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    Exact confidence limits after a group sequential single arm trial
    Lloyd, C (John Wiley and Sons, 2021-05-10)
    Group sequential single arm designs are common in phase II trials as well as attribute testing and acceptance sampling. After the trial is completed, especially if the recommendation is to proceed to further testing, there is interest in full inference on treatment efficacy. For a binary response, there is the potential to construct exact upper and lower confidence limits, the first published method for which is Jennison and Turnbull (1983). We place their method within the modern theory of exact confidence limits and provide a new general result that ensures that the exact limits are consistent with the test result, an issue that has been largely ignored in the literature. Amongst methods based on the minimal sufficient statistic, we propose two exact methods that out‐perform Jennison and Turnbull's method across 10 selected designs. One of these we prefer and recommend for practical and theoretical reasons. We also investigate a method based on inverting Fisher's combination test, as well as a pure tie‐breaking variant of it. For the range of designs considered, neither of these methods result in large enough improvements in efficiency to justify violation of the sufficiency principle. For any nonadaptive sequential design, an R‐package is provided to select a method and compute the inference from a given realization.
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    Reply to Drs Almendra‐Arao and Sotres‐Ramos regarding Barnard's concept of convexity and possible extensions
    Ripamonti, E ; Lloyd, C (Wiley, 2020-05-01)
    On p. 130 of his 1947's seminal paper, Barnard1 wrote: “we propose that in our ordering, the two points which, respectively, have the same abscissa or the same ordinate as (a, b), and which lie further from the diagonal PR, shall be considered as indicating wider differences than (a, b) itself. Thus, […] the points immediately above and immediately to the left of the point ‘x’ are reckoned to indicate wider differences than the point ‘x’ itself. This condition implies that the set of points indicating differences as wide or wider than (a, b) will have a shape property vaguely related to convexity, and we call it the ‘C condition’.” It appears clear from this quotation how Barnard himself did not refer to a mathematical definition of a convex set strictu sensu but, instead, was providing the reader with the intuition of convexity linked to his definition. In this spirit, since our paper2 was a review paper for non‐experts, our intent was to point out that there is no obvious connection with the usual notion of a convex set, which is usually a condition that ensures convergence of algorithms to a unique optimum. We were not aware of the extended notions of convexity geometry or polyomino convexity that have appeared in the literature, and we thank the authors for pointing out that Barnard convex sets do satisfy these definitions. We would suggest that in future, when quoting Barnard convexity, it will be noted that it is not related to closure under linear combination. In any case, we prefer to emphasise the monotonicity properties of the generating statistic and how this affects computation of the exact or quasi‐exact P value.