### Abstract

The problem of testing the separability of a covariance matrix against an unstructured variance-covariance matrix is studied in the context of multivariate repeated measures data using Rao's score test (RST). The RST statistic is developed with the first component of the separable structure as a first-order autoregressive (AR(1)) correlation matrix or an unstructured (UN) covariance matrix under the assumption of multivariate normality. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Monte Carlo simulations are then used to study the comparative behavior of the null distribution of the RST statistic, as well as that of the LRT statistic, in terms of sample size considerations, and for the estimation of the empirical percentiles. Our findings are compared with existing results where the first component of the separable structure is a compound symmetry (CS) correlation matrix. It is also shown by simulations that the empirical null distribution of the RST statistic converges faster than the empirical null distribution of the LRT statistic to the limiting χ^{2} distribution. The tests are implemented on a real dataset from medical studies.

Original language | English (US) |
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Pages (from-to) | 192-215 |

Number of pages | 24 |

Journal | Biometrical Journal |

Volume | 59 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

- Empirical null distribution
- Likelihood ratio test
- Maximum likelihood estimates
- Rao’s score test
- Separable covariance structure

### ASJC Scopus subject areas

- Statistics and Probability
- Medicine(all)
- Statistics, Probability and Uncertainty

### Cite this

*Biometrical Journal*,

*59*(1), 192-215. DOI: 10.1002/bimj.201600044

**A comparison of likelihood ratio tests and Rao's score test for three separable covariance matrix structures.** / Filipiak, Katarzyna; Klein, Daniel; Roy, Anuradha.

Research output: Contribution to journal › Article

*Biometrical Journal*, vol 59, no. 1, pp. 192-215. DOI: 10.1002/bimj.201600044

}

TY - JOUR

T1 - A comparison of likelihood ratio tests and Rao's score test for three separable covariance matrix structures

AU - Filipiak,Katarzyna

AU - Klein,Daniel

AU - Roy,Anuradha

PY - 2017/1/1

Y1 - 2017/1/1

N2 - The problem of testing the separability of a covariance matrix against an unstructured variance-covariance matrix is studied in the context of multivariate repeated measures data using Rao's score test (RST). The RST statistic is developed with the first component of the separable structure as a first-order autoregressive (AR(1)) correlation matrix or an unstructured (UN) covariance matrix under the assumption of multivariate normality. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Monte Carlo simulations are then used to study the comparative behavior of the null distribution of the RST statistic, as well as that of the LRT statistic, in terms of sample size considerations, and for the estimation of the empirical percentiles. Our findings are compared with existing results where the first component of the separable structure is a compound symmetry (CS) correlation matrix. It is also shown by simulations that the empirical null distribution of the RST statistic converges faster than the empirical null distribution of the LRT statistic to the limiting χ2 distribution. The tests are implemented on a real dataset from medical studies.

AB - The problem of testing the separability of a covariance matrix against an unstructured variance-covariance matrix is studied in the context of multivariate repeated measures data using Rao's score test (RST). The RST statistic is developed with the first component of the separable structure as a first-order autoregressive (AR(1)) correlation matrix or an unstructured (UN) covariance matrix under the assumption of multivariate normality. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Monte Carlo simulations are then used to study the comparative behavior of the null distribution of the RST statistic, as well as that of the LRT statistic, in terms of sample size considerations, and for the estimation of the empirical percentiles. Our findings are compared with existing results where the first component of the separable structure is a compound symmetry (CS) correlation matrix. It is also shown by simulations that the empirical null distribution of the RST statistic converges faster than the empirical null distribution of the LRT statistic to the limiting χ2 distribution. The tests are implemented on a real dataset from medical studies.

KW - Empirical null distribution

KW - Likelihood ratio test

KW - Maximum likelihood estimates

KW - Rao’s score test

KW - Separable covariance structure

UR - http://www.scopus.com/inward/record.url?scp=84996540296&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84996540296&partnerID=8YFLogxK

U2 - 10.1002/bimj.201600044

DO - 10.1002/bimj.201600044

M3 - Article

VL - 59

SP - 192

EP - 215

JO - Biometrical Journal

T2 - Biometrical Journal

JF - Biometrical Journal

SN - 0323-3847

IS - 1

ER -