efficient sequential designs with binary data abstract a class of sequential designs for estimating the percentiles of a quantal response curve is proposed its updating rule is based on an efficient summary of all of the data available via a parametric model the logit mle version of the proposed designs can be viewed as a natural analog of the robbins monro procedure in the case of binary data it is shown to be asymptotically consistent optimal and nonparametric via its connection with the latter procedure for certain choices of initial designs the proposed method performs very well in a simulation study for sample sizes up to 35 a nonparametric sequential design via the spearman karber estimator for estimating the median is also proposed

design and analysis of phase i clinical trials the phase i clinical trial is a study intended to estimate the so called maximum tolerable dose mtd of a new drug although there exists more or less a standard type of design for such trials its development has been largely ad hoc as usually implemented the trial design has no intrinsic property that provides a generally satisfactory basis for estimation of the mtd in this paper the standard design and several simple alternatives are compared with regard to the conservativeness of the design and with regard to point and interval estimation of an mtd 33rd percentile with small sample sizes using a markov chain representation we found several designs to be nearly as conservative as the standard design in terms of the proportion of patients entered at higher dose levels in monte carlo simulations two two stage designs are found to provide reduced bias in maximum likelihood estimation of the mtd in less than ideal dose response settings of the three methods considered for determining confidence intervals the delta method a method based on fieller s theorem and a likelihood ratio method none was able to provide both usefully narrow intervals and coverage probabilities close to nominal

a d optimality based sensitivity test sensitivity tests are often used to estimate the parameters associated with latent continuous variables that cannot be measured for example each explosive specimen has a threshold the specimen will detonate if and only if an applied shock exceeds this value since there is no way to determine the threshold of an individual specimens are tested at various levels to determine parameters of the population a new test described here produces efficient estimates of the parameters of the distribution even with limited prior knowledge this test efficiently characterizes the entire distribution and desired percentiles of any population

the effect of incomplete knowledge of parameter values on single and multiple stage designs for logistic regression we consider the design of single and multiple stage dose response trials in which the probability of response is a logistic function of the dose knowledge of the parameters of the logistic at the time of planning the trial is represented by a gaussian prior distribution methods are presented for determining a design that approximately optimizes a measure of the accuracy of estimation averaged over the prior distribution changes in design due to uncertainty of parameter values are described as well as the changes in sample size required to produce a specified precision in multiple stage trials the initial stage is planned as if it were the only stage for succeeding stages the initial gaussian prior distribution is updated using outcomes at the previous stages at each such stage the design optimizes a chosen criterion averaged over the updated prior distribution the effectiveness of this methodology is evaluated by comparing the operating characteristics of two stage designs with those of single stage designs

efficient robbins monro procedure for binary data summary the robbins monro procedure does not perform well in the estimation of extreme quantiles because the procedure is implemented using asymptotic results which are not suitable for binary data here we propose a modification of the robbins monro procedure and derive the optimal procedure for binary data under some reasonable approximations the improvement obtained by using the optimal procedure for the estimation of extreme quantiles is substantial

coherence principles in dose finding studies of the methods this paper shows examples in which coherence is violated and discusses how the coherence principles may be applied to calibrate a two stage design and to deal with situations with delayed toxicity

a algorithm for convex minimization for convex minimization we introduce an algorithm based on space decomposition the method uses a bundle subroutine to generate a sequence of approximate proximal points when a primal dual track leading to a solution and zero subgradient pair exists these points approximate the primal track points and give the algorithm s or corrector steps the subroutine also approximates dual track points that are gradients needed for the method s newton predictor steps with the inclusion of a simple line search the resulting algorithm is proved to be globally convergent the convergence is superlinear if the primal dual track points and the objective s hessian are approximated well enough

completeness of the eigenfunctions for the caudrey beals coifman system the direct and the inverse scattering problem for the first order linear systems of the type l x i d dx q x j x 0 j h q x gj which generalizes the zakharov shabat system and the systems studied by caudrey beals and coifman cbc is analyzed herein here j is a regular complex constant element of the cartan subalgebra h g of the simple lie algebra g and the potential q x vanishes fast enough for x taking values in the image gj of adj the cbc results are generalized and the fundamental analytic solution m x for any choice of the irreducible finite dimensional representation v of g is constructed four pairwise equivalent minimal sets of scattering data for l invariant with respect to the choice of the representation of g are extracted from the asymptotics of m x for x from m x the resolvent of l is constructed in the adjoint representation vad and the completeness relation is proven for the eigenfunctions of l in vad it is also proven that the discrete spectrum

bayesian model averaging continual reassessment method in phase i clinical trials the continual reassessment method crm is a popular dose finding design for phase i clinical trials this method requires that practitioners prespecify the toxicity probability at each dose such prespecification can be arbitrary and different specifications of toxicity probabilities may lead to very different design properties to overcome the arbitrariness and further enhance the robustness of the design we propose using multiple parallel crm models each with a different set of prespecified toxicity probabilities in the bayesian paradigm we assign a discrete probability mass to each crm model as the prior model probability the posterior probabilities of toxicity can be estimated by the bayesian model averaging bma approach dose escalation or deescalation is determined by comparing the target toxicity rate and the bma estimates of the dose toxicity probabilities we examine the properties of the bma crm approach through extensive simulation studies and also compare this new method and its vari

characterization of the likelihood continual reassessment method this paper deals with the design of the likelihood continual reassessment method which is an increasingly widely used model based method for dose finding studies it is common to implement the method in a two stage approach whereby the model based stage is activated after an initial sequence of patients has been treated while this two stage approach is practically appealing it lacks a theoretical framework and it is often unclear how the design components should be specified this paper develops a general framework based on the coherence principle from which we derive a design calibration process a real clinical trial example is used to demonstrate that the proposed process can be implemented in a timely and reproducible manner while offering competitive operating characteristics we explore the operating characteristics of different models within this framework and show the performance to be insensitive to the choice of dose toxicity model

the line transect method a nonparametric estimator based on shape restrictions summary a new estimator for population density based on line transect surveys is presented its sampling behaviour is investigated through simulation studies advantages of the proposed estimator include its simple intuitive base its stable behaviour over a wide variety of detection curves the ease with which it can be computed and the availability of reliable confidence limits the simulation results point to a general improvement in efficiency over existing estimators

interval estimation of the median lethal dose summary assuming a logistic dose response curve one can construct a confidence interval for the ld50 from the asymptotic likelihood ratio test reasons are given for preferring this likelihood ratio interval to the established interval calculated by applying fieller s theorem to the maximum likelihood estimates

integrable equations and recursion operators related to the affine lie algebras ar 1 we have derived a family of equations related to the untwisted affine lie algebras ar 1 using a coxeter r 1 reduction they represent the third member of the hierarchy of soliton equations related to the algebra applying additional 2 involutions we also obtain the real hamiltonian forms of these equations we also give some particular examples and impose additional reductions

primal dual gradient structured functions second order results links to epi derivatives and partly smooth functions we give second order expansions for quite general nonsmooth functions from the cal v cal u space decomposition point of view the results depend on primal dual gradient structure which we relate to general concepts of second order epi derivatives and partly smooth functions expressions for the associated second order objects are given in terms of cal u subspace hessians

sequential implementation of stepwise procedures for identifying the maximum tolerated dose this article considers the problem of finding the maximum tolerated dose mtd of a drug in human trials the mtd is defined as the maximum test dose with toxicity probability less than or equal to a target toxicity rate we adopt the multiple test framework with step down tests used in an escalation stage and step up tests used in a deescalation stage to allow sequential dose assignments for ethical purposes by formulating the estimation problem as a testing problem the proposed procedures formally control the error probability of selecting an unsafe dose in addition we can control the probability of correctly selecting the mtd under a parameter subspace where no toxicity probability lies in an interval bracketed by the target toxicity rate and an unacceptably high toxicity rate the so called indifference zone this frequentist property which is currently lacking in the conduct of dose finding trials in humans is appealing from a regulatory standpoint we give the general expressions of the se

estimation of extreme quantiles based on sensitivity tests a comparative study in reliability applications it is often of interest to estimate extreme quantiles of the distribution of critical stimulus levels based on a limited number of sensitivity tests to evaluate and compare the effectiveness of available sequential design sensitivity tests in this setting several such tests were compared through monte carlo simulation samples of size 20 35 and 50 were used to estimate the 99 and 999 quantiles of the probit model each test was evaluated using initial parameter estimates equal to the parameters of the model and then using inaccurate estimates the effect of assuming the probit model when the true model was logit generalized logistic or type i extreme value was also studied our results show that when the model is correctly specified tests designed to estimate the model parameters and subsequently to estimate quantiles as a function of the model parameters provided more accurate quantile estimates than tests designed to estimate a specified quantile the tests that es

the generating operator for the n n linear system abstract the class of exactly integrable non linear evolution equations nlee related to the general first order n n linear problem is studied the set of independent scattering data j is determined and trace identities are obtained next we use a special integro differential operator and obtain the expansions of the potential and its variation in the eigenfunctions of these enable us to consider the inverse scattering method as a generalized fourier transform and to write down in a compact form the integrals of motion we prove that there exist a hierarchy of hamiltonian structures generated by at the end we calculate the symplectic form in terms of the scattering data variations

sequential designs in bioassay in this paper we present a sequential scheme for allocating doses so that asymptotically we achieve full efficiency for estimating ld this scheme is based on an algorithm due to fedorov 1969 and wynn 1970 and studied in the context of design by white unpublished ph d thesis imperial college 1975 wu and wynn 1978 and wu 1978a 1978b bayesian optimal designs for the logistic regression problem treated here are discussed in chaloner and larntz 1989 an excellent exposition of the basic sequential algorithms as well as a general discussion of optimal design is found in silvey 1980 in principle the method we employ is applicable in a wide variety of problems in which design parameters cannot be optimally allocated without the knowledge of the underlying unknown parameters of the distribution the versatility of this scheme and its adaptivity are its primary virtues suppose that observations are made sequentially at dose levels xl x2 with ni observations at dose xi suppose that the probability that a response is observed at dose x takes the form p l x o j where p x is some known distribution function and d and 0 are unknown parameters let pi be the proportion of responses observed at dose xi then the likelihood function based on pi p2 n p and on xi x2 x z is with zi xi 6

the estimation of extreme quantiles in logit bioassay summary suppose that we administer a low dose of a drug and if no response is observed increase this dose by a fixed amount which is administered to a second experimental subject this procedure continues until at some dose level a response is finally observed this whole sequence may be repeated we investigate the features of an experimental design of this kind for estimating the dose at which the expected response is small this design is highly efficient when the number of responses e g deaths is considered the major portion of the cost of the study and various alternatives having some advantage over maximum likelihood estimation are suggested

on mathcalvu theory for functions with primal dual gradient structure we consider a general class of convex functions having what we call primal dual gradient structure it includes finitely determined max functions and maximum eigenvalue functions as well as other infinitely defined max functions for a function in this class we discuss a space decomposition that allows us to identify a subspace on which the function appears to be smooth moreover using the special structure of such a function we compute smooth trajectories along which certain second order expansions can be obtained we also give an explicit expression for the hessian of a related lagrangian

a superlinear space decomposition algorithm for constrained nonsmooth convex program a class of constrained nonsmooth convex optimization problems that is piecewise c 2 convex objectives with smooth convex inequality constraints are transformed into unconstrained nonsmooth convex programs with the help of exact penalty function the objective functions of these unconstrained programs are particular cases of functions with primal dual gradient structure which has connection with vu space decomposition then a vu space decomposition method for solving this unconstrained program is presented this method is proved to converge with local superlinear rate under certain assumptions an illustrative example is given to show how this method works

a random walk rule for phase i clinical trials we describe a family of random walk rules for the sequential allocation of dose levels to patients in a dose response study or phase i clinical trial patients are sequentially assigned the next higher same or next lower dose level according to some probability distribution which may be determined by ethical considerations as well as the patient s response it is shown that one can choose these probabilities in order to center dose level assignments unimodally around any target quantile of interest estimation of the quantile is discussed the maximum likelihood estimator and its variance are derived under a two parameter logistic distribution and the maximum likelihood estimator is compared with other nonparametric estimators random walk rules have clear advantages they are simple to implement and finite and asymptotic distribution theory is completely worked out for a specific random walk rule we compute finite and asymptotic properties and give examples of its use in planning studies having the finite distribution theory available and tractable obviates the need for elaborate simulation studies to analyze the properties of the design the small sample properties of our rule as determined by exact theory compare favorably to those of the continual reassessment method determined by simulation

an alternative approach to the optimal design of an ld50 bioassay in this paper we propose an alternative approach to the optimal design of an ld50 bioassay we adopt a bayesian approach to make use of prior information about the location and scale parameters of the tolerance distribution function to select the design parameters number of doses total number of animals centre of doses space between doses and we adopt a frequentist approach using the spearman karber statistic to estimate the ld50 we define the optimal design as the one that produces the minimum expected mean squared error e mse with respect to the joint prior distribution of the parameters of the tolerance distribution for the design parameters investigated we found i the shape of the e mse is relatively smooth and continuous the magnitude of which is influenced by the underlying tolerance distribution ii the amount of prior information about the location and scale parameters independently and jointly affect the optimal design and iii as the amount of prior information decreases one requires more doses and or animals finally we show the proposed method is robust for an incorrectly assumed tolerance distribution function