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New stopping criteria for iterative root finding.

ABSTRACT: A set of simple stopping criteria is presented, which improve the efficiency of iterative root finding by terminating the iterations immediately when no further improvement of the roots is possible. The criteria use only the function evaluations already needed by the root finding procedure to which they are applied. The improved efficiency is achieved by formulating the stopping criteria in terms of fractional significant digits. Test results show that the new stopping criteria reduce the iteration work load by about one-third compared with the most efficient stopping criteria currently available. This is achieved without compromising the accuracy of the extracted roots.

SUBMITTER: Nikolajsen JL

PROVIDER: S-EPMC4448898 | biostudies-other | 2014 Oct

REPOSITORIES: biostudies-other

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New stopping criteria for iterative root finding.

Nikolajsen Jorgen L JL

Royal Society open science 20141015 2

A set of simple stopping criteria is presented, which improve the efficiency of iterative root finding by terminating the iterations immediately when no further improvement of the roots is possible. The criteria use only the function evaluations already needed by the root finding procedure to which they are applied. The improved efficiency is achieved by formulating the stopping criteria in terms of fractional significant digits. Test results show that the new stopping criteria reduce the iterat ...[more]

PMID: 26064544

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Project description:BackgroundIn clinical trials with fixed study designs, statistical inference is only made when the trial is completed. In contrast, group sequential designs allow an early stopping of the trial at interim, either for efficacy when the treatment effect is significant or for futility when the treatment effect seems too small to justify a continuation of the trial. Efficacy stopping boundaries based on alpha spending functions have been widely discussed in the statistical literature, and there is also solid work on the choice of adequate futility stopping boundaries. Still, futility boundaries are often chosen with little or completely without theoretical justification, in particular in investigator initiated trails. Some authors contributed to fill this gap. In here, we rely on an idea of Schüler et al. (2017) who discuss optimality criteria for futility boundaries for the special case of trials with (multiple) time-to-event endpoints. Their concept can be adopted to define "optimal" futility boundaries (with respect to given performance indicators) for continuous endpoints.MethodsWe extend Schülers' definition for "optimal" futility boundaries to the most common study situation of a single continuous primary endpoint compared between two groups. First, we introduce the analytic algorithm to derive these futility boundaries. Second, the new concept is applied to a real clinical trial example. Finally, the performance of a study design with an "optimal" futility boundary is compared to designs with arbitrarily chosen futility boundaries.ResultsThe presented concept of deriving futility boundaries allows to control the probability of wrongly stopping for futility, that means stopping for futility even if the treatment effect is promizing. At the same time, the loss in power is also controlled by this approach. Moreover, "optimal" futility boundaries improve the probability of correctly stopping for futility under the null hypothesis of no difference between two groups.ConclusionsThe choice of futility boundaries should be thoroughly investigated at the planning stage. The sometimes met, arbitrary choice of futility boundaries can lead to a substantial negative impact on performance. Applying futility boundaries with predefined optimization criteria increases efficiency of group sequential designs. Other optimization criteria than proposed in here might be incorporated.

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New stopping criteria for iterative root finding.

Publications

New stopping criteria for iterative root finding.

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