How to estimate population variance matrices with a Prescribed Pattern of Zeros?
Didier Concordet, Djalil Chafaï
UMR181 Physiopathologie et Toxicologie Expérimentales, INRA, ENVT, Toulouse France
Background: One of the main goal of population PK/PD studies is to describe the population distribution by observation of concentrations. In parametric models, this distribution is often assumed Gaussian up to a monotone transformation. In many real situations, kineticists knowledge of the drug mechanism imposes some independence pattern on the individual parameters. This simply means that the variance matrix contains a prescribed pattern of zeros (PPZ). Estimation of such matrices is problematic due to the positive definiteness constraint in the optimization. Pinheiro and Bates [1] studied different parameterizations that ensure the definite positiveness of the estimate. In particular they suggested the usage of a Cholesky like parameterization. Unfortunately, except for the case where the variance matrix is block diagonal matrix (up to coordinates permutation), Cholesky like parameterizations do not preserve the structure of the PPZ and are thus useless. Another common method is to estimate the variance matrix in two steps. First, estimation is performed without any constraint, then zeros are plugged according to the PPZ into the estimate provided by the first step. Unfortunately, by "forcing the zeros" by this way, nothing guarantees that the obtained estimate is still a positive matrix, and even when it is positive definite, it is not the maximum likelihood in general. Recently, the Iterative Conditional Fitting (ICF) method has been proposed to deal with the analysis of graphical models [2]. We propose a method that couples ICF and EM algorithm. It enables to reach the maximum likelihood estimator of the population variance matrix whatever the PPZ.
Method: The ICF method is mainly based on the specific properties of the Schur complement of a matrix. The Schur complement appears naturally in the distribution of the conditional distribution of a Gaussian vector with respect to another Gaussian vector. Writing the EM contrast using these properties leads to a standard least-squares problem that has to be solved at each EM iteration. For homoscedastic models, the least-squares problem is quadratic with respect to the components of variance to be estimated. This nice property is lost when considering heteroscedastic models.
Results: Simulations were performed with a four parameters sigmoid model that contains a PPZ to compare the statistical properties of the zeros forced and the proposed estimators. If both estimators are consistent, the EM+ICF estimator has smaller bias and variance that the zero forced estimator. Surprisingly, the mean population estimate was better (smaller bias and variance) when the variance was estimated with EM+ICF suggesting that the mean and variance estimations are heavily dependent. This sheds light on approaches, like the zero forced method, that relies on estimating the full variance matrix first and modify it by forcing the PPZ: since all the non zeros entries are estimated under the assumption that the variance matrix has no zeros, they could be poorly estimated. This is consistent with the results obtained by Ye and Pan [3] that conclude, in another context, that misspecification of the working variance structure may lead to a large loss of efficiency of the estimators of the mean parameters.
Conclusion: In the framework of parametric nonlinear mixed-effects models, the method we propose enables estimation of population variance matrices with PPZ. Simulations suggest that the EM+ICF estimator reaches efficiency where traditional methods fail. However, this optimality result is only valid when the pattern of zeros is a priori known. Finding a reasonable pattern of zeros is another problem that could be addressed using multiple likelihood ratio-tests that fully needs an EM+ICF method to be performed.
References:
[1] Pinheiro JC and Bates DM. Unconstrained Parametrizations for Variance-Covariance Matrices, Statistics and Computing, 6 (1996), 289-296.
[2] Chaudhuri S, Drton M and Richardson TS. Estimation of a covariance matrix with zeros. Biometrika, 94(2007),199-216.
[3] Ye H and Pan J. Modelling of covariance structures in generalised estimating equations for longitudinal data. Biometrika, 93(2006),927--=941.
