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In the era of precision medicine, more and more clinical trials are now driven or guided by biomarkers, which are patient characteristics objectively measured and evaluated as indicators of normal biological processes, pathogenic processes, or pharmacologic responses to therapeutic interventions. With the overarching objective to optimize and personalize disease management, biomarker-guided clinical trials increase the efficiency by appropriately utilizing prognostic or predictive biomarkers in the design. However, the efficiency gain is often not quantitatively compared to the traditional all-comers design, in which a faster enrollment rate is expected (e.g. due to no restriction to biomarker positive patients) potentially leading to a shorter duration. To accurately predict biomarker-guided trial duration, we propose a general framework using mixture distributions accounting for heterogeneous population. Extensive simulations are performed to evaluate the impact of heterogeneous population and the dynamics of biomarker characteristics and disease on the study duration. Several influential parameters including median survival time, enrollment rate, biomarker prevalence and effect size are identitied. Re-assessments of two publicly available trials are conducted to empirically validate the prediction accuracy and to demonstrate the practical utility. The R package \emph{detest} is developed to implement the proposed method and is publicly available on CRAN.

Let $G_w$ be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of $G_w$ are called positive inertia index, negative inertia index and nullity of $G_w$, and denoted by $i_{+}(G_w)$, $i_{-}(G_w)$, $i_{0}(G_w)$, respectively. In this paper, sharp lower bound on the positive (resp. negative) inertia index of weighted bicyclic graphs of order $n$ with pendant vertices is obtained. Moreover, all the weighted bicyclic graphs of order $n$ with at most two positive, two negative and at least $n-4$ zero eigenvalues are identified, respectively., Comment: 12 pages, 5 figures, 2 tables. arXiv admin note: text overlap with arXiv:1307.0059 by other authors