Statistical Tests for Large Tree-Structured Data

• Published in: Journal of the American Statistical Association Vol. 112; no. 520; pp. 1733 - 1743
• Main Authors: Bharath, Karthik, Kambadur, Prabhanjan, Dey, Dipak. K., Rao, Arvind, Baladandayuthapani, Veerabhadran
• Format: Journal Article
• Language: English
• Published: United States Taylor & Francis 01.01.2017; Taylor & Francis Group,LLC; Taylor & Francis Ltd
• Subjects: Asymptotic properties; Brain cancer; Brain tumors; Conditioned Galton-Watson trees; Consistent statistical models; Dyck path; Goodness of fit; Goodness-of-fit tests; Heterogeneity; Invariants; Magnetic resonance imaging; Random variables; Regression analysis; Statistical inference; Statistical methods; Statistical models; Statistical tests; Statistics; Structured data; Theory and Methods; Trees; Tumors; Conditioned Galton-Watson trees; Goodness-of-fit tests; Dyck path; Consistent statistical models
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.2016.1240081

We develop a general statistical framework for the analysis and inference of large tree-structured data, with a focus on developing asymptotic goodness-of-fit tests. We first propose a consistent statistical model for binary trees, from which we develop a class of invariant tests. Using the model for binary trees, we then construct tests for general trees by using the distributional properties of the continuum random tree, which arises as the invariant limit for a broad class of models for tree-structured data based on conditioned Galton-Watson processes. The test statistics for the goodness-of-fit tests are simple to compute and are asymptotically distributed as χ 2 and F random variables. We illustrate our methods on an important application of detecting tumor heterogeneity in brain cancer. We use a novel approach with tree-based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients. Supplementary materials for this article are available online.