A likelihood ratio test for separability of covariances

Matthew W. Mitchell*, Marc G. Genton, Marcia L. Gumpertz

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

76 Scopus citations


We propose a formal test of separability of covariance models based on a likelihood ratio statistic. The test is developed in the context of multivariate repeated measures (for example, several variables measured at multiple times on many subjects), but can also apply to a replicated spatio-temporal process and to problems in meteorology, where horizontal and vertical covariances are often assumed to be separable. Separable models are a common way to model spatio-temporal covariances because of the computational benefits resulting from the joint space-time covariance being factored into the product of a covariance function that depends only on space and a covariance function that depends only on time. We show that when the null hypothesis of separability holds, the distribution of the test statistic does not depend on the type of separable model. Thus, it is possible to develop reference distributions of the test statistic under the null hypothesis. These distributions are used to evaluate the power of the test for certain nonseparable models. The test does not require second-order stationarity, isotropy, or specification of a covariance model. We apply the test to a multivariate repeated measures problem.

Original languageEnglish (US)
Pages (from-to)1025-1043
Number of pages19
Issue number5
StatePublished - May 2006
Externally publishedYes


  • Kronecker product
  • Multivariate regression
  • Multivariate repeated measures
  • Nonstationary
  • Separable covariance
  • Spatio-temporal process

ASJC Scopus subject areas

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty


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