Stage-Wise Adaptive Designs

An expert introduction to stage-wise adaptive designs in all areas of statistics

Stage-Wise Adaptive Designs presents the theory and methodology of stage-wise adaptive design across various areas of study within the field of statistics, from sampling surveys and time series analysis to generalized linear models and decision theory. Providing the necessary background material along with illustrative S-PLUS functions, this book serves as a valuable introduction to the problems of adaptive designs.

The author begins with a cohesive introduction to the subject and goes on to concentrate on generalized linear models, followed by stage-wise sampling procedures in sampling surveys. Adaptive forecasting in the area of time series analysis is presented in detail, and two chapters are devoted to applications in clinical trials. Bandits problems are also given a thorough treatment along with sequential detection of change-points, sequential applications in industrial statistics, and software reliability.

S-Plus functions are available to accompany particular computations, and all examples can be worked out using R, which is available on the book's related FTP site. In addition, a detailed appendix outlines the use of these software functions, while an extensive bibliography directs readers to further research on the subject matter.

Assuming only a basic background in statistical topics, Stage-Wise Adaptive Designs is an excellent supplement to statistics courses at the upper-undergraduate and graduate levels. It also serves as a valuable reference for researchers and practitioners in the fields of statistics and biostatistics.

Customarily, one thinks of adaptive statistics only in the context of clinical trials. However, clinical trials are just one field of statistics in which adaptive designs are applied. Actually, adaptive methods can be found in all fields of statistics, like sampling surveys, time series analysis, generalized linear models, decision theory, design of experiments, biostatistics, among others. The objective of this book is to present the theory and methodology of stagewise adaptive designs in all applicable fields of statistics.

Preface. 1. Synopsis. 1.1 Multi-Stage and Sequential Estimation. 1.2 Adaptive Designs for Generalized Linear Models. 1.3 Adaptive Methods for Sampling from Finite Populations. 1.4 Adaptive Prediction and Forecasting in Time Series Analysis. 1.5 Adaptive Search of an MTD in Cancer Phase I Clinical Trials. 1.6 Adaptive and Sequential Procedures in Phase III of Clinical Trials. 1.7 Sequential Allocation of Resources: The One-Armed and Multi-Armed Bandits Applied to Clinical Trials. 1.8 Sequential Detection of Change-Points. 1.9 Sequential Methods in Industrial Testing, Reliability and Design of Experiments. 2. Multi-Stage and Sequential Estimation. 2.1 Steins Two-Stage Procedure. 2.2 Modifications to Attain Asymptotic Efficiency. 2.3 Two-Stage Sampling From Exponential Distributions. 2.3.1 Fixed-Width Confidence Interval For The Location Parameter of An Exponential Distribution. 2.3.2 Two-Stage Sampling For a Bounded Risk Point Estimation of The Exponential Parameter. 2.4 Sequential Fixed-Width Interval Estimation. 2.5 The Distributions of Stopping Variables of Sequential Sampling From Exponential Distributions. 2.5.1 General Theory. 2.5.2 The Characteristics of Rays Procedure. 2.5.3 The Risk of Some Sequential Point Estimators. 2.6 Sequential Fixed-Width Intervals for the Log-Odds In Bernoulli Trials. 2.6.1 The Problem. 2.6.2 The Distribution of N(ë). 2.6.3 Functionals of ??N(ë). 2.7 Bayesian Sequential Estimation. 2.7.1 General Theory. 2.7.2 Estimating the Scale Parameter of the Exponential Distribution. 3. Adaptive Designs For Generalized Linear Models. 3.1 Exponential Example. 3.2 Adaptive Designs For The Fisher Information. 3.3 Adaptive Bayesian Designs. 3.4 Adaptive Designs For Inverse Regression. 3.4.1 Non-Bayesian Adaptive Designs. 3.4.2 Bayesian Adaptive Designs, å Known. 3.5 Stochastic Approximation. 4. Adaptive Methods For Sampling From Finite Populations. 4.1 Basic Theory. 4.1.1 The Design Approach. 4.1.2 The Modeling Approach. 4.2 Two-Stage and Sequential Estimation of the Population Mean. 4.2.1 Design Approach: SRSWR. 4.2.2 Design Approach: SRSWOR. 4.2.3 Modeling Approach. 4.3 Adaptive Allocation of Stratified SRS. 4.3.1 Basic Theory. 4.3.2 A Two-Stage Procedure For a Fixed-Width Interval Estimation of Y N Under Stratified Sampling. 4.4 Adaptive Search For Special Units. 4.5 Adaptive Estimation Of The Size of A Finite Population. 4.6 Applications In Software Reliability. 4.6.1 Sequential Stopping For Time Domain Models. 4.6.2 Sequential Stopping For Data Domain Models. 4.7 Sampling Inspection Schemes. 4.7.1 Two-Stage Sampling For Attributes. 4.7.2 Sequential Sampling For Attributes. 4.8 Dynamic Bayesian Prediction. 5. Adaptive Prediction and Forecasting in Time Series Analysis. 5.1 Basic Tools of T.S. Analysis. 5.2 Linear Predictors for Covariance Stationary T.S. 5.2.1 Optimal Linear Predictors. 5.2.2 Minimal PMSE Predictors For AR(p) T.S. 5.2.3 Prediction With Unknown Covariance Structure. 5.2.4 ARIMA Forecasting. 5.3 Quadratic LSE Predictors For Non-Stationary T.S. 5.4 Moving Averages Predictors For Non-Stationary T.S. 5.4.1 Linear MAS Predictors. 5.5 Predictors For General Trends With Exponential Discounting. 5.5.1 Recursive Computations With Shifted Origin. 5.5.2 Linear Trend. 5.5.3 Linear Trend With Cyclical Components. 5.6 Smoothing Model For Seasonal Data: The Winter Multiplicative Model. 5.7 Dynamic Linear Models. 5.7.1 Recursive Computations for The Normal Random Walk DLM, Constant. Variances. 5.7.2 Incorporating External Information. 5.7.3 The General DLM With Applications. 5.7.4 DLM for ARMA(p, q) T.S. 5.8 Asymptotic Behavior of DLM. 5.9 Linear Control of DLM. 5.9.1 Deterministic Linear Control. 5.9.2 Stochastic Linear Control. 6. Adaptive Search of an MTD in Cancer Phase I Clinical Trials. 6.1 Up-And-Down Adaptive Designs. 6.2 Bayesian Adaptive Search: The Continuous Reassessment Method. 6.3 Efficient Dose Escalation With Overdose Control. 6.4 Patient Specific Dosing. 6.5 Toxicity Versus Efficacy. 7. Adaptive and Sequential Procedures In Clinical Trials, Phases II and III. 7.1 Randomization In Clinical Trials. 7.2 Adaptive Randomization Procedures. 7.2.1 Random Allocation Rule. 7.2.2 Truncated Binomial Design. 7.2.3 Efrons Biased Coin Design. 7.2.4 Weis Urn Design. 7.2.5 Response Adaptive Designs. 7.3 Fixed Width Sequential Estimation of the Success Probability in Bernoulli Trials. 7.4 Sequential Procedure For Estimating The Probability of Success in Bernoulli Trials With Prescribed Proportional Closeness. 7.5 Sequential Comparison of Success Probabilities. 7.6 Group Sequential Methods. 7.7 Dynamic Determination of Stage-Wise Samples Size. 7.7.1 A Truncated Three-Stage Procedure for Power (TTSP). 7.7.2 The Bartoff-Lai GLR Procedure. 8. Sequential Allocation of Resources. 8.1 Bernoulli Bandits. 8.2 Gittins Dynamic Allocation Indices. 8.3 Sequential Allocations In Clinical Trials. 8.4 Bernoulli Bandits With Change-Point. 8.4.1 Introduction. 8.4.2 Optimizing The Final Cycle. 8.4.3 The Surveillance Cycle. 8.4.4 Multiple Surveillance Cycles. 8.5 Sequential Designs for Estimating The Common Mean of Two Normal Distributions: One Variance Known. 9. Sequential Detection of Change-Points. 9.1 Bayesian Detection When The Distribution Before and After The Change Are Known. 9.1.1 The Problem. 9.1.2 The Bayesian Framework. 9.1.3 Application in System Reliability. 9.1.4 Optimal Stopping For Detecting A Change-Point In The Intensity of A Poisson Process. 9.2 Bayesian Detection When The Distributions Before and After The Change Are Unknown. 9.2.1 The Bayesian Framework. 9.2.2 Optimal Stopping Rules. 9.2.3 Detecting A Change In The Success Probabilities of Binomial Trials: An Example. 9.3 CUSUM Procedures For Sequential Detection. 9.3.1 The Structure of CUSUM Procedures. 9.3.2 Asymptotic Minimaxity of CUSUM Procedures. 9.3.3 Exact Distribution Of Stopping Variables In CUSUM Procedures. 9.4 Tracking Algorithms For Processes With Change-Points. 9.4.1 General Comments. 9.4.2 Tracking a Process With Change-Points. 9.4.3 Recursive Non-Linear Estimation With Moving Windows. 9.4.4 Specific Cases. 9.4.5 Case Studies. 9.5 Recursive Estimation With Change-Points. 9.5.1 The Reaction of Recursive Estimators to Change-Points. 9.5.2 Recursive Estimation With The Kalman Filter. 9.5.3 Detecting Change-Points in Recursive Estimation. 9.5.4 Adjustment of the Kalman-Filter For Change-Points. 9.5.5 A Special Case. 9.6 Additional Theoretical Contributions. 10. Sequential Methods In Industrial Testing. 10.1 Sequential Testing (SPRT). 10.1.1 The Wald Sequential Probability Ratio Test. 10.1.2 Exact Distributions of N in The Exponential Case. 10.2 The Characteristics of Sequential Procedures In Reliability Estimation and Testing. 10.2.1 Reliability Estimation. 10.2.2 Reliability Testing. 10.2.3 The Total Operating Time of Repairable System. 10.3 Some Comments On Sequential Design of Experiments. 10.4 Sequential Testing of Software Reliability. 10.4.1 The Complete Bayesian Model. 10.4.2 Empirical Bayes: Adaptive Approach When -and à are Unknown. Bibliography. Appendix: SPLUS/R Programs. Author Index. Subject Index.