Robust transformation mixed‐effects models for longitudinal continuous proportional data

Auxiliary covariates are often encountered in biomedical research settings where the primary exposure variable is measured only for a subgroup of study subjects. This article is concerned with generalized linear mixed models in the presence of auxiliary covariate information for clustered data. We propose a novel semiparametric estimation method based on a pairwise likelihood function and develop an estimating equation-based inference procedure by treating both the error structure and random effects as nuisance parameters. This method is robust against misspecification of either error structure or random-effects distribution and allows for dependence between random effects and covariates. We show that the resulting estimators are consistent and asymptotically normal. Extensive simulation studies evaluate the finite sample performance of the proposed estimators and demonstrate their advantage over the validation set based method and the existing method. We illustrate the method with two real data examples.

ABSTRACTIt often happens in longitudinal studies that repeated measurements of markers are observed with various data features of a heterogeneous population comprising of several subclasses, left-censoring due to a limit of detection (LOD) and covariates measured with error. Moreover, repeatedly measured markers in time may be associated with a time-to-event of interest. Inferential procedures may become very complicated when one analyzes data with these features together. This article explores a finite mixture of hierarchical joint models of event times and longitudinal measures with an attempt to alleviate departures from homogeneous characteristics, tailor observations below LOD as missing values, mediate accuracy from measurement error in covariate and overcome shortages of confidence in specifying a parametric time-to-event model with a nonparametric distribution. The Bayesian joint modeling is employed to not only estimate all parameters in mixture of joint models, but also evaluate probabilities of class membership. A real data example is analyzed to demonstrate the methodology by jointly modeling the viral dynamics and the time to decrease in CD4/CD8 ratio in the presence of CD4 cell counts with measurement error and the analytic results are reported by comparing potential models for various scenarios. Supplementary materials for this article are available online.

A semiparametric mixed effects regression model is proposed for the analysis of clustered or longitudinal data with continuous, ordinal, or binary outcome. The common assumption of Gaussian random effects is relaxed by using a predictive recursion method (Newton and Zhang, 1999) to provide a nonparametric smooth density estimate. A new strategy is introduced to accelerate the algorithm. Parameter estimates are obtained by maximizing the marginal profile likelihood by Powell's conjugate direction search method. Monte Carlo results are presented to show that the method can improve the mean squared error of the fixed effects estimators when the random effects distribution is not Gaussian. The usefulness of visualizing the random effects density itself is illustrated in the analysis of data from the Wisconsin Sleep Survey. The proposed estimation procedure is computationally feasible for quite large data sets.

Chapter 12 - Specific Methods

In Prequential analysis, an inference method is viewed as a forecasting system, and the quality of the inference method is based on the quality of its predictions. This is an alternative approach to more traditional statistical methods that focus on the inference of parameters of the data generating distribution. In this paper, we introduce adaptive combined average predictors (ACAPs) for the Prequential analysis of complex data. That is, we use convex combinations of two different model averages to form a predictor at each time step in a sequence. A novel feature of our strategy is that the models in each average are re-chosen adaptively at each time step. To assess the complexity of a given data set, we introduce measures of data complexity for continuous response data. We validate our measures in several simulated contexts prior to using them in real data examples. The performance of ACAPs is compared with the performances of predictors based on stacking or likelihood weighted averaging in several model classes and in both simulated and real data sets. Our results suggest that ACAPs achieve a better trade off between model list bias and model list variability in cases where the data is very complex. This implies that the choices of model class and averaging method should be guided by a concept of complexity matching, i.e. the analysis of a complex data set may require a more complex model class and averaging strategy than the analysis of a simpler data set. We propose that complexity matching is akin to a bias-variance tradeoff in statistical modeling.

ABSTRACTVarying-coefficient models are very useful for longitudinal data analysis. In this paper, we focus on varying-coefficient models for longitudinal data. We develop a new estimation procedure using Cholesky decomposition and profile least squares techniques. Asymptotic normality for the proposed estimators of varying-coefficient functions has been established. Monte Carlo simulation studies show excellent finite-sample performance. We illustrate our methods with a real data example.

Longitudinal designs for biomarker traits are very appealing for a number of reasons. First, the collection of repeated biomarker measurements over time allows investigators to characterize both the overall and individual trajectories in these measurements. This helps investigators to understand overall effects as well as how individuals vary around these average effects. Distinguishing individual heterogeneity from average trend is impossible without longitudinal data. This distinction is important for understanding individual between-subject in its own right, as well as in developing predictors of disease outcome from longitudinal marker data. There are many statistical issues that are unique to longitudinal data analysis ranging from correctly specifying realistic models which characterize average changes over time and individual variation, accounting for observations that are taken at unequally spaced observation times which may differ between individuals, to accounting for missing data due to occasionally missed measurements or loss to follow-up. When the reason for missing data is not related to the underlying observation process, most longitudinal data analysis methods can be routinely applied. However, complex modelling is necessary when the reason for missing an observation or dropout is directly related to the underlying outcome process [1–2]. Statistical techniques that appropriately analyze continuous and discrete longitudinal data have been discussed by many authors. Diggle, et al. [3] provides a good summary of these techniques for the practitioner, while Fitzmaurice et al. [4] discusses recent advances from a more technical perspective. The three papers in this section present innovative approaches for solving important methodological problems in biomarker studies. Most standard longitudinal data analysis techniques assume that the measurement process is independent of the outcome process. However, when the outcome is related to the underlying observation process, failure to properly account for the observation process can result in bias [5]. Schildcrout et al.[6] proposed an approach for regression modelling for longitudinal binary data where the outcome is measured with probabilities determined by an auxiliary variable that is related, but not identical to the binary outcome. Modelling multivariate longitudinal biomarker data is an increasingly common issue with the advent of an increasing number of important biomarker techniques (e.g., multiple cytokines in a multiplex assay or multiple reproductive hormones that describe complex biological phenomenon). Although various authors have proposed techniques for analyzing multivariate longitudinal data [7–8 and others], little work has focused on estimating multivariate linear mixed models under known constraints in the relationship between the multivariate outcomes. Roy et al [9] proposes such an approach for analyzing multivariate reproductive hormones over the menstrual cycle, where established relationships have solid biological justification. Finally, the collection of longitudinal biomarker data as disease predictors is becoming an increasingly important design in diagnostic medicine. Pfeiffer et al. [10] proposes flexible non-parametric methodology for combining biomarkers that exploits the longitudinal structure in formulating a score that is used for predicting a binary outcome.

Conventional approaches for handling missingness in substance use disorder trials commonly rely upon a single deterministic "worst value" imputation that posits a perfect relationship between missingness and drug use ("missing value = presumed drug use"); this yields biased estimates of treatment effects and their standard errors. Instead, deterministic imputations should be replaced by probabilistic versions that encode researchers prior beliefs that those with missing data are more likely to be using drugs at those occasions. Motivated by this problem, we present a method for handling non-monotone missing binary data in longitudinal studies. Specifically, we consider a joint model that combines a not missing at random (NMAR) selection model with a generalized linear mixed model for longitudinal binary data. The selection model links the distribution of a missing outcome to the corresponding distribution of the outcome for those observed at that occasion via a fixed and known sensitivity parameter. The mixed model for longitudinal binary data assumes the random effects have bridge distributions; the latter yields regression parameters that have both subject-specific and marginal interpretations. This approach is completely transparent about what is being assumed about missing data and can be used as the basis for sensitivity analysis.

ABSTRACTThe paper presents 3D diffraction imaging based on the spectral decomposition of a different combination of selective or partial images. These images are obtained by the pre‐stack asymmetric migration procedure, which is weighted data summation. Spectral decomposition is done in the Fourier domain with respect to spatial dip and azimuth angles. Numerical examples with the application of different workflows for the synthetic and real data examples demonstrate detailed reliable reconstruction of the fractured zones and reliable reconstruction of fracture orientation on synthetic and real 3D data examples.

In multivariate longitudinal HIV/AIDS studies, multi-outcome repeated measures on each patient over time may contain outliers, and the viral loads are often subject to a upper or lower limit of detection depending on the quantification assays. In this article, we consider an extension of the multivariate nonlinear mixed-effects model by adopting a joint multivariate-$t$ distribution for random effects and within-subject errors and taking the censoring information of multiple responses into account. The proposed model is called the multivariate-$t$ nonlinear mixed-effects model with censored responses (MtNLMMC), allowing for analyzing multi-outcome longitudinal data exhibiting nonlinear growth patterns with censorship and fat-tailed behavior. Utilizing the Taylor-series linearization method, a pseudo-data version of expectation conditional maximization either (ECME) algorithm is developed for iteratively carrying out maximum likelihood estimation. We illustrate our techniques with two data examples from HIV/AIDS studies. Experimental results signify that the MtNLMMC performs favorably compared to its Gaussian analogue and some existing approaches.

In this paper, we propose Bayesian analysis methods dealing with longitudinal data involving repeated binary outcomes on subjects with dropouts. The proposed Bayesian methods implement probit models with random effects to capture heterogeneity and hypersphere decomposition to model the correlation matrix for serial correlation of repeated responses. We investigate the model robustness against misspecifications of the probit models along with techniques to handle missing data. The parameters of the proposed models are estimated by implementing an Markov chain Monte Carlo (MCMC) algorithm, and simulations were performed to provide a comparison with other models and validate the choice of prior distributions. The simulations show that when suitable correlation structures are specified, the proposed approach improves estimation of the regression parameters in terms of the mean percent relative error and the mean squared error. Finally, two real data examples are provided to illustrate the proposed approach.

Abstract. The goodness‐of‐fit of the distribution of random effects in a generalized linear mixed model is assessed using a conditional simulation of the random effects conditional on the observations. Provided that the specified joint model for random effects and observations is correct, the marginal distribution of the simulated random effects coincides with the assumed random effects distribution. In practice, the specified model depends on some unknown parameter which is replaced by an estimate. We obtain a correction for this by deriving the asymptotic distribution of the empirical distribution function obtained from the conditional sample of the random effects. The approach is illustrated by simulation studies and data examples.

In analysis of longitudinal data, it is not uncommon that observation times of repeated measurements are subject-specific and correlated with underlying longitudinal outcomes. Taking account of the dependence between observation times and longitudinal outcomes is critical under these situations to assure the validity of statistical inference. In this article, we propose a flexible joint model for longitudinal data analysis in the presence of informative observation times. In particular, the new procedure considers the shared random-effect model and assumes a time-varying coefficient for the latent variable, allowing a flexible way of modeling longitudinal outcomes while adjusting their association with observation times. Estimating equations are developed for parameter estimation. We show that the resulting estimators are consistent and asymptotically normal, with variance-covariance matrix that has a closed form and can be consistently estimated by the usual plug-in method. One additional advantage of the procedure is that it provides a unified framework to test whether the effect of the latent variable is zero, constant, or time-varying. Simulation studies show that the proposed approach is appropriate for practical use. An application to a bladder cancer data is also given to illustrate the methodology.

Most deconvolution algorithms try to transform the seismic wavelet into spikes by designing inverse filters that remove an estimated seismic wavelet from seismic data. We assume that seismic trace subtle discontinuities are associated with acoustic impedance contrasts and can be characterized by wavelet transform spectral ridges, also called modulus maxima lines (WTMML), allowing us to improve seismic resolution by using the wavelet transform. Specifically, we apply the complex Morlet continuous wavelet transform (CWT) to each seismic trace and compute the WTMMLs. Then, we reconstruct the seismic trace with the inverse continuous wavelet transform from the computed WTMMLs with a broader band complex Morlet wavelet than that used in the forward CWT. Because the reconstruction process preserves amplitude and phase along different scales, or frequencies, the result looks like a deconvolution method. Considering this high-resolution seismic representation as a reflectivity approximation, we estimate the relative acoustic impedance (RAI) by filtering and trace integrating it. Conventional deconvolution algorithms assume the seismic wavelet to be stochastic, while the CWT is implicitly time varying such that it can be applied to both depth and time-domain data. Using synthetic and real seismic data, we evaluated the effectiveness of the methodology on detecting seismic events associated with acoustic impedance changes. In the real data examples, time and in-depth RAI results, show good correlation with real P-impedance band-pass data computed using more rigorous commercial inversion software packages that require well logs and low-frequency velocity model information.

Single index models are natural extensions of linear models and overcome the so-called curse of dimensionality. They are very useful for longitudinal data analysis. In this paper, we develop a new efficient estimation procedure for single index models with longitudinal data, based on Cholesky decomposition and local linear smoothing method. Asymptotic normality for the proposed estimators of both the parametric and nonparametric parts will be established. Monte Carlo simulation studies show excellent finite sample performance. Furthermore, we illustrate our methods with a real data example.