Statistical monitoring of clinical trials. A unified approach.

*(English)*Zbl 1121.62098
Statistics for Biology and Health. New York, NY: Springer (ISBN 0-387-30059-7/hbk). xiii, 258 p. (2006).

This volume presents a comprehensive manual how to perform (repeated) interim analyses in clinical trials in different testing situations. This concept of interim analyses exhibits statistical monitoring for efficacy of the examined treatment options. The authors present a unifying approach for the design of interim analyses for many different test situations (“\(B\) value” approach). This approach makes use of the fact that in many settings the sequence of test statistics possesses the same correlation structure as the Brownian motion and converges for increasing sample sizes to a Brownian motion process depending on the information fraction so far obtained in the trial (sample size fraction or event fraction for survival data).

In chapter 2, the general framework of \(B\)-values is introduced considering two topics: hypothesis-testing and parameter estimation. The authors begin with the simple setting in which test statistic and treatment effect estimator are a sum and mean, respectively, of independent and identically distributed (iid) random variables. Then they show that in more complicated settings the test statistics and effect estimates constitute so-defined sum processes (\(S\)-processes) and estimation processes (\(E\)-processes) which behave like a sum resp. a sample mean process. This is proved for comparison of means (continuous outcomes) as well as for rate comparison (dichotomous outcomes) or event rate comparison (survival data). The \(S\)- and \(E\)-processes can be converted into Brownian motion processes. So in all these settings Brownian motion can be used to approximate the joint distribution of the sequence of test statistics and estimates. This can also be seen if less simple treatment effect estimators are used (maximum likelihood estimators, minimum variance unbiased estimators) and with adjustment for covariates (normal linear and mixed models).

In the following chapters the representation in terms of Brownian motion is used to calculate different types of power (conditional, unconditional, predictive, chapter 3), to formulate classical group-sequential procedures (Pocock, Haybittle and O’Brien and Fleming, chapter 4) as well as the spending function approach of Lan and DeMets (chapter 5). Chapter 6 deals with practical survival monitoring in the case of staggered entry. The theory of counting processes and the central limit theorem, developed in the 1980’s, allow a unified approach with Brownian motion under the proportional hazard assumption. For group-sequential settings, where the \(B\)-value approach holds, the authors derive p-values for the final analysis by using \(B\)-values and Z-scores to define orderings of the outcome space (chapter 7). These orderings are also used for computation of confidence intervals thus allowing inferences to be made after a group-sequential trial. However, the authors mention in several chapters that settings exist where a Brownian motion is not appropriate to describe the test situation – namely for not iid observations, non proportional hazards and small sample sizes this approach may not be applicable. For these situations some options how to proceed are proposed in chapter 8. In clinical trials not only the statistical results for testing efficacy but also safety concerns have to be considered, therefore the question is treated how to include this type of data in the monitoring process (chapter 9).

Chapter 10 shows how Bayesian methods can be applied to B-values. A whole chapter is also dedicated to the issue of adaptive sample size calculation incorporating data about nuisance or the treatment effect, where the latter is seen with some criticism (chapter 11).

The book presents a very clear and comprehensible overview of multiple kinds of data monitoring and interim analyses in clinical trials. All topics are illustrated with numerous numerical examples or case studies. Some proofs for formerly stated results are given in detail in the appendices of the individual chapters and practical application of the presented methods are facilitated by some \(S\)- (\(R\)-) Code and a detailed description of selected group-sequential software which can be found in the appendix of the volume.

In chapter 2, the general framework of \(B\)-values is introduced considering two topics: hypothesis-testing and parameter estimation. The authors begin with the simple setting in which test statistic and treatment effect estimator are a sum and mean, respectively, of independent and identically distributed (iid) random variables. Then they show that in more complicated settings the test statistics and effect estimates constitute so-defined sum processes (\(S\)-processes) and estimation processes (\(E\)-processes) which behave like a sum resp. a sample mean process. This is proved for comparison of means (continuous outcomes) as well as for rate comparison (dichotomous outcomes) or event rate comparison (survival data). The \(S\)- and \(E\)-processes can be converted into Brownian motion processes. So in all these settings Brownian motion can be used to approximate the joint distribution of the sequence of test statistics and estimates. This can also be seen if less simple treatment effect estimators are used (maximum likelihood estimators, minimum variance unbiased estimators) and with adjustment for covariates (normal linear and mixed models).

In the following chapters the representation in terms of Brownian motion is used to calculate different types of power (conditional, unconditional, predictive, chapter 3), to formulate classical group-sequential procedures (Pocock, Haybittle and O’Brien and Fleming, chapter 4) as well as the spending function approach of Lan and DeMets (chapter 5). Chapter 6 deals with practical survival monitoring in the case of staggered entry. The theory of counting processes and the central limit theorem, developed in the 1980’s, allow a unified approach with Brownian motion under the proportional hazard assumption. For group-sequential settings, where the \(B\)-value approach holds, the authors derive p-values for the final analysis by using \(B\)-values and Z-scores to define orderings of the outcome space (chapter 7). These orderings are also used for computation of confidence intervals thus allowing inferences to be made after a group-sequential trial. However, the authors mention in several chapters that settings exist where a Brownian motion is not appropriate to describe the test situation – namely for not iid observations, non proportional hazards and small sample sizes this approach may not be applicable. For these situations some options how to proceed are proposed in chapter 8. In clinical trials not only the statistical results for testing efficacy but also safety concerns have to be considered, therefore the question is treated how to include this type of data in the monitoring process (chapter 9).

Chapter 10 shows how Bayesian methods can be applied to B-values. A whole chapter is also dedicated to the issue of adaptive sample size calculation incorporating data about nuisance or the treatment effect, where the latter is seen with some criticism (chapter 11).

The book presents a very clear and comprehensible overview of multiple kinds of data monitoring and interim analyses in clinical trials. All topics are illustrated with numerous numerical examples or case studies. Some proofs for formerly stated results are given in detail in the appendices of the individual chapters and practical application of the presented methods are facilitated by some \(S\)- (\(R\)-) Code and a detailed description of selected group-sequential software which can be found in the appendix of the volume.

Reviewer: Christina Wunder (Heidelberg)

##### MSC:

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

60J65 | Brownian motion |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62N03 | Testing in survival analysis and censored data |

62N02 | Estimation in survival analysis and censored data |