Approximate Likelihood Inference Research Articles

Approximate Likelihood Inference in Generalized Linear Models with Censored Covariates

In many surveys and clinical trials, we obtain measurements on covariates or biomarkers that are left-censored due to the limit of detection. In such cases, it is necessary to correct for the left-censoring when studying covariate effects in regression models. The expectation-maximization (EM) algorithm is widely used for the likelihood inference in generalized linear models with censored covariates. The EM method, however, requires intensive computation involving high-dimensional integration with respect to the covariates when the dimension of the censored covariates is large. To reduce such computational difficulties, we propose and explore a Monte Carlo EM method based on the Metropolis algorithm. The finite-sample properties of the proposed estimators are studied using Monte Carlo simulations. An application is also provided using actual data obtained from a health and nutrition examination survey. Journal of Statistical Research 2021, Vol. 55, No. 2, pp. 359-375

Ell _2-penalized approximate likelihood inference in logit mixed models for regional prevalence estimation under covariate rank-deficiency

Regional prevalence estimation requires the use of suitable statistical methods on epidemiologic data with substantial local detail. Small area estimation with medical treatment records as covariates marks a promising combination for this purpose. However, medical routine data often has strong internal correlation due to diagnosis-related grouping in the records. Depending on the strength of the correlation, the space spanned by the covariates can become rank-deficient. In this case, prevalence estimates suffer from unacceptable uncertainty as the individual contributions of the covariates to the model cannot be identified properly. We propose an area-level logit mixed model for regional prevalence estimation with a new fitting algorithm to solve this problem. We extend the Laplace approximation to the log-likelihood by an ell _2-penalty in order to stabilize the estimation process in the presence of covariate rank-deficiency. Empirical best predictors under the model and a parametric bootstrap for mean squared error estimation are presented. A Monte Carlo simulation study is conducted to evaluate the properties of our methodology in a controlled environment. We further provide an empirical application where the district-level prevalence of multiple sclerosis in Germany is estimated using health insurance records.

Robust prediction of domain compositions from uncertain data using isometric logratio transformations in a penalized multivariate Fay–Herriot model

Assessing regional population compositions is an important task in many research fields. Small area estimation with generalized linear mixed models marks a powerful tool for this purpose. However, the method has limitations in practice. When the data are subject to measurement errors, small area models produce inefficient or biased results since they cannot account for data uncertainty. This is particularly problematic for composition prediction, since generalized linear mixed models often rely on approximate likelihood inference. Obtained predictions are not reliable. We propose a robust multivariate Fay–Herriot model to solve these issues. It combines compositional data analysis with robust optimization theory. The nonlinear estimation of compositions is restated as a linear problem through isometric logratio transformations. Robust model parameter estimation is performed via penalized maximum likelihood. A robust best predictor is derived. Simulations are conducted to demonstrate the effectiveness of the approach. An application to alcohol consumption in Germany is provided.

Joint estimation of heterogeneous exponential Markov Random Fields through an approximate likelihood inference

In biomedical research, increasing attention has been paid to the discovery of regulatory relationships among heterogeneous biological features. We present a new statistical framework to jointly learn multiple heterogeneous exponential Markov Random Fields. We establish an approximate likelihood inference problem regularized by an embedded group lasso penalty, and propose an efficient algorithm in the Alternating Direction Method of Multipliers framework. We also establish structure recovery consistency for the proposed joint network learning. The practical merits of the proposed integrative structural learning method are demonstrated through simulations and real applications to discovering regulatory relationships among heterogeneous biological variables from distinct but related types of cancer.

Approximate likelihood inference in spatial generalized linear mixed models with closed skew normal latent variables

Spatial generalized linear mixed models are used commonly for modeling discrete spatial responses, where spatial correlation of the data is considered via latent variables which follow the normal distribution. From a computational point of view, the normal assumption for latent variables are considered just for the convenience of calculations. In this paper, the closed skew normal distribution which is more flexible and includes the normal distribution is considered for the spatial latent variables. A new approximate algorithm is introduced to obtain maximum likelihood estimates of the parameters. Prediction of the latent variables at a new unsampled location is obtained using a new approximation minimum mean square error method. The performance of the proposed method is illustrated on a simulation study and on a real data set.

A joint model for mixed and truncated longitudinal data and survival data, with application to HIV vaccine studies.

In HIV vaccine studies, a major research objective is to identify immune response biomarkers measured longitudinally that may be associated with risk of HIV infection. This objective can be assessed via joint modeling of longitudinal and survival data. Joint models for HIV vaccine data are complicated by the following issues: (i) left truncations of some longitudinal data due to lower limits of quantification; (ii) mixed types of longitudinal variables; (iii) measurement errors and missing values in longitudinal measurements; (iv) computational challenges associated with likelihood inference. In this article, we propose a joint model of complex longitudinal and survival data and a computationally efficient method for approximate likelihood inference to address the foregoing issues simultaneously. In particular, our model does not make unverifiable distributional assumptions for truncated values, which is different from methods commonly used in the literature. The parameters are estimated based on the h-likelihood method, which is computationally efficient and offers approximate likelihood inference. Moreover, we propose a new approach to estimate the standard errors of the h-likelihood based parameter estimates by using an adaptive Gauss-Hermite method. Simulation studies show that our methods perform well and are computationally efficient. A comprehensive data analysis is also presented.

Approximate likelihood inference in generalized linear latent variable models based on the dimension-wise quadrature

We propose a new method to perform approximate likelihood inference in latent variable models. Our approach provides an approximation of the integrals involved in the likelihood function through a reduction of their dimension that makes the computation feasible in situations in which classical and adaptive quadrature based methods are not applicable. We derive new theoretical results on the accuracy of the obtained estimators. We show that the proposed approximation outperforms several existing methods in simulations, and it can be successfully applied in presence of multidimensional longitudinal data when standard techniques are not applicable or feasible.

The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes

The Pearson diffusions is a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. Explicit optimal martingale estimating functions are found, and the corresponding estimators are shown to be consistent and asymptotically normal. The discussion covers GMM, quasi-likelihood, and non-linear weighted least squares estimation too, and it is discussed how explicit likelihood or approximate likelihood inference is possible for the Pearson diffusions. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well-known instances are the Ornstein-Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy-tailed and skew marginals are included. Special attention is given to a skew t-type distribution. Explicit formula for the conditional moments and the polynomial eigenfunctions are derived. The analytical tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions, and stochastic volatility models with Pearson volatility process. For the non-Markov models explicit optimal prediction based estimating functions are found and shown to yield consistent and asymptotically normal estimators.

A computationally efficient method for nonlinear mixed-effects models with nonignorable missing data in time-varying covariates

Nonlinear mixed-effects (NLME) models are widely used for longitudinal data analyses. Time-dependent covariates are often introduced to partially explain inter-individual variation. These covariates often have missing data, and the missingness may be nonignorable. Likelihood inference for NLME models with nonignorable missing data in time-varying covariates can be computationally very intensive and may even offer computational difficulties such as nonconvergence. We propose a computationally very efficient method for approximate likelihood inference. The method is illustrated using a real data example.

ON PSEUDO-LIKELIHOOD INFERENCE IN THE BINARY LONGITUDINAL MIXED MODEL

ABSTRACT Binary logistic and Poisson mixed models are used to analyse over/under-dispersed proportion and count data, respectively. It is, however, well known that a full likelihood analysis for such mixed models is hampered by the need for numerical integrations. To overcome such integration problems, recently Sutradhar and Qu (On Approximate Likelihood Inference in Poisson Mixed Model. The Canadian Journal of Statistics 1998, 26, 169–186) has introduced a small variance component (for random effects) based likelihood approximation (LA) approach to estimate the parameters of the Poisson mixed models and have shown that their LA approach performs better as compared to other leading approaches. More recently, Sutradhar and Das (A Higher-Order Approximation to the Likelihood Inference in the Poisson Mixed Model. Statistics and Probability Letters 2001, 52, 59–67) further improved the LA approach of Sutradhar and Qu to accommodate larger values of the variance component. These likelihood approximation techniques developed for Poisson mixed models are however not applicable to the binary mixed models. In this paper, we propose a multivariate binary distribution based pseudo-likelihood approach for the estimation of the parameters of the binary mixed models. We, in fact, do this in a wider binary longitudinal mixed model set up, binary mixed model being a special case. More specifically, two types of binary longitudinal mixed models are considered. Under the first model, conditional on certain independent random effects, repeated binary responses are assumed to follow a Bahadur type multivariate binary distribution, so that, unconditionally, the responses in the cluster follow a longitudinal binary mixed model. Under the second model, however, the binary responses in the cluster are assumed to be conditionally independent, conditional on certain correlated random effects, so that, unconditionally, responses in the cluster also follow a binary longitudinal mixed model. It is of primary interest to estimate the regression and the variance component parameters of the binary longitudinal mixed model, longitudinal correlation parameters being nuisance. The performance of the proposed pseudo-likelihood based estimators is examined through a simulation study. A comparison is also made with a highly competitive generalized estimating equation (GEE) approach, especially for the estimation of the variance component of the random effects.

On approximate likelihood inference in a poisson mixed model

AbstractA two‐step estimation approach is proposed for the fixed‐effect parameters, random effects and their variance σ2 of a Poisson mixed model. In the first step, it is proposed to construct a small σ2‐based approximate likelihood function of the data and utilize this function to estimate the fixed‐effect parameters and σ2. In the second step, the random effects are estimated by minimizing their posterior mean squared error. Methods of Waclawiw and Liang (1993) based on so‐called Stein‐type estimating functions and of Breslow and Clayton (1993) based on penalized quasilikelihood are compared with the proposed likelihood method. The results of a simulation study on the performance of all three approaches are reported.