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

optimal designs for bivariate logistic regression summary in drug testing experiments the primary responses of interest are efficacy and toxicity these can be modeled as a bivariate quantal response using the gumbel model for bivariate logistic regression d optimal and q optimal experimental designs are developed for this model the q optimal design minimizes the average asymptotic prediction variance of p l 0 d the probability of efficacy without toxicity at dose d over a desired range of doses the robustness of these designs to parameter misspecification is discussed in addition d efficiencies of q optimal designs and q efficiencies of doptimal designs are presented an extension of the general equivalence theorem to the multivariate case is applied to these designs

predictive variable selection in generalized linear models here we extend predictive method for model selection of laud and ibrahim to the generalized linear model this prescription avoids the need to directly specify prior probabilities of models and prior densities for the parameters instead a prior prediction for the response induces the required priors we propose normal and conjugate priors for generalized linear models each using a single prior prediction for the mean response to induce suitable priors for each variable subset model in this way an informative prior is used to select a subset of variables in addition to producing a ranking of models by size of the predictive criterion the standard deviation of the criterion is used as a calibration number to produce a set of equally good models a straightforward markov chain monte carlo algorithm is used to accomplish the necessary computations we illustrate this method with real and simulated datasets and compare results with the bayes factors and the akaike information and bayes information model

semiparametric dose finding methods summary we describe a new class of dose finding methods to be used in early phase clinical trials under some added parametric conditions the class reduces to the family of continual reassessment method crm designs under some relaxation of the underlying structure the method is equivalent to the cumulative cohort design the modified toxicity probability interval method or bayesian optimal interval design classes of methods which are non parametric in nature whereas the crm class can be viewed as being strongly parametric the class proposed is characterized as being semiparametric since it corresponds to the crm with a nuisance parameter performance is good matching that of the crm class and improving on it in some cases the structure allows theoretical questions to be more easily investigated and to understand better how different classes of methods relate to one another

optimal sensor array configuration in remote image formation determination of optimal sensor configuration is an important issue in many remote imaging modalities such as tomographic and interferometric imaging in this paper a statistical optimality criterion is defined and a search is performed over the space of candidate sensor locations to determine the configuration that optimizes the criterion over all candidates to make the search process computationally feasible a modified version of a previously proposed suboptimal backward greedy algorithm is used a statistical framework is developed which allows for inclusion of several widely used image constraints computational complexity of the proposed algorithm is discussed and a fast implementation is described furthermore upper bounds on the sum of the squared error of the proposed algorithm are derived connections of the method to the deterministic backward greedy algorithm for the subset selection problem are presented and two application examples are described five compelling optimality criteria are considered and their performance is investigated through numerical experiments for a tomographic imaging scenario in all cases it is verified that the configuration designed by the proposed algorithm performs better than wisely chosen alternatives

berry esseen theorems for quadratic forms of gaussian stationary processes letx1 x2 be a gaussian stationary process with spectral densityf and let z j frac 1 sqrt t left sum limits l 1 t j 1 left x l x l j 1 ex l x l j 1 right right j 1 p the present paper gives a berry esseen type theorem for the joint distribution of z1 zp

cramer type large deviations for linear combinations of order statistics let t n 1 1 cnx be a linear combination of order statistics and put tn t e t vvar tn sufficient conditions on the cn and on the moments of the underlying distribution are established under which the ratio p t x 1 d x tends to 1 either uniformly in the range a c x c cavihh a 2 0 c 0 moderate deviation theorem or uniformly in the range a c x c o n a 2 0 cramer type large deviation theorem the proof relies on helmers approximation method and on the corresponding results for u statistics

sampling design for gaussian detection problems we propose an approach for the design of sampling schemes for gaussian hypothesis testing problems our approach for this design is based on the class of ali silvey see j royal stat soc series b vol 28 p 131 43 1996 distance measures closed forms for the bhattacharyya distance the i divergence the j divergence and the chernoff distance between the class conditional densities are obtained for the sampling design problem in the strong signal case a new member of the class of ali silvey distance measures that is suitable for the detection problem in the weak signal case is also derived sampling schemes are determined to maximize those four distance measures as well as the new distance measure for the strong signal case and the weak signal case respectively the detection performance of our sampling schemes is compared with those of various other sampling schemes by means of numerical examples comparisons show that the sampling design based on ali silvey distance measures result in superior performance

an extension of the continual reassessment method using decision theory the primary goal of a phase i trial is to find the maximally tolerated dose mtd of a treatment the mtd is usually defined in terms of a tolerable probability q of toxicity our objective is to find the highest dose with toxicity risk that does not exceed q a criterion that is often desired in designing phase i trials this criterion differs from that of finding the dose with toxicity risk closest to q that is used in methods such as the continual reassessment method we use the theory of decision processes to find optimal sequential designs that maximize the expected number of patients within the trial allocated to the highest dose with toxicity not exceeding q among the doses under consideration the proposed method is very general in the sense that criteria other than the one considered here can be optimized and that optimal dose assignment can be defined in terms of patients within or outside the trial it includes as an important special case the continual reassessment method numerical study indicates the strategy compares favourably with other phase i designs

a berry esseen bound for symmetric statistics the rate of convergence of the distribution function of a symmetric function of n independent and identically distributed random variables to its normal limit is investigated under appropriate moment conditions the rate is shown to be o left n frac 1 2 right this theorem generalizes many known results for special cases and two examples are given possible further extensions are indicated

incorporating toxicity considerations into the design of two stage phase ii clinical trials phase ii study designs are proposed that evaluate both clinical response and toxicity and that are similar in structure to simon s two stage designs sample sizes and decision criteria are chosen to minimize the maximum expected accrual given that the treatment is unacceptable either in terms of clinical response or toxicity this is achieved subject to control of error rates either uniformly over all possible correlation structures linking response and toxicity or alternatively under an assumption of independence between response and toxicity in the latter case bounds on the error rates show that effective control is still uniformly achieved even if the independence assumption is relaxed

selection of observations in signal reconstruction in some signal reconstruction problems the observation equations can be used as a priori information for selecting the best combination of observations before acquiring them in the present correspondence the authors define a selection criterion and propose efficient methods for optimizing the criterion with respect to the combination of observations the examples illustrate the value of optimized sampling using the proposed methods

small sample confidence sets for the mtd in a phase i clinical trial properties of procedures for interval estimation of the maximum tolerable dose mtd in a phase i clinical trial are examined using a two stage stochastic sampling scheme storer 1989 as a paradigm for development although the likelihood function for the data arising from such a scheme is identical to one arising under a conditional binomial sampling assumption intervals based on the exact distribution of the usual large sample statistics under this assumption do not offer improvement over unadjusted intervals however consideration of the distribution of these statistics based on the true stochastic sampling scheme can lead to the construction of intervals with correct coverage probabilities that do not depend on the true values of the model parameters membership in the confidence set can be evaluated by monte carlo simulation at a restricted maximum likelihood estimate of the nuisance slope parameter despite upward bias in its estimation from the up and down sampling scheme the lack of information in the small sample setting is reflected in a large proportion of confidence intervals that include infinite values for the mtd especially when the dose response curve is shallow intervals based on a likelihood ratio criterion perform best in this regard

efficient and adaptive nonparametric test for the two sample problem the notion of efficient test for a euclidean parameter in a semiparametric model was introduced by stein proc third berkeley symp math statist probab 1 1956 187 195 such tests are locally most powerful for a wide class of infinite dimensional nuisance parameters the first formal application of this notion to a suitably parametrized two sample problem was provided by hajek ann math statist 33 1962 1124 1147 however this and subsequent solutions appear to be not well suited for practical applications this article aims to show that an adaptive two sample test introduced recently by janic wroblewska and ledwina scand j statist 27 2000 281 297 is locally most powerful under a more realistic setting

cramer type large deviations for simple linear rank statistics a cramer type large deviation theorem for simple linear rank statistics is obtained range 0 x 0 n 1 mathord left vphantom 1 6 right kern nulldelimiterspace 6 the method of proof consists in approximating the simple linear rank statistic by a sum of independent uniformly bounded random variables and then applying a cramer type large deviation theorem on this sum

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

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

prototype image constraints for set theoretic image restoration constraints based on prototype images are developed and used in set theoretic image restoration a prototype can be obtained as a result of applying a predetermined operator to the observed image in this case the operator and the bound which limits the variation of the restored image from the prototype are the two defining quantities of a prototype constraint general guidelines for rigorously estimating the defining bound of a prototype constraint under certain simplifying conditions are discussed the authors provide two examples of prototype constraints where the prototypes are obtained by the wiener filtering operator and a local averaging operator the projection onto convex sets algorithm using the prototype constraints is applied to both monochrome and color images degraded by out of focus blur at different noise levels the results show significant improvement over the wiener restoration in reducing the restoration artifacts

markov chain modelling of bioassay toxicity procedures many bioassay procedures to test a new substance for toxicity like some phase i clinical trials can be modelled as first order markov chains with absorbing states we provide a general framework for such procedures we show how one may calculate under any dose response curve assumption final absorption probabilities and a number of total average costs such as time required until trial completion number of subjects treated number of subjects exhibiting unaceptable levels of toxicity and amount of medication administered one may use the above information associated with the trial for an a priori comparison between competing designs under consideration we provide an example of such a comparison

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

optimal designs for clinical trials with dichotomous responses we consider two stage designs for clinical trials that involve two treatments with dichotomous responses the first is the information gathering stage the treatment chosen as the better from first stage and prior data is used exclusively in the second stage determination of treatment allocation in the first stage results from weighing the anticipated gain in information with effective treatment the objective is to maximize the expected number of successes in the entire trial this is in contrast to randomized controlled trials with the restricted objective of obtaining information concerning treatment differences we allow the length of the first stage to be arbitrary and fixed in advance or optimized as a function of prior information and the patient horizon we can regard this patient horizon as either the number of patients in the trial or the number who have the condition under treatment we consider two forms of prior information both success probabilities known but the better of the two treatments is unknown and one success probability known whereas the other has an arbitrary distribution in many instances of the latter case the optimal first stage size is of the order of the square root of the patient horizon

bayesian dose finding in phase i ii clinical trials using toxicity and efficacy odds ratios a bayesian adaptive design is proposed for dose finding in phase i ii clinical trials to incorporate the bivariate outcomes toxicity and efficacy of a new treatment without specifying any parametric functional form for the drug dose response curve we jointly model the bivariate binary data to account for the correlation between toxicity and efficacy after observing all the responses of each cohort of patients the dosage for the next cohort is escalated deescalated or unchanged according to the proposed odds ratio criteria constructed from the posterior toxicity and efficacy probabilities a novel class of prior distributions is proposed through logit transformations which implicitly imposes a monotonic constraint on dose toxicity probabilities and correlates the probabilities of the bivariate outcomes we conduct simulation studies to evaluate the operating characteristics of the proposed method under various scenarios the new bayesian design based on the toxicity efficacy odds ratio trade offs exhibits good properties and treats most patients at the desirable dose levels the method is illustrated with a real trial design for a breast medical oncology study

optimal k space sampling in mrsi for images with a limited region of support magnetic resonance spectroscopic imaging requires a great deal of time to gather the data necessary to achieve satisfactory resolution when the image has a limited region of support ros it is possible to reconstruct the image from a subset of k space samples therefore the authors desire to choose the best possible combination of a small number of k space samples to guarantee the quality of the reconstructed image sequential forward selection sfs is appealing as an optimization method because the previously selected sample can be observed while the next sample is selected however when the number of selected k space samples is less than the number of unknowns at the beginning of the selection process the optimality criterion is undefined and the resulting sfs algorithm cannot be used here the authors present a modified form of the criterion that overcomes this problem and develop an sfs algorithm for the new criterion then the authors develop an efficient computational strategy for this algorithm as well as for the standard sfs algorithm the combined algorithm efficiently selects a reduced set of k space samples from which the ros can be reconstructed with minimal noise amplification

sample size determination for phase ii clinical trials based on bayesian decision theory this paper describes an application of bayesian decision theory to the determination of sample size for phase ii clinical studies the approach uses the method of backward induction to obtain group sequential designs that are optimal with respect to some specified gain function a gain function is proposed focussing on the financial costs of and potential profits from the drug development programme on the basis of this gain function the optimal procedure is also compared with an alternative bayesian procedure proposed by thall and simon the latter method which tightly controls type i error rate is shown to lead to an expected gain considerably smaller than that from the optimal test gain functions with respect to which thall and simon s boundary is optimal are sought and it is shown that these can only be of the form considered that is with constant cost for phase iii study and cost of the phase ii study proportional to the sample size if potential profit increases over time

determinant and exchange algorithms for observation subset selection in many applications involving image reconstruction signal observation time is limited this emphasizes the requirement for optimal observation selection algorithms a selection criterion using the trace of a matrix forms the basis of two existing algorithms the sequential backward selection and sequential forward selection algorithms neither is optimal although both generally perform well here we introduce a trace row exchange criterion to further improve the quality of the selected subset and introduce another observation selection criterion based upon the determinant of a matrix

a bayesian predictive two stage design for phase ii clinical trials in this paper we propose a bayesian two stage design for phase ii clinical trials which represents a predictive version of the single threshold design std recently introduced by tan and machin the std two stage sample sizes are determined specifying a minimum threshold for the posterior probability that the true response rate exceeds a pre specified target value and assuming that the observed response rate is slightly higher than the target unlike the std we do not refer to a fixed experimental outcome but take into account the uncertainty about future data in both stages the design aims to control the probability of getting a large posterior probability that the true response rate exceeds the target value such a probability is expressed in terms of prior predictive distributions of the data the performance of the design is based on the distinction between analysis and design priors recently introduced in the literature the properties of the method are studied when all the design parameters vary

distribution free tests of independence in high dimensions we consider the testing of mutual independence among all entries in a formula see text dimensional random vector based on formula see text independent observations we study two families of distribution free test statistics which include kendall s tau and spearman s rho as important examples we show that under the null hypothesis the test statistics of these two families converge weakly to gumbel distributions and we propose tests that control the type i error in the high dimensional setting where formula see text we further show that the two tests are rate optimal in terms of power against sparse alternatives and that they outperform competitors in simulations especially when formula see text is large

sequential algorithms for observation selection some signal reconstruction problems allow for flexibility in the selection of observations and hence the signal formation equation in such cases we have the opportunity to determine the best combination of observations before acquiring the data we present and analyze two classes of sequential algorithms to select observations sequential backward selection sbs and sequential forward selection sfs although both are suboptimal they perform consistently well we analyze the computational complexity of various forms of sbs and sfs and develop upper bounds on the sum of squared errors sse of the solutions obtained by sbs and sfs

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

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

designs for single or multiple agent phase i trials phase i trials of cytotoxic agents in oncology are usually dose finding studies that involve a single cytotoxic agent many statistical methods have been proposed for these trials all of which are based on the assumption of a monotonic dose toxicity curve for single agent trials this is a valid assumption in many trials however investigators are interested in finding the maximally tolerated dose based on escalating multiple cytotoxic agents when there are multiple agents monotonicity of the dose toxicity curve is not clearly defined in this article we present a design for phase i trials in which the toxicity probabilities follow a partial order meaning that there are pairs of treatments for which the ordering of the toxicity probabilities is not known at the start of the trial we compare the new design to existing methods for simple orders and investigate the properties of the design for two partial orders

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

consistency of continual reassessment method under model misspecification summary this paper studies large sample behaviour of the continual reassessment method used to determine the target dose levels in phase i clinical trials a sequential maximum likelihood procedure is investigated for local estimation of a given targeted toxicity level consistency and asymptotic normality for the maximum likelihood estimates are established under model misspecification and it is shown that in general the recommended dose level will converge to the target level there are however situations in which the recommended dose may converge to a level close to the target level but not necessarily the closest this contrasts with the conclusions of o quigley pepe fisher 1990 and we describe this situation as well as providing the conditions under which convergence is assured

a bayesian decision approach for sample size determination in phase ii trials stallard 1998 biometrics 54 279 294 recently used bayesian decision theory for sample size determination in phase ii trials his design maximizes the expected financial gains in the development of a new treatment however it results in a very high probability 0 65 of recommending an ineffective treatment for phase iii testing on the other hand the expected gain using his design is more than 10 times that of a design that tightly controls the false positive error thall and simon 1994 biometrics 50 337 349 stallard s design maximizes the expected gain per phase ii trial but it does not maximize the rate of gain or total gain for a fixed length of time because the rate of gain depends on the proportion of treatments forwarding to the phase iii study we suggest maximizing the rate of gain and the resulting optimal one stage design becomes twice as efficient as stallard s one stage design furthermore the new design has a probability of only 0 12 of passing an ineffective treatment to phase iii study

directed walk designs for dose response problems with competing failure modes summary we examine adaptive allocation designs for the problem of determining the optimal therapeutic dose for subjects in early phase clinical trials a su bject can fail due to lack of efficacy or due to a toxic reaction successful subjects will have both a posit ive response and no toxic side effects thus we seek to maximize the product of the non toxicity and effica cy dose response curves we are interested in sampling rules that perform well along several criteria including the ethical criterion that as often as possible experimental subjects be treated at or close to th e maximum in question statistically we wish to identify the optimum dose with high probability at the close of the experiment here we propose designs that combine new allocation policies directed walks with new smoothed shape constrained curve fitting techniques these are compared with a variety of other curve fitting techniques and with up and down and equal allocation rules

analysis of error in reconstruction of two dimensional signals from irregularly spaced samples we consider the problem of estimating a band limited two dimensional 2 d signal based on a finite set of irregularly spaced samples we derive the minimum mean squared error estimator of the form of a sum of weighted interpolating functions that are identical in shape but are centered at the irregularly spaced sample points and show that this estimator is identical to the minimum energy band limited interpolator that has been previously obtained by others the relation between the mean squared error and the set of irregularly spaced sampling points is studied using the minimax principle the maximum of the mean squared error over a class of signals is minimized by proper choice of the set of sampling points three criteria for evaluating the performance of the sampling point set are derived from a candidate group of sets of irregularly spaced sampling points these criteria may be used to select that point set which is optimal for recovery over a class of signals

heterogeneity in phase i clinical trials prior elicitation and computation using the continual reassessment method heterogeneity in a phase i clinical trial patient population may lead to distinctly different dose response relationships along covariate values for a given target probability of toxicity this implies different maximum tolerated doses mtds for each distinct subpopulation within the framework of o quigley pepe and fisher s 1990 continual reassessment method we propose the notion of average and patient specific mtds by augmenting the dose response model with other covariates to account for such differences a method to elicit prior distributions on the dose and other covariate parameters are proposed based on the predictive approach of ibrahim and laud 1994 laud and ibrahim 1995 and ibrahim ryan and chen 1998 this approach relies on prior predictions for the response vector y 0 and a quantity a 0 specifying uncertainty in y 0 then y 0 and a 0 are used to specify a prior for the regression coefficients in a semi automatic fashion the elicitation scheme for y 0 uses results from previous phase i cancer clinical trials the average and patient specific mtds and an elicitation method are demonstrated in logistic regression examples

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

on the equivalence of set theoretic and maxent map estimation we establish an equivalence between two conceptually different methods of signal estimation under modeling uncertainty viz set theoretic st estimation and maximum entropy maxent map estimation the first method assumes constraints on the signal to be estimated and the second assumes constraints on a probability distribution for the signal we provide broad conditions under which these two estimation paradigms produce the same signal estimate we also show how the maxent formalism can be used to provide solutions to three important problems how to select sizes of constraint sets in st estimation the analysis highlights the role of shrinkage how to choose the values of parameters in regularized restoration when using multiple regularization functionals how to trade off model complexity and goodness of fit in a model selection problem

on the recovery and resolution of exponential relaxation rates from experimental data ii the optimum choice of experimental sampling points for laplace transform inversion in a previous paper bertero boccacci pike proc r soc lond a 383 15 1982 we investigated the benefits of a priori knowledge of finite support of the solution in the inversion of the laplace transform and determined the number of exponential components that could be recovered from given data in the presence of quantified amounts of noise the support of the data was assumed to be unrestricted in the present paper these calculations are extended to cover the case of a necessarily finite number of experimental data points we consider two cases uniform distribution and geometric distribution of data points in both cases the condition number of singular value inversions is minimized with respect to the placing of the data points for fixed numbers of data samples the results allow optimum placing of the experimental data points as a function of the ratio of given upper and lower bounds of the support of the solution the number of exponentials recoverable is less than in the unrestricted case but for 8 we find that with 32 linearly spaced data points optimally placed this reduction is rather small geometrical sampling of experimental data reduces further the number of sample points required for inversion for 8 we show that with 5 geometrically spaced data points points optimally placed the ill conditioning in the restoration of 2 to 4 exponential components is smaller than with 32 linearly spaced data samples this has important implications for the design of a number of scientific experiments and instruments