coupled error spending functions for parallel bivariate sequential tests sequential procedures are developed to facilitate marginal monitoring of bivariate response vectors in clinical trials the general approach is based on an extension of the error spending function methodology of lan and demets 1983 biometrika 70 659 663 and is sufficiently flexible that one may elect to fix the experimental type i error rate cook 1994 controlled clinical trials 15 3 187 200 or the marginal type i error rates sample size calculations are described to ensure power requirements are satisfied for marginal tests of significance reformulating the procedures in terms of repeated confidence intervals jennison and turnbull 1989 journal of the royal statistical society series b 51 305 361 lends added flexibility to the monitoring process the developments are discussed in the context of responses with a bivariate normal distribution data from an asthma intervention trial are used for illustrative purposes

on adaptive extensions of group sequential trials for clinical investigations in group sequential trials it is important to obtain adequate data to assess overall benefits and risks of an experimental treatment for patients to achieve this goal we provide a general formal framework for adaptively extending a group sequential trial to stop at any interim analysis time often after a significance boundary for a clinical endpoint is crossed for statistical inference we propose to order the sample space by a class of well ordered group sequential tests on this basis we develop a unified sequential statistical inference approach that is applicable to both interim monitoring and final analysis we also show that the new ordering provides the foundation for the repeated confidence intervals procedure of jennison and turnbull

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

design of group sequential clinical trials with multiple endpoints abstract group sequential testing for randomized clinical trials designed with multiple endpoints is considered previously computed tables for single endpoints are still useful with a change in interpretation of certain parameters the advantage in setting sample size based on multiple endpoints is that the sample size required when a trial is designed using more than one endpoint is smaller than the sample size based on any single endpoint if the two calculations are undertaken using matching significance levels and powers an example is provided and possible extensions of the work are discussed

group sequential methods for clinical trials allowing early acceptance of ho and incorporating costs summary when interim accumulated results from a clinical trial clearly indicate that the null hypothesis should be accepted or rejected ethical and cost considerations may dictate terminating the trial and making the appropriate decision pocock 1977 has described a group sequential procedure for clinical trials that permits early termination only for rejecting the null hypothesis this paper extends pocock s procedure to allow early termination with acceptance of the null hypothesis theoretical and simulation results demonstrate that designs allowing early acceptance of the null hypothesis can have lower expected costs than designs that do not even when the alternative hypothesis is true

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

music type electromagnetic imaging of a collection of small three dimensional inclusions in this paper we consider the localization of a collection of small three dimensional bounded homogeneous inclusions via time harmonic electromagnetic means typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization s at some distance from the collection possibly also lying in the far field the inclusions somewhat apart or closely spaced are buried within a homogeneous medium and are of arbitrary contrast of permittivity conductivity and permeability vis ag vis this embedding medium the problem is formulated as an inverse scattering problem for the full maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix no specific application is studied at this stage but characterization of obstacles in subsoils nondestructive evaluation of man made structures and medical imaging are primary fields of application envisaged the proposed approach uses a music multiple signal classification type algorithm and it yields fast numbering accurate localization and estimates of the electromagnetic and geometric parameters polarization tensors of the inclusions the mathematical machinery is detailed first some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes for the latter known results are retrieved then the viability of this algorithm which would be easily extended to planarly layered environments by introduction of their green s functions is documented by a variety of numerical results from synthetic noiseless and severely noisy field data

confidence intervals for a binomial parameter following a multistage test with application to mil std 105d and medical trials this paper describes a method for constructing confidence intervals for a binomial parameter upon termination of a sequential or multistage test tables are presented for use with the mil std 105d multiple sampling plans for acceptance sampling also given are tables for use with some three stage schemes that have been proposed in connection with biomedical trials the results are compared with confidence intervals calculated as if the sampling plan had been one with a fixed sample size

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

analysis of the time reversal operator for a small spherical scatterer in an electromagnetic field the time reversal operator tro for a planar array of crossed dipole elements illuminating a small conducting and or dielectric sphere is investigated in order to determine the general properties of an electromagnetic time reversing array system the behavior of such a system for a given frequency is analyzed by studying the eigenvalues and eigenvectors of the tro each eigenvector specifies a set of complex driving currents for the array elements that produce received voltages that are proportional to the conjugates of the drive currents the proportionality constant is equal to the square root of the associated eigenvalue and is the same for all elements the eigenvalues and eigenvectors can be determined by performing a singular value decomposition svd on the multistatic data matrix of the array the eigenvalues of the tro are the squares of the singular values and the eigenvectors are identical to the singular vectors we have shown that the maximum number of singular vectors associated with the sphere is equal to the number of orthogonal orientations of the dipole moments induced in the sphere when irradiated by the array so there is a maximum of six for a conducting sphere but only three are significant when the conductivity is small and the sphere may be considered being just a dielectric numerical results are presented for linear and circular arrays to show the general behavior of the system

asymmetric group sequential boundaries for monitoring clinical trials a group sequential method is proposed for the periodic analysis of accumulating data in clinical trials which are essentially one sided as in treatment placebo studies the proposed boundaries retain the early stopping properties of the procedure described previously by demets ware 1980 when treatment appears to be ineffective or harmful but incorporate the more stringent requirements of the model proposed by o brien fleming 1979 for early rejection of the null hypothesis in favour of treatment

closed testing procedures for group sequential clinical trials with multiple endpoints summary a simple approach is given for conducting closed testing in clinical trials with multiple end points in which group sequential monitoring is planned the approach allows a flexible stopping time the earliest and latest stopping times are described the paradigm is applicable both to clinical trials with multiple endpoints and to the one sided multiple comparison problem of several treatments versus a control the approach leads to enhancements of previous methods and suggestions for new methods an example of a respiratory disease trial with four endpoints is given

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

on the design of synthetic case control studies a design is proposed for case control within cohort studies in this design controls are sampled without replacement from failure free members of the cohort at each distinct failure time upon selection a subject ceases to be eligible for control selection at later failure times also if a subject failing at time t had been selected as a control at t less than t then the matched controls at t are selected to have also been at risk at t in these circumstances correlation exists between score statistic contributions at t and t an estimator is developed for this correlation a small simulation study compares the design just described to other possible synthetic case control designs

exact statistical inference for group sequential trials summary this paper considers clinical trials comparing two treatments with dichotomous responses where the data are examined periodically for early evidence of treatment difference the existing group sequential methods for such trials are based on the large sample normal approximation to the joint distribution of the estimators of treatment difference over interim analyses we demonstrate through extensive numerical studies that for small and even moderate sized trials these approximate procedures may lead to tests with supranominal size mainly when unpooled estimators of variance are used and confidence intervals with under nominal coverage probability we then study exact methods for group sequential testing repeated interval estimation and interval estimation following sequential testing the new procedures can accommodate any treatment allocation rules an example using real data is provided in clinical trials with sequential patient entry fixed sample size designs are sometimes unjustified on ethical grounds and fully sequential procedures are often impractical one compromise is the group sequential design under which multiple looks are performed during the trial it is widely recognized that the use of the standard single stage f1xedsample size test or confidence interval at each look would lead to an overall type i error or noncoverage rate significantly higher than the nominal level a number of methods have been proposed to determine proper critical values for maintaining a prespecified overall significance level in sequential testing to construct repeated confidence intervals with a given overall coverage probability and to obtain a valid confidence interval following a sequential trial an excellent review on the existing methodology was given by jennison and turnbull 1989 in this paper we are mainly concerned with the commonly used group sequential trials comparing two treatments a and b with relatively quick dichotomous responses success and failure the data are reviewed periodically with cumulative mk subjects receiving treatment a and nk subjects receiving treatment b by the kth look let random variables xk and yk denote the numbers of successes out of the msk and nk subjects respectively we assume that the maximum number of looks k is fixed in advance though some of the procedures developed here can be extended to the case where k is not fixed in advance

a unified method for monitoring and analysing controlled trials group sequential methods are becoming increasingly popular for monitoring and analysing large controlled trials especially clinical trials they not only allow trialists to monitor the data as it accumulates but also reduce the expected sample size such methods are traditionally based on preserving the overall type i error by increasing the conservatism of the hypothesis tests performed at any single analysis using methods which are based on hypothesis testing in this way makes point estimation and the calculation of confidence intervals difficult and controversial we describe a class of group sequential procedures based on a single parameter which reflects initial scepticism towards unexpectedly large effects these procedures have good expected and maximum sample sizes and lead to natural point and interval estimates of the treatment difference hypothesis tests point estimates and interval estimates calculated using this procedure are consistent with each other and tests and estimates made at the end of the trial are consistent with interim tests and estimates this class of sequential tests can be considered in both a traditional group sequential manner or as a bayesian solution to the problem

comparison of different methods for decision making in bioequivalence assessment if the regulatory requirements are symmetrical the use of symmetrical confidence intervals as a decision rule for bioequivalence assessment leads as shown by simulations to better level properties and an inferior power compared to a rule based on shortest confidence intervals a choice between these two approaches will have to depend on a loss function for asymmetric regulatory requirements symmetrical confidence intervals should not be used however a decision can still be based on posterior probabilities pr theta epsilon r1 r2 x or shortest confidence intervals for purposes of inference presentation and interpretation of results we think that the use of symmetrical confidence intervals alone can be misleading and we therefore recommend that the posterior probabilities and densities or at least the shortest confidence intervals be given

multistatic response matrix of spherical scatterers and the back propagation of singular fields in view of the development of multiple signal classification type non iterative imaging procedures based on the singular value decomposition of the multistatic response msr matrix of a collection of inclusions the full 3 d electromagnetic case with arbitrary contrasts of permeability and permittivity including perfectly electric conducting or perfectly magnetic conducting limit cases is studied the structure of the msr matrix of a single inclusion or a set of well separated ones is analyzed emphasis is on a far field situation in which one considers an electric dipole array operated in the transmit receive mode at a single frequency back propagated electric and magnetic fields associated to the singular vectors of the msr matrix for a single spherical inclusion or again for well separated ones are given in closed form and their leading order values are exhibited numerical illustrations cross sectional maps of back propagated fields computed from a singular value decomposition of the msr matrix are presented for one and two inclusions to illustrate this behavior as a function of the geometric and electromagnetic parameters of the configuration in a possibly noisy case

sequential rank tests with repeated measurements in clinical trials abstract for comparing responses in two groups of subjects observed repeatedly we propose a group sequential procedure based on linear rank statistics the asymptotic normality of the sequentially computed linear rank statistics is obtained and construction of the group sequential boundaries is based on this distribution theory by virtue of this asymptotic approximation the proposed procedure can be applied to interim analyses with either continuous or discrete repeated measurements even for staggered patient entry simulation results suggest the theory is approximately correct it can also be useful for testing the equality of two changes and rates of change as well as the equality of two means of the responses this procedure is illustrated with a real example

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

a representation of partially ordered preferences this essay considers decision theoretic foundations for robust bayesian statistics we modify the approach of ramsey de finetti savage and anscombe and aumann in giving axioms for a theory of robust preferences we establish that preferences which satisfy axioms for robust preferences can be represented by a set of expected utilities in the presence of two axioms relating to state independent utility robust preferences are represented by a set of probability utility pairs where the utilities are almost state independent in a sense which we make precise our goal is to focus on preference alone and to extract whatever probability and or utility information is contained in the preference relation when that is merely a partial order this is in contrast with the usual approach to bayesian robustness that begins with a class of priors or likelihoods and a single loss function in order to derive preferences from these probability utility assumptions

on the determination of subjective probability by choices the paper explores the uniqueness properties of the subjective probabilities in two axiomatizations of state dependent preferences karni schmeidler and vind s ksv 1983 system depends on selecting an arbitrary auxiliary probability and as such does not guarantee the uniqueness of the derived subjective probability however an axiom system initially designed by karni and schmeidler ks 1981 and further elaborated upon here does guarantee the desired uniqueness as well as a useful property of stability of the derived solution when the preference relation displays state independence even the ks probabilities may not agree with those derived from the classic anscombe aumann aa 1963 theorem however we claim that in this case the ks rather than the aa probabilities are the appropriate representation of the agent s beliefs

the power function of the studentised range test in this paper we investigate the power function of the studentised range test for comparing the means of normal populations in the one way fixed effects analysis of variance model the main results provide rigorous proofs of certain least favourable configurations of population means these results are important in the calculation of the sample sizes required to guarantee power levels under certain restrictions on the ranges of the population means

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

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

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

the analysis of multiple endpoints in clinical trials treatment comparisons in randomized clinical trials usually involve several endpoints such that conventional significance testing can seriously inflate the overall type i error rate one option is to select a single primary endpoint for formal statistical inference but this is not always feasible another approach is to apply bonferroni correction i e multiply each p value by the total number of endpoints its conservatism for correlated endpoints is examined for multivariate normal data a third approach is to derive an appropriate global test statistic and this paper explores one such test applicable to any set of asymptotically normal test statistics quantitative binary and survival endpoints are all considered within this general framework two examples are presented and the relative merits of the proposed strategies are discussed

incomplete preferences under uncertainty indecisiveness in beliefs versus tastes we investigate the classical anscombe aumann model of decision making under uncertainty without the completeness axiom we distinguish between the dual traits of indecisiveness in beliefs and indecisiveness in tastes the former is captured by the knightian uncertainty model the latter by the single prior expected multi utility model we characterize axiomatically the latter model then we show that under independence and continuity these two models can be jointly characterized by means of a partial completeness property

sequential comparison of changes with repeated measurements data abstract there are many clinical trials in which the patients enter sequentially and a response variable is measured repeatedly over time for each patient a group sequential procedure is proposed for comparing the rates of change between two treatment groups some existing procedures for testing the equality of means between two treatment groups with repeated measurements data such as those proposed by armitage stratton and worthington and by geary can be interpreted as special cases of the proposed procedure under a linear mixed effects model the asymptotic joint distribution of the sequentially computed statistics is derived construction of the group sequential boundaries is based on this distribution theory the extension to nonnormal distributions of errors is also discussed the proposed procedure allows for using either the sequential method of slud and wei or that of lan and demets this procedure is then applied to repeated bone density measurements on a sample of 74 middle aged women and

multiplicative models and cohort analysis abstract three methods of cohort analysis are presented for a statistical model wherein the explanatory or exposure variables act multiplicatively on age calendar year specific death rates the first method which assumes that the baseline rates are known from national vital statistics is a multiple regression analysis of the standardized mortality ratio the second method is a variant of cox s proportional hazards analysis in which the baseline rates are treated as unknown nuisance parameters the third method consists of case control sampling from the risk sets formed in the course of applying cox s model it requires substantially less computation than do the other two in illustrative analysis of respiratory cancer deaths among a cohort of smelter workers all three approaches yield roughly equivalent estimates of the relative risk associated with arsenic exposure the discussion centers on the tradeoff between efficiency and bias in the selection of a particular method of analysis and on practica

interim analyses based on median survival times summary for a typical two arm survival study analyses are usually performed periodically to evaluate for example the treatment difference in this paper we propose to use differences or ratios of estimated median survival times repeatedly during a study to evaluate the relative merit of two treatment groups this new procedure provides quantitative assessments of the treatment difference and is completely model free our method is illustrated with a recent aids clinical study which evaluates the efficacy of a reduced dose of azt

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

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

estimation in flexible two stage designs adaptive test designs for clinical trials allow for a wide range of data driven design adaptations using all information gathered until an interim analysis the basic principle is to use a test statistics which is invariant with respect to the design adaptations under the null hypothesis this allows for a control of the type i error rate for the primary hypothesis even for adaptations not specified a priori in the study protocol estimation is usually another important part of a clinical trial however is more difficult in adaptive designs in this research paper we give an overview of point and interval estimates for flexible designs and compare methods for typical sample size rules we also make some proposals for confidence intervals which have nominal coverage probability also after an unforeseen design adaptation and which contain the maximum likelihood estimate and the usual unadjusted confidence interval

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

group sequential comparison of changes ad hoc versus more exact method there are many clinical trials in which we are interested in comparing changes in responses between two treatment groups sequentially investigators might assume a simple linear model take the average of the ordinary least square estimators of slope within each treatment group and use the standardized difference between these two averages as the test statistic at each interim analysis ad hoc construction of the boundary values for interim analysis might be based on group sequential methods which assume that the interim test statistics have independent increments even though it may not be true when a response variable is measured repeatedly over time for each subject this ad hoc method is very simple and appealing and thus has been used in a clinical trial setting lee and demets 1991 journal of the american statistical association 86 757 762 have proposed a more exact group sequential method for comparing rates of change under the assumption that the response follows the linear mixed effects model they have derived the asymptotic joint distribution of the sequentially computed statistics construction of group sequential boundaries is based on this distribution theory by simulation studies we first study the robustness of the more exact method to violations of the typical assumptions in addition we compare the ad hoc method with the more exact method and examine how well these two methods work for various situations the relationship between two different information times fisher information and a surrogate information are also discussed from the simulation studies

objective and subjective rationality in a multiple prior model a decision maker dm is characterized by two binary relations the first reflects choices that are rational in an objective sense the dm can convince others that she is right in making them the second relation models choices that are rational in a subjective sense the dm cannot be convinced that she is wrong in making them in the context of decision under uncertainty we propose axioms that the two notions of rationality might satisfy these axioms allow a joint representation by a single set of prior probabilities and a single utility index it is objectively rational to choose f in the presence of g if and only if the expected utility of f is at least as high as that of g given each and every prior in the set it is subjectively rational to choose f rather than g if and only if the minimal expected utility of f with respect to all priors in the set is at least as high as that of g in other words the objective and subjective rationality relations admit respectively a representation a la bewley 2002 and a la gilboa and schmeidler 1989 our results thus provide a bridge between these two classic models as well as a novel foundation for the latter

on adjusting p values for multiplicity response 1 the development of proc multtest while controversial the use of multiplicity adjustments has gained acceptance recently in varied fields of scientific endeavor and their associated publications multiplicity adjustments have been considered important in pharmaceutical safety determinations involving multiple endpoints such as in adverse events analysis of clinical trials and in animal carcinogenicity studies in situations where the compound tested is completely safe it is likely to observe false positive indications of one or more untoward outcomes when unadjusted testing methods are used multiplicity adjustment is also important in epidemiology and other complicated areas of data analysis as it offers protection against conclusions that are driven by excessive data mining multiplicity concerns in toxicology and clinical trials prompted the development of the sas procedure proc multtest which calculates adjusted p values for a user supplied family of tests in a wide variety of applications biometrics readers will be interested to know that this software is readily available to calculate many but not all of the multiplicity adjusted p values described by wright 1992 in addition the software incorporates many improvements and enhancements such as the ability to incorporate correlations and nonnormal distributions the development of proc multtest started in may 1987 one of us westfall presented a resampling approach to calculating adjusted p values for multiple tests in multivariate binomial models with special application to the animal carcinogenicity problem the talk was entitled multivariate binomial testing and was given as the keynote address for the midwestern biopharmaceutical statistics workshop mbsw held in muncie indiana similar approaches to the same problem were developed concurrently and independently all of which have been published farrar and crump 1988 heyse and rom 1988 westfall and young 1989 an interesting biometrics connection should be mentioned here dr young thought that westfall s 1985 publication concerning simultaneous inference with multivariate binary data might be applied to the multiplicity problem in animal carcinogenicity studies dr young therefore invited dr westfall to speak at the mbsw conference on this application the initial precursor to proc multtest called proc mbin was developed in 1988 we wrote the specifications for the software and the coding was performed by youling lin at texas tech university under the auspices of the pharmaceutical manufacturing association a consortium of drug companies offered financial and intellectual support for the project specific individuals and companies who contributed are listed below in the acknowledgements section the development proceeded through a series of meetings at national and regional statistics conferences attendees included representatives from the funding organizations and from the united states food and drug administration from the outset it was decided that the primary form of output from the software would be the adjusted p value for the reasons wright describes in his opening paragraph the procedure proc mbin was described by westfall lin and young 1989 and was donated to sas institute inc the software performed multiplicity adjustments for multiple tests z score or exact permutation tests in multivariate possibly stratified multiple group binary outcome situations the main feature of the output of this software was the use of adjusted p values to summarize many statistical tests single step resampling bootstrap or permutation bonferroni and sidak methods were used to calculate the adjustments the tabular form of the output was much like tables 2 and 3 of wright showing unadjusted p values which we called raw p values side by side with various types of adjusted p values

exact confidence intervals following a group sequential trial a comparison of methods summary with interest in designing group sequential clinical trials increasing methodology for analysing the data arising in such trials is needed we discuss several approaches to constructing confidence intervals for the parameter of interest in a group sequential trial after the trial has ended we compare these met1fods to one already in the literature and point out interesting differences

repeated confidence intervals for the median survival time summary we describe methods for the construction of a confidence interval for median survival time based on right censored data these methods are extended to the construction of repeated confidence intervals for the median based on accumulating data here the overall probability that all intervals contain the true median is guaranteed at a fixed level the use of repeated confidence intervals for median survival time in post marketing surveillance is discussed a confidence interval for median survival time provides a useful summary of the survival experience of a group of patients if confidence intervals are calculated repeatedly as data accumulates the probability that at least one interval fails to contain the median may be much higher than the error rate for a single interval and if these confidence intervals are used in a decision making process the probability of an incorrect decision increases accordingly jennison turnbull 1984 have proposed methods for calculating repeated confidence intervals appropriate to such situations similar ideas have also been discussed by lai 1984 in 2 we propose a new form of single sample nonparametric confidence interval this interval has asymptotically correct coverage probability and monte carlo simulations suggest it is superior to its competitors for small sample sizes repeated confidence intervals for the median are presented in 3 and their stnall sample size performance is assessed by monte carlo simulation an example of their use is given in 4 all the methods considered can easily be modified to give confidence intervals for other quantiles or for the survival probability at a fixed time

estimators of the mantel haenszel variance consistent in both sparse data and large strata limiting models summary this paper proposes a new estimator of the variance of the mantel haenszel estimator of common odds ratio that is easily computed and consistent in both sparse data and large strata limiting models monte carlo experiments compare its performance to that of previously proposed variance estimators

a confidence interval for the median survival time a nonparametric asymptotic confidence interval for the median survival time is developed for the case where data are subject to arbitrary right censoring this is accomplished by inverting a generalization of the sign test for censored data a simulation study shows that this nonparametric confidence interval performs well for a variety of underlying survival functions the procedure is applied to data from a clinical trial that compared f our dosage regimens of 5 uorouracil

exact calculations for sequential t x2 and f tests summary sequential and group sequential procedures are proposed for monitoring repeated t x2or f statistics these can be used to construct sequential hypothesis tests or repeated confidence intervals when the parameter of interest is a normal mean with unknown variance or a multivariate normal mean with variance matrix known or known up to a scale factor exact methods for calculating error probabilities and sample size distributions are described and tables of critical values needed to implement the procedures are provided

sequential analysis of matched dichotomous data from prospective case control studies sequential analysis of randomized controlled clinical trials and epidemiological prospective matched case control studies can have ethical or economical advantages above a fixed sample size approach it offers the possibility to stop early when enough evidence for an apparent effect of the risk factor or lack of the expected effect is achieved in clinical trials it is well accepted to stop the trial early in favour of the alternative hypothesis in epidemiological studies in general the need is not felt to stop early in case of a clear exposure effect little attention has been paid however to early stopping and accepting the null hypothesis in metabolic epidemiological studies where analysis destroys the biological material the question of efficient use of samples for example those stored in a biobank becomes crucial also a slow accrual of cases or the costs of follow up of a cohort nested study can make it desirable to stop a study early once it becomes clear that no relevant exposure effect will be found matching can further reduce the amount of information necessary to reach a conclusion we derived test statistics z efficient score and v fisher s information for the sequential analysis of studies with dichotomous data where each case can be matched to one or more controls a variable matching ratio is allowed these test statistics can be entered into the software pest to monitor the course of the study the double sequential probability ratio test and the double triangular test were evaluated with simulated data for odds ratios equal to 1 5 2 0 and 2 5 and various type i and type ii error probabilities both under h 0 and under h 1 our simulations resulted in average and median values for the amount of information v that are far less than those for a fixed sample size study efficiency gain ranges from 32 per cent to 60 per cent the proposed sequential analysis was applied in an investigation on the possible relationship between the polymorphism of the mthfr gene and rectal cancer in a cohort of women with cases matched by age to one and to three controls a sequential analysis of matched data can lead to early stopping in favour of h 0 or h 1 thus conserving valuable resources for future testing a sequentially designed study can be more economical and less arbitrary than a study that makes use of conditional power or conditional coverage probability calculations to decide early stopping

multistatic response matrix of a 3 d inclusion in half space and music imaging recent work on the retrieval of 3 d bounded dielectric and or magnetic inclusions in free space is extended to burial in a half space though emphasis is on the case of a single inclusion enabling the derivation of closed form mathematical results in illuminating fashion the approach extends to the case of an unknown number of well separated inclusions within an asymptotic field formulation derived from exact contrast source vector integral formulations satisfied by the time harmonic fields and using proper reciprocity relationships of the dyadic green s functions the multistatic response matrix of the inclusion is constructed from the leading order term of the fields its singular value structure is analyzed in detail for a dielectric or a magnetic contrast or both contrasts this is performed in the case of a unique electric dipole array operated in the transmit receive mode at a single frequency a multiple signal classification type algorithm follows from the decomposition yielding a cost functional the magnitude of which peaks at the inclusion center numerical results illustrate the above as a function of the geometric and electromagnetic parameters of the configuration imaging of a spherical inclusion is in particular investigated from severely noisy synthetic data as well as of two inclusions embedded within a non conductive or conductive half space

efficient group sequential tests with unpredictable group sizes summary we derive group sequential tests to minimize expected sample size a parametric family of group sequential tests is proposed containing tests which are nearly optimal in terms of minimizing expected sample size amongst all group sequential tests these parametric tests can be implemented when group sizes are unequal and unpredictable they are shown to achieve error rates close to their nominal level in a wide range of situations and to be efficient when compared with other tests based on the same group sizes and achieving the same error rates we consider group sequential procedures for choosing between two hypotheses concerning the mean of a normal distribution with known variance the group sequential approach pocock 1977 offers a practicable alternative to fully sequential tests which require accumulating data to be examined as each new observation is recorded it is also desirable to ensure the maximum sample size needed in a long term experiment is not much greater than that of the corresponding fixed sample size test and we shall take the specification of the maximum allowable sample size and the number of groups of observations as our starting point in deriving sequential procedures in 2 we formulate the testing problem and discuss its relevance to clinical trials in 3 we describe the derivation of boundaries which are optimal in the sense of minimizing various expected sample sizes subject to a fixed number of equally sized groups and a fixed maximum sample size in 4 we introduce a parametric family of group sequential tests containing tests which are nearly optimal by the criteria of 3 these tests are easily adapted to allow unequal and unpredictable group sizes we show in 5 that they remain efficient when actual group sizes are quite different from those anticipated programs used for the calculations are available from the author

decision analysis with incomplete utility and probability information an approach to the solution of decision analysis problems under uncertainty with imprecise and incomplete information is presented the methodology is designed for cases in which payoffs conditional on the state of nature are known precisely but only limited or imprecise probability and utility information is available regarding a decision maker s beliefs and tastes a decision maker provides conditional payoffs optionally bounds on state probabilities bounds on the certainty equivalent for a simple lottery any known relationships between probabilities of states of nature and a series of strict preferences between pairs of vectors of conditional payoffs we assume an exponential utility function with unknown parameter the method proceeds by sequentially eliciting preferences new bounds on probabilities and or the certainty equivalent and new relationships among probabilities until the problem is solved the methodology is demonstrated through a two state and a three state example which illust

confidence intervals for a normal mean following a group sequential test this paper presents methodology based on the likelihood ratio test developed for construction of confidence intervals for a normal mean following a group sequential test examples show that the confidence intervals produced by this method have accurate nominal probability of coverage and have generally shorter average length than that produced by tsiatis rosner and mehta 1984 biometrics 40 797 803

exact confidence intervals following a group sequential test a numerical method is used to compute confidence intervals which have exact coverage probabilities for the mean of a normal distribution following a group sequential test this method which uses an ordering of the sample space similar to that employed by siegmund 1978 biometrika 65 341 349 is contrasted with the usual confidence interval for the mean

stopping rules for clinical trials incorporating clinical opinion a method is described of eliciting a range of equivalence i e a range of differences between two treatments over which a group of clinical trial participants would have no clear preference for either treatment this range of equivalence is then incorporated into a formal stopping rule for the trial using an extension of the group sequential design tables for the implementation of the design are presented the method is discussed in the context of other sequential trial designs

interim analyses using repeated confidence bands summary clinical trials often include interim analyses that compare treatment groups with respect to the mean function of a response process sometimes it is unclear how the mean functions of the groups might differ and thus one cannot confidently prespecify a simple metric upon which a stopping rule or repeated confidence interval can be based this motivated us to extend the repeated confidence intervals approach for a finite dimensional parameter jennison turnbull 1989 to the use of repeated confidence bands for the mean function of a response process formal tests of hypotheses are easily constructed from the repeated confidence bands we also describe how inferences for the mean function can be adaptively restricted to different subsets of its domain at different interim analyses an example is given involving an aids clinical trial

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

separating probability elicitation from utilities abstract this article deals with the separation of probability elicitation from utilities we show that elicited probabilities can be related to utilities not just through the explicit or implicit payoffs related to the elicitation process but also through other stakes the expert may have in the events of interest we study three elicitation procedures lotteries scoring rules and promissory notes and show how the expert s utility function and stakes in the events can influence the resulting probabilities particularly extreme results are obtained in an example involving a market at equilibrium the applicability of a no stakes condition and some implications for probability elicitation are discussed let represent an expert s probability for an event a and let p denote the elicited probability from some elicitation procedure we determine the value of p that maximizes the expert s expected utility when utility is linear in money p for all of the procedures studied here under nonlinear utility

decision making with imprecise probabilities and utilities by means of statistical preference and stochastic dominance abstract a problem of decision making under uncertainty in which the choice must be made between two sets of alternatives instead of two single ones is considered a number of choice rules are proposed and their main properties are investigated focusing particularly on the generalizations of stochastic dominance and statistical preference the particular cases where imprecision is present in the utilities or in the beliefs associated to two alternatives are considered

adaptive sample size calculations in group sequential trials a method for group sequential trials that is based on the inverse normal method for combining the results of the separate stages is proposed without exaggerating the type i error rate this method enables data driven sample size reassessments during the course of the study it uses the stopping boundaries of the classical group sequential tests furthermore exact test procedures may be derived for a wide range of applications the procedure is compared with the classical designs in terms of power and expected sample size

the shape of incomplete preferences incomplete preferences provide the epistemic foundation for models of imprecise subjective probabilities and utilities that are used in robust bayesian analysis and in theories of bounded rationality this paper presents a simple axiomatization of incomplete preferences and characterizes the shape of their representing sets of probabilities and utilities deletion of the completeness assumption from the axiom system of anscombe and aumann yields preferences represented by a convex set of state dependent expected utilities of which at least one must be a probability utility pair a strengthening of the state independence axiom is needed to obtain a representation purely in terms of a set of probability utility pairs

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

a bayesian approach to bioequivalence for the 2 x 2 changeover design bioequivalence trials are carried out to compare two or more formulations of a drug containing the same active ingredient in order to determine whether the different formulation give rise to comparable blood levels we consider the 2 x 2 changeover experiment the compares two formulations one of which is considered the standard for a single univariate characteristic of the plasma concentration time curve a criterion for bioequivalence is proposed based on the posterior probability that the difference in formulation means is less than a specific percentage of the mean of the standard the sensitivity of this posterior probability to alternative priors is investigated differences in carry over effects can be incorporated within the bayesian framework without restoring to the all or nothing approach implied by a preliminary test the use of sequential experimentation is discussed

decomposition of the time reversal operator detection and selective focusing on two scatterers the decomposition of the time reversal operator dort method is a selective detection and focusing technique using an array of transmit receive transducers it relies on the theory of iterative time reversal mirrors which was presented by prada et al c prada j l thomas and m fink j acoust soc am 97 62 71 1995 the time reversal operator was defined as k k where is the frequency means complex conjugate and k is the transfer matrix of the array of l transducers insonifying a time invariant scattering medium it was shown that this time reversal operator can be diagonalized and that for ideally resolved scatterers of different reflectivities each of its eigenvectors of nonzero eigenvalue provides the phase law to be applied to the transducers in order to focus on one of the scatterers the dort method consists in determining these eigenvectors and using them for the selective focusing this paper presents a complete analysis of this method in the case of two scatterers the

group sequential distribution free methods for the analysis of multivariate observations many studies involve the collection of multivariate observations such as repeated measures on two groups of subjects who are recruited over time i e with staggered entry of subjects various marginal distribution free multivariate methods have been proposed for the analyses of such multivariate observations where some measures may be missing at random using the multivariate u statistic of wei and johnson 1985 biometrika 72 359 364 we describe the group sequential analysis of such a study where the multivariate observations are observed sequentially both within and among subjects we describe a multivariate generalization of the hodges and lehmann 1963 annals of mathematical statistics 34 598 611 estimator of a location shift that can be obtained via the multivariate u statistic with the mann whitney wilcoxon kernel we then describe large sample group sequential interval estimators and tests based on an aggregate estimate of the location shift combined over all of the repeated measures we also describe how the same steps could be employed to perform a group sequential analysis based on any one of the variety of marginal multivariate methods that have been proposed these methods are applied to a real life example

indeterminate probabilities on finite sets this paper presents a quasi bayesian model of subjective uncertainty in which beliefs which are represented by lower and upper probabilities qualified by numerical confidence weights the representation is derived from a system of axioms of binary preferences which differs from standard axiom systems insofar as completeness is not assumed and transitivity is weakened confidence weighted probabilities may be elicited through the acceptance of bets with limited stakes a generalization of the operational method of de finetti the model is applicable to the reconciliation of inconsistent probability judgments and to the sensitivity analysis of bayesian decision models

state dependent utilities several axiom systems for preference among acts lead to the existence of a unique probability and a state independent utility such that acts are ranked according to their expected utilities these axioms have been used as a foundation for bayesian decision theory and the subjective probability calculus in this paper we note that the uniqueness of the probability is relative to the choice of what counts as a constant outcome although it is sometimes clear what should be considered constant there are many cases in which there are several possible choices each choice can lead to a different unique probability and utility by focusing attention on state dependent utilities we determine conditions under which a truly unique probability and utility can be determined from an agent s expressed preferences among acts suppose that an agent s preference can be represented in terms of a probability p and a utility u that is the agent prefers one act to another if and only if the expected utility of the one act is higher than that of the other there are many other equivalent representations in terms of probabilities q which are mutually absolutely continuous with p and state dependent utilities v which differ from u by possibly different positive affine transformations in each state of nature an example is described in which two different but equivalent state independent utility representations exist for the same preference structure what differs between the two representations is which acts count as constants the acts involve receiving different amounts of one or the other of two currencies and the states are different exchange rates between the currencies it is easy to see how it would not be possible for constant amounts of both currencies to simultaneously have constant values across the different states savage foundations of statistics john wiley new york 1954 sec 5 5 discovered a situation in which two seemingly equivalent preference structures are represented by different pairs of probability and utility savage attributed the phenomenon to the construction of a small world we show that the small world problem is just another example of two different but equivalent representations treating different acts as constants finally we prove a theorem similar to one of karni decision making under uncertainty harvard university press cambridge 1985 that shows how to elicit a unique state dependent utility and does not assume that there are prizes with constant value to do this we define a new hypothetical kind of act in which both the prize to be awarded and the state of nature are determined by an auxiliary experiment

a comparative study of conditional maximum likelihood estimation of a common odds ratio the finite sample properties of various point estimators of a common odds ratio from multiple 2 x 2 tables have been considered in a number of simulation studies however the conditional maximum likelihood estimator has received only limited attention that omission is partially rectified here for cases of relatively small numbers of tables and moderate to large within table sample sizes the conditional maximum likelihood estimator is found to be superior to the unconditional maximum likelihood estimator and equal or superior to the mantel haenszel estimator in both bias and precision

coalitional expected multi utility theory this paper begins by observing that any reflexive binary preference relation over risky prospects that satisfies the independence axiom admits a form of expected utility representation we refer to this representation notion as the coalitional minmax expected utility representation by adding the remaining properties of the expected utility theorem namely continuity completeness and transitivity one by one we find how this representation gets sharper and sharper thereby deducing the versions of this classical theorem in which any combination of these properties is dropped from its statement this approach also allows us to weaken transitivity in this theorem rather than eliminate it entirely say to quasitransitivity or acyclicity apart from providing a unified dissection of the expected utility theorem these results are relevant for the growing literature on boundedly rational choice in which revealed preference relations often lack the properties of completeness and or transitivity but often satisfy the independence axiom they are also especially suitable for the yet overlooked case in which the decision maker is made up of distinct individuals and consequently transitivity is routinely violated finally and perhaps more importantly we show that our representation theorems allow us to answer many economic questions that are posed in terms of nontransitive incomplete preferences say about the maximization of preferences the existence of nash equilibrium the preference for portfolio diversification and the possibility of the preference reversal phenomenon

a group sequential procedure for all pairwise comparisons of k treatments based on the range statistic in this paper a group sequential procedure for all private comparisons of the means of k independent normal populations with a common known variance is proposed a repeated range test is defined and its critical points are tabulated the power function is studied and minimum group size needed to achieve a desirable power level is provided the test can also be converted into a set of repeated simultaneous confidence intervals from which all pairwise comparisons can be made an example of the application of this new interference method is given

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

exact two sample permutation tests based on the randomized play the winner rule in comparing two treatments in a clinical trial the randomized play the winner rule tends to assign more study subjects to the better treatment it is applicable when patients have delayed responses to treatments it is not deterministic and is less vulnerable to experimental bias than other adaptive designs in this paper an efficient algorithm is provided to construct exact permutation tests for testing the equality of the two treatments with the randomized play the winner rule the test procedure is illustrated with a real life example the example shows that the design used in the trial should not be ignored in the analysis