On the Behaviour of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models

In linear mixed models, model selection frequently includes the selection of random effects. Two versions of the Akaike information criterion (AIC) have been used, based either on the marginal or on the conditional distribution. We show that the marginal AIC is no longer an asymptotically unbiased estimator of the Akaike information, and in fact favours smaller models without random effects. For the conditional AIC, we show that ignoring estimation uncertainty in the random effects covariance matrix, as is common practice, induces a bias that leads to the selection of any random effect not predicted to be exactly zero. We derive an analytic representation of a corrected version of the conditional AIC, which avoids the high computational cost and imprecision of available numerical approximations. An implementation in an R package is provided. All theoretical results are illustrated in simulation studies, and their impact in practice is investigated in an analysis of childhood malnutrition in Zambia.

FUNCTIONAL PRINCIPAL COMPONENTS MODEL FOR HIGH-DIMENSIONAL BRAIN IMAGING

We establish a fundamental equivalence between singular value decomposition (SVD) and functional principal components analysis (FPCA) models. The constructive relationship allows to deploy the numerical efficiency of SVD to fully estimate the components of FPCA, even for extremely high-dimensional functional objects, such as brain images. As an example, a functional mixed effect model is fitted to high-resolution morphometric (RAVENS) images. The main directions of morphometric variation in brain volumes are identified and discussed.

Checking Assumptions in Latent Class Regression Models via a Markov Chain Monte Carlo Estimation Approach: An Application to Depression and Socio-Economic Status

Latent class regression models are useful tools for assessing associations between covariates and latent variables. However, evaluation of key model assumptions cannot be performed using methods from standard regression models due to the unobserved nature of latent outcome variables. This paper presents graphical diagnostic tools to evaluate whether or not latent class regression models adhere to standard assumptions of the model: conditional independence and non-differential measurement. An integral part of these methods is the use of a Markov Chain Monte Carlo estimation procedure. Unlike standard maximum likelihood implementations for latent class regression model estimation, the MCMC approach allows us to calculate posterior distributions and point estimates of any functions of parameters. It is this convenience that allows us to provide the diagnostic methods that we introduce. As a motivating example we present an analysis focusing on the association between depression and socioeconomic status, using data from the Epidemiologic Catchment Area study. We consider a latent class regression analysis investigating the association between depression and socioeconomic status measures, where the latent variable depression is regressed on education and income indicators, in addition to age, gender, and marital status variables. While the fitted latent class regression model yields interesting results, the model parameters are found to be invalid due to the violation of model assumptions. The violation of these assumptions is clearly identified by the presented diagnostic plots. These methods can be applied to standard latent class and latent class regression models, and the general principle can be extended to evaluate model assumptions in other types of models.

Studying Effects of Primary Care Physicians and Patients on the Trade-Off Between Charges for Primary Care and Specialty Care Using a Hierarchical Multivariate Two-Part Model

Objective. To examine effects of primary care physicians (PCPs) and patients on the association between charges for primary care and specialty care in a point-of-service (POS) health plan. Data Source. Claims from 1996 for 3,308 adult male POS plan members, each of whom was assigned to one of the 50 family practitioner-PCPs with the largest POS plan member-loads. Study Design. A hierarchical multivariate two-part model was fitted using a Gibbs sampler to estimate PCPs' effects on patients' annual charges for two types of services, primary care and specialty care, the associations among PCPs' effects, and within-patient associations between charges for the two services. Adjusted Clinical Groups (ACGs) were used to adjust for case-mix. Principal Findings. PCPs with higher case-mix adjusted rates of specialist use were less likely to see their patients at least once during the year (estimated correlation: –.40; 95% CI: –.71, –.008) and provided fewer services to patients that they saw (estimated correlation: –.53; 95% CI: –.77, –.21). Ten of 11 PCPs whose case-mix adjusted effects on primary care charges were significantly less than or greater than zero (p < .05) had estimated, case-mix adjusted effects on specialty care charges that were of opposite sign (but not significantly different than zero). After adjustment for ACG and PCP effects, the within-patient, estimated odds ratio for any use of primary care given any use of specialty care was .57 (95% CI: .45, .73). Conclusions. PCPs and patients contributed independently to a trade-off between utilization of primary care and specialty care. The trade-off appeared to partially offset significant differences in the amount of care provided by PCPs. These findings were possible because we employed a hierarchical multivariate model rather than separate univariate models.

MULTIPLE MODEL EVALUATION ABSENT THE GOLD STANDARD VIA MODEL COMBINATION

We describe a method for evaluating an ensemble of predictive models given a sample of observations comprising the model predictions and the outcome event measured with error. Our formulation allows us to simultaneously estimate measurement error parameters, true outcome — aka the gold standard — and a relative weighting of the predictive scores. We describe conditions necessary to estimate the gold standard and for these estimates to be calibrated and detail how our approach is related to, but distinct from, standard model combination techniques. We apply our approach to data from a study to evaluate a collection of BRCA1/BRCA2 gene mutation prediction scores. In this example, genotype is measured with error by one or more genetic assays. We estimate true genotype for each individual in the dataset, operating characteristics of the commonly used genotyping procedures and a relative weighting of the scores. Finally, we compare the scores against the gold standard genotype and find that Mendelian scores are, on average, the more refined and better calibrated of those considered and that the comparison is sensitive to measurement error in the gold standard.

PENALIZED FUNCTIONAL REGRESSION

We develop fast fitting methods for generalized functional linear models. An undersmooth of the functional predictor is obtained by projecting on a large number of smooth eigenvectors and the coefficient function is estimated using penalized spline regression. Our method can be applied to many functional data designs including functions measured with and without error, sparsely or densely sampled. The methods also extend to the case of multiple functional predictors or functional predictors with a natural multilevel structure. Our approach can be implemented using standard mixed effects software and is computationally fast. Our methodology is motivated by a diffusion tensor imaging (DTI) study. The aim of this study is to analyze differences between various cerebral white matter tract property measurements of multiple sclerosis (MS) patients and controls. While the statistical developments proposed here were motivated by the DTI study, the methodology is designed and presented in generality and is applicable to many other areas of scientific research. An online appendix provides R implementations of all simulations.

Fixed-Width Output Analysis for Markov Chain Monte Carlo

Markov chain Monte Carlo is a method of producing a correlated sample in order to estimate features of a complicated target distribution via simple ergodic averages. A fundamental question in MCMC applications is when should the sampling stop? That is, when are the ergodic averages good estimates of the desired quantities? We consider a method that stops the MCMC sampling the first time the width of a confidence interval based on the ergodic averages is less than a user-specified value. Hence calculating Monte Carlo standard errors is a critical step in assessing the output of the simulation. In particular, we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution. We describe sufficient conditions for the strong consistency and asymptotic normality of both methods and investigate their finite sample properties in a variety of examples.

Nonparametric Estimation of the Bivariate Recurrence Time Distribution

This paper considers statistical models in which two different types of events, such as the diagnosis of a disease and the remission of the disease, occur alternately over time and are observed subject to right censoring. We propose nonparametric estimators for the joint distribution of bivariate recurrence times and the marginal distribution of the first recurrence time. In general, the marginal distribution of the second recurrence time cannot be estimated due to an identifiability problem, but a conditional distribution of the second recurrence time can be estimated non-parametrically. In literature, statistical methods have been developed to estimate the joint distribution of bivariate recurrence times based on data of the first pair of censored bivariate recurrence times. These methods are efficient in the current model because recurrence times of higher orders are not used. Asymptotic properties of the estimators are established. Numerical studies demonstrate the estimator performs well with practical sample sizes. We apply the proposed method to a Denmark psychiatric case register data set for illustration of the methods and theory.

Kernel Estimation of Rate Function for Recurrent Event Data

Recurrent event data are largely characterized by the rate function but smoothing techniques for estimating the rate function have never been rigorously developed or studied in statistical literature. This paper considers the moment and least squares methods for estimating the rate function from recurrent event data. With an independent censoring assumption on the recurrent event process, we study statistical properties of the proposed estimators and propose bootstrap procedures for the bandwidth selection and for the approximation of confidence intervals in the estimation of the occurrence rate function. It is identified that the moment method without resmoothing via a smaller bandwidth will produce curve with nicks occurring at the censoring times, whereas there is no such problem with the least squares method. Furthermore, the asymptotic variance of the least squares estimator is shown to be smaller under regularity conditions. However, in the implementation of the bootstrap procedures, the moment method is computationally more efficient than the least squares method because the former approach uses condensed bootstrap data. The performance of the proposed procedures is studied through Monte Carlo simulations and an epidemiological example on intravenous drug users.

Large Sample Theory for Semiparametric Regression Models with Two-Phase, Outcome Dependent Sampling

Outcome-dependent, two-phase sampling designs can dramatically reduce the costs of observational studies by judicious selection of the most informative subjects for purposes of detailed covariate measurement. Here we derive asymptotic information bounds and the form of the efficient score and influence functions for the semiparametric regression models studied by Lawless, Kalbfleisch, and Wild (1999) under two-phase sampling designs. We show that the maximum likelihood estimators for both the parametric and nonparametric parts of the model are asymptotically normal and efficient. The efficient influence function for the parametric part aggress with the more general information bound calculations of Robins, Hsieh, and Newey (1995). By verifying the conditions of Murphy and Van der Vaart (2000) for a least favorable parametric submodel, we provide asymptotic justification for statistical inference based on profile likelihood.

Stochastic Models Based on Molecular Hybridization Theory for Short Oligonucleotide Microarrays

High density oligonucleotide expression arrays are a widely used tool for the measurement of gene expression on a large scale. Affymetrix GeneChip arrays appear to dominate this market. These arrays use short oligonucleotides to probe for genes in an RNA sample. Due to optical noise, non-specific hybridization, probe-specific effects, and measurement error, ad-hoc measures of expression, that summarize probe intensities, can lead to imprecise and inaccurate results. Various researchers have demonstrated that expression measures based on simple statistical models can provide great improvements over the ad-hoc procedure offered by Affymetrix. Recently, physical models based on molecular hybridization theory, have been proposed as useful tools for prediction of, for example, non-specific hybridization. These physical models show great potential in terms of improving existing expression measures. In this paper we demonstrate that the system producing the measured intensities is too complex to be fully described with these relatively simple physical models and we propose empirically motivated stochastic models that compliment the above mentioned molecular hybridization theory to provide a comprehensive description of the data. We discuss how the proposed model can be used to obtain improved measures of expression useful for the data analysts.