A cautionary note on generalized linear models for covariance of unbalanced longitudinal data

Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positive-definiteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positive-definiteness constraint and use a generalized linear model setup to jointly model the mean and covariance using covariates (Pourahmadi, 2000). However, this approach may not be directly applicable when the longitudinal data are unbalanced, as coherent regression models for the dependence across all times and subjects may not exist. Within the existing generalized linear model framework, we show how to overcome this and other challenges by embedding the covariance matrix of the observed data for each subject in a larger covariance matrix and employing the familiar EM algorithm to compute the maximum likelihood estimates of the parameters and their standard errors. We illustrate and assess the methodology using real data sets and simulations.     

Instrumental variable-based empirical likelihood inferences for varying-coefficient models with error-prone covariates

This paper presents the empirical likelihood inferences for a class of varying-coefficient models with error-prone covariates. We focus on the case that the covariance matrix of the measurement errors is unknown and neither repeated measurements nor validation data are available. We propose an instrumental variable-based empirical likelihood inference method and show that the proposed empirical log-likelihood ratio is asymptotically chi-squared. Then, the confidence intervals for the varying-coefficient functions are constructed. Some simulation studies and a real data application are used to assess the finite sample performance of the proposed empirical likelihood procedure.     

A bivariate distribution with gamma and beta marginals with application to drought data

The first known bivariate distribution with gamma and beta marginals is introduced. Various representations are derived for its joint probability density function (pdf), joint cumulative distribution function (cdf), product moments, conditional pdfs, conditional cdfs, conditional moments, joint moment generating function, joint characteristic function and entropies. The method of maximum likelihood and the method of moments are used to derive the associated estimation procedures as well as the Fisher information matrix, variance–covariance matrix and the profile likelihood confidence intervals. An application to drought data from Nebraska is provided. Some other applications are also discussed. Finally, an extension of the bivariate distribution to the multivariate case is proposed.     

Robust Bayesian synthetic likelihood via a semi-parametric approach

Bayesian synthetic likelihood (BSL) is now a well-established method for performing approximate Bayesian parameter estimation for simulation-based models that do not possess a tractable likelihood function. BSL approximates an intractable likelihood function of a carefully chosen summary statistic at a parameter value with a multivariate normal distribution. The mean and covariance matrix of this normal distribution are estimated from independent simulations of the model. Due to the parametric assumption implicit in BSL, it can be preferred to its nonparametric competitor, approximate Bayesian computation, in certain applications where a high-dimensional summary statistic is of interest. However, despite several successful applications of BSL, its widespread use in scientific fields may be hindered by the strong normality assumption. In this paper, we develop a semi-parametric approach to relax this assumption to an extent and maintain the computational advantages of BSL without any additional tuning. We test our new method, semiBSL, on several challenging examples involving simulated and real data and demonstrate that semiBSL can be significantly more robust than BSL and another approach in the literature.     

Unbiased Estimator for a Covariance Matrix Under Two-Step Monotone Incomplete Sample

In this article, we consider an inference for a covariance matrix under two-step monotone incomplete sample. The maximum likelihood estimator of the mean vector is unbiased but that of the covariance matrix is biased. We derive an unbiased estimator for the covariance matrix using some fundamental properties of the Wishart matrix. The properties of the estimators are investigated and the accuracies are checked by a numerical simulation.     

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.     

Sensitivity analysis in covariance structure analysis with equality constraints

Influence functions are derived for covariance structure analysis with equality constraints, where the parameters are estimated by minimizing a discrepancy function between the assumed covariance matrix and the sample covariance matrix. As a special case maximum likelihood exploratory factor analysis is studied precisely with a numerical example. Comparison is made with the the results of Tanaka and Odaka (1989), who have proposed a sensitivity analysis procedure in maximum likelihood exploratory factor analysis using the perturbation expansion of a certain function of eigenvalues and eigenvectors of a real symmetric matrix. Also the present paper gives a generalization of Tanaka, Watadani and Moon (1991) to the case with equality constraints.     

Cox Regression with Incomplete Covariate Measurements using the EM-algorithm

Ibrahim (1990) used the EM-algorithm to obtain maximum likelihood estimates of the regression parameters in generalized linear models with partially missing covariates. The technique was termed EM by the method of weights. In this paper, we generalize this technique to Cox regression analysis with missing values in the covariates. We specify a full model letting the unobserved covariate values be random and then maximize the observed likelihood. The asymptotic covariance matrix is estimated by the inverse information matrix. The missing data are allowed to be missing at random but also the non-ignorable non-response situation may in principle be considered. Simulation studies indicate that the proposed method is more efficient than the method suggested by Paik & Tsai (1997). We apply the procedure to a clinical trials example with six covariates with three of them having missing values.     

On discrimination and classification with multivariate repeated measures data

We study the problem of classification with multiple q-variate observations with and without time effect on each individual. We develop new classification rules for populations with certain structured and unstructured mean vectors and under certain covariance structures. The new classification rules are effective when the number of observations is not large enough to estimate the variance–covariance matrix. Computational schemes for maximum likelihood estimates of required population parameters are given. We apply our findings to two real data sets as well as to a simulated data set.     

Bayesian goodness-of-fit test for censored data

We propose a Bayesian computation and inference method for the Pearson-type chi-squared goodness-of-fit test with right-censored survival data. Our test statistic is derived from the classical Pearson chi-squared test using the differences between the observed and expected counts in the partitioned bins. In the Bayesian paradigm, we generate posterior samples of the model parameter using the Markov chain Monte Carlo procedure. By replacing the maximum likelihood estimator in the quadratic form with a random observation from the posterior distribution of the model parameter, we can easily construct a chi-squared test statistic. The degrees of freedom of the test equal the number of bins and thus is independent of the dimensionality of the underlying parameter vector. The test statistic recovers the conventional Pearson-type chi-squared structure. Moreover, the proposed algorithm circumvents the burden of evaluating the Fisher information matrix, its inverse and the rank of the variance–covariance matrix. We examine the proposed model diagnostic method using simulation studies and illustrate it with a real data set from a prostate cancer study.     

Case-cohort analysis with semiparametric transformation models

Semiparametric transformation models provide flexible regression models for survival analysis, including the Cox proportional hazards and the proportional odds models as special cases. We consider the application of semiparametric transformation models in case-cohort studies, where the covariate data are observed only on cases and on a subcohort randomly sampled from the full cohort. We first propose an approximate profile likelihood approach with full-cohort data, which amounts to the pseudo-partial likelihood approach of Zucker [2005. A pseudo-partial likelihood method for semiparametric survival regression with covariate errors. J. Amer. Statist. Assoc. 100, 1264–1277]. Simulation results show that our proposal is almost as efficient as the nonparametric maximum likelihood estimator. We then extend this approach to the case-cohort design, applying the Horvitz–Thompson weighting method to the estimating equations from the approximated profile likelihood. Two levels of weights can be utilized to achieve unbiasedness and to gain efficiency. The resulting estimator has a closed-form asymptotic covariance matrix, and is found in simulations to be substantially more efficient than the estimator based on martingale estimating equations. The extension to left-truncated data will be discussed. We illustrate the proposed method on data from a cardiovascular risk factor study conducted in Taiwan.     

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.     

Incremental modelling for compositional data streams

ABSTRACT

Incremental modelling of data streams is of great practical importance, as shown by its applications in advertising and financial data analysis. We propose two incremental covariance matrix decomposition methods for a compositional data type. The first method, exact incremental covariance decomposition of compositional data (C-EICD), gives an exact decomposition result. The second method, covariance-free incremental covariance decomposition of compositional data (C-CICD), is an approximate algorithm that can efficiently compute high-dimensional cases. Based on these two methods, many frequently used compositional statistical models can be incrementally calculated. We take multiple linear regression and principle component analysis as examples to illustrate the utility of the proposed methods via extensive simulation studies.     

On a convergent stochastic estimation algorithm for frailty models

A maximum likelihood estimation procedure is presented for the frailty model. The procedure is based on a stochastic Expectation Maximization algorithm which converges quickly to the maximum likelihood estimate. The usual expectation step is replaced by a stochastic approximation of the complete log-likelihood using simulated values of unobserved frailties whereas the maximization step follows the same lines as those of the Expectation Maximization algorithm. The procedure allows to obtain at the same time estimations of the marginal likelihood and of the observed Fisher information matrix. Moreover, this stochastic Expectation Maximization algorithm requires less computation time. A wide variety of multivariate frailty models without any assumption on the covariance structure can be studied. To illustrate this procedure, a Gaussian frailty model with two frailty terms is introduced. The numerical results based on simulated data and on real bladder cancer data are more accurate than those obtained by using the Expectation Maximization Laplace algorithm and the Monte-Carlo Expectation Maximization one. Finally, since frailty models are used in many fields such as ecology, biology, economy, …, the proposed algorithm has a wide spectrum of applications.     

Adaptive robust estimation in joint mean–covariance regression model for bivariate longitudinal data

The estimation of the covariance matrix is important in the analysis of bivariate longitudinal data. A good estimator for the covariance matrix can improve the efficiency of the estimators of the mean regression coefficients. Furthermore, the covariance estimation itself is also of interest, but it is a challenging job to model the covariance matrix of bivariate longitudinal data due to the complex structure and positive definite constraint. In addition, most of existing approaches are based on the maximum likelihood, which is very sensitive to outliers or heavy-tail error distributions. In this article, an adaptive robust estimation method is proposed for bivariate longitudinal data. Unlike the existing likelihood-based methods, the proposed method can adapt to different error distributions. Specifically, at first, we utilize the modified Cholesky block decomposition to parameterize the covariance matrices. Secondly, we apply the bounded Huber's score function to develop a set of robust generalized estimating equations to estimate the parameters both in the mean and the covariance models simultaneously. A data-driven approach is presented to select the parameter c in the Huber's score function, which can ensure that the proposed method is robust and efficient. A simulation study and a real data analysis are conducted to illustrate the robustness and efficiency of the proposed approach.     

Missing values: sparse inverse covariance estimation and?an?extension to sparse regression

We propose an ℓ ₁-regularized likelihood method for estimating the inverse covariance matrix in the high-dimensional multivariate normal model in presence of missing data. Our method is based on the assumption that the data are missing at random (MAR) which entails also the completely missing at random case. The implementation of the method is non-trivial as the observed negative log-likelihood generally is a complicated and non-convex function. We propose an efficient EM algorithm for optimization with provable numerical convergence properties. Furthermore, we extend the methodology to handle missing values in a sparse regression context. We demonstrate both methods on simulated and real data.     

Variational Bayes with synthetic likelihood

Synthetic likelihood is an attractive approach to likelihood-free inference when an approximately Gaussian summary statistic for the data, informative for inference about the parameters, is available. The synthetic likelihood method derives an approximate likelihood function from a plug-in normal density estimate for the summary statistic, with plug-in mean and covariance matrix obtained by Monte Carlo simulation from the model. In this article, we develop alternatives to Markov chain Monte Carlo implementations of Bayesian synthetic likelihoods with reduced computational overheads. Our approach uses stochastic gradient variational inference methods for posterior approximation in the synthetic likelihood context, employing unbiased estimates of the log likelihood. We compare the new method with a related likelihood-free variational inference technique in the literature, while at the same time improving the implementation of that approach in a number of ways. These new algorithms are feasible to implement in situations which are challenging for conventional approximate Bayesian computation methods, in terms of the dimensionality of the parameter and summary statistic.     

Covariance Matrix Estimation via Network Structure

In this article, we employ a regression formulation to estimate the high-dimensional covariance matrix for a given network structure. Using prior information contained in the network relationships, we model the covariance as a polynomial function of the symmetric adjacency matrix. Accordingly, the problem of estimating a high-dimensional covariance matrix is converted to one of estimating low dimensional coefficients of the polynomial regression function, which we can accomplish using ordinary least squares or maximum likelihood. The resulting covariance matrix estimator based on the maximum likelihood approach is guaranteed to be positive definite even in finite samples. Under mild conditions, we obtain the theoretical properties of the resulting estimators. A Bayesian information criterion is also developed to select the order of the polynomial function. Simulation studies and empirical examples illustrate the usefulness of the proposed methods.     

Estimation and Inference in Two-Step Econometric Models

A commonly used procedure in a wide class of empirical applications is to impute unobserved regressors, such as expectations, from an auxiliary econometric model. This two-step (T-S) procedure fails to account for the fact that imputed regressors are measured with sampling error, so hypothesis tests based on the estimated covariance matrix of the second-step estimator are biased, even in large samples. We present a simple yet general method of calculating asymptotically correct standard errors in T-S models. The procedure may be applied even when joint estimation methods, such as full information maximum likelihood, are inappropriate or computationally infeasible. We present two examples from recent empirical literature in which these corrections have a major impact on hypothesis testing.     

Impact of missing data on type 1 error rates in non‐inferiority trials

In this paper, a simulation study is conducted to systematically investigate the impact of different types of missing data on six different statistical analyses: four different likelihood‐based linear mixed effects models and analysis of covariance (ANCOVA) using two different data sets, in non‐inferiority trial settings for the analysis of longitudinal continuous data. ANCOVA is valid when the missing data are completely at random. Likelihood‐based linear mixed effects model approaches are valid when the missing data are at random. Pattern‐mixture model (PMM) was developed to incorporate non‐random missing mechanism. Our simulations suggest that two linear mixed effects models using unstructured covariance matrix for within‐subject correlation with no random effects or first‐order autoregressive covariance matrix for within‐subject correlation with random coefficient effects provide well control of type 1 error (T1E) rate when the missing data are completely at random or at random. ANCOVA using last observation carried forward imputed data set is the worst method in terms of bias and T1E rate. PMM does not show much improvement on controlling T1E rate compared with other linear mixed effects models when the missing data are not at random but is markedly inferior when the missing data are at random. Copyright © 2009 John Wiley & Sons, Ltd.