We propose a novel adaptive design for clinical trials with time-to-event outcomes and covariates (which may consist of or include biomarkers). Our method is based on the expected entropy of the posterior distribution of a proportional hazards model.
The expected entropy is evaluated as a function of a patients covariates, and the information gained due to a patient is defined as the decrease in the corresponding entropy. Candidate patients are only recruited onto the trial if they are likely to provide sufficient information. Patients with covariates that are deemed uninformative are filtered out. A special case is where all patients are recruited, and we determine the optimal treatment arm allocation. This adaptive design has the advantage of potentially elucidating the relationship between covariates, treatments, and survival probabilities using fewer patients, albeit at the cost of rejecting some candidates. We assess the performance of our adaptive design using data from the German Breast Cancer Study group and numerical simulations of a biomarker validation trial.
The analysis of high dimensional survival data is challenging, primarily due to the problem of overfitting which occurs when spurious relationships are inferred from data that subsequently fail to exist in test data. Here we propose a novel method of
extracting a low dimensional representation of covariates in survival data by combining the popular Gaussian Process Latent Variable Model (GPLVM) with a Weibull Proportional Hazards Model (WPHM). The combined model offers a flexible non-linear probabilistic method of detecting and extracting any intrinsic low dimensional structure from high dimensional data. By reducing the covariate dimension we aim to diminish the risk of overfitting and increase the robustness and accuracy with which we infer relationships between covariates and survival outcomes. In addition, we can simultaneously combine information from multiple data sources by expressing multiple datasets in terms of the same low dimensional space. We present results from several simulation studies that illustrate a reduction in overfitting and an increase in predictive performance, as well as successful detection of intrinsic dimensionality. We provide evidence that it is advantageous to combine dimensionality reduction with survival outcomes rather than performing unsupervised dimensionality reduction on its own. Finally, we use our model to analyse experimental gene expression data and detect and extract a low dimensional representation that allows us to distinguish high and low risk groups with superior accuracy compared to doing regression on the original high dimensional data.
We present a catalog of 1921 spectroscopic redshifts measured in the fields of the massive galaxy clusters MACSJ0416.1--2403 ($z=0.397$), MACSJ0717.5+3745 ($z=0.546$), and MACSJ1149.5+2223 ($z=0.544$), i.e., three of the four clusters selected by STS
cI as the targets of the Frontier Fields (FF) initiative for studies of the distant Universe via gravitational lensing. Compiled in the course of the MACS project (Massive Cluster Survey) that detected the FF clusters, this catalog is provided to the community for three purposes: (1) to allow the identification of cluster members for studies of the galaxy population of these extreme systems, (2) to facilitate the removal of unlensed galaxies and thus reduce shear dilution in weak-lensing analyses, and (3) to improve the calibration of photometric redshifts based on both ground- and spacebased observations of the FF clusters.
We apply Gaussian process (GP) regression, which provides a powerful non-parametric probabilistic method of relating inputs to outputs, to survival data consisting of time-to-event and covariate measurements. In this context, the covariates are regar
ded as the `inputs and the event times are the `outputs. This allows for highly flexible inference of non-linear relationships between covariates and event times. Many existing methods, such as the ubiquitous Cox proportional hazards model, focus primarily on the hazard rate which is typically assumed to take some parametric or semi-parametric form. Our proposed model belongs to the class of accelerated failure time models where we focus on directly characterising the relationship between covariates and event times without any explicit assumptions on what form the hazard rates take. It is straightforward to include various types and combinations of censored and truncated observations. We apply our approach to both simulated and experimental data. We then apply multiple output GP regression, which can handle multiple potentially correlated outputs for each input, to competing risks survival data where multiple event types can occur. By tuning one of the model parameters we can control the extent to which the multiple outputs (the time-to-event for each risk) are dependent thus allowing the specification of correlated risks. Simulation studies suggest that in some cases assuming dependence can lead to more accurate predictions.
We investigate the Plateau and isoperimetric problems associated to Feffermans measure for strongly pseudoconvex real hypersurfaces in $mathbb C^n$ (focusing on the case $n=2$), showing in particular that the isoperimetric problem shares features of
both the euclidean isoperimetric problem and the corresponding problem in Blaschkes equiaffine geometry in which the key inequalities are reversed. The problems are invariant under constant-Jacobian biholomorphism, but we also introduce a non-trivial modified isoperimetric quantity invariant under general biholomorphism.
We show that the efficiency of a natural pairing between certain projectively invariant Hardy spaces on dual strongly C-linearly convex real hypersurfaces in complex projective space is measured by the norm of the corresponding Leray transform.
The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in $mathbb C^2$. Our class is self-dual; it contains some domains with less than $C^2$-smooth boundary and also some domains with smooth boundary a
nd degenerate Levi form. $L^2$-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains.
