Shelemyahu Zacks

Cancer phase I clinical trials: efficient dose escalation with overdose control

We describe an adaptive dose escalation scheme for use in cancer phase I clinical trials. The method is fully adaptive, makes use of all the information available at the time of each dose assignment, and directly addresses the ethical need to control the probability of overdosing. It is designed to approach the maximum tolerated dose as fast as possible subject to the constraint that the predicted proportion of patients who receive an overdose does not exceed a specified value. We conducted simulations to compare the proposed method with four up-and-down designs, two stochastic approximation methods, and with a variant of the continual reassessment method. The results showed the proposed method effective as a means to control the frequency of overdosing. Relative to the continual reassessment method, our scheme overdosed a smaller proportion of patients, exhibited fewer toxicities and estimated the maximum tolerated dose with comparable accuracy. When compared to the non-parametric schemes, our method treated fewer patients at either subtherapeutic or severely toxic dose levels, treated more patients at optimal dose levels and estimated the maximum tolerated dose with smaller average bias and mean squared error. Hence, the proposed method is promising alternative to currently used cancer phase I clinical trial designs.

Prediction theory for finite populations

A large number of papers have appeared in the past 20 years on estimating and predicting characteristics of finite populations. This monograph is designed to present this modern theory in a systematic and consistent manner. The authors approach is that of superpopulation models in which values of the population elements are considered as random variables having joint distributions. Throughout, the emphasis is on the analysis of data rather than on the design of samples. Topics covered include: optimal predictors for various superpopulation models, Bayes, minimax, and maximum likelihood predictors, classical and Bayesian prediction internals, model robustness, and models with measurement errors. Each chapter contains numerous examples, and exercises which extend and illustrate the themes in the text. As a result, this book will be ideal for all those research workers seeking an up-to-date and well-referenced introduction to the subject.

Applications of Catastrophe Theory for Statistical Modeling in the Biosciences

Abstract Although catastrophe theory has been applied with mixed success to many problems in the biosciences, very few of these applications have used any form of statistical modeling. We present examples of the applications of statistical catastrophe theory in the analysis of experimental data. These include examples of hysteresis effects, bifurcation effects, and the full cusp catastrophe model. The methods of statistical catastrophe theory draw upon the theories of parameter estimation for multiparameter exponential families, nonlinear time-series analysis, and stochastic differential equations. We discuss the application of these methods to both canonical and noncanonical catastrophe models.

Optimal Bayesian-feasible dose escalation for cancer phase I trials

We present an adaptive dose escalation scheme for cancer phase I clinical trials which is based on a parametric quantal response model. The dose escalation is Bayesian-feasible, Bayesian-optimal and consistent. It is designed to approach the maximum tolerated dose as fast as possible subject to the constraint that the predicted probability of assigning doses higher than the maximum tolerated dose is equal to a specified value.

First-exit times for compound poisson processes for some types of positive and negative jumps

We consider the one-sided and the two-sided first-exit problem for a compound Poisson process with linear deterministic decrease between positive and negative jumps. This process (X(t)) t≥0 occurs as the workload process of a single-server queueing system with random workload removal, which we denote by M/G u /G d /1, where G u (G d ) stands for the distribution of the upward (downward) jumps; other applications are to cash management, dams, and several related fields. Under various conditions on G u and G d (assuming e.g. that one of them is hyperexponential, Erlang or Coxian), we derive the joint distribution of τ y =inf{t≥0|X(t)∉(0,y)}, y>0, and X(τ y ) as well as that of T=inf{t≥0|X(t)≤0} and X(T). We also determine the distribution of sup{X(t)|0≤t≤T}.

UPPER FIRST-EXIT TIMES OF COMPOUND POISSON PROCESSES REVISITED

For a compound Poisson process (CPP) with only positive jumps, an elegant formula connects the density of the hitting time for a lower straight line with that of the process itself at time t, h(x; t), considered as a function of time and position jointly. We prove an analogous (albeit more complicated) result for the first time the CPP crosses an upper straight line. We also consider the conditional density of the CPP at time t, given that the upper line has not been reached before t. Finally, it is shown how to compute certain moment integrals of h.

A Markovian growth-collapse model

We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate function r(x). We deal with the stationary distribution of such a GCP, (X t ) t≥0, and the distributions of the hitting times T a = inf{t ≥ 0 : X t = a}, a > 0. After presenting the general theory of these GCPs, several important special cases are studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[T a , σ] in this case (where σ is the time of the first jump), and apply it to determine the long-run average cost of running a certain Markov-modulated disaster-ridden system.

Contributions to the theory of first-exit times of some compound processes in queueing theory

We consider compound processes that are linear with constant slope between i.i.d. jumps at time points forming a renewal process. These processes are basic in queueing, dam and risk theory. For positive and for negative slope we derive the distribution of the first crossing time of a prespecified level. The related problem of busy periods of single‐server queueing systems is also studied.

Bayes procedures for detecting a shift in the probability of success in a series of Bernoulli trials

Abstract The determination of a stopping rule for the detection of the time of an increase in the success probability of a sequence of independent Bernoulli trials is discussed. Both success probabilities are assumed unknown. A Bayesian approach is applied; the distribution of the location of the shift in the success probability is assumed geometric and the success probabilities are assumed to have known joint prior distribution. The costs involved are penalties for late or early stoppings. The nature of the optimal dynamic programming solution is discussed and a procedure for obtaining a suboptimal stopping rule is determined. The results indicate that the detection procedure is quite effective.

The probability distribution and the expected value of a stopping variable associated with one-sided cusum procedures for non-negative integer valued random variables

The structure of a stopping variable N based on one-sided CUSUM procedures is analyzed. Stopping occurs when a Markovian sequence of maxima of partial sums {M } crosses a certain boundary. On the basis of a recursive relationship between the Mn+1 and Mn a recursive equation is derived for the determination of the defective distributions Kn(x) = P{M ≤ x, N ≤n} . This recursive equation yields a recursive algorithm for the determination of P {N > n} . The paper studies the case when the basic random variables are non-negative integers-valued. In these cases the values of P{N > n} and E{N} can be determined by solving proper systems of linear equations.