On robust regression with high-dimensional predictors.

We study regression M-estimates in the setting where p, the number of covariates, and n, the number of observations, are both large, but p ≤ n. We find an exact stochastic representation for the distribution of ß = argmin(ß∈ℝ(p)) Σ(i=1)(n) ρ(Y(i) - X(i')ß) at fixed p and n under various assumptions on the objective function ρ and our statistical model. A scalar random variable whose deterministic limit rρ(κ) can be studied when p/n → κ > 0 plays a central role in this representation. We discover a nonlinear system of two deterministic equations that characterizes rρ(κ). Interestingly, the system shows that rρ(κ) depends on ρ through proximal mappings of ρ as well as various aspects of the statistical model underlying our study. Several surprising results emerge. In particular, we show that, when p/n is large enough, least squares becomes preferable to least absolute deviations for double-exponential errors.

Characterization of Numb expression in astrocytomas.

During early neurodevelopment, asymmetric segregation of Numb in mitotic progenitor cells influences the fate of daughter cells, whereby one daughter retains the progenitor phenotype while the other proceeds along a differentiation pathway. Numb has also been reported to function as a tumor suppressor in breast cancers and medulloblastomas. Given its role in maintaining neural progenitor pools in animal models and its reported role as a tumor suppressor, Numb could potentially contribute to astrocytoma oncogenesis. We characterized Numb expression in both human astrocytoma tissue samples and glioblastoma cell lines. We found that Numb is expressed in all grades of astrocytomas, being predominantly cytoplasmic in higher-grade astrocytomas but nuclear in pilocytic astrocytomas. Numb is also present in normal neurons, but not in normal astrocytes. In cultured glioblastoma cells, Numb concentrates in the perinuclear region and process tips. Numb expression in astrocytomas recapitulates that of progenitor cells during neurodevelopment, and suggests a role for Numb in astrocytoma oncogenesis.

RhoA protein expression correlates positively with degree of malignancy in astrocytomas.

Astrocytic tumors are the most common intracranial neoplasms. Their prognoses correlate with a conventional morphological grading system that suffers from diagnostic subjectivity and hence, inter-observer inconsistency. A molecular marker that provides an objective reference for classification and prognostication of astrocytic tumors would be useful in diagnostic pathology. RhoA, a GTPase protein involved in cell migration and adhesion, has been shown to be upregulated in a variety of human cancers. Based on direct analysis of clinical materials, our study demonstrates increased expression of RhoA in high-grade astrocytomas. This observation may be relevant to astrocytoma biology and the development of potential therapeutics against high-grade astrocytomas. Of more immediate consequence, utilization of this marker may aid in the routine pathological grading (and hence prognostication) of astrocytomas.