Empirical Bayes estimation of random effects parameters in mixed effects logistic regression models

We extend an approach for estimating random effects parameters under a random intercept and slope logistic regression model to include standard errors, thereby including confidence intervals. The procedure entails numerical integration to yield posterior empirical Bayes (EB) estimates of random effects parameters and their corresponding posterior standard errors. We incorporate an adjustment of the standard error due to Kass and Steffey (KS; 1989, Journal of the American Statistical Association 84, 717-726) to account for the variability in estimating the variance component of the random effects distribution. In assessing health care providers with respect to adult pneumonia mortality, comparisons are made with the penalized quasi-likelihood (PQL) approximation approach of Breslow and Clayton (1993, Journal of the American Statistical Association 88, 9-25) and a Bayesian approach. To make comparisons with an EB method previously reported in the literature, we apply these approaches to crossover trials data previously analyzed with the estimating equations EB approach of Waclawiw and Liang (1994, Statistics in Medicine 13, 541-551). We also perform simulations to compare the proposed KS and PQL approaches. These two approaches lead to EB estimates of random effects parameters with similar asymptotic bias. However, for many clusters with small cluster size, the proposed KS approach does better than the PQL procedures in terms of coverage of nominal 95% confidence intervals for random effects estimates. For large cluster sizes and a few clusters, the PQL approach performs better than the KS adjustment. These simulation results agree somewhat with those of the data analyses.     

Structured additive regression for categorical space-time data: a mixed model approach

Motivated by a space-time study on forest health with damage state of trees as the response, we propose a general class of structured additive regression models for categorical responses, allowing for a flexible semiparametric predictor. Nonlinear effects of continuous covariates, time trends, and interactions between continuous covariates are modeled by penalized splines. Spatial effects can be estimated based on Markov random fields, Gaussian random fields, or two-dimensional penalized splines. We present our approach from a Bayesian perspective, with inference based on a categorical linear mixed model representation. The resulting empirical Bayes method is closely related to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to inverse smoothing parameters, are estimated using (approximate) restricted maximum likelihood. In simulation studies we investigate the performance of different choices for the spatial effect, compare the empirical Bayes approach to competing methodology, and study the bias of mixed model estimates. As an application we analyze data from the forest health survey.     

A joint model for longitudinal measurements and survival data in the presence of multiple failure types

Summary .   In this article we study a joint model for longitudinal measurements and competing risks survival data. Our joint model provides a flexible approach to handle possible nonignorable missing data in the longitudinal measurements due to dropout. It is also an extension of previous joint models with a single failure type, offering a possible way to model informatively censored events as a competing risk. Our model consists of a linear mixed effects submodel for the longitudinal outcome and a proportional cause-specific hazards frailty submodel ( Prentice et al., 1978 , Biometrics 34, 541–554) for the competing risks survival data, linked together by some latent random effects. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their standard errors using a profile likelihood method. The developed method works well in our simulation studies and is applied to a clinical trial for the scleroderma lung disease.     

Improved Estimation of the Noncentrality Parameter Distribution from a Large Number of t‐Statistics,with Applications to False Discovery Rate Estimation in Microarray Data Analysis

Summary Given a large number of t‐statistics, we consider the problem of approximating the distribution of noncentrality parameters (NCPs) by a continuous density. This problem is closely related to the control of false discovery rates (FDR) in massive hypothesis testing applications, e.g., microarray gene expression analysis. Our methodology is similar to, but improves upon, the existing approach by Ruppert, Nettleton, and Hwang (2007, Biometrics, 63, 483–495). We provide parametric, nonparametric, and semiparametric estimators for the distribution of NCPs, as well as estimates of the FDR and local FDR. In the parametric situation, we assume that the NCPs follow a distribution that leads to an analytically available marginal distribution for the test statistics. In the nonparametric situation, we use convex combinations of basis density functions to estimate the density of the NCPs. A sequential quadratic programming procedure is developed to maximize the penalized likelihood. The smoothing parameter is selected with the approximate network information criterion. A semiparametric estimator is also developed to combine both parametric and nonparametric fits. Simulations show that, under a variety of situations, our density estimates are closer to the underlying truth and our FDR estimates are improved compared with alternative methods. Data‐based simulations and the analyses of two microarray datasets are used to evaluate the performance in realistic situations.     

A partial likelihood approach to smooth estimation of dynamic covariate effects using penalised splines

Survival data are often modelled by the Cox proportional hazards model, which assumes that covariate effects are constant over time. In recent years however, several new approaches have been suggested which allow covariate effects to vary with time. Non-proportional hazard functions, with covariate effects changing dynamically, can be fitted using penalised spline (P-spline) smoothing. By utilising the link between P-spline smoothing and generalised linear mixed models, the smoothing parameters steering the amount of smoothing can be selected. A hybrid routine, combining the mixed model approach with a classical Akaike criterion, is suggested. This approach is evaluated with simulations and applied to data from the West of Scotland Coronary Prevention Study.     

On Method of Moments Estimation in Linear Mixed Effects Models with Measurement Error on Covariates and Response with Application to a Longitudinal Study of Gene-Environment Interaction

We study a linear mixed effects model for longitudinal data, where the response variable and covariates with fixed effects are subject to measurement error. We propose a method of moment estimation that does not require any assumption on the functional forms of the distributions of random effects and other random errors in the model. For a classical measurement error model we apply the instrumental variable approach to ensure identifiability of the parameters. Our methodology, without instrumental variables, can be applied to Berkson measurement errors. Using simulation studies, we investigate the finite sample performances of the estimators and show the impact of measurement error on the covariates and the response on the estimation procedure. The results show that our method performs quite satisfactory, especially for the fixed effects with measurement error (even under misspecification of measurement error model). This method is applied to a real data example of a large birth and child cohort study.     

A Monte Carlo EM algorithm for generalized linear mixed models with flexible random effects distribution

A popular way to represent clustered binary, count, or other data is via the generalized linear mixed model framework, which accommodates correlation through incorporation of random effects. A standard assumption is that the random effects follow a parametric family such as the normal distribution; however, this may be unrealistic or too restrictive to represent the data. We relax this assumption and require only that the distribution of random effects belong to a class of 'smooth' densities and approximate the density by the seminonparametric (SNP) approach of Gallant and Nychka (1987). This representation allows the density to be skewed, multi-modal, fat- or thin-tailed relative to the normal and includes the normal as a special case. Because an efficient algorithm to sample from an SNP density is available, we propose a Monte Carlo EM algorithm using a rejection sampling scheme to estimate the fixed parameters of the linear predictor, variance components and the SNP density. The approach is illustrated by application to a data set and via simulation.     

Fixed and Random Effects Selection in Mixed Effects Models

Summary We consider selecting both fixed and random effects in a general class of mixed effects models using maximum penalized likelihood (MPL) estimation along with the smoothly clipped absolute deviation (SCAD) and adaptive least absolute shrinkage and selection operator (ALASSO) penalty functions. The MPL estimates are shown to possess consistency and sparsity properties and asymptotic normality. A model selection criterion, called the IC_Q statistic, is proposed for selecting the penalty parameters ( Ibrahim, Zhu, and Tang, 2008 , Journal of the American Statistical Association 103, 1648–1658). The variable selection procedure based on IC_Q is shown to consistently select important fixed and random effects. The methodology is very general and can be applied to numerous situations involving random effects, including generalized linear mixed models. Simulation studies and a real data set from a Yale infant growth study are used to illustrate the proposed methodology.     

Assessing differential gene expression with small sample sizes in oligonucleotide arrays using a mean-variance model

The identification of the genes that are differentially expressed in two-sample microarray experiments remains a difficult problem when the number of arrays is very small. We discuss the implications of using ordinary t-statistics and examine other commonly used variants. For oligonucleotide arrays with multiple probes per gene, we introduce a simple model relating the mean and variance of expression, possibly with gene-specific random effects. Parameter estimates from the model have natural shrinkage properties that guard against inappropriately small variance estimates, and the model is used to obtain a differential expression statistic. A limiting value to the positive false discovery rate (pFDR) for ordinary t-tests provides motivation for our use of the data structure to improve variance estimates. Our approach performs well compared to other proposed approaches in terms of the false discovery rate.     

On estimation and prediction for spatial generalized linear mixed models

We use spatial generalized linear mixed models (GLMM) to model non-Gaussian spatial variables that are observed at sampling locations in a continuous area. In many applications, prediction of random effects in a spatial GLMM is of great practical interest. We show that the minimum mean-squared error (MMSE) prediction can be done in a linear fashion in spatial GLMMs analogous to linear kriging. We develop a Monte Carlo version of the EM gradient algorithm for maximum likelihood estimation of model parameters. A by-product of this approach is that it also produces the MMSE estimates for the realized random effects at the sampled sites. This method is illustrated through a simulation study and is also applied to a real data set on plant root diseases to obtain a map of disease severity that can facilitate the practice of precision agriculture.     

Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate

We present the application of a nonparametric method to performing functional principal component analysis for functional curve data that consist of measurements of a random trajectory for a sample of subjects. This design typically consists of an irregular grid of time points on which repeated measurements are taken for a number of subjects. We introduce shrinkage estimates for the functional principal component scores that serve as the random effects in the model. Scatterplot smoothing methods are used to estimate the mean function and covariance surface of this model. We propose improved estimation in the neighborhood of and at the diagonal of the covariance surface, where the measurement errors are reflected. The presence of additive measurement errors motivates shrinkage estimates for the functional principal component scores. Shrinkage estimates are developed through best linear prediction and in a generalized version, aiming at minimizing one-curve-leave-out prediction error. The estimation of individual trajectories combines data obtained from that individual as well as all other individuals. We apply our methods to new data regarding the analysis of the level of 14C-folate in plasma as a function of time since dosing of healthy adults with a small tracer dose of 14C-folic acid. A time transformation was incorporated to handle design irregularity concerning the time points on which the measurements were taken. The proposed methodology, incorporating shrinkage and data-adaptive features, is seen to be well suited for describing population kinetics of 14C-folate-specific activity and random effects, and can also be applied to other functional data analysis problems.     

Application of the covariance function approach with an iterative two-stage algorithm to the estimation of parameters of a random regression test day model for dairy production traits

The covariance function approach with an iterative two-stage algorithm of LIU et al. (2000) was applied to estimate parameters for the Polish Black-and-White dairy population based on a sample of 338 808 test day records for milk, fat, and protein yields. A multiple trait sire model was used to estimate covariances of lactation stages. A third-order Legendre polynomial was subsequently fitted to the estimated (co)variances to derive (co)variances of random regression coefficients for both additive genetic and permanent environment effects. Daily and 305-day heritability estimates obtained are consistent with several studies which used both fixed and random regression test day models. Genetic correlations between any two days in milk (DIM) of the same lactation as well as genetic correlations between the same DIM of two lactations were within a biologically acceptable range. It was shown that the applied estimation procedure can utilise very large data sets and give plausible estimates of (co)variance components.     

Comments about Joint Modeling of Cluster Size and Binary and Continuous Subunit-Specific Outcomes

In longitudinal studies and in clustered situations often binary and continuous response variables are observed and need to be modeled together. In a recent publication Dunson, Chen, and Harry (2003, Biometrics 59, 521-530) (DCH) propose a Bayesian approach for joint modeling of cluster size and binary and continuous subunit-specific outcomes and illustrate this approach with a developmental toxicity data example. In this note we demonstrate how standard software (PROC NLMIXED in SAS) can be used to obtain maximum likelihood estimates in an alternative parameterization of the model with a single cluster-level factor considered by DCH for that example. We also suggest that a more general model with additional cluster-level random effects provides a better fit to the data set. An apparent discrepancy between the estimates obtained by DCH and the estimates obtained earlier by Catalano and Ryan (1992, Journal of the American Statistical Association 87, 651-658) is also resolved. The issue of bias in inferences concerning the dose effect when cluster size is ignored is discussed. The maximum-likelihood approach considered herein is applicable to general situations with multiple clustered or longitudinally measured outcomes of different type and does not require prior specification and extensive programming.     

A model-based approach to study nearest-neighbor influences reveals complex substitution patterns in non-coding sequences

In this article, we present a likelihood-based framework for modeling site dependencies. Our approach builds upon standard evolutionary models but incorporates site dependencies across the entire tree by letting the evolutionary parameters in these models depend upon the ancestral states at the neighboring sites. It thus avoids the need for introducing new and high-dimensional evolutionary models for site-dependent evolution. We propose a Markov chain Monte Carlo approach with data augmentation to infer the evolutionary parameters under our model. Although our approach allows for wide-ranging site dependencies, we illustrate its use, in two non-coding datasets, in the case of nearest-neighbor dependencies (i.e., evolution directly depending only upon the immediate flanking sites). The results reveal that the general time-reversible model with nearest-neighbor dependencies substantially improves the fit to the data as compared to the corresponding model with site independence. Using the parameter estimates from our model, we elaborate on the importance of the 5-methylcytosine deamination process (i.e., the CpG effect) and show that this process also depends upon the 5' neighboring base identity. We hint at the possibility of a so-called TpA effect and show that the observed substitution behavior is very complex in the light of dinucleotide estimates. We also discuss the presence of CpG effects in a nuclear small subunit dataset and find significant evidence that evolutionary models incorporating context-dependent effects perform substantially better than independent-site models and in some cases even outperform models that incorporate varying rates across sites.     

A general maximum likelihood analysis of variance components in generalized linear models

This paper describes an EM algorithm for nonparametric maximum likelihood (ML) estimation in generalized linear models with variance component structure. The algorithm provides an alternative analysis to approximate MQL and PQL analyses (McGilchrist and Aisbett, 1991, Biometrical Journal 33, 131-141; Breslow and Clayton, 1993; Journal of the American Statistical Association 88, 9-25; McGilchrist, 1994, Journal of the Royal Statistical Society, Series B 56, 61-69; Goldstein, 1995, Multilevel Statistical Models) and to GEE analyses (Liang and Zeger, 1986, Biometrika 73, 13-22). The algorithm, first given by Hinde and Wood (1987, in Longitudinal Data Analysis, 110-126), is a generalization of that for random effect models for overdispersion in generalized linear models, described in Aitkin (1996, Statistics and Computing 6, 251-262). The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully nonparametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters can be sensitive to the specification of a parametric form for the mixing distribution. The nonparametric analysis can be extended straightforwardly to general random parameter models, with full NPML estimation of the joint distribution of the random parameters. This can produce substantial computational saving compared with full numerical integration over a specified parametric distribution for the random parameters. A simple method is described for obtaining correct standard errors for parameter estimates when using the EM algorithm. Several examples are discussed involving simple variance component and longitudinal models, and small-area estimation.     

Semiparametric frailty models for clustered failure time data

We consider frailty models with additive semiparametric covariate effects for clustered failure time data. We propose a doubly penalized partial likelihood (DPPL) procedure to estimate the nonparametric functions using smoothing splines. We show that the DPPL estimators could be obtained from fitting an augmented working frailty model with parametric covariate effects, whereas the nonparametric functions being estimated as linear combinations of fixed and random effects, and the smoothing parameters being estimated as extra variance components. This approach allows us to conveniently estimate all model components within a unified frailty model framework. We evaluate the finite sample performance of the proposed method via a simulation study, and apply the method to analyze data from a study of sexually transmitted infections (STI).     

Conditional-cumulant-of-exposure method in logistic missing covariate regression

We consider estimation in logistic regression where some covariate variables may be missing at random. Satten and Kupper (1993, Journal of the American Statistical Association 88, 200-208) proposed estimating odds ratio parameters using methods based on the probability of exposure. By approximating a partial likelihood, we extend their idea and propose a method that estimates the cumulant-generating function of the missing covariate given observed covariates and surrogates in the controls. Our proposed method first estimates some lower order cumulants of the conditional distribution of the unobserved data and then solves a resulting estimating equation for the logistic regression parameter. A simple version of the proposed method is to replace a missing covariate by the summation of its conditional mean and conditional variance given observed data in the controls. We note that one important property of the proposed method is that, when the validation is only on controls, a class of inverse selection probability weighted semiparametric estimators cannot be applied because selection probabilities on cases are zeroes. The proposed estimator performs well unless the relative risk parameters are large, even though it is technically inconsistent. Small-sample simulations are conducted. We illustrate the method by an example of real data analysis.     

QUANTITATIVE GENETIC MODELING AND INFERENCE IN THE PRESENCE OF NONIGNORABLE MISSING DATA

Natural selection is typically exerted at some specific life stages. If natural selection takes place before a trait can be measured, using conventional models can cause wrong inference about population parameters. When the missing data process relates to the trait of interest, a valid inference requires explicit modeling of the missing process. We propose a joint modeling approach, a shared parameter model, to account for nonrandom missing data. It consists of an animal model for the phenotypic data and a logistic model for the missing process, linked by the additive genetic effects. A Bayesian approach is taken and inference is made using integrated nested Laplace approximations. From a simulation study we find that wrongly assuming that missing data are missing at random can result in severely biased estimates of additive genetic variance. Using real data from a wild population of Swiss barn owls Tyto alba, our model indicates that the missing individuals would display large black spots; and we conclude that genes affecting this trait are already under selection before it is expressed. Our model is a tool to correctly estimate the magnitude of both natural selection and additive genetic variance.     

An Approximate EM Algorithm for Nonlinear Mixed Effects Models

In this article, we construct an approximate EM algorithm to estimate the parameters of a nonlinear mixed effects model. The iterative procedure can be viewed as an iterative method of moments procedure for estimating the variance components and an iterative reweighted least squares estimates for estimating the fixed effects. Therefore, it is valid without the normality assumptions on the random components. A computationally simple method of moments estimates of the model parameters are used as the starting values for our iterative procedure. A simulation study was conducted to compare the performances of the proposed procedure with the procedure proposed by Lindstrom and Bates (1990) for some normal models and nonnormal models.