Anthony C. Atkinson

This open access book presents robust statistical methods and procedures through the monitoring approach, with an emphasis on applications to linear regression. Illustrating the theory, it explores both large and small-sample properties. The performance of the forward search and of the monitoring of static robust estimators for regression data are illuminated through numerous data analyses using MATLAB and R. The book describes the results of many years’ work of the authors in the development of powerful methods of robust regression analysis. Robust methods are designed to analyse contaminated data. The well-established static robust methods estimate model features, such as parameter estimates, assuming the amount of contamination in the data is known. These methods are described in detail in Chapter 2 for estimation in a simple sample. The extension to regression is presented in Chapter 3, with an emphasis on S-estimation and related procedures as well as on least trimmed squares. The monitoring methods of Chapter 4, including the forward search, find the appropriate level of robustness for each data set and so avoid biased estimation from the inclusion of outliers and inefficiency due to the deletion of uncontaminated observations. This analysis is followed by examples which illustrate the use of the interactive graphical analyses associated with the authors’ FSDA toolbox. Numerical comparisons of the size and power of outlier tests appear in Chapter 5. Later chapters illustrate applications to response transformation in regression and to non-parametric regression. Extensions of the robust multiple regression model include Bayesian, heteroskedastic, time series and compositional regression, together with the clustering of regression models. Finally, several approaches to model selection are investigated and robust analyses of regression data are presented that illustrate the use of the techniques introduced earlier. Exercises are given at the end of each chapter, with solutions at the end of the book. The MATLAB code can be reproduced using MATLAB Online, without the need for a license, or via the language-agnostic Jupyter notebook environment, after installing the MATLAB kernel. Online computer code is available for all examples and exercises, together with a series of YouTube videos. Aimed at professional statisticians and researchers concerned with insightful data analysis, as well as postgraduate students, the book may also serve as a text for a modern interactive robust regression course.

Randomised Response-Adaptive Designs in Clinical Trials presents methods for the randomised allocation of treatments to patients in sequential clinical trials. Emphasizing the practical application of clinical trial designs, the book is designed for medical and applied statisticians, clinicians, and statisticians in training. After introducing clinical trials in drug development, the authors assess a simple adaptive design for binary responses without covariates. They discuss randomisation and covariate balance in normally distributed responses and cover many important response-adaptive designs for binary responses. The book then develops response-adaptive designs for continuous and longitudinal responses, optimum designs with covariates, and response-adaptive designs with covariates. It also covers response-adaptive designs that are derived by optimising an objective function subject to constraints on the variance of estimated parametric functions. The concluding chapter explores future directions in the development of adaptive designs.

Randomised Response-Adaptive Designs in Clinical Trials presents methods for the randomised allocation of treatments to patients in sequential clinical trials. Emphasizing the practical application of clinical trial designs, the book is designed for medical and applied statisticians, clinicians, and statisticians in training. After introducing clinical trials in drug development, the authors assess a simple adaptive design for binary responses without covariates. They discuss randomisation and covariate balance in normally distributed responses and cover many important response-adaptive designs for binary responses. The book then develops response-adaptive designs for continuous and longitudinal responses, optimum designs with covariates, and response-adaptive designs with covariates. It also covers response-adaptive designs that are derived by optimising an objective function subject to constraints on the variance of estimated parametric functions. The concluding chapter explores future directions in the development of adaptive designs.

Why We Wrote This Book This book is about using graphs to explore and model continuous multi­ variate data. Such data are often modelled using the multivariate normal distribution and, indeed, there is a literatme of weighty statistical tomes presenting the mathematical theory of this activity. Our book is very dif­ ferent. Although we use the methods described in these books, we focus on ways of exploring whether the data do indeed have a normal distribution. We emphasize outlier detection, transformations to normality and the de­ tection of clusters and unsuspected influential subsets. We then quantify the effect of these departures from normality on procedures such as dis­ crimination and duster analysis. The normal distribution is central to our book because, subject to our exploration of departures, it provides useful models for many sets of data. However, the standard estimates of the parameters, especially the covari­ ance matrix of the observations, are highly sensitive to the presence of outliers. This is both a blessing and a curse. It is a blessing because, if we estimate the parameters with the outliers excluded, their effect is appre­ ciable and apparent if we then include them for estimation. It is however a curse because it can be hard to detect which observations are outliers. We use the forward search for this purpose.

Why We Wrote This Book This book is about using graphs to explore and model continuous multi­ variate data. Such data are often modelled using the multivariate normal distribution and, indeed, there is a literatme of weighty statistical tomes presenting the mathematical theory of this activity. Our book is very dif­ ferent. Although we use the methods described in these books, we focus on ways of exploring whether the data do indeed have a normal distribution. We emphasize outlier detection, transformations to normality and the de­ tection of clusters and unsuspected influential subsets. We then quantify the effect of these departures from normality on procedures such as dis­ crimination and duster analysis. The normal distribution is central to our book because, subject to our exploration of departures, it provides useful models for many sets of data. However, the standard estimates of the parameters, especially the covari­ ance matrix of the observations, are highly sensitive to the presence of outliers. This is both a blessing and a curse. It is a blessing because, if we estimate the parameters with the outliers excluded, their effect is appre­ ciable and apparent if we then include them for estimation. It is however a curse because it can be hard to detect which observations are outliers. We use the forward search for this purpose.