The xyz algorithm for fast interaction search in high-dimensional data

When performing regression on a data set with p variables, it is often of interest to go beyond using main linear effects and include interactions as products between individual variables. For small-scale problems, these interactions can be computed explicitly but this leads to a computational complexity of at least O(p2) if done naively. This cost can be prohibitive if p is very large. We introduce a new randomised algorithm that is able to discover interactions with high probability and under mild conditions has a runtime that is subquadratic in p. We show that strong interactions can be discovered in almost linear time, whilst finding weaker interactions requires O(pα) operations for 1<α<2 depending on their strength. The underlying idea is to transform interaction search into a closest pair problem which can be solved efficiently in subquadratic time. The algorithm is called xyz and is implemented in the language R. We demonstrate its efficiency for application to genome-wide association studies, where more than 1011 interactions can be screened in under 280 seconds with a single-core 1.2 GHz CPU.ISSN:1532-4435ISSN:1533-792

Right singular vector projection graphs: fast high dimensional covariance matrix estimation under latent confounding

We consider the problem of estimating a high dimensional p×p covariance matrix Σ, given n observations of confounded data with covariance Σ + ΓΓT, where Γ is an unknown p×q matrix of latent factor loadings. We propose a simple and scalable estimator based on the projection onto the right singular vectors of the observed data matrix, which we call right singular vector projection (RSVP). Our theoretical analysis of this method reveals that, in contrast with approaches based on the removal of principal components, RSVP can cope well with settings where the smallest eigenvalue of ΓTΓ is relatively close to the largest eigenvalue of Σ, as well as when the eigenvalues of ΓTΓ are diverging fast. RSVP does not require knowledge or estimation of the number of latent factors q, but it recovers Σ only up to an unknown positive scale factor. We argue that this suffices in many applications, e.g. if an estimate of the correlation matrix is desired. We also show that, by using subsampling, we can further improve the performance of the method. We demonstrate the favourable performance of RSVP through simulation experiments and an analysis of gene expression data sets collated by the GTEX consortium.ISSN:1369-7412ISSN:0035-9246ISSN:1467-986

Blood neurofilament light levels predict non-relapsing progression following anti-CD20 therapy in relapsing and primary progressive multiple sclerosis: findings from the ocrelizumab randomised, double-blind phase 3 clinical trials.

BackgroundNeurofilament light chain (NfL), a neuronal cytoskeletal protein that is released upon neuroaxonal injury, is associated with multiple sclerosis (MS) relapsing activity and has demonstrated some prognostic ability for future relapse-related disease progression, yet its value in assessing non-relapsing disease progression remains unclear.MethodsWe examined baseline and longitudinal blood NfL levels in 1421 persons with relapsing MS (RMS) and 596 persons with primary progressive MS (PPMS) from the pivotal ocrelizumab MS trials. NfL treatment-response and risk for disease worsening (including disability progression into the open-label extension period and slowly expanding lesions [SELs] on brain MRI) at baseline and following treatment with ocrelizumab were evaluated using time-to-event analysis and linear regression models.FindingsIn persons from the RMS control arms without acute disease activity and in the entire PPMS control arm, higher baseline NfL was prognostic for greater whole brain and thalamic atrophy, greater volume expansion of SELs, and clinical progression. Ocrelizumab reduced NfL levels vs. controls in persons with RMS and those with PPMS, and abrogated the prognostic value of baseline NfL on disability progression. Following effective suppression of relapse activity by ocrelizumab, NfL levels at weeks 24 and 48 were significantly associated with long-term risk for disability progression, including up to 9 years of observation in RMS and PPMS.InterpretationHighly elevated NfL from acute MS disease activity may mask a more subtle NfL abnormality that reflects underlying non-relapsing progressive biology. Ocrelizumab significantly reduced NfL levels, consistent with its effects on acute disease activity and disability progression. Persistently elevated NfL levels, observed in a subgroup of persons under ocrelizumab treatment, demonstrate potential clinical utility as a predictive biomarker of increased risk for clinical progression. Suppression of relapsing biology with high-efficacy immunotherapy provides a window into the relationship between NfL levels and future non-relapsing progression.FundingF. Hoffmann-La Roche Ltd

Blood neurofilament light levels predict non-relapsing progression following anti-CD20 therapy in relapsing and primary progressive multiple sclerosis: findings from the ocrelizumab randomised, double-blind phase 3 clinical trials

BACKGROUND: Neurofilament light chain (NfL), a neuronal cytoskeletal protein that is released upon neuroaxonal injury, is associated with multiple sclerosis (MS) relapsing activity and has demonstrated some prognostic ability for future relapse-related disease progression, yet its value in assessing non-relapsing disease progression remains unclear. METHODS: We examined baseline and longitudinal blood NfL levels in 1421 persons with relapsing MS (RMS) and 596 persons with primary progressive MS (PPMS) from the pivotal ocrelizumab MS trials. NfL treatment-response and risk for disease worsening (including disability progression into the open-label extension period and slowly expanding lesions [SELs] on brain MRI) at baseline and following treatment with ocrelizumab were evaluated using time-to-event analysis and linear regression models. FINDINGS: In persons from the RMS control arms without acute disease activity and in the entire PPMS control arm, higher baseline NfL was prognostic for greater whole brain and thalamic atrophy, greater volume expansion of SELs, and clinical progression. Ocrelizumab reduced NfL levels vs. controls in persons with RMS and those with PPMS, and abrogated the prognostic value of baseline NfL on disability progression. Following effective suppression of relapse activity by ocrelizumab, NfL levels at weeks 24 and 48 were significantly associated with long-term risk for disability progression, including up to 9 years of observation in RMS and PPMS. INTERPRETATION: Highly elevated NfL from acute MS disease activity may mask a more subtle NfL abnormality that reflects underlying non-relapsing progressive biology. Ocrelizumab significantly reduced NfL levels, consistent with its effects on acute disease activity and disability progression. Persistently elevated NfL levels, observed in a subgroup of persons under ocrelizumab treatment, demonstrate potential clinical utility as a predictive biomarker of increased risk for clinical progression. Suppression of relapsing biology with high-efficacy immunotherapy provides a window into the relationship between NfL levels and future non-relapsing progression