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Researchers are sometimes interested in predicting a distal or external outcome (such as smoking cessation at follow-up) from the trajectory of an intensively recorded longitudinal variable (such as urge to smoke). This can be done in a semiparametric way via scalar-on-function regression. However, the resulting fitted coefficient regression functi...
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Semiparametric joint models of longitudinal and competing risks data are computationally costly and their current implementations do not scale well to massive biobank data. This paper identifies and addresses some key computational barriers in a semiparametric joint model for longitudinal and competing risks survival data. By developing and impleme...
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... An increase in rain is associated with more error throughout the year, whereas temperature is associated with error from Week 16 to Week 40, or from May to the end of September. The coefficient plots can be interpreted following the rules in [28]. ...
With billions of people facing moderate or severe food insecurity, the resilience of the global food supply will be of increasing concern due to the effects of climate change and geopolitical events. In this paper we describe a framework to better identify food security hotspots using a combination of remote sensing, deep learning, crop yield modeling, and causal modeling of the food distribution system. While we feel that the methods are adaptable to other regions of the world, we focus our analysis on the wheat breadbasket of northern India, which supplies a large percentage of the world's population. We present a quantitative analysis of deep learning domain adaptation methods for wheat farm identification based on curated remote sensing data from France. We model climate change impacts on crop yields using the existing crop yield modeling tool WOFOST and we identify key drivers of crop simulation error using a longitudinal penalized functional regression. A description of a system dynamics model of the food distribution system in India is also presented, along with results of food insecurity identification based on seeding this model with the predicted crop yields.
... where the intercept 0 and random error i are both scalar, and the coefficients j (t), j = 1, … , p are curves. The coefficient of each time point t of the functional predictor X j (T) represents joint associations considering other points of time X j (t), t ≠ t and other points X j � (T), j � ≠ j , i.e., j (t ) = (Y, X j (t )|X j � (t)) , where (⋅) denotes the linear conditional correlation coefficient, which represents joint associations considering other points of time t ≠ t and other points X j � (t), j � ≠ j (Dziak et al. 2019). ...
Variable selection involves selecting truly important predictors from p-dimensional multivariate functional predictors in functional predictive models. In this paper, a variable selection method is designed for scalar-on-function predictions entangled with nonlinear joint associations among scalar response and multiple functional predictors. First, a nonparametric functional nonlinear conditional correlation coefficient, namely, the FunNCC coefficient, is proposed to measure complex dependencies, including the nonmonotonic marginal dependence, along with the conditional associations of redundancy, complement, and interaction. Then, a model-free feature ordering and selection method is designed, where the FunNCC is utilized to rank relevance, enabling the selection of a subset of predictors with the strongest joint dependence. Since this method allows for quantitatively evaluating the contributions of predictors in explaining responses, it achieves moderate model interpretability. Finally, extensive simulation studies and two real-data cases involving air pollution regression and hand gesture recognition are conducted to evaluate the finite sample performance of the proposed method, and the results show that the proposed FunNCC and variable selection methods outperform state-of-the-art baselines.
... To investigate whether patients' in-session trajectory of valence, as continuously evaluated by their facial expressions, could predict their reports of in-session negative and positive emotions and changes in depressive symptoms, we used scalar-on-function regression models (Dziak et al., 2019;Reiss et al., 2017). These models allow for a data-driven examination of which aspects of the within-sessions dynamics predict the overall assessment of the session by the patient. ...
The main diagnostic criteria for major depressive disorder (MDD) are consistent experiences of high levels of negative emotions and low levels of positive emotions. Therefore, modification of these emotions is essential in the treatment of MDD. In the current study, we harnessed a computational approach to explore whether experiencing negative emotions during psychological treatment is related to subsequent changes in these emotions. Facial expressions were automatically extracted from 175 sessions of 58 patients with MDD. Within sessions, a U-shaped trajectory of change in valence was observed in which patients expressed an increase in negative emotions in the middle of the session. Between sessions, a consistent increase in valence was observed. A trajectory of within-sessions decrease followed by an increase in valence was positively associated with greater perceived positive emotions and subsequent decreases in depressive symptoms. These findings highlight the importance of targeting negative emotions during treatment to achieve more favorable outcomes.
... In addition, we are able to draw the following considerations: (1) a general significant association between the acceleration data and the investigated clinical assessments occurs, confirming that wearable sensors are able to recognize infants' motor impairments, quantified by the clinical scales; (2) our approach is able to discern an asymmetry (given by the difference between the arms movement) by correlating the sensors' data with a validated clinical assessment, suggesting that is possible to rely solely on the sensor data with respect to the studied clinical evaluations (in particular, the brain damage severity); (3) the negative correlation between brain injury and model outcome means that higher clinical scores correspond to a minor severity, as confirmed by the clinical experience, while the negative correlation between prematurity and model outcome is in contrast with the clinical experience, as reported in [25]; (4) according to [26] the weight curves have an interesting significance because they could help to determine certain periods that are more influential in the determination of the model outcome. ...
Background:
Nowadays, wearable sensors are widely used to quantify physical and motor activity during daily life, and they also represent innovative solutions for healthcare. In the clinical framework, the assessment of motor behaviour is entrusted to clinical scales, but they are dependent on operator experience. Thanks to their intrinsic objectivity, sensor data are extremely useful to provide support to clinicians. Moreover, wearable sensors are user-friendly and compliant to be used in an ecological environment (i.e., at home). This paper aims to propose an innovative approach useful to predict clinical assessment scores of infants' motor activity.
Materials and methods:
Starting from data acquired by accelerometers placed on infants' wrists and trunk during playtime, we exploit the method of functional data analysis to implement new models combining quantitative data and clinical scales. In particular, acceleration data, transformed into activity indexes and combined with baseline clinical data, represent the input dataset for functional linear models.
Conclusions:
Despite the small number of data samples available, results show correlation between clinical outcome and quantitative predictors, indicating that functional linear models could be able to predict the clinical evaluation. Future works will focus on a more refined and robust application of the proposed method, based on the acquisition of more data for validating the presented models.
Trial registration number:
ClincalTrials.gov; NCT03211533. Registered: July, 7th 2017. ClincalTrials.gov; NCT03234959. Registered: August, 1st 2017.
... In this study, our focus is restricted to the scalar-on-function regression model (SoFRM), which investigates the functional relationship between a scalar response and functional predictor variables. SoFRM has received significant attention in the literature and has been successfully applied in many fields, such as bioscience, meteorology, chemometrics, and engineering (see, e.g., previous studies 1, 3,[6][7][8][9][10][11][12][13]. ...
Scalar‐on‐function regression, where the response is scalar‐valued and the predictor consists of random functions, is one of the most important tools for exploring the functional relationship between a scalar response and functional predictor/s. The functional partial least squares method improves estimation accuracy for estimating the regression coefficient function compared to other existing methods, such as least squares, maximum likelihood, and maximum penalized likelihood. The functional partial least squares method is often based on the SIMPLS or NIPALS algorithm, but these algorithms can be computationally slow for analyzing a large dataset. In this study, we propose two modified functional partial least squares methods to efficiently estimate the regression coefficient function under the scalar‐on‐function regression. In the proposed methods, the infinite‐dimensional functional predictors are first projected onto a finite‐dimensional space using a basis expansion method. Then, two partial least squares algorithms, based on re‐orthogonalization of the score and loading vectors, are used to estimate the linear relationship between scalar response and the basis coefficients of the functional predictors. The finite‐sample performance and computing speed are evaluated using a series of Monte‐Carlo simulation studies and a sugar process dataset.
... In contrast with ordinary regression coefficients of principal component scores (Table 2), which provide the association between the full daily activity profile and health outcomes, the fPCR approach treated regression coefficients as smooth functions of time and computed an estimated coefficient at each time-point. As it was mentioned in Dziak et al. [14], a motivation of incorporating coefficient curves is to look for a period of time during which the predictors are more strongly associated with outcomes. However, we urge caution when interpreting time intervals, as our results are based on pointwise 95% intervals, and thus could be subject to increased Type 1 error when considering multiple time-points. ...
Accelerometers are widely used for tracking human movement and provide minute-level (or even 30 Hz level) physical activity (PA) records for detailed analysis. Instead of using day-level summary statistics to assess these densely sampled inputs, we implement functional principal component analysis (FPCA) approaches to study the temporal patterns of PA data from 245 overweight/obese women at three visits over a 1-year period. We apply longitudinal FPCA to decompose PA inputs, incorporating subject-specific variability, and then test the association between these patterns and obesity-related health outcomes by multiple mixed effect regression models. With the proposed methods, the longitudinal patterns in both densely sampled inputs and scalar outcomes are investigated and connected. The results show that the health outcomes are strongly associated with PA variation, in both subject and visit-level. In addition, we reveal that timing of PA during the day can impact changes in outcomes, a finding that would not be possible with day-level PA summaries. Thus, our findings imply that the use of longitudinal FPCA can elucidate temporal patterns of multiple levels of PA inputs. Furthermore, the exploration of the relationship between PA patterns and health outcomes can be useful for establishing weight-loss guidelines.
... As a result, both effect estimates and directionality of β must be interpreted with great care. Similar warnings regarding interpretability were issued by Dziak et al. (2019), although they do not suggest general purpose tools to resolve these challenges. As such, we are motivated to produce posterior summarization techniques that enable more interpretable and reliable inferences for SOFR. ...
Pollutant exposures during gestation are a known and adverse factor for birth and health outcomes. However, the links between prenatal air pollution exposures and educational outcomes are less clear, in particular the critical windows of susceptibility during pregnancy. Using a large cohort of students in North Carolina, we study prenatal \mbox{PM}_{2.5} exposures recorded at near-continuous resolutions and linked to 4th end-of-grade reading scores. We develop a locally-adaptive Bayesian regression model for scalar responses with functional and scalar predictors. The proposed model pairs a B-spline basis expansion with dynamic shrinkage priors to capture both smooth and rapidly-changing features in the regression surface. The local adaptivity is manifested in more accurate point estimates and more precise uncertainty quantification than existing methods on simulated data. The model is accompanied by a highly scalable Gibbs sampler for fully Bayesian inference on large datasets. In addition, we describe broad limitations with the interpretability of scalar-on-function regression models, and introduce new decision analysis tools to guide the model interpretation. Using these methods, we identify a period within the third trimester as the critical window of susceptibility to \mbox{PM}_{2.5} exposure.
... Also, in Lindquist's substantive example, the equivalent of the index t was space (voxel position along an axis through the brain in a functional brain imaging study) rather than time. A time-specific indirect effect α t X β t M with actual time-ordered data could be more difficult to interpret because of temporal dependence (see Dziak et al., 2019; VanderWeele & Tchetgen Tchetgen, 2017) and because time-specific snapshots may not adequately represent time-lagged effects (see Maxwell, Cole, & Mitchell, 2011); we address this further in the Discussion. Figure 2. Notice that the time-specific effect of any one of the M t variables on Y is complicated by having to adjust for earlier and/or later values of the mediator. ...
... The effect of the mediator on the outcome, CO-confirmed 7-day point prevalence abstinence at 4 weeks post-TQD, is shown in Figure 5. Results from functional regression are more difficult to explain with a precise interpretation than those from a time-varying effect model (Dziak et al., 2019). Nevertheless, it is evident from the confidence intervals in Figure 5 that for the most part, the confidence intervals include 0, with the exception of the end of the first week post-TQD. ...
... Functional regression coefficients can be somewhat difficult to interpret on a time-specific basis with longitudinal data, at least without additional model assumptions. The mediator values at different time points are not independent, so that the most statistically predictive time period need not be the most causally important time period (see de Stavola et al., 2021;Dziak et al., 2019;Green & Popham, 2016). Furthermore, even without intensive longitudinal data, researchers face the challenge of time-varying con-founding (de Stavola et al., 2021;VanderWeele & Tchetgen Tchetgen, 2017). ...
The increase in the use of mobile and wearable devices now allow dense assessment of mediating processes over time. For example, a pharmacological intervention may have an effect on smoking cessation via reductions in momentary withdrawal symptoms. We define and identify the causal direct and indirect effects in terms of potential outcomes on the mean difference and odds ratio scales, and present a method for estimating and testing the indirect effect of a randomized treatment on a distal binary variable as mediated by the nonparametric trajectory of an intensively measured longitudinal variable (e.g., from ecological momentary assessment). Coverage of a bootstrap test for the indirect effect is demonstrated via simulation. An empirical example is presented based on estimating later smoking abstinence from patterns of craving during smoking cessation treatment. We provide an R package, funmediation, to conveniently apply this technique. We conclude by discussing possible extensions to multiple mediators and directions for future research.
Compared with the conditional mean regression-based scalar-on-function regression model, the scalar-on-function quantile regression is robust to outliers in the response variable. However, it is susceptible to outliers in the functional predictor (called leverage points). This is because the influence function of the regression quantiles is bounded in the response variable but unbounded in the predictor space. The leverage points may alter the eigenstructure of the predictor matrix, leading to poor estimation and prediction results. This study proposes a robust procedure to estimate the model parameters in the scalar-on-function quantile regression method and produce reliable predictions in the presence of both outliers and leverage points. The proposed method is based on a functional partial quantile regression procedure. We propose a weighted partial quantile covariance to obtain functional partial quantile components of the scalar-on-function quantile regression model. After the decomposition, the model parameters are estimated via a weighted loss function, where the robustness is obtained by iteratively reweighting the partial quantile components. The estimation and prediction performance of the proposed method is evaluated by a series of Monte-Carlo experiments and an empirical data example. The results are compared favorably with several existing methods. The method is implemented in an R package robfpqr.
