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Basics of Multivariate Analysis in Neuroimaging Data
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MM Algorithms For Variance Components Models.

Hua Zhou1, Liuyi Hu2, Jin Zhou3

  • 1Department of Biostatistics, University of California, Los Angeles, CA.

Journal of Computational and Graphical Statistics : a Joint Publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America
|October 9, 2019
PubMed
Summary
This summary is machine-generated.

A new minorization-maximization (MM) algorithm offers a simpler and faster method for variance components estimation in statistical models. This iterative approach is competitive for large datasets and extends to complex mixed models.

Keywords:
global convergencelinear mixed model (LMM)matrix convexitymaximum a posteriori (MAP) estimationminorization-maximization (MM)multivariate responsepenalized estimationvariance components model

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Area of Science:

  • Statistics
  • Numerical Analysis
  • Biostatistics

Background:

  • Variance components estimation and mixed model analysis are crucial in many scientific fields.
  • Maximum likelihood estimation (MLE) and restricted maximum likelihood estimation (REML) for these models present significant numerical challenges.
  • Existing methods require substantial computational resources and expertise.

Purpose of the Study:

  • To introduce a novel, efficient, and easily implementable iterative algorithm for variance components estimation.
  • To extend the algorithm's applicability to various complex statistical models, including linear mixed models and multivariate response models.
  • To establish the theoretical convergence properties and compare the algorithm's performance against established methods like the EM algorithm.

Main Methods:

  • Development of an iterative algorithm based on the minorization-maximization (MM) principle.
  • Global convergence analysis to a Karush-Kuhn-Tucker (KKT) point.
  • Numerical and theoretical comparisons with the Expectation-Maximization (EM) algorithm.

Main Results:

  • The proposed MM algorithm is straightforward to implement and performs competitively on large-scale data.
  • The algorithm successfully extends to linear mixed models, multivariate responses, missing data, maximum a posteriori (MAP) estimation, and penalized estimation.
  • Global convergence to a KKT point is established.
  • The MM algorithm demonstrates faster convergence than the EM algorithm when the number of variance components exceeds two and covariance matrices are positive definite.

Conclusions:

  • The MM algorithm provides a robust and efficient alternative for variance components estimation, addressing the limitations of traditional MLE and REML methods.
  • Its versatility allows application to a wide range of complex statistical modeling scenarios.
  • The algorithm offers a significant computational advantage over the EM algorithm in specific conditions, enhancing statistical analysis efficiency.