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ItemExact confidence limits after a group sequential single arm trialLloyd, C (John Wiley and Sons, 20210510)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.

ItemThe size accuracy of combination testsLloyd, CJ (WILEY, 201709)Summary Adaptive clinical trials typically involve several independent stages. The P‐values from each stage are synthesized through a so‐called combination function which ensures that the overall test will be valid if the stagewise tests are valid. In practice however, approximate and possibly invalid stagewise tests are used. This paper studies how imperfections of the stagewise tests feed through into the combination test. Several general results are proven including some for discrete models. An approximation formula which directly links the combined size accuracy to the component size accuracy is given. In the wider context of adaptive clinical trials, the main conclusion is that the basic tests used should be size accurate at nominal sizes both much larger and also much smaller than nominal desired size. For binary outcomes, the implication is that the parametric bootstrap should be used.

ItemAccurate pvalues for adaptive designs with binary endpointsHeritier, S ; Lloyd, CJ ; Lo, SN (WILEY, 20170730)

ItemTests for noninferiority trials with binomial endpoints: A guide to modern and quasiexact methods for biomedical researchersRipamonti, E ; Lloyd, CJ (Wiley, 20190501)Applied statisticians and pharmaceutical researchers are frequently involved in the design and analysis of clinical trials where at least one of the outcomes is binary. Treatments are judged by the probability of a positive binary response. A typical example is the noninferiority trial, where it is tested whether a new experimental treatment is practically not inferior to an active comparator with a prespecified margin δ. Except for the special case of δ = 0, no exact conditional test is available although approximate conditional methods (also called second‐order methods) can be applied. However, in some situations, the approximation can be poor and the logical argument for approximate conditioning is not compelling. The alternative is to consider an unconditional approach. Standard methods like the pooled z‐test are already unconditional although approximate. In this article, we review and illustrate unconditional methods with a heavy emphasis on modern methods that can deliver exact, or near exact, results. For noninferiority trials based on either rate difference or rate ratio, our recommendation is to use the so‐called E‐procedure, based on either the score or likelihood ratio statistic. This test is effectively exact, computationally efficient, and respects monotonicity constraints in practice. We support our assertions with a numerical study, and we illustrate the concepts developed in theory with a clinical example in pulmonary oncology; R code to conduct all these analyses is available from the authors.

ItemA scenario analysis of future Hong Kong age and labour force profiles and its implicationsLloyd, CJ ; Kwok, R ; Yip, PSF (Wiley, 20190601)The consequences of reduced fertility and mortality on the age distribution are an issue for most developed countries, but especially for the ‘Asian tiger’ economies. We use functional data analysis forecasting techniques to project the population of Hong Kong. Our projections include error estimates that allow for forecasting error as well as exogenous variations of fertility and migration numbers. We separate out the effects of pure demographic shifts from projected behavioural changes in labour force participation.This enables us to look at the kinds of changes in labour force participation that would be required to offset the aging effects that we estimate.

ItemReply to Drs Almendra‐Arao and Sotres‐Ramos regarding Barnard's concept of convexity and possible extensionsRipamonti, E ; Lloyd, C (Wiley, 20200501)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.

ItemA new method of identifying target groups for pronatalist policy applied to AustraliaChen, M ; Lloyd, CJ ; Yip, PSF ; van Wouwe, JP (PUBLIC LIBRARY SCIENCE, 20180209)A country's total fertility rate (TFR) depends on many factors. Attributing changes in TFR to changes of policy is difficult, as they could easily be correlated with changes in the unmeasured drivers of TFR. A case in point is Australia where both pronatalist effort and TFR increased in lock step from 2001 to 2008 and then decreased. The global financial crisis or other unobserved confounders might explain both the reducing TFR and pronatalist incentives after 2008. Therefore, it is difficult to estimate causal effects of policy using econometric techniques. The aim of this study is to instead look at the structure of the population to identify which subgroups most influence TFR. Specifically, we build a stochastic model relating TFR to the fertility rates of various subgroups and calculate elasticity of TFR with respect to each rate. For each subgroup, the ratio of its elasticity to its group size is used to evaluate the subgroup's potential cost effectiveness as a pronatalist target. In addition, we measure the historical stability of group fertility rates, which measures propensity to change. Groups with a high effectiveness ratio and also high propensity to change are natural policy targets. We applied this new method to Australian data on fertility rates broken down by parity, age and marital status. The results show that targeting parity 3+ is more costeffective than lower parities. This study contributes to the literature on pronatalist policies by investigating the targeting of policies, and generates important implications for formulating costeffective policies.