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

embedding formulae in diffraction theory embedding formulae are remarkable as they allow one to decompose scattering problems apparently dependent upon several angular variables angles of incidence and observation into those dependent upon fewer angular variables in terms of facilitating rapid computations across considerable parameter regimes this is a considerable advantage our aim is to derive embedding formulae for scattering and diffraction problems in acoustics electromagnetism and elasticity here we construct a general approach to formulating and using embedding formulae we do this using complementary approaches overly singular states and a physical interpretation in terms of sources the crucial point we identify is the form of the auxiliary state used in the reciprocal theorem this is unphysically singular at the edge and is reminiscent of weight function methods used in fracture mechanics illustrative implementations of our approach are given using wiener hopf techniques for semi infinite model problems in both elasticity and acoustics we also demonstrate our approach using a numerical example from acoustics and we make connections with high frequency asymptotic methods

hidden markov models applied to on line handwritten isolated character recognition hidden markov models are used to model the generation of handwritten isolated characters models are trained on examples using the baum welch optimization routine then given the models for the alphabet unknown characters can be classified using maximum likelihood classification experiments have been conducted and an average error rate of 6 9 was achieved over the alphabet consisting of the lowercase english alphabet

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

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

the bias of the sample proportion following a group sequential phase ii clinical trial group sequential testing procedures have seen wide use in phase ii clinical trials the sample proportion p of responders is the commonly used estimator for the binomial response probability p it can be shown that p is the maximum likelihood estimator mle of p it is well known that mle can be in general whitehead and p in particular dupont biased estimators if ther computation follows a group sequential procedure in this paper we numerically investigate the bias of p we find that the magnitude of the bias of p is less than 0 025 in all cases we investigated we apply the idea in whitehead to propose a bias adjusted estimator that reduces the bias substantially and reduces the mean square error as well in a certain range of p we also evaluate the uniformly minimum variance unbiased umvu estimator if one does not mind a bias of 0 025 one may find the sample proportion a suitable estimator for p because of its simplicity and easy explanation if one is concerned with bias the bias adjusted estimator may be a good choice

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

embedding formulae for diffraction by rational wedge and angular geometries embedding is the process of taking the far field directivity from a diffraction problem or problems involving line sources or multipoles placed at a sharp edge and then constructing the far field for the same geometry for more general incidence using only this canonical problem s thus far embedding has been limited to planar parallel scattering surfaces for instance collections of parallel cracks or slits it appeared that there was a fundamental limitation to embedding disallowing its use for angular structures in this article we overcome this limitation and demonstrate the use of embedding upon wedge diffraction problems and upon a simple polygonal shape a limitation is that the final formulae are for rational wedge angles

tests and confidence intervals for secondary endpoints in sequential clinical trials in a sequential clinical trial whose stopping rule depends on the primary endpoint inference on secondary endpoints is an important long standing problem ignoring the possibility of early stopping based on the primary endpoint may result in substantial bias to address this problem a commonly used approach is to develop bias correction by estimating the bias in the case of bivariate normal outcomes and appealing to joint asymptotic normality of the statistics associated with the primary and secondary endpoints we propose herein a new approach that uses resampling and a novel ordering scheme in the sample space of sequential statistics observed up to a stopping time this approach is shown to provide accurate inference in complex clinical trials including time sequential trials with survival endpoints and covariates

application of hidden markov models for signature verification this paper describes a technique for on line signature verification using hidden markov models hmms signatures are captured and digitized in real time using a graphic tablet for each signature a hmm is constructed using a set of sample signatures described by the normalized directional angle function of the distance along the signature trajectory the baum welch algorithm is used for both training and classification experimental results based on 496 signatures from 31 subjects are presented which show that hmm technique is very potential for signature verification

on the estimation of the binomial probability in multistage clinical trials due to the optional sampling effect in a sequential design the maximum likelihood estimator mle following sequential tests is generally biased in a typical two stage design employed in a phase ii clinical trial in cancer drug screening a fixed number of patients are enrolled initially the trial may be terminated for lack of clinical efficacy of treatment if the observed number of treatment responses after the first stage is too small otherwise an additional fixed number of patients are enrolled to accumulate additional information on efficacy as well as on safety there have been numerous suggestions for design of such two stage studies here we establish that under the two stage design the sufficient statistic i e stopping stage and the number of treatment responses for the parameter of the binomial distribution is also complete then based on the rao blackwell theorem we derive the uniformly minimum variance unbiased estimator umvue as the conditional expectation of an unbiased estimator which in this case is simply the maximum likelihood estimator based only on the first stage data given the complete sufficient statistic our results generalize to a multistage design we will illustrate features of the umvue based on two stage phase ii clinical trial design examples and present results of numerical studies on the properties of the umvue in comparison to the usual mle

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

group sequential tests for delayed responses with discussion summary group sequential methods are used routinely to monitor clinical trials and to provide early stopping when there is evidence of a treatment effect a lack of an effect or concerns about patient safety in many studies the response of clinical interest is measured some time after the start of treatment and there are subjects at each interim analysis who have been treated but are yet to respond we formulate a new form of group sequential test which gives a proper treatment of these pipeline subjects these tests can be applied even when the continued accrual of data after the decision to stop the trial is unexpected we illustrate our methods through a series of examples we define error spending versions of these new designs which handle unpredictable group sizes and provide an information monitoring framework that can accommodate nuisance parameters such as an unknown response variance by studying optimal versions of our new designs we show how the benefits of lower expected sample size that are normally achieved by a group sequential test are reduced when there is a delay in response the loss of efficiency for larger delays can be ameliorated by incorporating data on a correlated short term end point fitting a joint model for the two end points but still making inferences on the original longer term end point we derive p values and confidence intervals on termination of our new tests

supplementary analysis of probabilities at the termination of a group sequential phase ii trial we consider estimation of various probabilities after termination of a group sequential phase ii trial a motivating example is that the stopping rule of a phase ii oncologic trial is determined solely based on response to a drug treatment and at the end of the trial estimating the rate of toxicity and response is desirable the conventional maximum likelihood estimator sample proportion of a probability is shown to be biased and two alternative estimators are proposed to correct for bias a bias reduced estimator obtained by using whitehead s bias adjusted approach and an unbiased estimator from the rao blackwell method of conditioning all three estimation procedures are shown to have certain invariance property in bias moreover estimators of a probability and their bias and precision can be evaluated through the observed response rate and the stage at which the trial stops thus avoiding extensive computation

unbiased estimation of secondary parameters following a sequential test we consider within a brownian motion framework estimation of secondary parameters following a sequential test whitehead 1986b proposed a method for reducing the bias of maximum likelihood estimators in this paper we present unbiased estimators to eliminate the bias

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

unbiased estimation following a group sequential test it is shown that in a group sequential test about the drift 0 of a brownian motion x t stopped at time t the sufficient statistic t x t is not complete for 0 there exist infinitely many unbiased estimators of 0 and none has uniformly minimum variance a truncation adaptable criterion is proposed and the uniformly minimum variance estimator among all truncation adaptable unbiased estimators is found this estimator is identical to estimators of ferebee 1983 and emerson fleming 1990

an optimal three stage design for phase ii clinical trials a phase ii clinical trial in cancer therapeutics is usually a single arm study to determine whether an experimental treatment e holds sufficient promise to warrant further testing when the criterion of treatment efficacy is a binary endpoint response no response with probability of response p we propose a three stage optimal design for testing h0 p p0 versus h1 p p1 where p1 and p0 are response rates such that e does or does not merit further testing at given levels of statistical significance and power 1 the proposed design is essentially a combination of earlier proposals by gehan and simon the design stops with rejection of h1 at stage 1 when there is an initial moderately long run of consecutive treatment failures otherwise there is continuation to stage 2 and possibly stage 3 which have decision rules analogous to those in stages 1 and 2 of simon s design thus rejection of h1 is possible at any stage but acceptance only at the final stage the design is optimal in the sense that expected sample size is minimized when p p0 subject to the practical constraint that the minimum stage 1 sample size is at least 5 the proposed design has greatest utility when the true response rate of e is small it is desirable to stop early if there is a moderately long run of early treatment failures and it is practical to implement a three stage design compared to simon s optimal two stage design the optimal three stage design has the following features stage 1 is the same size or smaller and has the possibility of stopping earlier when 0 successes are observed the expected sample size under the null hypothesis is smaller stages 1 and 2 generally have more patients than stage 1 of the two stage design but a higher probability of early termination under h0 and the total sample size and criteria for rejection of h1 at stage 3 are similar to the corresponding values at the end of stage 2 in the two stage optimal design

confidence intervals for secondary parameters following a sequential test in sequential studies formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study however the completed data set may warrant a more detailed analysis involving the estimation of parameters corresponding to effects that were not used to determine when to stop and yet correlated with those that were this paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used the method assumes that information accumulates on the primary and secondary parameters at proportional rates this requirement will be valid in many potential applications but only in limited situations in survival analysis

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

weinstein s diffraction problem embedding formula and spectral equation in parabolic approximation a short wave problem of reflection and radiation by an open end of a two dimensional planar waveguide is studied the incident mode is assumed to have frequency close to the cut off the problem is studied in the parabolic approximation a recently developed approach based on the embedding formula and the spectral equation for the directivity of an edge green s function is applied to the problem

training hidden markov models with multiple observations a combinatorial method hidden markov models hmm are stochastic models capable of statistical learning and classification they have been applied in speech recognition and handwriting recognition because of their great adaptability and versatility in handling sequential signals on the other hand as these models have a complex structure and also because the involved data sets usually contain uncertainty it is difficult to analyze the multiple observation training problem without certain assumptions for many years researchers have used the training equations of levinson 1983 in speech and handwriting applications simply assuming that all observations are independent of each other this paper presents a formal treatment of hmm multiple observation training without imposing the above assumption in this treatment the multiple observation probability is expressed as a combination of individual observation probabilities without losing generality this combinatorial method gives one more freedom in making different dependence independence assumptions by generalizing baum s auxiliary function into this framework and building up an associated objective function using the lagrange multiplier method it is proven that the derived training equations guarantee the maximization of the objective function furthermore we show that levinson s training equations can be easily derived as a special case in this treatment

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

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

diffraction of elastic waves by a penny shaped crack analytical and numerical results the diffraction of time harmonic stress waves by a penny shaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials martin proc r soc lond a 378 263 1981 has proved that the corresponding linear boundary value problem has precisely one solution and that this solution can be constructed by solving a two dimensional fredholm integral equation of the second kind however this integral equation has a complicated matrix kernel and the components of its vector solution are coupled the main purpose of the present paper is to show how martin s integral equation can be explicitly solved in terms of a sequence of functions each of which satisfies a very simple scalar integral equation of the second kind this simplification may be made for any incident wave for an incident plane wave further simplifications are possible we show that the solution at an arbitrary angle of incidence can be derived from the solution at a particular angle of incidence namely grazing incidence the resulting computational procedure is especially attractive if only the stressintensity factors or the far field displacements are required finally we present some numerical results for the scattering of a p wave at normal incidence and an sv wave at oblique incidence and compare these with those of other authors

scattering by cracks beneath fluid solid interfaces abstract the scattering of incident plane elastic or fluid body waves and interfacial waves by an arbitrarily orientated subsurface crack is considered the crack lies in an infinite elastic half space that is coupled to an overlying fluid half space material parameters relevant for water metal and water rock combinations are taken and far field scattering patterns are given these demonstrate beam formation along critical angles for light or moderate fluid coupling it is shown that the beams form along different critical angles depending upon the magnitude of the coupling in addition reciprocity relations relating the far field scattering coefficients viewed along an angle and generated by one type of plane wave incident along to the scattering coefficient viewed along generated by another plane wave incident along are found reciprocity relations involving interfacial waves are also given power flow theorems are derived these relate the time averaged scattered power to a combination of far field scattering coefficients this is used to determine the proportion of scattered power converted into the different types of scattered wave the reciprocity and power flow theorems provide a powerful consistency check upon the numerical accuracy of the results the boundary value problem is recast as a system of coupled integro differential equations for the unknown jump in displacement across the crack faces these integral equations are solved in an efficient and fast numerical manner by performing the integrations over the crack faces analytically thus reducing the computational effort substantially

continuous and group sequential conditional probability ratio tests for phase ii clinical trials we present continuous and group sequential designs for phase ii clinical trials based on the sequential conditional probability ratio test scprt the scprt is derived from a conditional likelihood ratio where the conditioning is on what the corresponding reference fixed sample size test rfsst would achieve in other words we obtain the sequential design by controlling the maximum probability that the scprt does not agree with the rfsst we shall discuss the difference between scprt and stochastic curtailment which also uses the concept of conditional distribution we show that the power function of the scprt is virtually the same as that of the rfsst and its average sample numbers asns are close to those of wald s sequential probability ratio test sprt whereas its maximum sample size is no greater than that of the rfsst thus the scprt has all the desirable properties such as allowing the use of the rfsst at the last analysis of the fleming procedure for phase ii trials the scprt however preserves the power function of the rfsst better and gives us the option for continuous monitoring our recommendation therefore is to use a group scprt boundary for interim analyses performed as scheduled embedded in a continuous scprt boundary for unplanned interim analyses and analyses at times based on data trends we provide as well a bias adjusted estimator of the success rate after sequential stopping we illustrate the method with several examples the method applies to any single arm clinical trial with binary endpoints such as the classic paired design

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

inference about a secondary process following a sequential trial summary we consider the following sequential testing problem a group sequential or fullysequential test is carried out for a primary parameter using a score process or an effective score process to eliminate nuisance parameters after stopping the possibility of additional parameters is considered and appropriate tests and estimators are desired that recognise the sequential stopping rule we formulate an asymptotic multi dimensional gaussian process form of such problems and then construct tests and confidence procedures optimality conditions are given and an example is summarised

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