Semiparametric Mixed Models Research Articles

Bayesian inference in semiparametric mixed models for longitudinal data.

We consider Bayesian inference in semiparametric mixed models (SPMMs) for longitudinal data. SPMMs are a class of models that use a nonparametric function to model a time effect, a parametric function to model other covariate effects, and parametric or nonparametric random effects to account for the within-subject correlation. We model the nonparametric function using a Bayesian formulation of a cubic smoothing spline, and the random effect distribution using a normal distribution and alternatively a nonparametric Dirichlet process (DP) prior. When the random effect distribution is assumed to be normal, we propose a uniform shrinkage prior (USP) for the variance components and the smoothing parameter. When the random effect distribution is modeled nonparametrically, we use a DP prior with a normal base measure and propose a USP for the hyperparameters of the DP base measure. We argue that the commonly assumed DP prior implies a nonzero mean of the random effect distribution, even when a base measure with mean zero is specified. This implies weak identifiability for the fixed effects, and can therefore lead to biased estimators and poor inference for the regression coefficients and the spline estimator of the nonparametric function. We propose an adjustment using a postprocessing technique. We show that under mild conditions the posterior is proper under the proposed USP, a flat prior for the fixed effect parameters, and an improper prior for the residual variance. We illustrate the proposed approach using a longitudinal hormone dataset, and carry out extensive simulation studies to compare its finite sample performance with existing methods.

Variable Selection for Semiparametric Mixed Models in Longitudinal Studies

We propose a double-penalized likelihood approach for simultaneous model selection and estimation in semiparametric mixed models for longitudinal data. Two types of penalties are jointly imposed on the ordinary log-likelihood: the roughness penalty on the nonparametric baseline function and a nonconcave shrinkage penalty on linear coefficients to achieve model sparsity. Compared to existing estimation equation based approaches, our procedure provides valid inference for data with missing at random, and will be more efficient if the specified model is correct. Another advantage of the new procedure is its easy computation for both regression components and variance parameters. We show that the double-penalized problem can be conveniently reformulated into a linear mixed model framework, so that existing software can be directly used to implement our method. For the purpose of model inference, we derive both frequentist and Bayesian variance estimation for estimated parametric and nonparametric components. Simulation is used to evaluate and compare the performance of our method to the existing ones. We then apply the new method to a real data set from a lactation study.

Analysis of cross‐over designs with serial correlation within periods using semi‐parametric mixed models

The use of semi-parametric mixed models has proven useful in a wide variety of settings. Here, we focus on the application of the methodology in the particular case of a cross-over design with relatively long sequences of repeated measurements within each treatment period and for each subject. Other than an overall measure of the difference between each one of the experimental groups and the control group, specific time point comparisons may also be of interest. To that effect, we propose the use of flexible semi-parametric mixed models, enabling the construction of simulation-based simultaneous confidence bands. The bands take into account both between- and within-subject variabilities, while simultaneously correcting for multiple time point comparisons. Owing to the relatively long sequences of measurements per subject, the presence of serially correlated errors is anticipated and investigated. We illustrate how several formulations of semi-parametric mixed models can be fitted and the construction of simulation-based simultaneous confidence bands using SAS PROC MIXED.

Modelling Price Paths in On-Line Auctions: Smoothing Sparse and Unevenly Sampled Curves by Using Semiparametric Mixed Models

Summary On-line auctions pose many challenges for the empirical researcher, one of which is the effective and reliable modelling of price paths. We propose a novel way of modelling price paths in eBay's on-line auctions by using functional data analysis. One of the practical challenges is that the functional objects are sampled only very sparsely and unevenly. Most approaches rely on smoothing to recover the underlying functional object from the data, which can be difficult if the data are irregularly distributed. We present a new approach that can overcome this challenge. The approach is based on the ideas of mixed models. Specifically, we propose a semiparametric mixed model with boosting to recover the functional object. As well as being able to handle sparse and unevenly distributed data, the model also results in conceptually more meaningful functional objects. In particular, we motivate our method within the framework of eBay's on-line auctions. On-line auctions produce monotonic increasing price curves that are often correlated across auctions. The semiparametric mixed model accounts for this correlation in a parsimonious way. It also manages to capture the underlying monotonic trend in the data without imposing model constraints. Our application shows that the resulting functional objects are conceptually more appealing. Moreover, when used to forecast the outcome of an on-line auction, our approach also results in more accurate price predictions compared with standard approaches. We illustrate our model on a set of 183 closed auctions for Palm M515 personal digital assistants.

Semiparametric Mixed Models for Increment-Averaged Data With Application to Carbon Sequestration in Agricultural Soils

Adoption of conservation tillage practice in agriculture offers the potential to mitigate greenhouse gas emissions. Studies comparing conservation tillage methods to traditional tillage pair fields under the two management systems and obtain soil core samples from each treatment. Cores are divided into multiple increments, and matching increments from one or more cores are aggregated and analyzed for carbon stock. These data represent not the actual value at a specific depth, but rather the total or average over a depth increment. A semiparametric mixed model is developed for such increment-averaged data. The model uses parametric fixed effects to represent covariate effects, random effects to capture correlation within studies, and an integrated smooth function to describe effects of depth. The depth function is specified as an additive model, estimated with penalized splines using standard mixed model software. Smoothing parameters are automatically selected using restricted maximum likelihood. The methodology is applied to the problem of estimating a change in carbon stock due to a change in tillage practice.

Streamlined variance calculations for semiparametric mixed models

Semiparametric mixed model analysis benefits from variability estimates such as standard errors of effect estimates and variability bars to accompany curve estimates. We show how the underlying variance calculations can be done extremely efficiently compared with the direct naïve approach. These streamlined calculations are linear in the number of subjects, representing a two orders of magnitude improvement.

A boosting approach to flexible semiparametric mixed models

In linear mixed models the influence of covariates is restricted to a strictly parametric form. With the rise of semi- and non-parametric regression also the mixed model has been expanded to allow for additive predictors. The common approach uses the representation of additive models as mixed models. An alternative approach that is proposed in the present paper is likelihood based boosting. Boosting originates in the machine learning community where it has been proposed as a technique to improve classification procedures by combining estimates with reweighted observations. Likelihood based boosting is a general method which may be seen as an extension of L2 boost. In additive mixed models the advantage of boosting techniques in the form of componentwise boosting is that it is suitable for high dimensional settings where many explanatory variables are present. It allows to fit additive models for many covariates with implicit selection of relevant variables and automatic selection of smoothing parameters. Moreover, boosting techniques may be used to incorporate the subject-specific variation of smooth influence functions by specifying 'random slopes' on smooth effects. This results in flexible semiparametric mixed models which are appropriate in cases where a simple random intercept is unable to capture the variation of effects across subjects.

Simple fitting of subject‐specific curves for longitudinal data

We present a simple semiparametric model for fitting subject-specific curves for longitudinal data. Individual curves are modelled as penalized splines with random coefficients. This model has a mixed model representation, and it is easily implemented in standard statistical software. We conduct an analysis of the long-term effect of radiation therapy on the height of children suffering from acute lymphoblastic leukaemia using penalized splines in the framework of semiparametric mixed effects models. The analysis revealed significant differences between therapies and showed that the growth rate of girls in the study cannot be fully explained by the group-average curve and that individual curves are necessary to reflect the individual response to treatment. We also show how to implement these models in S-PLUS and R in the appendix.

Variance component testing in semiparametric mixed models

It is of considerable interest to test for heteroscedasticity in statistical studies. In this paper, we investigate such a problem under the framework of a semiparametric mixed model. A score test is proposed for the hypothesis that all the variance components are zero. We establish the asymptotic property of the test, and examine its performance in a simulation study. The test is illustrated with the analysis of a longitudinal study of measurements of serum creatinine.

Semiparametric Nonlinear Mixed-Effects Models and Their Applications

Nonlinear mixed effects models (NLMMs) and self-modeling nonlinear regression (SEMOR) models are often used to fit repeated measures data. They use a common function shared by all subjects to model variation within each subject and some fixed and/or random parameters to model variation between subjects. The parametric NLMM may be too restrictive, and the semiparametric SEMOR model ignores correlations within each subject. In this article we propose a class of semiparametric nonlinear mixed effects models (SNMMs) that extend NLMMs, SEMOR models, and many other existing models in a natural way. A SNMM assumes that the mean function depends on some parameters and nonparametric functions. The parameters provide an interpretable data summary. The nonparametric functions provide flexibility to allow the data to decide some unknown or uncertain components, such as the shape of the mean response over time. A second-stage model with fixed and random effects is used to model the parameters. Smoothing splines are used to model the nonparametric functions. Covariate effects on parameters can be built into the second-stage model, and covariate effects on nonparametric functions can be constructed using smoothing spline ANOVA decompositions. Laplace approximations to the marginal likelihood and penalized marginal likelihood are used to estimate all parameters and nonparametric functions. We propose and compare two estimation procedures, and also show how to construct approximate Bayesian confidence intervals for the nonparametric functions based on a Bayesian formulation of SNMMs. We evaluate the proposed estimation and inference procedures through a simulation study. Applications of SNMMs are illustrated with analyses of Canadian temperature data.

Bayesian Inference for Generalized Additive Mixed Models Based on Markov Random Field Priors

SUMMARY Most regression problems in practice require flexible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a unified approach for Bayesian inference via Markov chain Monte Carlo simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as the usual covariates with fixed effects, metrical covariates with non-linear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates, are all treated within the same general framework by assigning appropriate Markov random field priors with different forms and degrees of smoothness. We applied the approach in several case-studies and consulting cases, showing that the methods are also computationally feasible in problems with many covariates and large data sets. In this paper, we choose two typical applications.

Micro-models of criminal careers: A synthesis of the criminal careers and life course approaches via semiparametric mixed poisson regression models, with empirical applications

Much recent research and debate in criminology have centered around how to conceptualize and model longitudinal sequences of delinquent and criminal acts committed by individuals. Two approaches dominate this controversy. One originates in thecriminal careers paradigm, which emphasizes a potentialheterogeneity of offending groups in the general population—thus leading to a distinction between incidence and prevalence of criminal offending, a focus on the onset, persistence, and desistence of criminal careers, and the possibility that criminals are a distinctive group with constant high rates of offending. Another approach places criminal events within a broader context ofstudies of the life course by explicitly substituting the conceptualization of “social events” for that of “criminal careers”. With respect to analytical models, this approach emphasizes a potentialheterogeneity of offenders with respect to order of criminal events from first to second to higher orders and thus suggests an analysis of the “risks” or “hazards” of offending by order of offense. Some extant commentaries on the criminal careers and life course approaches to conceptualizing and modeling longitudinal sequences of delinquent and criminal events committed by individuals have emphasized their differences and incompatibilities. In contrast, we apply recently developed semiparametric mixed Poisson regression techniques to develop conditions under which the two conceptual/modeling approaches are formally equivalent. We also modify the semiparametric mixed Poisson regression model of criminal careers to incorporate information on order of the delinquent/criminal event and develop an empirical application. This modification demonstrates the complementarity of the criminal careers and life course approaches, even though they have somewhat different foci.

MISSING TIME‐SERIES DATA AND THE IMPACT OF SENTENCING GUIDELINES IN MINNESOTA: CAN THE DEBATE BE ADJUDICATED?

CriminologyVolume 34, Issue 2 p. 281-288 MISSING TIME-SERIES DATA AND THE IMPACT OF SENTENCING GUIDELINES IN MINNESOTA: CAN THE DEBATE BE ADJUDICATED? KENNETH C. LAND, KENNETH C. LAND Duke UniversitySearch for more papers by this authorRICHARD McCLEARY, RICHARD McCLEARY University of California, IrvineSearch for more papers by this author KENNETH C. LAND, KENNETH C. LAND Duke UniversitySearch for more papers by this authorRICHARD McCLEARY, RICHARD McCLEARY University of California, IrvineSearch for more papers by this author First published: May 1996 https://doi.org/10.1111/j.1745-9125.1996.tb01208.xCitations: 3 Kenneth C. Land is the John Franklin Crowell Professor and Chairman of Sociology at Duke University. His recent research interests include hierarchical models analyses of victimization rates and fear of crime, the development of micromodels of criminal careers, and statistical methods for semiparametric mixed Poisson regression models. Richard McCleary is Professor of Social Ecology at the University of California, Irvine. His current research interests are risk factors for intentional injury and methods for quasi-experimental time series research designs. AboutPDF ToolsExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume34, Issue2May 1996Pages 281-288 RelatedInformation