Probabilistic Fréchet means for time varying persistence diagrams

© 2015, Institute of Mathematical Statistics. All rights reserved.In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in (Dp, Wp), the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fréchet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fréchet mean definition to the realm of time-varying persistence diagrams, better known as vineyards. We fix this problem by altering the original definition of Fréchet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each N a map (Dp)^N→ℙ(Dp). We show that this map is Hölder continuous on finite diagrams and thus can be used to build a useful statistic on vineyards.

Standardizing clinical trials workflow representation in UML for international site comparison.

BACKGROUND: With the globalization of clinical trials, a growing emphasis has been placed on the standardization of the workflow in order to ensure the reproducibility and reliability of the overall trial. Despite the importance of workflow evaluation, to our knowledge no previous studies have attempted to adapt existing modeling languages to standardize the representation of clinical trials. Unified Modeling Language (UML) is a computational language that can be used to model operational workflow, and a UML profile can be developed to standardize UML models within a given domain. This paper's objective is to develop a UML profile to extend the UML Activity Diagram schema into the clinical trials domain, defining a standard representation for clinical trial workflow diagrams in UML. METHODS: Two Brazilian clinical trial sites in rheumatology and oncology were examined to model their workflow and collect time-motion data. UML modeling was conducted in Eclipse, and a UML profile was developed to incorporate information used in discrete event simulation software. RESULTS: Ethnographic observation revealed bottlenecks in workflow: these included tasks requiring full commitment of CRCs, transferring notes from paper to computers, deviations from standard operating procedures, and conflicts between different IT systems. Time-motion analysis revealed that nurses' activities took up the most time in the workflow and contained a high frequency of shorter duration activities. Administrative assistants performed more activities near the beginning and end of the workflow. Overall, clinical trial tasks had a greater frequency than clinic routines or other general activities. CONCLUSIONS: This paper describes a method for modeling clinical trial workflow in UML and standardizing these workflow diagrams through a UML profile. In the increasingly global environment of clinical trials, the standardization of workflow modeling is a necessary precursor to conducting a comparative analysis of international clinical trials workflows.

Development of a decision aid to inform patients' and families' renal replacement therapy selection decisions.

BACKGROUND: Few educational resources have been developed to inform patients' renal replacement therapy (RRT) selection decisions. Patients progressing toward end stage renal disease (ESRD) must decide among multiple treatment options with varying characteristics. Complex information about treatments must be adequately conveyed to patients with different educational backgrounds and informational needs. Decisions about treatment options also require family input, as families often participate in patients' treatment and support patients' decisions. We describe the development, design, and preliminary evaluation of an informational, evidence-based, and patient-and family-centered decision aid for patients with ESRD and varying levels of health literacy, health numeracy, and cognitive function. METHODS: We designed a decision aid comprising a complementary video and informational handbook. We based our development process on data previously obtained from qualitative focus groups and systematic literature reviews. We simultaneously developed the video and handbook in "stages." For the video, stages included (1) directed interviews with culturally appropriate patients and families and preliminary script development, (2) video production, and (3) screening the video with patients and their families. For the handbook, stages comprised (1) preliminary content design, (2) a mixed-methods pilot study among diverse patients to assess comprehension of handbook material, and (3) screening the handbook with patients and their families. RESULTS: The video and handbook both addressed potential benefits and trade-offs of treatment selections. The 50-minute video consisted of demographically diverse patients and their families describing their positive and negative experiences with selecting a treatment option. The video also incorporated health professionals' testimonials regarding various considerations that might influence patients' and families' treatment selections. The handbook was comprised of written words, pictures of patients and health care providers, and diagrams describing the findings and quality of scientific studies comparing treatments. The handbook text was written at a 4th to 6th grade reading level. Pilot study results demonstrated that a majority of patients could understand information presented in the handbook. Patient and families screening the nearly completed video and handbook reviewed the materials favorably. CONCLUSIONS: This rigorously designed decision aid may help patients and families make informed decisions about their treatment options for RRT that are well aligned with their values.

Persistent Homology Analysis of Brain Artery Trees.

New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries.