This paper presents methods for calculating confidence intervals for estimates of sampling uncertainty (s(samp)) and analytical uncertainty (s(anal)) using the chi-squared distribution. These uncertainty estimates are derived from application of the duplicate method, which recommends a minimum of eight duplicate samples. The methods are applied to two case studies--moisture in butter and nitrate in lettuce. Use of the recommended minimum of eight duplicate samples is justified for both case studies as the confidence intervals calculated using greater than eight duplicates did not show any appreciable reduction in width. It is considered that eight duplicates provide estimates of uncertainty that are both acceptably accurate and cost effective.

In clinical trials to compare two or more treatments with dichotomous responses, group-sequential designs may reduce the total number of patients involved in the trial and response-adaptive designs may result in fewer patients being assigned to the inferior treatments. In this paper, we combine group-sequential and response-adaptive designs, extending recent work on sample size re-estimation in trials to compare two treatments with normally distributed responses, to analogous binary response trials. We consider the use of two parameters of interest in the group-sequential design, the log odds ratio and the simple difference between the probabilities of success. In terms of the adaptive sampling rules, we study two urn models, the drop-the-loser rule and the randomized Pólya urn rule, and compare their properties with those of two sequential maximum likelihood estimation rules, which minimize the expected number of treatment failures. We investigate two ways in which adaptive urn designs can be used in conjunction with group-sequential designs. The first method updates the urn at each interim analysis and the second method continually updates the urn after each patient response, assuming immediate patient responses. Our simulation results show that the group-sequential design, which uses the drop-the-loser rule, applied fully sequentially, is the most effective method for reducing the expected number of treatment failures and the average sample number, whilst still maintaining the nominal error rates, over a range of success probabilities.

We consider a clinical trial model comparing an experimental treatment with a control treatment when the responses are binary. For fixed significance level and power, we compare the expected number of treatment failures for two designs--the randomized play-the-winner rule and the triangular test. The former is an example of an adaptive design while the latter is an example of a fully sequential design. We show how to determine the sample size for the randomized play-the-winner rule and how to choose the stopping boundaries for the triangular test so that the two designs have similar power functions. With this choice of design parameters, simulation indicates that the triangular test is generally more effective at reducing the expected number of treatment failures, particularly when there is a large difference between the two probabilities of success. The expected number of treatment failures can be further reduced if the triangular test is applied using the randomized play-the-winner rule to assign each patient to one of the two treatments.

Key words and Phrases: adapttve normal h e a r model; autoregresstv~ model; fundanenta: dentaty of seqtieiitzal analysts; m a z z m u n !tkelthood estzmator: zlrlhozun zrarzabtlzty; very weak erpanszon. ABSTRACT A linear model is considered in which the design variables may be functions of previous responses and /or auxiliary randomisation. T h e model is observed t successive times, where t is a stopping time, and interest lies in estimating the parameters of the model. Approximations are derived for the bias and variance of the maximum likelihood estimators of the parameters a t time t . T h e derivations involve differentiating the fundamental identity of sequential analysis. The accuracy of the approximations is assessed by simulation for a multi-armed clinical trial model proposed by Coad (1995), two autoregressive models and the sequential des~gn of Ford and Silvey (1980). Very weak expansions are used t o justify the approximations. Downloaded by [Columbia University] at 15:

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