A Bayesian approach to functional-based multilevel modeling of longitudinal data: applications to environmental epidemiology

Nonlinear function-on-scalar regression via functional universal approximation

We consider general nonlinear function-on-scalar (FOS) regression models, where the functional response depends on multiple scalar predictors in a general unknown nonlinear form. Existing methods either assume specific model forms (e.g., additive models) or directly estimate the nonlinear function in a space with dimension equal to the number of scalar predictors, which can only be applied to models with a few scalar predictors. To overcome these shortcomings, motivated by the classic universal approximation theorem used in neural networks, we develop a functional universal approximation theorem which can be used to approximate general nonlinear FOS maps and can be easily adopted into the framework of functional data analysis. With this theorem and utilizing smoothness regularity, we develop a novel method to fit the general nonlinear FOS regression model and make predictions. Our new method does not make any specific assumption on the model forms, and it avoids the direct estimation of nonlinear functions in a space with dimension equal to the number of scalar predictors. By estimating a sequence of bivariate functions, our method can be applied to models with a relatively large number of scalar predictors. The good performance of the proposed method is demonstrated by empirical studies on various simulated and real datasets.

Robust probabilistic classification applicable to irregularly sampled functional data

A robust probabilistic classifier for functional data is developed to predict class membership based on functional input measurements and to provide a reliable probability estimates for class membership. The method combines a Bayes classifier and semi-parametric mixed effects model with robust tuning parameter to make the method robust to outlying curves, and to improve the accuracy of the risk or uncertainty estimates, which is crucial in medical diagnostic applications. The approach applies to functional data with varying ranges and irregular sampling without making parametric assumptions on the within-curve covariance. Simulation studies evaluate the proposed method and competitors in terms of sensitivity to heavy tailed functional distributions and outlying curves. Classification performance is evaluated by both error rate and logloss, the latter of which imposes heavier penalties on highly confident errors than on less confident errors. Runtime experiments on the R implementation indicate that the proposed method scales well computationally. Illustrative applications include data from quantitative ultrasound analysis and phoneme recognition.

Bayesian Semiparametric Functional Mixed Models for Serially Correlated Functional Data, with Application to Glaucoma Data

Glaucoma, a leading cause of blindness, is characterized by optic nerve damage related to intraocular pressure (IOP), but its full etiology is unknown. Researchers at UAB have devised a custom device to measure scleral strain continuously around the eye under fixed levels of IOP, which here is used to assess how strain varies around the posterior pole, with IOP, and across glaucoma risk factors such as age. The hypothesis is that scleral strain decreases with age, which could alter biomechanics of the optic nerve head and cause damage that could eventually lead to glaucoma. To evaluate this hypothesis, we adapted Bayesian Functional Mixed Models to model these complex data consisting of correlated functions on spherical scleral surface, with nonparametric age effects allowed to vary in magnitude and smoothness across the scleral surface, multi-level random effect functions to capture within-subject correlation, and functional growth curve terms to capture serial correlation across IOPs that can vary around the scleral surface. Our method yields fully Bayesian inference on the scleral surface or any aggregation or transformation thereof, and reveals interesting insights into the biomechanical etiology of glaucoma. The general modeling framework described is very flexible and applicable to many complex, high-dimensional functional data.

Testing Gait with Ankle-Foot Orthoses in Children with Cerebral Palsy by Using Functional Mixed-Effects Analysis of Variance

Existing statistical methods extract insufficient information from 3-dimensional gait data, rendering clinical interpretation of impaired movement patterns sub-optimal. We propose an alternative approach based on functional data analysis that may be worthy of exploration. We apply this to gait data analysis using repeated-measurements data from children with cerebral palsy who had been prescribed fixed ankle-foot orthoses as an example. We analyze entire gait curves by means of a new functional F test with comparison to multiple pointwise F tests and also to the traditional method - univariate repeated-measurements analysis of variance of joint angle minima and maxima. The new test maintains the nominal significance level and can be adapted to test hypotheses for specific phases of the gait cycle. The main findings indicate that ankle-foot orthoses exert significant effects on coronal and sagittal plane ankle rotation; and both sagittal and horizontal plane foot rotation. The functional F test provided further information for the stance and swing phases. Differences between the results of the different statistical approaches are discussed, concluding that the novel method has potential utility and is worthy of validation through larger scale patient and clinician engagement to determine whether it is preferable to the traditional approach.

A longitudinal cohort study of body mass index and childhood exposure to secondhand tobacco smoke and air pollution: the Southern California Children's Health Study

BACKGROUND

Childhood body mass index (BMI) and obesity prevalence have been associated with exposure to secondhand smoke (SHS), maternal smoking during pregnancy, and vehicular air pollution. There has been little previous study of joint BMI effects of air pollution and tobacco smoke exposure.

METHODS

Information on exposure to SHS and maternal smoking during pregnancy was collected on 3,318 participants at enrollment into the Southern California Children's Health Study. At study entry at average age of 10 years, residential near-roadway pollution exposure (NRP) was estimated based on a line source dispersion model accounting for traffic volume, proximity, and meteorology. Lifetime exposure to tobacco smoke was assessed by parent questionnaire. Associations with subsequent BMI growth trajectory based on annual measurements and attained BMI at 18 years of age were assessed using a multilevel modeling strategy.

RESULTS

Maternal smoking during pregnancy was associated with estimated BMI growth over 8-year follow-up (0.72 kg/m2 higher; 95% CI: 0.14, 1.31) and attained BMI (1.14 kg/m2 higher; 95% CI: 0.66, 1.62). SHS exposure before enrollment was positively associated with BMI growth (0.81 kg/m2 higher; 95% CI: 0.36, 1.27) and attained BMI (1.23 kg/m2 higher; 95% CI: 0.86, 1.61). Growth and attained BMI increased with more smokers in the home. Compared with children without a history of SHS and NRP below the median, attained BMI was 0.80 kg/m2 higher (95% CI: 0.27, 1.32) with exposure to high NRP without SHS; 0.85 kg/m2 higher (95% CI: 0.43, 1.28) with low NRP and a history of SHS; and 2.15 kg/m2 higher (95% CI: 1.52, 2.77) with high NRP and a history of SHS (interaction p-value 0.007). These results suggest a synergistic effect.

CONCLUSIONS

Our findings strengthen emerging evidence that exposure to tobacco smoke and NRP contribute to development of childhood obesity and suggest that combined exposures may have synergistic effects.

Identification of biomarkers for tuberculosis susceptibility via integrated analysis of gene expression and longitudinal clinical data

Tuberculosis (TB) is an infectious disease caused by the bacteria Mycobacterium tuberculosis (Mtb) that affects millions of people worldwide. The majority of individuals who are exposed to Mtb develop latent infections, in which an immunological response to Mtb antigens is present but there is no clinical evidence of disease. Because currently available tests cannot differentiate latent individuals who are at low risk from those who are highly susceptible to developing active disease, there is considerable interest in the identification of diagnostic biomarkers that can predict reactivation of latent TB. We present results from our analysis of a controlled longitudinal experiment in which a group of rhesus macaques were exposed to a low dose of Mtb to study their progression to latent infection or active disease. Subsets of the animals were then euthanized at scheduled time points, and granulomas taken from their lungs were assayed for gene expression using microarrays. The clinical profiles associated with the animals following Mtb exposure revealed considerable variability, and we developed models for the disease trajectory for each subject using a Bayesian hierarchical B-spline approach. Disease severity estimates were derived from these fitted curves and included as covariates in linear models to identify genes significantly associated with disease progression. Our results demonstrate that the incorporation of clinical data increases the value of information extracted from the expression profiles and contributes to the identification of predictive biomarkers for TB susceptibility.

Traffic-related air pollution and obesity formation in children: a longitudinal, multilevel analysis

BACKGROUND

Biologically plausible mechanisms link traffic-related air pollution to metabolic disorders and potentially to obesity. Here we sought to determine whether traffic density and traffic-related air pollution were positively associated with growth in body mass index (BMI = kg/m2) in children aged 5-11 years.

METHODS

Participants were drawn from a prospective cohort of children who lived in 13 communities across Southern California (N = 4550). Children were enrolled while attending kindergarten and first grade and followed for 4 years, with height and weight measured annually. Dispersion models were used to estimate exposure to traffic-related air pollution. Multilevel models were used to estimate and test traffic density and traffic pollution related to BMI growth. Data were collected between 2002-2010 and analyzed in 2011-12.

RESULTS

Traffic pollution was positively associated with growth in BMI and was robust to adjustment for many confounders. The effect size in the adjusted model indicated about a 13.6% increase in annual BMI growth when comparing the lowest to the highest tenth percentile of air pollution exposure, which resulted in an increase of nearly 0.4 BMI units on attained BMI at age 10. Traffic density also had a positive association with BMI growth, but this effect was less robust in multivariate models.

CONCLUSIONS

Traffic pollution was positively associated with growth in BMI in children aged 5-11 years. Traffic pollution may be controlled via emission restrictions; changes in land use that promote jobs-housing balance and use of public transit and hence reduce vehicle miles traveled; promotion of zero emissions vehicles; transit and car-sharing programs; or by limiting high pollution traffic, such as diesel trucks, from residential areas or places where children play outdoors, such as schools and parks. These measures may have beneficial effects in terms of reduced obesity formation in children.

Organized physical activity in young school children and subsequent 4-year change in body mass index

OBJECTIVE

To determine whether participation in organized outdoor team sports and structured indoor nonschool activity programs in kindergarten and first grade predicted subsequent 4-year change in body mass index (BMI; calculated as weight in kilograms divided by height in meters squared) during the adiposity rebound period of childhood.

DESIGN

Longitudinal cohort study.

SETTING

Forty-five schools in 13 communities across Southern California.

PARTICIPANTS

Largely Hispanic and non-Hispanic white children (N = 4550) with a mean (SD) age at study entry of 6.60 (0.65) years.

MAIN EXPOSURES

Parents completed questionnaires assessing physical activity, demographic characteristics, and other relevant covariates at baseline. Data on built and social environmental variables were linked to the neighborhood around children's homes using geographical information systems.

MAIN OUTCOME MEASURES

Each child's height and weight were measured annually during 4 years of follow-up.

RESULTS

After adjusting for several confounders, BMI increased at a rate 0.05 unit/year slower for children who participated in outdoor organized team sports at least twice per week compared with children who did not. For participation in each additional indoor nonschool structured activity class, lesson, and program, BMI increased at a rate 0.05 unit/year slower, and the attained BMI level at age 10 years was 0.48 units lower.

CONCLUSION

Engagement in organized sports and activity programs as early as kindergarten and the first grade may result in smaller increases in BMI during the adiposity rebound period of childhood.

Automobile traffic around the home and attained body mass index: a longitudinal cohort study of children aged 10-18 years

OBJECTIVES

The objective of this study is to examine the relationship between measured traffic density near the homes of children and attained body mass index (BMI) over an eight-year follow up.

METHODS

Children aged 9-10 years were enrolled across multiple communities in Southern California in 1993 and 1996 (n=3318). Children were followed until age 18 or high school graduation to collect longitudinal information, including annual height and weight measurements. Multilevel growth curve models were used to assess the association between BMI levels at age 18 and traffic around the home.

RESULTS

For traffic within 150 m around the child's home, there were significant positive associations with attained BMI for both sexes at age 18. With the 300 m traffic buffer, associations for both male and female growth in BMI were positive, but significantly elevated only in females. These associations persisted even after controlling for numerous potential confounding variables.

CONCLUSIONS

This analysis yields the first evidence of significant effects from traffic density on BMI levels at age 18 in a large cohort of children. Traffic is a pervasive exposure in most cities, and our results identify traffic as a major risk factor for the development of obesity in children.