Estimating negative variance components from Gaussian and non-Gaussian data: A mixed models approach

Negative variance components and intercept-slope correlations greater than one in magnitude: How do such "non-regular" random intercept and slope models arise, and what should be done when they do?

Statistical models with random intercepts and slopes (RIAS models) are commonly used to analyze longitudinal data. Fitting such models sometimes results in negative estimates of variance components or estimates on parameter space boundaries. This can be an unlucky chance occurrence, but can also occur because certain marginal distributions are mathematically identical to those from RIAS models with negative intercept and/or slope variance components and/or intercept-slope correlations greater than one in magnitude. We term such parameters "pseudo-variances" and "pseudo-correlations," and the models "non-regular." We use eigenvalue theory to explore how and when such non-regular RIAS models arise, showing: (i) A small number of measurements, short follow-up, and large residual variance increase the parameter space for which data (with a positive semidefinite marginal variance-covariance matrix) are compatible with non-regular RIAS models. (ii) Non-regular RIAS models can arise from model misspecification, when non-linearity in fixed effects is ignored or when random effects are omitted. (iii) A non-regular RIAS model can sometimes be interpreted as a regular linear mixed model with one or more additional random effects, which may not be identifiable from the data. (iv) Particular parameterizations of non-regular RIAS models have no generality for all possible numbers of measurements over time. Because of this lack of generality, we conclude that non-regular RIAS models can only be regarded as plausible data-generating mechanisms in some situations. Nevertheless, fitting a non-regular RIAS model can be acceptable, allowing unbiased inference on fixed effects where commonly recommended alternatives such as dropping the random slope result in bias.

A Robust Hierarchical Estimation Scheme for Vehicle State Based on Maximum Correntropy Square-Root Cubature Kalman Filter

Accurate acquisition of vehicle dynamics state information is essential for vehicle active safety control systems. However, these states cannot be easily measured, and the measurement is expensive. Conventional Kalman filters perform well for vehicle state estimation in Gaussian environments but exhibit low accuracy and robustness under practical non-Gaussian noise. Vehicle model parameter ingestion, inaccurate tire force calculation, and non-Gaussian noise from on-board sensors cause great challenges to the estimation of vehicle driving states. Therefore, this paper presents a robust hierarchical estimation scheme for vehicle driving state based on the maximum correntropy square-root cubature Kalman filter (MCSCKF) using easily measurable on-board sensor information. First, the vehicle mass is dynamically updated based on the recursive least squares (FRLS) method with a forgetting factor. Then, an adaptive sliding mode observer (ASMO) is designed to estimate the longitudinal and lateral tire forces. Ultimately, the vehicle states are estimated based on the MCSCKF under non-Gaussian noise. Two typical operating situations are carried out to verify the validity of the proposed estimation scheme. The results prove that the proposed estimation scheme can estimate the vehicle's driving state accurately compared to other common methods. And the MCSCKF algorithm has better accuracy and robustness than the traditional Kalman filters for vehicle state estimation in non-Gaussian situations.

Modeling Dependence Structures for Response Times in a Bayesian Framework

A multivariate generalization of the log-normal model for response times is proposed within an innovative Bayesian modeling framework. A novel Bayesian Covariance Structure Model (BCSM) is proposed, where the inclusion of random-effect variables is avoided, while their implied dependencies are modeled directly through an additive covariance structure. This makes it possible to jointly model complex dependencies due to for instance the test format (e.g., testlets, complex constructs), time limits, or features of digitally based assessments. A class of conjugate priors is proposed for the random-effect variance parameters in the BCSM framework. They give support to testing the presence of random effects, reduce boundary effects by allowing non-positive (co)variance parameters, and support accurate estimation even for very small true variance parameters. The conjugate priors under the BCSM lead to efficient posterior computation. Bayes factors and the Bayesian Information Criterion are discussed for the purpose of model selection in the new framework. In two simulation studies, a satisfying performance of the MCMC algorithm and of the Bayes factor is shown. In comparison with parameter expansion through a half-Cauchy prior, estimates of variance parameters close to zero show no bias and undercoverage of credible intervals is avoided. An empirical example showcases the utility of the BCSM for response times to test the influence of item presentation formats on the test performance of students in a Latin square experimental design.

A Weibull-count approach for handling under- and overdispersed longitudinal/clustered data structures

A Weibull-model-based approach is examined to handle under- and overdispersed discrete data in a hierarchical framework. This methodology was first introduced by Nakagawa and Osaki (1975, IEEE Transactions on Reliability, 24, 300–301), and later examined for under- and overdispersion by Klakattawi et al. (2018, Entropy, 20, 142) in the univariate case. Extensions to hierarchical approaches with under- and overdispersion were left unnoted, even though they can be obtained in a simple manner. This is of particular interest when analysing clustered/longitudinal data structures, where the underlying correlation structure is often more complex compared to cross-sectional studies. In this article, a random-effects extension of the Weibull-count model is proposed and applied to two motivating case studies, originating from the clinical and sociological research fields. A goodness-of-fit evaluation of the model is provided through a comparison of some well-known count models, that is, the negative binomial, Conway–Maxwell–Poisson and double Poisson models. Empirical results show that the proposed extension flexibly fits the data, more specifically, for heavy-tailed, zero-inflated, overdispersed and correlated count data. Discrete left-skewed time-to-event data structures are also flexibly modelled using the approach, with the ability to derive direct interpretations on the median scale, provided the complementary log–log link is used. Finally, a large simulated set of data is created to examine other characteristics such as computational ease and orthogonality properties of the model, with the conclusion that the approach behaves best for highly overdispersed cases.

An overdispersion model in meta-analysis

We propose a general approach based on the concept of overdispersion for specification of a random effects model (REM) in meta-analysis. This approach is similar to that used in generalized linear models, and includes the traditional REM as a particular case. A key feature of the model is the interpretation of the multiplicative factor as an intra-class correlation parameter. We provide several motivating examples, discuss statistical inference, and compare the new and standard methods on two examples of published meta-analyses. Estimation of the overdispersion parameter in the proposed model is compared in simulations to that of the traditional between-studies variance in the case of normal means. For small values of heterogeneity, the coverage of the confidence intervals for the overdispersion parameter is more stable.

Adaptive Fitting of Linear Mixed-Effects Models with Correlated Random-effects

Linear mixed-effects model has been widely used in longitudinal data analyses. In practice, the fitting algorithm can fail to converge due to boundary issues of the estimated random-effects covariance matrix G, i.e., being near-singular, non-positive definite, or both. Current available algorithms are not computationally optimal because the condition number of matrix G is unnecessarily increased when the random-effects correlation estimate is not zero. We propose an adaptive fitting (AF) algorithm using an optimal linear transformation of the random-effects design matrix. It is a data-driven adaptive procedure, aiming at reducing subsequent random-effects correlation estimates down to zero in the optimal transformed estimation space. Simulations show that AF significantly improves the convergent properties, especially under small sample size, relative large noise and high correlation settings. One real data for Insulin-like Growth Factor (IGF) protein is used to illustrate the application of this algorithm implemented with software package R (nlme).