Group Sequential Analysis Research Articles

Robust tests in group sequential analysis: One- and two-sided hypotheses in the linear model

Consider the linear modelY=Xθ+E in the usual matrix notation where the errors are independent and identically distributed. We develop robust tests for a large class of one- and two-sided hypotheses about θ when the data are obtained and tests are carried out according to a group sequential design. To illustrate the nature of the main results, let\(\hat \theta\) and\(\tilde \theta\) be anM- and the least squares estimator of θ respectively which are asymptotically normal about θ with covariance matrices σ2(XtX)−1 and τ2(XtX)−1 respectively. Let the Wald-type statistics based on\(\hat \theta\) and\(\tilde \theta\) be denoted byRW andW respectively. It is shown thatRW andW have the same asymptotic null distributions; here the limit is taken with the number of groups fixed but the numbers of observations in the groups increase proportionately. Our main result is that the asymptotic Pitman efficiency ofRW relative toW is (σ2/τ2). Thus, the asymptotic efficiency-robustness properties of\(\hat \theta\) relative to\(\tilde \theta\) translate to asymptotic power-robustness ofRW relative toW. Clearly, this is an attractive result since we already have a large literature which shows that\(\hat \theta\) is efficiency-robust compared to\(\tilde \theta\). The results of a simulation study show that with realistic sample sizes,RW is likely to have almost as much power asW for normal errors, and substantially more power if the errors have long tails. The simulation results also illustrate the advantages of group sequential designs compared to a fixed sample design, in terms of sample size requirements to achieve a specified power.

Group Sequential Distribution-Free Methods for the Analysis of Multivariate Observations

Many studies involve the collection of multivariate observations, such as repeated measures, on two groups of subjects who are recruited over time, i.e., with staggered entry of subjects. Various marginal distribution-free multivariate methods have been proposed for the analyses of such multivariate observations where some measures may be missing at random. Using the multivariate U statistic of Wei and Johnson (1985, Biometrika 72, 359-364), we describe the group sequential analysis of such a study where the multivariate observations are observed sequentially--both within and among subjects. We describe a multivariate generalization of the Hodges and Lehmann (1963, Annals of Mathematical Statistics 34, 598-611) estimator of a location shift that can be obtained via the multivariate U statistic with the Mann-Whitney-Wilcoxon kernel. We then describe large-sample group sequential interval estimators and tests based on an aggregate estimate of the location shift combined over all of the repeated measures. We also describe how the same steps could be employed to perform a group sequential analysis based on any one of the variety of marginal multivariate methods that have been proposed. These methods are applied to a real-life example.

Product-Type Probability Bounds of Higher Order

GIaz and Johnson [14] introduce ith-order product-type approximations, βi, i =1,…n − 1, for Pn = P(X1 ≤c1, X2 ≤ c2,…Xn≤ cn) and show that Pn≥ βn−1 ≥ βn−2 ≥… ≥ β2 ≥ β1 when X is MTP2. In this article, it is shown thatunder weaker positive dependence conditions. For multivariate normal distributions, these conditions reduce to cov(Xi,Xj) ≥ 0 for 1 ≤ i < j ≤ n and cov(Xi,Xj| Xj−1) ≥ 0 for 1 ≤ i < j − 1, j = 3,…,n. This is applied to group sequential analysis with bivariate normal responses. Conditions for Pn ≥ β3 ≥ β2 ≥β1 are also derived. Bound conditions are also obtained that ensure that product-type approximations are nested lower bounds to upper orthant probabilities P(X1 > C1,…,Xn > cn). It is shown that these conditions are satisfied for the multivariate exponential distribution of Marshall and Olkin [20].

A New Procedure for Group Sequential Analysis in Clinical Trials

Since Pocock (1977, Biometrika 64, 191-199), many methods have been developed for group sequential analysis of clinical trials. However, these methods remain underemployed partly because of inconsistencies of sequential testing [Berry (1987, The Statistician 36, 181-189)]. This paper considers a new approach, which, by requiring that a succession of interim analyses be significant at the alpha level, both preserves the overall significance level alpha and does not present some of the inconsistencies of the previous methods. Results are obtained for a normal or binary response and for survival data. A comparison with the usual group sequential testing is also presented.

Sample size determination for group sequential clinical trials with immediate response.

The use function approach to group sequential methods has been explored previously. Any group sequential design requires specifying the frequency and times of repeated analyses, but only the use function approach allows deviations from those specified in the design, without affecting the type I error in the analysis. This paper illustrates how the use function provides a simple and flexible design procedure and how the initially projected maximum sample size can be calculated for randomized clinical trials in which responses are known relatively soon after patient entry and for which there is an early stopping rule built into the study protocol. Also the consequence of using the proposed design procedure is investigated in terms of the operating characteristics of the subsequent group sequential analyses.

Interim analysis in clinical trials

The early development of experimental design discouraged a sequential approach to the analysis of data, yet this seems a natural form of scientific enquiry. Clinical trial investigators should continuously monitor the quality of their techniques, but will often wish to delegate data monitoring to an independent group. The history and functions of data monitoring committees (DMCs) are reviewed. DMCs come in many shapes and sizes. They will need to consider many aspects of the data before making recommendations to the investigators, who have ultimate responsibility for early termination or protocol changes. Statistical issues form part of the assessment, and will involve management, safety and efficacy. Two broad approaches to early stopping are (i) the demonstration of strong evidence that a treatment effect falls above or below some critical value (not necessarily zero); (ii) stochastic curtailment based on prediction of final results. The latter is examined somewhat critically. Most trials will involve group sequential analyses at discrete time points. The effect of repeated data inspection on (i) is well known, although its relevance is debatable. Bayesian and likelihood methods do not entirely remove the difficulty.

Some extensions to a new approach for interim analysis in clinical trials

There already exists many methods for group sequential analysis of clinical trials. A new approach suggests to preserve the overall type I error alpha by requiring that a succession of interim analyses be significant at the alpha level. In this paper, this procedure is extended. A rule for early acceptance of H0 is presented, with an extension for flexible number and times of analyses. A real example is given.

Single dose cefotaxime plus metronidazole versus three dose cefuroxime plus metronidazole as prophylaxis against wound infection in colorectal surgery: multicentre prospective randomised study.

To establish whether a single preoperative dose of cefotaxime plus metronidazole was as effective as a standard three dose regimen of cefuroxime plus metronidazole in preventing wound infection after colorectal surgery. Prospective randomised allocation to one of two prophylactic antibiotic regimens in a parallel group trial. Group sequential analyses of each 250 patients were performed. 14 District general and teaching hospitals. 1018 Adults having colorectal operations were randomised, of whom 943 were evaluated. Demographic features, conditions requiring surgery, and operative procedures were similar in the two groups. Most patients had surgery for carcinoma of the colon or rectum. Group 1 received cefotaxime 1 g intravenously plus metronidazole 500 mg intravenously preoperatively. Group 2 received cefuroxime 1.5 g intravenously plus metronidazole 500 mg intravenously preoperatively, followed by cefuroxime 750 mg intravenously plus metronidazole 500 mg intravenously eight hours and 16 hours postoperatively. Development of surgical wound infection (as evidenced by the presence of pus), death, or discharge from hospital. Wound condition was scored on a five point scale on alternate days until discharge or for up to 20 days postoperatively. Wound infection rates were: group 1, 32/453 (7.1%; 95% confidence interval 4.7% to 9.4%); group 2, 33/454 (7.3%; 95% confidence interval 4.9% to 9.6%). Death rates (group 1: 26/470 (5.5%); group 2: 31/471 (6.6%], the incidence of postoperative complications, the median duration of hospital stay (12 days), and antibiotic tolerance were all similar in the two groups. Pooled data from groups 1 and 2 showed that wound infections were more frequent when minor faecal contamination had occurred at operation and when the duration of operation exceeded 90 minutes (greater than 90 min 11.2% of cases; less than 90 min 4.8%) and were associated with an extended hospital stay. A single preoperative dose of cefotaxime plus metronidazole is an efficacious as a three dose regimen of cefuroxime plus metronidazole in preventing wound infection after colorectal surgery and has practical advantages in eliminating the need for postoperative antibiotics.

A bayesian group sequential analysis procedure

In this work a method is developed for determining the expected sample size, 2nN, required by a group sequential test using a Bayesian approach. This method is proved to be superior to some recently developed methods. It gives a specific technique to determine the maximum number of groups, N, as well as the group, size, 2n. The proposed method allows a very early termination of the experiment when the alternative hypothesis is true, i.e. when i there is real difference between the treatments under consideration.

A simulation study of three sequential methods for the comparison of two treatment groups when the response criterion is censored.

Three recent sequential methods, group sequential analysis (GSA), the sequential probability ratio test (SPRT) and the triangular test (TT) are well suited to randomized clinical trials with a censored response criterion, as they do not require matched pairs of patients. We undertook a simulation study to investigate their statistical properties and to compare these three methods with the fixed-sample design. Our results suggest that the three methods have the expected statistical properties for size and power; they allow an important reduction of the average number of events before stopping, except with GSA when there is no treatment difference; the triangular test (closed design) appears the optimal design, as the variance of the number of events is smaller than with the sequential probability ratio test (open design) and analysis after every twenty new events does not alter the statistical properties of these sequential methods and enhances their usefulness.