of solutions but also periodic and solitary waves if we set the parameters in the obtained wider set of solutions as special values then some previously known solutions can be recovered the method appears to be easier and faster by means of a symbolic computation system

note on asymptotic prime divisors and symbolic powers of primary ideals at the close of a recent survey paper on asymptotic prime divisors 12 it was commented that there were no results in the paper concerning symbolic powers of pr imary ideals since the results and methods in these two obviously related areas do not seem to blend together the purpose of the present paper is to show that at least some results in these two areas do blend together this is done by proving four theorems on asymptotic prime divisors and five related theorems on symbolic powers of primary ideals the five results on symbolic powers are rather negative in tone since they show that if all large powers of a primary ideal a noetherian ring a are primary then a must be a rather nice ring to begin with the four theorems on asymptotic prime divisors mentioned in the preceding paragraph are proved in sect 2 and the theorems on symbolic powers are proved in sect 3 to efficiently state and prove these results a fair amount of notation is needed so this section will be closed by summarizing this notation and giving a brief remark concerning it

confidence intervals following group sequential tests in clinical trials tsiatis rosner and mehta 1984 biometrics 40 797 803 proposed a procedure for constructing confidence intervals following group sequential tests of a normal mean this method is first extended for group sequential tests for which the sample sizes between interim analyses are not identical or the times are not equally spaced then properties of this confidence interval estimation procedure are studied by simulation the extension accommodates the flexible procedure by lan and demets 1983 biometrika 70 659 663 for constructing discrete group sequential boundaries to form a structure for monitoring and estimation following a class of group sequential tests finally it is demonstrated how to combine the procedures by lan and demets and by tsiatis rosner and mehta using a fortran program

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

asymptotic sequences and rees rings abstract let b1 bs be an asymptotic sequence over an ideal i in a noetherian ring r and let bi b1 bi r then it is shown that certain sequences closely related to these elements are asymptotic sequences over r r bi over ti r r i and over u r r i where r r j is the rees ring of r with respect to the ideal j these results then imply that certain other sequences are asymptotic sequences over the corresponding ideals in the associated graded rings and in the monadic transformation rings as an application of the results it is shown that if r is local then each permutation of b1 bs is an asymptotic sequence over i and that b1 bs are an asymptotic sequence in r

asymptotic prime divisors and integral extension rings let r b be rings such that r is noetherian and b is an intermediate ring between a noetherian integral extension ring c of r and the integral closure of c then it is shown that if i is a finitely generated ideal in b then the sets ass b i k a are finite and stabilize for all large k where i k a is the integral closure in b of i k moreover if i jb for some ideal j in r and if p e ass r i k a then there exists p e ass b i k b a such that p r p and the converse holds if minimal prime ideals in c lie over minimal prime ideals in r and k is large from this it is shown that most of the theorems concerning asymptotic prime divisors asymptotic sequences and asymptotic grade that are known to hold for ideals in r continue to hold for finitely generated ideals in b

p values after repeated significance testing a simple approximation method the p value after a repeated significance test is a useful measure of the strength of evidence against the null hypothesis its computation however requires a computer intensive numerical integration method the p value is not conceptually straightforward because it depends on how the sample space is ordered which can be arbitrary we look at two orderings of the sample space one proposed by tsiatis et al and the other by rosner and tsiatis and chang although studies have shown that the latter ordering gives more reasonable confidence intervals than the former the former gives a conservative and therefore more reasonable p value both however should yield an identical p value in most applications in this paper we present a simple method of approximating p values we provide tables to implement the method for two to ten stages with alpha 0 1 0 05 and 0 01 for the pocock and o brien fleming procedures the proposed method can be applied to both orderings

p value adjustment in sequential clinical trials many randomized clinical trials utilize a group sequential monitoring procedure for assessing treatment efficacy during the course of the trial for a group sequential trial in which the null hypothesis is ultimately rejected the nominal p value generally overstates the statistical significance of the result and some adjustment is required several orderings of the sample space have been proposed for performing this adjustment each with unique operating characteristics in this article we compare four proposed methods with respect to the degree to which each captures the strength of evidence against the null hypothesis implied by the data we conclude that the ordering of the sample space induced by the observed z score has the most desirable operating characteristics

on the asymptotic cograde of an ideal abstract let i be an ideal in a local ring r m and let s i be the asymptotic cograde of i the maximum length of an asymptotic sequence over i rees has shown that s i altitude r l i where l i is the analytic spread of i we prove several more bounds on s i i s i min little depth p p is an asymptotic prime divisor of i ii s i grade m i the maximum length of an asymptotic sequence contained in m i iii s i grade m grade i iv grade m l i s i and v if r is quasi unmixed then grade r i n s i for all large n also it is shown that the asymptotic prime divisors of i b 1 b s r contain those of i when b 1 b s are an asymptotic sequence over i and that the residue classes modulo i of such a sequence are an asymptotic sequence in r i finally several sufficient conditions for l i to be equal to l i p for some asymptotic prime divisor p of i are given

point estimation following group sequential tests kim and demets 1987 biometrics 43 857 864 described an exact procedure for constructing confidence intervals for a normal mean following group sequential tests when the boundaries were generated based on the notion of use functions proposed by lan and demets 1983 biometrika 70 659 663 in this article three point estimators for a normal mean following group sequential tests are considered and their properties are investigated by monte carlo simulation based on the simulation results some suggestions are made as to the choice of group sequential designs and use function

p values for tests using a repeated significance test design summary repeated significance testing can be used as a multistage testing procedure which has advantages over single sample tests and purely sequential tests repeated significance tests having a constant nominal level of significance have designs which are more easily understood by researchers and still have good power relative to designs using varying nominal levels of significance in this paper we present a method for defining an overall p value for repeated significance tests with constant significance levels which can aid researchers in interpreting the observed results of such tests a table of p values for various designs is given

p values for group sequential testing summary group sequential testing procedures are widely employed in long term clinical trials we discuss several methods for computing the p value following a group sequential trial based on different outcome space orderings we use a criterion defined in this paper to compare these methods and point out interesting differences

the length heuristic for simultaneous hypothesis tests we observe a sequence of test statistics t t 1 t 2 t j each of which is approximately n 0 1 under the null hypothesis h 0 but which are correlated with each other not being certain which t j is best we use the test statistic t max max t 1 t j for a given observed value of t max say c what is the significance probability pr t max c define the length of t to be j l 2 arccos p j 1 j where p j 1 j is the null hypothesis correlation between t j 1 and t j hotelling s theorem on the volume of tubes leads to a length based bound on pr t max c that usually beats the bonferroni bound it is easy to improve hotelling s bound several examples show that in favourable circumstances the improved bounds can be good approximations to the actual value of pr t max c

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