Functional Mixed Models Research Articles

Greater striatopallidal adaptive coding during cue–reward learning and food reward habituation predict future weight gain

Animal experiments indicate that after repeated pairings of palatable food receipt and cues that predict palatable food receipt, dopamine signaling increases in response to predictive cues, but decreases in response to food receipt. Using functional MRI and mixed effects growth curve models with 35 females (M age=15.5±0.9; M BMI=24.5±5.4) we documented an increase in BOLD response in the caudate (r=.42) during exposure to cues predicting impending milkshake receipt over repeated exposures, demonstrating a direct measure of in vivo cue–reward learning in humans. Further, we observed a simultaneous decrease in putamen (r=−.33) and ventral pallidum (r=−.45) response during milkshake receipt that occurred over repeated exposures, putatively reflecting food reward habitation. We then tested whether cue–reward learning and habituation slopes predicted future weight over 2-year follow-up. Those who exhibited the greatest escalation in ventral pallidum responsivity to cues and the greatest decrease in caudate response to milkshake receipt showed significantly larger increases in BMI (r=.39 and −.69 respectively). Interestingly, cue–reward learning propensity and food reward habituation were not correlated, implying that these factors may constitute qualitatively distinct vulnerability pathways to excess weight gain. These two individual difference factors may provide insight as to why certain people have shown obesity onset in response to the current obesogenic environment in western cultures, whereas others have not.

Trends in stratospheric ozone profiles using functional mixed models

Abstract. This paper is devoted to the modeling of altitude-dependent patterns of ozone variations over time. Umkehr ozone profiles (quarter of Umkehr layer) from 1978 to 2011 are investigated at two locations: Boulder (USA) and Arosa (Switzerland). The study consists of two statistical stages. First we approximate ozone profiles employing an appropriate basis. To capture primary modes of ozone variations without losing essential information, a functional principal component analysis is performed. It penalizes roughness of the function and smooths excessive variations in the shape of the ozone profiles. As a result, data-driven basis functions (empirical basis functions) are obtained. The coefficients (principal component scores) corresponding to the empirical basis functions represent dominant temporal evolution in the shape of ozone profiles. We use those time series coefficients in the second statistical step to reveal the important sources of the patterns and variations in the profiles. We estimate the effects of covariates – month, year (trend), quasi-biennial oscillation, the solar cycle, the Arctic oscillation, the El Niño/Southern Oscillation cycle and the Eliassen–Palm flux – on the principal component scores of ozone profiles using additive mixed effects models. The effects are represented as smooth functions and the smooth functions are estimated by penalized regression splines. We also impose a heteroscedastic error structure that reflects the observed seasonality in the errors. The more complex error structure enables us to provide more accurate estimates of influences and trends, together with enhanced uncertainty quantification. Also, we are able to capture fine variations in the time evolution of the profiles, such as the semi-annual oscillation. We conclude by showing the trends by altitude over Boulder and Arosa, as well as for total column ozone. There are great variations in the trends across altitudes, which highlights the benefits of modeling ozone profiles.

Functional Linear Models for Association Analysis of Quantitative Traits

Functional linear models are developed in this paper for testing associations between quantitative traits and genetic variants, which can be rare variants or common variants or the combination of the two. By treating multiple genetic variants of an individual in a human population as a realization of a stochastic process, the genome of an individual in a chromosome region is a continuum of sequence data rather than discrete observations. The genome of an individual is viewed as a stochastic function that contains both linkage and linkage disequilibrium (LD) information of the genetic markers. By using techniques of functional data analysis, both fixed and mixed effect functional linear models are built to test the association between quantitative traits and genetic variants adjusting for covariates. After extensive simulation analysis, it is shown that the F-distributed tests of the proposed fixed effect functional linear models have higher power than that of sequence kernel association test (SKAT) and its optimal unified test (SKAT-O) for three scenarios in most cases: (1) the causal variants are all rare, (2) the causal variants are both rare and common, and (3) the causal variants are common. The superior performance of the fixed effect functional linear models is most likely due to its optimal utilization of both genetic linkage and LD information of multiple genetic variants in a genome and similarity among different individuals, while SKAT and SKAT-O only model the similarities and pairwise LD but do not model linkage and higher order LD information sufficiently. In addition, the proposed fixed effect models generate accurate type I error rates in simulation studies. We also show that the functional kernel score tests of the proposed mixed effect functional linear models are preferable in candidate gene analysis and small sample problems. The methods are applied to analyze three biochemical traits in data from the Trinity Students Study.

A Study of Mexican Free-Tailed Bat Chirp Syllables: Bayesian Functional Mixed Models for Nonstationary Acoustic Time Series

We describe a new approach to analyze chirp syllables of free-tailed bats from two regions of Texas in which they are predominant: Austin and College Station. Our goal is to characterize any systematic regional differences in the mating chirps and assess whether individual bats have signature chirps. The data are analyzed by modeling spectrograms of the chirps as responses in a Bayesian functional mixed model. Given the variable chirp lengths, we compute the spectrograms on a relative time scale interpretable as the relative chirp position, using a variable window overlap based on chirp length. We use two-dimensional wavelet transforms to capture correlation within the spectrogram in our modeling and obtain adaptive regularization of the estimates and inference for the regions-specific spectrograms. Our model includes random effect spectrograms at the bat level to account for correlation among chirps from the same bat and to assess relative variability in chirp spectrograms within and between bats. The modeling of spectrograms using functional mixed models is a general approach for the analysis of replicated nonstationary time series, such as our acoustical signals, to relate aspects of the signals to various predictors, while accounting for between-signal structure. This can be done on raw spectrograms when all signals are of the same length and can be done using spectrograms defined on a relative time scale for signals of variable length in settings where the idea of defining correspondence across signals based on relative position is sensible. Supplementary materials for this article are available online.

Robust Classification of Functional and Quantitative Image Data Using Functional Mixed Models

This article introduces new methods for performing classification of complex, high-dimensional functional data using the functional mixed model (FMM) framework. The FMM relates a functional response to a set of predictors through functional fixed and random effects, which allows it to account for various factors and between-function correlations. The methods include training and prediction steps. In the training steps we train the FMM model by treating class designation as one of the fixed effects, and in the prediction steps we classify the new objects using posterior predictive probabilities of class. Through a Bayesian scheme, we are able to adjust for factors affecting both the functions and the class designations. While the methods can be used in any FMM framework, we provide details for two specific Bayesian approaches: the Gaussian, wavelet-based FMM (G-WFMM) and the robust, wavelet-based FMM (R-WFMM). Both methods perform modeling in the wavelet space, which yields parsimonious representations for the functions, and can naturally adapt to local features and complex nonstationarities in the functions. The R-WFMM allows potentially heavier tails for features of the functions indexed by particular wavelet coefficients, leading to a down-weighting of outliers that makes the method robust to outlying functions or regions of functions. The models are applied to a pancreatic cancer mass spectroscopy data set and compared with other recently developed functional classification methods.

Statistical Methods for Proteomic Biomarker Discovery based on Feature Extraction or Functional Modeling Approaches.

In recent years, developments in molecular biotechnology have led to the increased promise of detecting and validating biomarkers, or molecular markers that relate to various biological or medical outcomes. Proteomics, the direct study of proteins in biological samples, plays an important role in the biomarker discovery process. These technologies produce complex, high dimensional functional and image data that present many analytical challenges that must be addressed properly for effective comparative proteomics studies that can yield potential biomarkers. Specific challenges include experimental design, preprocessing, feature extraction, and statistical analysis accounting for the inherent multiple testing issues. This paper reviews various computational aspects of comparative proteomic studies, and summarizes contributions I along with numerous collaborators have made. First, there is an overview of comparative proteomics technologies, followed by a discussion of important experimental design and preprocessing issues that must be considered before statistical analysis can be done. Next, the two key approaches to analyzing proteomics data, feature extraction and functional modeling, are described. Feature extraction involves detection and quantification of discrete features like peaks or spots that theoretically correspond to different proteins in the sample. After an overview of the feature extraction approach, specific methods for mass spectrometry (Cromwell) and 2D gel electrophoresis (Pinnacle) are described. The functional modeling approach involves modeling the proteomic data in their entirety as functions or images. A general discussion of the approach is followed by the presentation of a specific method that can be applied, wavelet-based functional mixed models, and its extensions. All methods are illustrated by application to two example proteomic data sets, one from mass spectrometry and one from 2D gel electrophoresis. While the specific methods presented are applied to two specific proteomic technologies, MALDI-TOF and 2D gel electrophoresis, these methods and the other principles discussed in the paper apply much more broadly to other expression proteomics technologies.

Robust, Adaptive Functional Regression in Functional Mixed Model Framework

Functional data are increasingly encountered in scientific studies, and their high dimensionality and complexity lead to many analytical challenges. Various methods for functional data analysis have been developed, including functional response regression methods that involve regression of a functional response on univariate/multivariate predictors with nonparametrically represented functional coefficients. In existing methods, however, the functional regression can be sensitive to outlying curves and outlying regions of curves, so is not robust. In this article, we introduce a new Bayesian method, robust functional mixed models (R-FMM), for performing robust functional regression within the general functional mixed model framework, which includes multiple continuous or categorical predictors and random effect functions accommodating potential between-function correlation induced by the experimental design. The underlying model involves a hierarchical scale mixture model for the fixed effects, random effect, and residual error functions. These modeling assumptions across curves result in robust nonparametric estimators of the fixed and random effect functions which down-weight outlying curves and regions of curves, and produce statistics that can be used to flag global and local outliers. These assumptions also lead to distributions across wavelet coefficients that have outstanding sparsity and adaptive shrinkage properties, with great flexibility for the data to determine the sparsity and the heaviness of the tails. Together with the down-weighting of outliers, these within-curve properties lead to fixed and random effect function estimates that appear in our simulations to be remarkably adaptive in their ability to remove spurious features yet retain true features of the functions. We have developed general code to implement this fully Bayesian method that is automatic, requiring the user to only provide the functional data and design matrices. It is efficient enough to handle large datasets, and yields posterior samples of all model parameters that can be used to perform desired Bayesian estimation and inference. Although we present details for a specific implementation of the R-FMM using specific distributional choices in the hierarchical model, 1D functions, and wavelet transforms, the method can be applied more generally using other heavy-tailed distributions, higher dimensional functions (e.g., images), and using other invertible transformations as alternatives to wavelets. Supplementary materials for this article are available online.

Automated analysis of quantitative image data using isomorphic functional mixed models, with application to proteomics data

Image data are increasingly encountered and are of growing importance in many areas of science. Much of these data are quantitative image data, which are characterized by intensities that represent some measurement of interest in the scanned images. The data typically consist of multiple images on the same domain and the goal of the research is to combine the quantitative information across images to make inference about populations or interventions. In this paper, we present a unified analysis framework for the analysis of quantitative image data using a Bayesian functional mixed model approach. This framework is flexible enough to handle complex, irregular images with many local features, and can model the simultaneous effects of multiple factors on the image intensities and account for the correlation between images induced by the design. We introduce a general isomorphic modeling approach to fitting the functional mixed model, of which the wavelet-based functional mixed model is one special case. With suitable modeling choices, this approach leads to efficient calculations and can result in flexible modeling and adaptive smoothing of the salient features in the data. The proposed method has the following advantages: it can be run automatically, it produces inferential plots indicating which regions of the image are associated with each factor, it simultaneously considers the practical and statistical significance of findings, and it controls the false discovery rate. Although the method we present is general and can be applied to quantitative image data from any application, in this paper we focus on image-based proteomic data. We apply our method to an animal study investigating the effects of opiate addiction on the brain proteome. Our image-based functional mixed model approach finds results that are missed with conventional spot-based analysis approaches. In particular, we find that the significant regions of the image identified by the proposed method frequently correspond to subregions of visible spots that may represent post-translational modifications or co-migrating proteins that cannot be visually resolved from adjacent, more abundant proteins on the gel image. Thus, it is possible that this image-based approach may actually improve the realized resolution of the gel, revealing differentially expressed proteins that would not have even been detected as spots by modern spot-based analyses.

Wavelet-based functional linear mixed models: an application to measurement error-corrected distributed lag models.

Frequently, exposure data are measured over time on a grid of discrete values that collectively define a functional observation. In many applications, researchers are interested in using these measurements as covariates to predict a scalar response in a regression setting, with interest focusing on the most biologically relevant time window of exposure. One example is in panel studies of the health effects of particulate matter (PM), where particle levels are measured over time. In such studies, there are many more values of the functional data than observations in the data set so that regularization of the corresponding functional regression coefficient is necessary for estimation. Additional issues in this setting are the possibility of exposure measurement error and the need to incorporate additional potential confounders, such as meteorological or co-pollutant measures, that themselves may have effects that vary over time. To accommodate all these features, we develop wavelet-based linear mixed distributed lag models that incorporate repeated measures of functional data as covariates into a linear mixed model. A Bayesian approach to model fitting uses wavelet shrinkage to regularize functional coefficients. We show that, as long as the exposure error induces fine-scale variability in the functional exposure profile and the distributed lag function representing the exposure effect varies smoothly in time, the model corrects for the exposure measurement error without further adjustment. Both these conditions are likely to hold in the environmental applications we consider. We examine properties of the method using simulations and apply the method to data from a study examining the association between PM, measured as hourly averages for 1-7 days, and markers of acute systemic inflammation. We use the method to fully control for the effects of confounding by other time-varying predictors, such as temperature and co-pollutants.

Bayesian Analysis of Mass Spectrometry Proteomic Data Using Wavelet‐Based Functional Mixed Models

In this article, we apply the recently developed Bayesian wavelet-based functional mixed model methodology to analyze MALDI-TOF mass spectrometry proteomic data. By modeling mass spectra as functions, this approach avoids reliance on peak detection methods. The flexibility of this framework in modeling nonparametric fixed and random effect functions enables it to model the effects of multiple factors simultaneously, allowing one to perform inference on multiple factors of interest using the same model fit, while adjusting for clinical or experimental covariates that may affect both the intensities and locations of peaks in the spectra. For example, this provides a straightforward way to account for systematic block and batch effects that characterize these data. From the model output, we identify spectral regions that are differentially expressed across experimental conditions, in a way that takes both statistical and clinical significance into account and controls the Bayesian false discovery rate to a prespecified level. We apply this method to two cancer studies.

Generalized Monotonic Functional Mixed Models with Application to Modelling Normal Tissue Complications

Summary Normal tissue complications are a common side effect of radiation therapy. They are the consequence of the dose of radiation that is received by the normal tissue surrounding the site of the tumour. Within a specified organ each voxel receives a certain dose of radiation, leading to a distribution of doses over the organ. It is often not known what aspect of the dose distribution drives the presence and severity of the complications. A summary measure of the dose distribution can be obtained by integrating a weighting function of dose (w(d)) over the density of dose. For biological reasons the weight function should be monotonic. We propose a generalized monotonic functional mixed model to study the dose effect on a clinical outcome by estimating this weight function non-parametrically by using splines and subject to the monotonicity constraint, while allowing for overdispersion and correlation of multiple obervations within the same subject. We illustrate our method with data from a head and neck cancer study in which the irradiation of the parotid gland results in loss of saliva flow.

Adaptive functional mixed NHPP models for the analysis of recurrent event panel data

An adaptive semi-parametric model for analyzing longitudinal panel count data is presented. Panel data refers here to data collected as the number of events occurring between specific followup times over a period or disjoint periods of observation of a subject. The counts are assumed to arise from a mixed nonhomogeneous Poisson process where frailties account for heterogeneity common to this type of data. The generating intensity of the counting process is assumed to be a smooth function modeled with penalized splines. A main feature is that the penalization used to control the amount of smoothing, usually assumed to be time homogeneous, is allowed to be time dependent so that the spline can more easily adapt to sharp changes in curvature regimes. Splines are also used to model time dependent covariate effects. Penalized quasi-likelihood (PQL; [Breslow, N.E., Clayton, D.G., 1993. Approximate inference in generalized linear mixed models. Journal of the American Statistical Association 88, 9–25]) is used to derive estimating equations for this adaptive spline model so that only low moment assumptions are required for inference. Both jackknife and bootstrap variance estimators are developed. The finite sample properties of the proposed estimating functions are investigated empirically by simulation. Comparisons with a model assuming a time homogeneous penalty are made. The methods are used in an analysis of data from an experiment to test the effectiveness of pheromones in disrupting the mating pattern of the cherry bark tortrix moth. Recommendations are provided on when the simpler model with a time homogeneous penalty may provide a fair approximation to data and where such an approach will be lacking, calling for the more complicated adaptive methods.

Bayesian generalized monotonic functional mixed models for the effects of radiation dose histograms on normal tissue complications

When treating cancer patients with radiation therapy, the normal tissue in an organ close to the tumour usually receives some dose of radiation. The dose is not of the same intensity throughout the organ. This radiation can cause normal tissue complications, so for treatment planning purposes, it is important to understand the relationship between the distribution of dose intensities in the organ and the occurrence of complications. One general summary measure of the dose effect is obtained by integrating a weighting function (w(d)) over the dose distribution. The weighting function w(d) should be monotone for biological reasons. Because the true shape of w(d) is not known, we estimate it non-parametrically subject to the monotonicity constraint. In our approach w(d) is written as a weighted sum of monotone basis functions. The weights in this sum are formulated as a mixture of point mass at zero and a Gamma random variable. A key feature of our method is that it allows for flat regions through the use of this mixture prior. The model is estimated using a Markov Chain Monte Carlo algorithm. We illustrate our method with data from a head and neck cancer study in which the irradiation of the parotid gland results in loss of saliva flow.

Two‐Stage Functional Mixed Models for Evaluating the Effect of Longitudinal Covariate Profiles on a Scalar Outcome

The Daily Hormone Study, a substudy of the Study of Women's Health Across the Nation (SWAN) consisting of more than 600 pre- and perimenopausal women, includes a scalar measure of total hip bone mineral density (BMD) together with repeated measures of creatinine-adjusted follicle stimulating hormone (FSH) assayed from daily urine samples collected over one menstrual cycle. It is of scientific interest to investigate the effect of the FSH time profile during a menstrual cycle on total hip BMD, adjusting for age and body mass index. The statistical analysis is challenged by several features of the data: (1) the covariate FSH is measured longitudinally and its effect on the scalar outcome BMD may be complex; (2) due to varying menstrual cycle lengths, subjects have unbalanced longitudinal measures of FSH; and (3) the longitudinal measures of FSH are subject to considerable among- and within-subject variations and measurement errors. We propose a measurement error partial functional linear model, where repeated measures of FSH are modeled using a functional mixed effects model and the effect of the FSH time profile on BMD is modeled using a partial functional linear model by treating the unobserved true subject-specific FSH time profile as a functional covariate. We develop a two-stage nonparametric regression calibration method using period smoothing splines. Using the connection between smoothing splines and mixed models, we show that a key feature of our approach is that estimation at both stages can be conveniently cast into a unified mixed model framework. A simple testing procedure for constant functional covariate effect is also proposed. The proposed methods are evaluated using simulation studies and applied to the SWAN data.

Using Wavelet-Based Functional Mixed Models to Characterize Population Heterogeneity in Accelerometer Profiles

We present a case study illustrating the challenges of analyzing accelerometer data taken from a sample of children participating in an intervention study designed to increase physical activity. An accelerometer is a small device worn on the hip that records the minute-by-minute activity levels throughout the day for each day it is worn. The resulting data are irregular functions characterized by many peaks representing short bursts of intense activity. We model these data using the wavelet-based functional mixed model. This approach incorporates multiple fixed-effects and random-effects functions of arbitrary form, the estimates of which are adaptively regularized using wavelet shrinkage. The method yields posterior samples for all functional quantities of the model, which can be used to perform various types of Bayesian inference and prediction. In our case study, a high proportion of the daily activity profiles are incomplete (i.e., have some portion of the profile missing), and thus cannot be modeled directly using the previously described method. We present a new method for stochastically imputing the missing data that allows us to incorporate these incomplete profiles in our analysis. Our approach borrows strength from both the observed measurements within the incomplete profiles and from other profiles, from the same child as well as from other children with similar covariate levels, while appropriately propagating the uncertainty of the imputation throughout all subsequent inference. We apply this method to our case study, revealing some interesting insights into children's activity patterns. We point out some strengths and limitations of using this approach to analyze accelerometer data.

Wavelet-based Functional Mixed Models

Increasingly, scientific studies yield functional data, in which the ideal units of observation are curves and the observed data consist of sets of curves that are sampled on a fine grid. We present new methodology that generalizes the linear mixed model to the functional mixed model framework, with model fitting done by using a Bayesian wavelet-based approach. This method is flexible, allowing functions of arbitrary form and the full range of fixed effects structures and between-curve covariance structures that are available in the mixed model framework. It yields nonparametric estimates of the fixed and random-effects functions as well as the various between-curve and within-curve covariance matrices. The functional fixed effects are adaptively regularized as a result of the non-linear shrinkage prior that is imposed on the fixed effects' wavelet coefficients, and the random-effect functions experience a form of adaptive regularization because of the separately estimated variance components for each wavelet coefficient. Because we have posterior samples for all model quantities, we can perform pointwise or joint Bayesian inference or prediction on the quantities of the model. The adaptiveness of the method makes it especially appropriate for modelling irregular functional data that are characterized by numerous local features like peaks.