A scalable approach for continuous time Markov models with covariates

Farhad Hatami; Alex Ocampo; Gordon Graham; Thomas E Nichols; Habib Ganjgahi

doi:10.1093/biostatistics/kxad012

A scalable approach for continuous time Markov models with covariates

Biostatistics. 2023 Jul 11:kxad012. doi: 10.1093/biostatistics/kxad012. Online ahead of print.

Authors

Farhad Hatami¹, Alex Ocampo², Gordon Graham², Thomas E Nichols^{3

4}, Habib Ganjgahi^{1

5}

Affiliations

¹ Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, Nuffield, Department of Medicine, University of Oxford and Department of Statistics, University of Oxford, Oxford, OX3 7LF, UK.
² Novartis Pharma AG, CH-4056 Basel, Switzerland.
³ Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, Nuffield Department of Population Health, University of Oxford, Oxford, OX3 7LF, UK.
⁴ Wellcome Centre for Integrative Neuroimaging, FMRIB, Nuffield Department of Clinical Neurosciences, University of Oxford, Oxford, OX3 9DU, UK.
⁵ Department of Statistics, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, UK.

PMID: 37433567
DOI: 10.1093/biostatistics/kxad012

Abstract

Existing methods for fitting continuous time Markov models (CTMM) in the presence of covariates suffer from scalability issues due to high computational cost of matrix exponentials calculated for each observation. In this article, we propose an optimization technique for CTMM which uses a stochastic gradient descent algorithm combined with differentiation of the matrix exponential using a Padé approximation. This approach makes fitting large scale data feasible. We present two methods for computing standard errors, one novel approach using the Padé expansion and the other using power series expansion of the matrix exponential. Through simulations, we find improved performance relative to existing CTMM methods, and we demonstrate the method on the large-scale multiple sclerosis NO.MS data set.

Keywords: Continuous-time Markov model; Multiple sclerosis; Multistate model; Padé; Scalable optimization; approximation.