a large sample study of cox s regression model strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in cox s regression model estimates are also derived for the underlying cumulative hazard function and survival distribution we establish the asymptotic normality of these estimates and calculate the limiting variances

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

inference for a nonlinear counting process regression model martingale and counting process techniques are applied to the problem of inference for general conditional hazard functions this problem was first studied by beran who introduced a class of estimators for the conditional cumulative hazard and survival functions in the special case of time independent covariates here the covariate can be time dependent the classical i i d assumptions are relaxed by replacing them with certain asymptotic stability assumptions and models involving recurrent failures are included this is done within the framework of a general nonparametric counting process regression model important examples of the model include right censored survival data semi markov processes an illness death process with duration dependence and age dependent birth and death processes

nonparametric sequential tests against ordered alternatives in multiple armed clinical trials summary in a multiple armed clinical trial where one agent is administered at several dose levels or where each treatment regimen is formed by adding more agents to the preceding combination it is desirable to use test statistics that are sensitive to alternatives of stochastically ordered distributions liu et al 1992 recently developed a class of such test statistics by taking linear combinations of the familiar two sample weighted log rank statistics in the present paper we derive the asymptotic joint distributions of their test statistics calculated at different points in time applications to sequential survival studies with staggered patient entry are emphasized a colon cancer example is provided

equivariant higher analytic torsion and equivariant euler characteristic we show that j lott s equivariant higher analytic torsion only depends on the equivariant euler characteristic we give an explicit formula 1 introduction let g be a compact connected lie group with lie algebra g let 1 g denote the ring of ad g invariant complex polynomials on g then 1 g 1 f e 1 g f 0 0 is a maximal ideal of 1 g by 1 g we denote the

parameter estimation following group sequential hypothesis testing summary parameter estimation techniques which fail to adjust for the interim analyses of group sequential test designs will introduce bias in much the same way that the repeated use of single sample hypothesis testing causes inflation of the type one statistical error rate methods based on the duality of hypothesis testing and interval estimation require definition of an ordering for the outcome space for the test statistic in this paper estimation following a group sequential hypothesis test for the mean of a normal distribution with known variance is investigated a proposed ordering of the sample space based on the maximum likelihood estimate of the mean is found to result in estimates which compare favourably with estimates computed from orderings investigated by tsiatis rosner mehta 1984 and chang o brien 1986 for a variety of group sequential designs the proposed ordering is then adapted for use when the sizes of groups accrued between analyses is random

a selberg integral for the lie algebra an a new q binomial theorem for macdonald polynomials is employed to prove an a n analogue of the celebrated selberg integral this confirms the mathfrak g rm a n case of a conjecture by mukhin and varchenko concerning the existence of a selberg integral for every simple lie algebra mathfrak g

construction of a continuous stopping boundary from an alpha spending function lan and demets 1983 biometrika 70 659 663 proposed a flexible method for monitoring accumulating data that does not require the number and times of analyses to be specified in advance yet maintains an overall type i error alpha their method amounts to discretizing a preselected continuous boundary by clumping the density of the boundary crossing time at discrete analysis times and calculating the resultant discrete time boundary values in this framework the cumulative distribution function of the continuous time stopping rule is used as an alpha spending function a key assumption that underlies this method is that future analysis times are not chosen on the basis of the current value of the statistic however clinical trials may be monitored more frequently when they are close to crossing the boundary in this situation the corresponding continuous time boundary should be used here we demonstrate how to construct a continuous stopping boundary from an alpha spending function this capability is useful in the design of clinical trials we use the beta blocker heart attack trial bhat and aids clinical trials group protocol 021 for illustration

sequential monitoring of survival data with the wilcoxon statistic when a spending function is used in sequential data monitoring of a clinical trial it is important to know the information fraction at the times of interim analysis in a maximum duration designed study the information fraction is unknown when data are monitored and it has to be estimated the modified wilcoxon statistic developed by peto and peto and modified by prentice is often used to compare two survival curves in a clinical trial we give guidelines for estimating the information fraction in a maximum duration trial when this statistic is employed when there is a relatively low event rate or the survival time is approximately exponential the information fraction for the peto peto prentice wilcoxon statistic is very close to that of the popular logrank statistic in other cases it would be helpful to estimate the information fraction as a function of elapsed calendar time we discuss both group sequential and continuous monitoring

equivariant analytic torsion for compact lie group actions abstract we define the equivariant analytic torsion for a compact lie group action and study its dependence on the geometric data

a nonparametric method for dealing with mismeasured covariate data abstract mismeasurement of covariate data is a frequent problem in statistical data analysis however when true and mismeasured data are obtained for a subsample of the observations it is possible to estimate the parameters relating the outcome to the covariate of interest maximum likelihood methods that rely on parametric models for the mismeasurement have not met with much success realistic models for the mismeasurement process are difficult to construct the form of the likelihood is often intractable and more important such methods are not robust to model misspecification we propose an easily implemented method that is nonparametric with respect to the mismeasurement process and that is applicable when mismeasurement is due to the problem of missing data errors in variables or use of imperfect surrogate covariates specifically denote the outcome variable by y the covariate data subject to mismeasurement by x and the remaining covariates including perhaps surrogates or mismeasured values

analysis of factorial survival experiments several new methodological issues that arise within two way factorial designs for survival experiments are discussed within the framework of asymptotic theory for the proportional hazards model with two binary treatment covariates these issues include the proper formulation of null hypotheses and alternatives the choice among log rank and adjusted or stratified log rank statistics the asymptotic correlation between test statistics for the separate main effects the asymptotic power under the various possible methods of analysis of tests to detect main effects and interactions the comparison of power to detect main effects within a 2 x 2 factorial design with power in a three group trial where no patients are randomized simultaneously to both treatments and the problems of analysis arising when accrual or exposure to one of the treatments is terminated early for ethical reasons

use of logrank tests and group sequential methods at fixed calendar times this paper presents simulations to determine the operating characteristics of several group sequential boundaries when applied to the repeated analysis of survival data at equal intervals of calendar time because the group sequential boundaries of pocock 1977 biometrika 64 191 199 and o brien and fleming 1979 biometrics 35 549 556 were constructed on the assumption that each interim analysis provides an equal increment of statistical information these boundaries are not theoretically appropriate for interim analysis at prescheduled calendar times nonetheless our simulations show that these boundaries yield size and power near nominal levels for repeated logrank analyses at equal intervals of calendar time

sequential linear rank tests for two sample censored survival data under extremely general patterns of patient arrival allocation to treatment and loss to follow up in randomized clinical trial settings the sequentially computed logrank statistic mantel 1966 is shown under the null hypothesis of identically distributed lifetimes to have exactly uncorrelated increments and is shown via rebolledo s 1980 martingale invariance principle to satisfy a functional central limit theorem justifying sequential logrank tests of jones and whitehead 1979 generalizations are made to other twosample rank tests for censored survival data and practical applicability to real randomized clinical trials is discussed

stratified partial likelihood estimation when multiple durations are generated by a single unit they may be related in a way that is not fully captured by the regressors the omitted unit specific variables might vary over the durations they might also be correlated with the variables in the regression component we propose an estimator that responds to these concerns and develop a specification test for detecting unobserved unit specific effects data from malaysia reveal that concentration of child mortality in some families is imperfectly explained by observed explanatory variables and that failure to control for unobserved heterogeneity seriously biases the parameter estimates

adaptive designs for the one sample log rank test summary traditional designs in phase iia cancer trials are single arm designs with a binary outcome for example tumor response in some settings however a time to event endpoint might appear more appropriate particularly in the presence of loss to follow up then the one sample log rank test might be the method of choice it allows to compare the survival curve of the patients under treatment to a prespecified reference survival curve the reference curve usually represents the expected survival under standard of the care in this work convergence of the one sample log rank statistic to brownian motion is proven using rebolledo s martingale central limit theorem while accounting for staggered entry times of the patients on this basis a confirmatory adaptive one sample log rank test is proposed where provision is made for data dependent sample size reassessment the focus is to apply the inverse normal method this is done in two different directions the first strategy exploits the independent increments property of the one sample log rank statistic the second strategy is based on the patient wise separation principle it is shown by simulation that the proposed adaptive test might help to rescue an underpowered trial and at the same time lowers the average sample number asn under the null hypothesis as compared to a single stage fixed sample design

flood frequency analysis using the cox regression model procedures for incorporating time varying exogenous information into flood frequency analyses are developed using the cox regression model for counting processes in this statistical model the probability of occurrence of a flood peak in a short interval t t dt depends in an explicit manner on the values at t of k covariate processes z1 zk specifically letting dn t be 1 if a flood peak occurs in t t dt and 0 otherwise dn t a t exp j 1kbjzj t dm t where a the baseline intensity is an unknown function b is a vector of unknown regression parameters and the error dm t is conditionally orthogonal to the past history two applications assessment of relative importance of physical processes such as snow melt or soil moisture storage on flood frequency at a site and derivation of time varying flood frequency estimates are considered

repeated significance testing in survival analysis using a litter matched design this paper shows that asymptotically a general class of linear rank procedures in a litter matched design form a gauss markov process over time this result is used to develop a procedure for repeated significance testing for time to response differences in litter matched studies in addition an important class of two sample linear rank procedures is shown to have uncorrelated increments over time an example is given using a litter matched cancer study empirical indications of the accuracy of a method for approximating multivariate normal critical values are also given

sequential log rank tests adjusting for covariates with the accelerated life model summary this paper presents a simple sequential method for comparing two treatments in randomized clinical trials which allows for the adjustment of other covariates with the accelerated life model the new procedure is the same as the ordinary sequential log rank test except that the test statistic is calculated on an adjusted time scale this covariate adjustment can lead to more efficient designs of group sequential trials especially when the effects of covariates being adjusted for are strong an aids example is provided

the l 1 method for robust nonparametric regression abstract consider the problem of estimating the regression function underlying a set of data that is contaminated by a heavy tailed error distribution the two standard approaches to such a problem are each flawed robust parametric least squares is appropriate only if there is good reason to believe that the underlying function has some particular form whereas most nonparametric regression methods are asymptotically equivalent to kernel regression methods which are not resistant against outliers existing algorithms for robustifying nonparametric regression procedures use either nonlinear optimization of an influence function or iterative solution of local polynomial fitting using reweighted least squares neither of these two approaches combines computational ease with asymptotic theoretical results furthermore application of the robust procedure has been limited almost exclusively to the case of a single explanatory variable with the response variable in this article a new hybrid method is propose

sequential analysis for censored regression data abstract motivated by buckley and james modification of the least squares procedure for censored regression data we derive a score process that incorporates the time evolution and handles staggered entry data in a natural way by censored regression data we mean that the responses are subject to a possible right censorship such cases often arise from clinical trials and industrial life tests the score process can be interpreted as a weighted comparison of transformed survival times this is especially suitable for the accelerated failure time regression model by expressing it in an appropriate form so that the counting process and its associate martingale theory can be applied we show that the score process is approximated by a mean 0 multidimensional gaussian process a consistent estimator of its covariance matrix function is provided based on the covariance matrix estimator and the sequential test of slud and wei a repeated significance test is then proposed usefulness of the procedure is il

equivariant de rham torsions the purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de rham theory this difference is the sum of the integral of a chern simons current and of a new invariant the v invariant of an odd dimensional manifold equipped with an action of a compact lie group the v invariant localizes on the critical manifolds of invariant morse bott functions the results in this paper are shown to be compatible with results of bunke and also our with previous results on analytic torsion forms

application of bayesian statistics to decision making during a clinical trial we describe the application of bayesian methods to the monitoring and analysis of a trial of treatment for patients with advanced colorectal carcinoma we discuss the choice of prior distribution and justify the use of a truncated normal distribution with a probability mass at zero difference the stopping rule based on the trials of the posterior distribution and a chosen range of equivalence yields an upper boundary very close to the pocock group sequential boundary the bayes stopping rule is quite sensitive to the amount of probability mass at zero in the prior distribution

fixed accuracy estimation of an autoregressive parameter abstract for a first order non explosive autoregressive process with unknown parameter beta epsilon 1 1 it is shown that if data are collected according to a particular stopping rule the least squares estimator of beta is asymptotically normally distributed uniformly in beta in the case of normal residuals the stopping rule may be interpreted as sampling until the observed fisher information reaches a preassigned level the situation is contrasted with the fixed sample size case where the estimator has a non normal limiting distribution when beta 1 author

monitoring a general class of two sample survival statistics with applications summary this paper considers a general class of statistics for testing the equality of two survival distributions in clinical trials with sequential monitoring the tests can be expressed as lebesgue stieltjes integrals of a weight function with respect to the difference between two survival distributions prominent members of this class include the two sample difference in kaplan meier estimates the test of medians brookmeyer crowley 1982 a truncated version of efron s 1967 test and the pepe fleming statistic pepe fleming 1989 1991 statistics in this class are shown to converge to a gaussian process indexed by information time under both null and local alternatives even if different statistics are used at different information times properly standardised statistics in a subclass converge to gaussian processes with independent increments so that the usual group sequential techniques for monitoring a clinical trial can be applied the design of a trial comparing two treatments with respect to mother to newborn transmission of hiv is used to illustrate practical aspects of monitoring

nonparametric rank based methods for group sequential monitoring of paired censored survival data summary this research gives methods for nonparametric sequential monitoring of paired censored survival data in the two sample problem using paired weighted log rank statistics with adjustments for dependence in survival and censoring outcomes the joint asymptotic closed form distribution of these sequentially monitored statistics has a dependent increments structure simulations validating operating characteristics of the proposed methods highlight power and size consequences of ignoring even mildly correlated data a motivating example is presented via the early treatment diabetic retinopathy study

group sequential tests with censored survival data adjusting for covariates summary group sequential tests for the analysis of survival data that compare two treatments and allow for the adjustment of other concomitant variables are discussed these methods allow early termination of a study if large treatment differences occur based on simulation results we compare the operating characteristics of these adjusted tests with the unadjusted sequentially computed log rank test

methods for the analysis of sampled cohort data in the cox proportional hazards model methods are provided for regression parameter and cumulative baseline hazard estimation in the cox proportional hazards model when the cohort is sampled according to a predictable sampling probability law the key to the methodology is to define counting processes which count joint failure and sampled risk sets occurrences and to derive the appropriate intensities for these counting processes leading to estimation methods parallel to those for full cohort data these techniques are illustrated for a number of sampling designs including three novel designs counter matching with additional randomly sampled controls quota sampling of controls and nested case control sampling with number of controls dependent on the failure s exposure status general asymptotic theory is developed for the maximum partial likelihood estimator and cumulative baseline hazard estimator and is used to derive the asymptotic distributions for estimators from a large class of designs

understanding cox s regression model a martingale approach abstract an informal discussion is given of how martingale techniques can be used to extend cox s regression model and to derive its large sample properties

repeated significance testing for a general class of statistics used in censored survival analysis abstract the asymptotic joint distribution of a class of sequentially computed statistics used in survival analysis is derived when the individuals under study enter serially and are subject to random loss to follow up these results are used for constructing repeated significance tests over real time

towards a general asymptotic theory for cox model with staggered entry a general asymptotic theory is established for the two parameter cox score process with staggered entry data it extends in several directions the existing theory developed by sellke and siegmund slud and gu and lai an essential tool employed here is a modern empirical process theory as elucidated in a recent monograph by pollard

implementation of group sequential logrank tests in a maximum duration trial summary to control the type i error probability in a group sequential procedure using the logrank test it is important to know the information times fractions at the times of interim analyses conducted for purposes of data monitoring for the logrank test the information time at an interim analysis is the fraction of the total number of events to be accrued in the entire trial in a maximum information trial design the trial is concluded when a prespecified total number of events has been accrued for such a design therefore the information time at each interim analysis is known however many trials are designed to accrue data over a fixed duration of follow up on a specified number of patients this is termed a maximum duration trial design under such a design the total number of events to be accrued is unknown at the time of an interim analysis for a maximum duration trial design therefore these information times need to be estimated a common practice is to assume that a fixed fraction of information will be accrued between any two consecutive interim analyses and then employ a pocock or o brien fleming boundary in this article we describe an estimate of the information time based on the fraction of total patient exposure which tends to be slightly negatively biased i e conservative if survival is exponentially distributed we then present a numerical exploration of the robustness of this estimate when nonexponential survival applies we also show that the lan demets 1983 biometrika 70 659 663 procedure for constructing group sequential boundaries with the desired level of type i error control can be computed using the estimated information fraction even though it may be biased finally we discuss the implications of employing a biased estimate of study information for a group sequential procedure

the asymptotic joint distribution of the efficient scores test for the proportional hazards model calculated over time the log rank test derived by mantel 1966 and peto 1972 is a special case of the efficient scores test for the proportional hazards model it is used frequently in the analysis of survival duration in clinical trials where patients entering the study are assigned at random to different treatments and are followed until they either fail or the study is terminated most survival studies are designed so that the log rank test is calculated at the termination of the study however it is not unusual for the investigator to monitor the experiment periodically it would be very useful to know the joint distribution of the test statistic at different points in time in order to adjust for the effect of repeated significance testing although the joint distribution has not been derived previously sequential procedures for the analysis of survival data in clinical trials have been given by several authors including armitage 1975 pp 140 6 jones whitehead 1979 and nagelkerke hart 1980 in this paper we establish the asymptotic joint distribution of the more general efficient scores test for the proportional hazards model calculated at different points in time and indicate how this can be used for repeated significance testing

competing risks regression for stratified data for competing risks data the fine gray proportional hazards model for subdistribution has gained popularity for its convenience in directly assessing the effect of covariates on the cumulative incidence function however in many important applications proportional hazards may not be satisfied including multicenter clinical trials where the baseline subdistribution hazards may not be common due to varying patient populations in this article we consider a stratified competing risks regression to allow the baseline hazard to vary across levels of the stratification covariate according to the relative size of the number of strata and strata sizes two stratification regimes are considered using partial likelihood and weighting techniques we obtain consistent estimators of regression parameters the corresponding asymptotic properties and resulting inferences are provided for the two regimes separately data from a breast cancer clinical trial and from a bone marrow transplantation registry illustrate the potential utility of the stratified fine gray model

boundary crossing probabilities and statistical applications abstract this paper surveys recent results involving boundary crossing probabilities and related statistical applications the first part is concerned with problems of sequential analysis especially repeated significance tests and their application to sequential clinical trials involving survival data the second part develops the probability theory motivated by the problems of part 1 a method for computing first passage distributions of brownian motion to linear boundaries is introduced and then modified to handle problems in discrete time and those involving nonlinear boundaries the third part is concerned with fixed sample statistical problems especially change point problems which involve boundary crossing probabilities examples are given of problems for which methods of part 2 appear adequate and of problems which require new methods

quasi maximum likelihood estimation of volatility with high frequency data this paper investigates the properties of the well known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise by extending the classic asymptotic results of quasi maximum likelihood estimation when trying to estimate the integrated volatility and the variance of noise this parametric approach remains consistent efficient and robust as a quasi estimator under misspecified assumptions moreover it shares the model free feature with nonparametric alternatives for instance realized kernels while being advantageous over them in terms of finite sample performance in light of quadratic representation this estimator behaves like an iterative exponential realized kernel asymptotically comparisons with a variety of implementations of the tukey hanning2 kernel are provided using monte carlo simulations and an empirical study with the euro us dollar future illustrates its application in practice

sequential analysis of the proportional hazards model abstract for the proportional hazards model of survival analysis an appropriate large sample theory is developed for cases of staggered entry and sequential analysis the principal techniques involve an approximation of the score process by a suitable martingale and a random rescaling of time based on the observed fisher information as a result we show that the maximum partial likelihood estimator behaves asymptotically like brownian motion author

sequential confidence regions for maximum likelihood estimates the goal of this paper is to develop a general framework for constructing sequential fixed size confidence regions based on maximum likelihood estimates asymptotic properties of the sequential procedure for setting up the confidence regions are analyzed under very broad assumptions on the underlying parametric model it is shown that the proposed sequential procedure is asymptotically optimal in the sense that it approximates the optimal fixed sample size procedure it is further shown that the cost of ignorance associated with the sequential procedure is bounded applications are made to estimation problems arising in prospective and retrospective studies

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

semiparametric efficiency and its implication on the design and analysis of group sequential studies abstract authors have shown that the time sequential joint distributions of many statistics used to analyze data arising from group sequential time to event and longitudinal studies are multivariate normal with an independent increments covariance structure in theorem 1 of this article we demonstrate that this limiting distribution arises naturally when one uses an efficient test statistic to test a single parameter in a semiparametric or parametric model because we are able to think of many of the statistics in the literature in this fashion the limiting distribution under investigation is just a special case of theorem 1 using this general structure we then develop an information based design and monitoring procedure that can be applied to any type of model for any type of group sequential study provided that there is a unique parameter of interest that can be efficiently tested

a method for sequential analysis of survival data with nonproportional hazards two tests are proposed for comparing the survival curves of patients randomised between an experimental treatment and a control treatment when it is anticipated that the two survival curves may not satisfy the assumption of proportional hazards the tests are particularly useful for the situation in which the survival curves are coincident or cross over early in the follow up period and then diverge the tests compare the probabilities of survival for longer than some fixed time since randomisation for the two groups of patients both methods take account of the right censored observations and both are associated with methods for estimating and setting confidence limits for treatment differences the first method is a mathematically direct approach based on the derivation of the efficient score statistic and fisher s information the second method is simpler being based on kaplan meier estimates and their variances conventional methods of sample size determination require the assumption of proportional hazards here a sequential approach is used as it is difficult to set the sample size in advance without strong assumptions about the relationship between the two survival curves simulation results giving information on the size and power of the proposed tests are provided and the tests are applied to data from a clinical trial in breast cancer

two sample repeated significance tests based on the modified wilcoxon statistic abstract the asymptotic distribution theory of sequentially computed modified wilcoxon scores is developed for two sample survival data with random staggered entry and random loss to follow up the asymptotic covariance indicates generally dependent modified wilcoxon increments contradicting the authors reading of jones and whitehead 1979 a repeated significance testing procedure is presented for testing the equality of two survival distributions based on the asymptotic theory the early stopping properties of this procedure are illustrated by a prostate cancer example

simultaneous group sequential analysis of rank based and weighted kaplan meier tests for paired censored survival data this research sequentially monitors paired survival differences using a new class of nonparametric tests based on functionals of standardized paired weighted log rank pwlr and standardized paired weighted kaplan meier pwkm tests during a trial these tests may alternately assume the role of the more extreme statistic by monitoring pemax the maximum between the absolute values of the standardized pwlr and pwkm one combines advantages of rank based rb and non rb paired testing paradigms simulations show that monitoring treatment differences using pemax maintains type i error and is nearly as powerful as using the more advantageous of the two tests in proportional hazards ph as well as non ph situations hence pemax preserves power more robustly than individually monitored pwlr and pwkm while maintaining a reasonably simple approach to design and analysis of results an example from the early treatment diabetic retinopathy study etdrs is given

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

design and analysis of group sequential logrank tests in maximum duration versus information trials when monitoring a clinical trial with failure time data using the logrank test and the type i error spending function approach the information time has to be estimated as a fraction of the maximum number of failures in maximum duration trials the denominator of this fraction is a random quantity and has to be estimated besides there are two candidates for the denominator one under the null hypothesis of no treatment difference and the other under the specified alternative hypothesis either way some adjustments are necessary in determining group sequential boundaries in order to maintain type i error at a desired significance level as a consequence the type i error spending function will be altered from the one chosen for the design thus affecting the operating characteristics of the subsequent group sequential logrank tests in maximum information trials however the maximum amount of information is fixed and thus the estimate of the information time is always unbiased the net effect is that computation of group sequential boundaries becomes straightforward with a potential saving in study durations as compared to maximum duration trials we will illustrate how adjustments are made in maximum duration trials to maintain type i error when the information times are estimated with the information horizons under the null and alternative hypotheses and present numerical explorations to compare robustness of two different estimates of the information times we then propose a design procedure for maximum information trials and investigate the properties of maximum information trials for different group sequential boundaries we also compare maximum information trials and maximum duration trials based on an example

a recursion and a combinatorial formula for jack polynomials the jack polynomials j x form a remarkable class of symmetric polynomials they are indexed by a partition and depend on a parameter one of their properties is that several classical families of symmetric functions can be obtained by specializing e g the monomial symmetric functions m the elementary functions e 0 the schur functions s 1 and nally the two classes of zonal polynomials 2 1 2 the jack polynomials can be de ned in various ways e g a as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b as simultaneous eigenfunctions of certain di erential operators the sekiguchi debiard operators recently opdam o constructed a similar family f x of nonsymmetric polynomials the index runs now through all compositions nn they are de ned in a completely similar fashion e g the sekiguchi debiard operators are being replaced by the cherednik di erential re ection operators see sect 3 it is becoming more and more clear that these polynomials are as important as their symmetric counterparts the purpose of this paper is to add to the existing characterizations of jack polynomials two further ones c a recursion formula among the f together with two formulas to obtain j from them d combinatorial formulas of both j and f in terms of certain generalized tableaux

sequential methods for parametric survival models summary the asymptotic sequential distribution is derived for score tests and maximum likelihood estimates in parametric survival models in the presence of nuisance parameters where individuals enter over time and the data are subject to right censoring it is shown that these sequential tests and estimates have an independent increments structure this allows the use of standard group sequential methods for early stopping

group sequential analysis incorporating covariate information abstract in this article we survey existing results concerning the joint distribution of the sequence of estimates of the parameter vector when a model is fitted to accumulating data and provide a unified theory that explains the independent increments structure commonly seen in group sequential test statistics our theory covers normal linear models including the case of correlated observations and asymptotic results extend to generalized linear models and the proportional hazards regression model for survival data the asymptotic results are derived using standard methods for the nonsequential case and they hold as long as these nonsequential techniques are applicable at each individual analysis in all cases the joint distribution of the sequence of parameter estimates has the same form exactly or asymptotically as that of the sequence of means of an increasing number of independent identically distributed normal variables thus our results provide the formal basis for extending the scope of s

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

sequential monitoring of clinical trials the role of information and brownian motion sequential monitoring has been a topic of major interest in clinical trials methodology over the past two decades this paper presents a unified conceptual framework for sequential monitoring that covers a wide variety of monitoring procedures in a wide variety of clinical trial settings the central elements of this framework consist of a suitable concept of statistical information and a scheme for using this concept as a basis for summarizing the accumulating results of a trial in a standardized form through a stochastic process that can be shown to approximate classical brownian motion the ideas are developed in a simple step by step fashion and illustrated by several practical examples

optimal two stage screening designs for survival comparisons summary a two stage design is presented which enables the screening at the first stage of several new experimental treatments for survival improvement over a standard regimen only promising treatments are carried forward to the second stage for definitive evaluation our procedure is to minimize the number of patients expected to be accrued to new regimens which do not offer a survival benefit over the standard regimen subject to constraints of specified alpha error power and a stopping rule wlhich allows specification of the magnitude of the hazards ratio which would warrant accrual beyond the first stage we explore the savings in patient accrual by comparing our design with a single stage design for a variety of situations our design may offer a substantial saving when the hazard rate is large relative to the patient accrual rate a situation often encountered in clinical trials in advanced cancer although computations are based on asymptotic results simulations verify that the approximations are adequate for most trials of interest in the development of treatments for advanced cancer a standard regimen with limited survival benefit may be available as well as several new promising regimens which have not been evaluated in a comparative trial to determine their possible survival benefit we offer an efficient two stage design which allows survival comparisons of the new regimens with a standard regimen but which permits early termination of accrual onto those new regimens which do not demonstrate a minimum prespecified survival advantage over the standard regimen in this way we hope to minimize the number of patients assigned to new regimens which are not promising the design also permits early accrual termination if a new treatment offers an early substantial survival advantage over the standard regimen 2 the design we propose a two stage screening design so that survival times of each of k experimental regimens will be compared to the survival of the standard treatment at possibly two time points t1 and t2 furthermore we assume proportional hazards such that

on the bias of maximum likelihood estimation following a sequential test summary tlhe bias of maximum likelihood estimates calculated at the end of a sequential procedure is investigated for the two sequential designs considered in detail the sequential probability ratio test and the triangular test this bias is appreciable a method of calculating an adjusted estimate with reduced bias is described and an approximation to the standard error of the new estimate is provided examples of the implementation of the method are given and its advantages and disadvantages relative to alternative approaches are discussed