Maximum likelihood estimates in the multivariate normal with patterned mean and covariance via the em algorithm

The maximum likelihood equations for a multivariate normal model with structured mean and structured covariance matrix may not have an explicit solution. In some cases the model's error term may be decomposed as the sum of two independent error terms, each having a patterned covariance matrix, such that if one of the unobservable error terms is artificially treated as "missing data", the EM algorithm can be used to compute the maximum likelihood estimates for the original problem. Some decompositions produce likelihood equations which do not have an explicit solution at each iteration of the EM algorithm, but within-iteration explicit solutions are shown for two general classes of models including covariance component models used for analysis of longitudinal data.     

Conditionally gaussian distributions and an application to kalman filtering with stochastic regressors

The work reviews theory of conditionally Gaussian distributions, especially so called theorems on normal correlation. Three theorems are given: the basic, the recursive, and the conditional theorem on normal correlation. They assume that (a,y), (a,x,y), or (a,y,z) has a Gaussian distribution, ussert that (a,y), (a,x,y), and (a,y,z), respectively, are Gaussian, and give formulas for the corresponding conditional mean vectors and variance covariance matrices. A proof is presented for the recursive and the conditional theorem.     

Scheffés mixed model for multivariate repeated measures:a relative efficiency evaluation

Scheffé’s mixed model, generalized for application to multivariate repeated measures, is known as the multivariate mixed model (MMM). The primary advantages the MMM are (1) the minimum sample size required to conduct an analysis is smaller than for competing procedures and (2) for certain covariance structures, the MMM analysis is more powerful than its competitors. The primary disadvantage is that the MMM makes a very restrictive covariance assumption; namely multivariate sphericity. This paper shows, first, that even minor departures from multivariate sphericity inflate the size of MMM based tests. Accordingly, MMM analyses, as computed in release 4.0 of SPSS MANOVA (SPSS Inc., 1990), can not be recommended unless it is known that multivariate sphericity is satisfied. Second, it is shown that a new Box-type (Box, 1954) Δ-corrected MMM test adequately controls test size unless departure from multivariate sphericity is severe or the covariance matrix departs substantially from a multiplicative-Kronecker structure. Third, power functions of adjusted MMM tests for selected covariance and noncentrality structures are compared to those of doubly multivariate methods that do not require multivariate sphericity. Based on relative efficiency evaluations, the adjusted MMM analyses described in this paper can be recommended only when sample sizes are very small or there is reason to believe that multivariate sphericity is nearly satisfied. Neither the e-adjusted analysis suggested in the SPSS MANOVA output (release 4.0) nor the adjusted analysis suggested by Boik (1988) can be recommended at all.     

Robust estimation of covariance parameters in partial linear model for longitudinal data

For longitudinal data, the within-subject dependence structure and covariance parameters may be of practical and theoretical interests. The estimation of covariance parameters has received much attention and been studied mainly in the framework of generalized estimating equations (GEEs). The GEEs method, however, is sensitive to outliers. In this paper, an alternative set of robust generalized estimating equations for both the mean and covariance parameters are proposed in the partial linear model for longitudinal data. The asymptotic properties of the proposed estimators of regression parameters, non-parametric function and covariance parameters are obtained. Simulation studies are conducted to evaluate the performance of the proposed estimators under different contaminations. The proposed method is illustrated with a real data analysis.     

Mixed models for data from thorough QT studies: part 2. One-step assessment of conditional QT prolongation

We investigate mixed analysis of covariance models for the 'one-step' assessment of conditional QT prolongation. Initially, we consider three different covariance structures for the data, where between-treatment covariance of repeated measures is modelled respectively through random effects, random coefficients, and through a combination of random effects and random coefficients. In all three of those models, an unstructured covariance pattern is used to model within-treatment covariance. In a fourth model, proposed earlier in the literature, between-treatment covariance is modelled through random coefficients but the residuals are assumed to be independent identically distributed (i.i.d.). Finally, we consider a mixed model with saturated covariance structure. We investigate the precision and robustness of those models by fitting them to a large group of real data sets from thorough QT studies. Our findings suggest: (i) Point estimates of treatment contrasts from all five models are similar. (ii) The random coefficients model with i.i.d. residuals is not robust; the model potentially leads to both under- and overestimation of standard errors of treatment contrasts and therefore cannot be recommended for the analysis of conditional QT prolongation. (iii) The combined random effects/random coefficients model does not always converge; in the cases where it converges, its precision is generally inferior to the other models considered. (iv) Both the random effects and the random coefficients model are robust. (v) The random effects, the random coefficients, and the saturated model have similar precision and all three models are suitable for the one-step assessment of conditional QT prolongation.     

Conditional covariance penalties for mixed models

The prediction error for mixed models can have a conditional or a marginal perspective depending on the research focus. We introduce a novel conditional version of the optimism theorem for mixed models linking the conditional prediction error to covariance penalties for mixed models. Different possibilities for estimating these conditional covariance penalties are introduced. These are bootstrap methods, cross-validation, and a direct approach called Steinian. The behavior of the different estimation techniques is assessed in a simulation study for the binomial-, the t-, and the gamma distribution and for different kinds of prediction error. Furthermore, the impact of the estimation techniques on the prediction error is discussed based on an application to undernutrition in Zambia.     

Reference priors for linear models with general covariance structures

We develop a new class of reference priors for linear models with general covariance structures. A general Markov chain Monte Carlo algorithm is also proposed for implementing the computation. We present several examples to demonstrate the results: Bayesian penalized spline smoothing, a Bayesian approach to bivariate smoothing for a spatial model, and prior specification for structural equation models.     

Checking identifiability of covariance parameters in linear mixed effects models

To build a linear mixed effects model, one needs to specify the random effects and often the associated parametrized covariance matrix structure. Inappropriate specification of the structures can result in the covariance parameters of the model not identifiable. Non-identifiability can result in extraordinary wide confidence intervals, and unreliable parameter inference. Sometimes software produces implication of model non-identifiability, but not always. In the simulation of fitting non-identifiable models we tried, about half of the times the software output did not look abnormal. We derive necessary and sufficient conditions of covariance parameters identifiability which does not require any prior model fitting. The results are easy to implement and are applicable to commonly used covariance matrix structures.     

A Class of Convolution‐Based Models for Spatio‐Temporal Processes with Non‐Separable Covariance Structure

Abstract. In this article, we propose a new parametric family of models for real‐valued spatio‐temporal stochastic processes S ( x , t ) and show how low‐rank approximations can be used to overcome the computational problems that arise in fitting the proposed class of models to large datasets. Separable covariance models, in which the spatio‐temporal covariance function of S ( x , t ) factorizes into a product of purely spatial and purely temporal functions, are often used as a convenient working assumption but are too inflexible to cover the range of covariance structures encountered in applications. We define positive and negative non‐separability and show that in our proposed family we can capture positive, zero and negative non‐separability by varying the value of a single parameter.     

Robust bootstrap estimates in heteroscedastic semi-varying coefficient models and applications in analyzing Australia CPI data

This article deals with the estimation of the parametric component, which is of primary interest, in the heteroscedastic semi-varying coefficient models. Based on the bootstrap technique, we present a procedure for estimating the parameters, which can provide a reliable approximation to the asymptotic distribution of the profile least-square (PLS) estimator. Furthermore, a bootstrap-type estimator of covariance matrix is developed, which is proved to be a consistent estimator of the covariance matrix. Moreover, some simulation experiments are conducted to evaluate the finite sample performance for the proposed methodology. Finally, the Australia CPI dataset is analyzed to demonstrate the application of the methods.     

Sensitivity analysis for the identifiability with application to latent random effect model for the mixed data

In this paper, we study the indentifiability of a latent random effect model for the mixed correlated continuous and ordinal longitudinal responses. We derive conditions for the identifiability of the covariance parameters of the responses. Also, we proposed sensitivity analysis to investigate the perturbation from the non-identifiability of the covariance parameters, it is shown how one can use some elements of covariance structure. These elements associate conditions for identifiability of the covariance parameters of the responses. Influence of small perturbation of these elements on maximal normal curvature is also studied. The model is illustrated using medical data.     

Estimation and identification considerations for the multiconsequence intervention model

A General Multiconsequence Intervention Model class that describes the simultaneous occurrence of a change in the process mean and covariance structure is introduced. When the covariance change is negligible, this model class reduces to intervention models described by Box and Tiao (1975). Maximum Likelihood Estimators for the parameters of the multiconsequence model class are developed for various important modeling situations that result from different a priori information about the form of the mean shift function form and the model parameters. As a consequence of these estimation results, an identification procedure for determining an appropriate dynamic mean shift form is suggested. The necessary hypothesis tests and corresponding confidence intervals.     

Bayesian analysis of joint mean and covariance models for longitudinal data

Efficient estimation of the regression coefficients in longitudinal data analysis requires a correct specification of the covariance structure. If misspecification occurs, it may lead to inefficient or biased estimators of parameters in the mean. One of the most commonly used methods for handling the covariance matrix is based on simultaneous modeling of the Cholesky decomposition. Therefore, in this paper, we reparameterize covariance structures in longitudinal data analysis through the modified Cholesky decomposition of itself. Based on this modified Cholesky decomposition, the within-subject covariance matrix is decomposed into a unit lower triangular matrix involving moving average coefficients and a diagonal matrix involving innovation variances, which are modeled as linear functions of covariates. Then, we propose a fully Bayesian inference for joint mean and covariance models based on this decomposition. A computational efficient Markov chain Monte Carlo method which combines the Gibbs sampler and Metropolis–Hastings algorithm is implemented to simultaneously obtain the Bayesian estimates of unknown parameters, as well as their standard deviation estimates. Finally, several simulation studies and a real example are presented to illustrate the proposed methodology.     

Simulation of truncated normal variables

We provide simulation algorithms for one-sided and two-sided truncated normal distributions. These algorithms are then used to simulate multivariate normal variables with convex restricted parameter space for any covariance structure.     

Positive and negative non-separability for space–time covariance models

Separable spatio-temporal covariance models, defined as the product of purely spatial and purely temporal covariance functions, are often used in practice, but frequently they only represent a convenient assumption. On the other hand, non-separable models are receiving a lot of attention, since they are more flexible to handle empirical covariances showed up in applications. Different forms of non-separability for space–time covariance functions have been recently defined in the literature. In this paper, the notion of positive and negative non-separability is further formalized in order to distinguish between pointwise and uniform non-separability. Various well-known non-separable space–time stationary covariance models are analyzed and classified by using the new definition of non-separability. In particular, wide classes of non-separable spatio-temporal covariance functions, able to capture positive and negative non-separability, are proposed and some examples of these classes are given. General results concerning the non-separability of spatial–temporal covariance functions obtained by a linear combination of spatial–temporal covariance functions and some stability properties are also presented. These results can be helpful to generate as well as to select appropriate covariance models for describing space–time data.     

A General Multivariate Threshold GARCH Model With Dynamic Conditional Correlations

We introduce a new multivariate GARCH model with multivariate thresholds in conditional correlations and develop a two-step estimation procedure that is feasible in large dimensional applications. Optimal threshold functions are estimated endogenously from the data and the model conditional covariance matrix is ensured to be positive definite. We study the empirical performance of our model in two applications using U.S. stock and bond market data. In both applications our model has, in terms of statistical and economic significance, higher forecasting power than several other multivariate GARCH models for conditional correlations.     

Generalized additive mixed models

Following the extension from linear mixed models to additive mixed models, extension from generalized linear mixed models to generalized additive mixed models is made, Algorithms are developed to compute the MLE's of the nonlinear effects and the covariance structures based on the penalized marginal likelihood. Convergence of the algorithms and selection of the smooth param¬eters are discussed.     

Bivariate Negative Binomial Generalized Linear Models for Environmental Count Data

We propose a new bivariate negative binomial model with constant correlation structure, which was derived from a contagious bivariate distribution of two independent Poisson mass functions, by mixing the proposed bivariate gamma type density with constantly correlated covariance structure (Iwasaki & Tsubaki, 2005), which satisfies the integrability condition of McCullagh & Nelder (1989, p. 334). The proposed bivariate gamma type density comes from a natural exponential family. Joe (1997) points out the necessity of a multivariate gamma distribution to derive a multivariate distribution with negative binomial margins, and the luck of a convenient form of multivariate gamma distribution to get a model with greater flexibility in a dependent structure with indices of dispersion. In this paper we first derive a new bivariate negative binomial distribution as well as the first two cumulants, and, secondly, formulate bivariate generalized linear models with a constantly correlated negative binomial covariance structure in addition to the moment estimator of the components of the matrix. We finally fit the bivariate negative binomial models to two correlated environmental data sets.     

CANONICAL VARIATE ANALYSIS WITH SPATIALLY-CORRELATED DATA

A general canonical variate model is derived when the observations are spatially correlated. For spatial covariance structures resulting from dependence of a pixel on its nearest neighbours, the solution reduces to an analysis of neighbour-corrected values. The usual analysis, in which spatial correlation is ignored, gives similar canonical vectors but over-estimates the canonical roots. A formula for approximating the reduction in the canonical roots to adjust for the spatial correlation is given.     

Singular Gaussian graphical models: Structure learning

ABSTRACT

The goal of this article is to introduce singular Gaussian graphical models and their conditional independence properties. In fact, we extend the concept of Gaussian Markov Random Field to the case of a multivariate normally distributed vector with a singular covariance matrix. We construct, then, the associated graph’s structure from the covariance matrix’s pseudo-inverse on the basis of a characterization of the pairwise conditional independence. The proposed approach can also be used when the covariance matrix is ill-conditioned, through projecting data on a smaller subspace. In this case, our method ensures numerical stability and consistency of the constructed graph and significantly reduces the inference problem’s complexity. These aspects are illustrated using numerical experiments.