Regression Trees for Longitudinal Data

While studying response trajectory, often the population of interest may be diverse enough to exist distinct subgroups within it and the longitudinal change in response may not be uniform in these subgroups. That is, the timeslope and/or influence of covariates in longitudinal profile may vary among these different subgroups. For example, Raudenbush (2001) used depression as an example to argue that it is incorrect to assume that all the people in a given population would be experiencing either increasing or decreasing levels of depression. In such cases, traditional linear mixed effects model (assuming common parametric form for covariates and time) is not directly applicable for the entire population as a group-averaged trajectory can mask important subgroup differences. Our aim is to identify and characterize longitudinally homogeneous subgroups based on the combination of baseline covariates in the most parsimonious way. This goal can be achieved via constructing regression tree for longitudinal data using baseline covariates as partitioning variables. We have proposed LongCART algorithm to construct regression tree for the longitudinal data. In each node, the proposed LongCART algorithm determines the need for further splitting (i.e. whether parameter(s) of longitudinal profile is influenced by any baseline attributes) via parameter instability tests and thus the decision of further splitting is type-I error controlled. We have obtained the asymptotic results for the proposed instability test and examined finite sample behavior of the whole algorithm through simulation studies. Finally, we have applied the LongCART algorithm to study the longitudinal changes in choline level among HIV patients

Structured penalties for functional linear models---partially empirical eigenvectors for regression

One of the challenges with functional data is incorporating spatial structure, or local correlation, into the analysis. This structure is inherent in the output from an increasing number of biomedical technologies, and a functional linear model is often used to estimate the relationship between the predictor functions and scalar responses. Common approaches to the ill-posed problem of estimating a coefficient function typically involve two stages: regularization and estimation. Regularization is usually done via dimension reduction, projecting onto a predefined span of basis functions or a reduced set of eigenvectors (principal components). In contrast, we present a unified approach that directly incorporates spatial structure into the estimation process by exploiting the joint eigenproperties of the predictors and a linear penalty operator. In this sense, the components in the regression are `partially empirical' and the framework is provided by the generalized singular value decomposition (GSVD). The GSVD clarifies the penalized estimation process and informs the choice of penalty by making explicit the joint influence of the penalty and predictors on the bias, variance, and performance of the estimated coefficient function. Laboratory spectroscopy data and simulations are used to illustrate the concepts.Comment: 29 pages, 3 figures, 5 tables; typo/notational errors edited and intro revised per journal review proces

Model-free estimation of time-varying correlation coefficients and their confidence intervals with an application to fMRI data

poster abstractOne of main interests in fMRI (functional magnetic resonance imaging) research is the study of associations between time series from different brain regions, so called functional connectivity (FC). Recently, it has become increasingly important to assess dynamic changes in FC, both during resting state and task-based fMRI experiments, as this is thought to provide the information needed to better understand the brain’s inner workings. Currently, the most common approach to estimate these dynamic changes is by computing the correlation coefficient between time series within a sliding-window. However, one of the disadvantages of this method is that it tends to overestimate the association between the time series obtained from different brain regions (Lindquist et al. 2014). Here we propose a new approach for estimating time-varying FC using the correlation between two time series and provide valid confidence bands for this estimator. We propose an algorithm based on the sliding-window approach which utilizes the multivariate linear process bootstrap. Both numerical results and an application to fMRI study will be presented

Assessing uncertainty in dynamic functional connectivity

Functional connectivity (FC) - the study of the statistical association between time series from anatomically distinct regions (Friston, 1994, 2011) - has become one of the primary areas of research in the field surrounding resting state functional magnetic resonance imaging (rs-fMRI). Although for many years researchers have implicitly assumed that FC was stationary across time in rs-fMRI, it has recently become increasingly clear that this is not the case and the ability to assess dynamic changes in FC is critical for better understanding of the inner workings of the human brain (Hutchison et al., 2013; Chang and Glover, 2010). Currently, the most common strategy for estimating these dynamic changes is to use the sliding-window technique. However, its greatest shortcoming is the inherent variation present in the estimate, even for null data, which is easily confused with true time-varying changes in connectivity (Lindquist et al., 2014). This can have serious consequences as even spurious fluctuations caused by noise can easily be confused with an important signal. For these reasons, assessment of uncertainty in the sliding-window correlation estimates is of critical importance. Here we propose a new approach that combines the multivariate linear process bootstrap (MLPB) method and a sliding-window technique to assess the uncertainty in a dynamic FC estimate by providing its confidence bands. Both numerical results and an application to rs-fMRI study are presented, showing the efficacy of the proposed method

LONGITUDINAL FUNCTIONAL MODELS WITH STRUCTURED PENALTIES

Collection of functional data is becoming increasingly common including longitudinal observations in many studies. For example, we use magnetic resonance (MR) spectra collected over a period of time from late stage HIV patients. MR spectroscopy (MRS) produces a spectrum which is a mixture of metabolite spectra, instrument noise and baseline profile. Analysis of such data typically proceeds in two separate steps: feature extraction and regression modeling. In contrast, a recently-proposed approach, called partially empirical eigenvectors for regression (PEER) (Randolph, Harezlak and Feng, 2012), for functional linear models incorporates a priori knowledge via a scientifically-informed penalty operator in the regression function estimation process. We extend the scope of PEER to the longitudinal setting with continuous outcomes and longitudinal functional covariates. The method presented in this paper: 1) takes into account external information; and 2) allows for a time-varying regression function. In the proposed approach, we express the time-varying regression function as linear combination of several time-invariant component functions; the time dependence enters into the regression function through their coefficients. The estimation procedure is easy to implement due to its mixed model equivalence. We derive the precision and accuracy of the estimates and discuss their connection with the generalized singular value decomposition. Real MRS data and simulations are used to illustrate the concepts

Study of sexual partner accrual patterns among adolescent women via Generalized Additive Mixed Models

poster abstractThe number of lifetime partners is a consistently identified epidemiological risk factor for sexually transmitted infections (STIs). Higher rate of partner accrual during adolescence has been associated with increased STI rates among adolescent women. To study sexual partner accrual pattern among adolescent females, we applied generalized additive mixed models (GAMM) to the data obtained from a longitudinal STI study. GAMM regression components included a bivariate function enabling separation of cohort (“age at study entry”) and longitudinal (“follow-up years”) effects on partner accrual while the correlation was accounted for by the subject-specific random components. Longitudinal effect partial derivative was used to estimate within-subject rates of partner accrual and their standard errors. The results show that slowing of partner accrual depends more on the prior sexual experience and less on the females’ chronological age. Our modeling approach combining the GAMM flexibility and the time covariates’ of interest definition enabled clear differentiation between the cohort (chronological age) and longitudinal (follow-up time) effects, thus providing the estimates of both between-subject differences and within-subject trajectories of partner accrual

Condom use as a function of number of coital events in new relationships

STUDY OBJECTIVE: To assess condom use as a function of number of coital events in newly formed sexual relationships. METHODS: Participants who reported at least one new partner during the 12-week study interval (n = 115; ages 18-29 years; 48% women; 90% African American) completed weekly sexually transmitted infections testing and 3 times daily electronic diary collection assessing individual and partner-specific affect, daily activities, sexual behavior, and condom use. We analyzed event-level condom use percentage and participant-level behavior response effects. generalized additive mixed models were used to estimate condom use probability accounting for within-partner and within-participant correlations via random effects. RESULTS: The average condom use probability at the first coital event in new relationships was 55% for men and 36% for women. Analyses showed that smooth shapes of estimated condom use probabilities were similar for both sexes and were fitted using generalized additive mixed models. Relatively higher condom use percentage was followed by a sharp decline during the first 9 coital events decreasing to 16% for men and 8% for women. More rapid decline in condom use among women was highly associated with higher levels of relationship and sexual satisfaction. CONCLUSIONS: The likelihood of condom use declines sharply for both men and women after the early accrual experience with a partner. Relationship and sexual satisfaction also influence declines in condom use, especially among women

Semiparametric Estimation of Task-Based Dynamic Functional Connectivity on the Population Level

Dynamic functional connectivity (dFC) estimates time-dependent associations between pairs of brain region time series as typically acquired during functional MRI. dFC changes are most commonly quantified by pairwise correlation coefficients between the time series within a sliding window. Here, we applied a recently developed bootstrap-based technique (Kudela et al., 2017) to robustly estimate subject-level dFC and its confidence intervals in a task-based fMRI study (24 subjects who tasted their most frequently consumed beer and Gatorade as an appetitive control). We then combined information across subjects and scans utilizing semiparametric mixed models to obtain a group-level dFC estimate for each pair of brain regions, flavor, and the difference between flavors. The proposed approach relies on the estimated group-level dFC accounting for complex correlation structures of the fMRI data, multiple repeated observations per subject, experimental design, and subject-specific variability. It also provides condition-specific dFC and confidence intervals for the whole brain at the group level. As a summary dFC metric, we used the proportion of time when the estimated associations were either significantly positive or negative. For both flavors, our fully-data driven approach yielded regional associations that reflected known, biologically meaningful brain organization as shown in prior work, as well as closely resembled resting state networks (RSNs). Specifically, beer flavor-potentiated associations were detected between several reward-related regions, including the right ventral striatum (VST), lateral orbitofrontal cortex, and ventral anterior insular cortex (vAIC). The enhancement of right VST-vAIC association by a taste of beer independently validated the main activation-based finding (Oberlin et al., 2016). Most notably, our novel dFC methodology uncovered numerous associations undetected by the traditional static FC analysis. The data-driven, novel dFC methodology presented here can be used for a wide range of task-based fMRI designs to estimate the dFC at multiple levels-group-, individual-, and task-specific, utilizing a combination of well-established statistical methods

Association of structural brain imaging markers with alcoholism incorporating structural connectivity information: a regularized statistical approach

poster abstractAbstract: Brain imaging studies collect multiple imaging data types, but most analyses are done for each modality separately. Statistical methods that simultaneously utilize and combine multiple data types can instead provide a more holistic view of brain function. Here we model associations between alcohol abuse phenotypes and imaging data while incorporating prior scientific knowledge. Specifically, we utilize cortical thickness and integrated rectified mean curvature measures obtained by FreeSurfer software [1] to predict the alcoholism-related phenotypes while incorporating prior information from the structural connectivity between cortical regions. The sample consisted of 148 young (21-35 years) social-to-heavy drinking male subjects from several alcoholism risk studies [2,3,4]. Structural connectivity model [5] was used to estimate the density of connections between 66 cortical regions based on Desikan-Killiany atlas [6]. We employed a functional linear model with a penalty operator to quantify the relative contributions of imaging markers obtained from high resolution structural MRI (cortical thickness and curvature) as predictors of drinking frequency and risk-relevant personality traits, while co-varying for age. Model parameters were estimated by a unified approach directly incorporating structural connectivity information into the estimation by exploiting the joint eigenproperties of the predictors and the penalty operator [7]. We found that the best predictive imaging markers of the Alcohol Use Disorders Identification Test (AUDIT) score were the average thickness of left frontal pole (-), right transverse temporal gyrus (+), left inferior parietal lobule (+), right supramarginal gyrus (-), right rostral middle frontal gyrus (+), right precentral gyrus (+), left superior parietal lobule (-), left lateral orbitofrontal cortex (+), left rostral middle frontal gyrus (+), left postcentral gyrus (+) and left supramarginal gyrus (-), where (+) denotes positive and (-) negative association. In summary, the use of structural connectivity information allowed the incorporation of different modalities in associating cortical measures and alcoholism risk