Varying Coefficient Regression Models: A Review and New Developments

Sustained mental health and functional responses to the COVID-19 pandemic in Black and White Veterans with psychosis or recent homelessness

The COVID-19 pandemic disproportionately impacted marginalized populations including Black Americans, people with serious mental illness, and individuals experiencing homelessness. Although the double disadvantage hypothesis would suggest that individuals with multiple minoritized statuses would experience worse psychosocial impacts from the pandemic, this may not be the case for vulnerable Black Veterans. The present study investigated the sustained mental health and functional responses to the pandemic in Black and White Veterans with psychosis or recent homelessness and in a control group of Veterans enrolled in the Department of Veterans Affairs healthcare services. Clinical interviews and questionnaires were administered remotely by telephone at five time points from May 2020 through July 2021, including a retrospective time point for March 2020 (i.e., before the pandemic started). Overall, there was a striking absence of systematic differences by race in the trajectories of psychiatric symptoms and functioning among Veterans during the study period. These findings are consistent with a report on initial responses to the pandemic that revealed only a few select differences by race among Veteran groups. The lack of racial disparities is inconsistent with the double disadvantage hypothesis. Although further investigation is needed, one possible interpretation is that the wrap-around services offered by the Veterans Health Administration may have mitigated expected differences by race among Veterans with psychosis or homelessness. Future research should continue to examine whether VA services mitigate disparities in mental health and psychosocial outcomes.

A new time-varying coefficient regression approach for analyzing infectious disease data

Since the beginning of the global pandemic of Coronavirus (SARS-COV-2), there has been many studies devoted to predicting the COVID-19 related deaths/hospitalizations. The aim of our work is to (1) explore the lagged dependence between the time series of case counts and the time series of death counts; and (2) utilize such a relationship for prediction. The proposed approach can also be applied to other infectious diseases or wherever dynamics in lagged dependence are of primary interest. Different from the previous studies, we focus on time-varying coefficient models to account for the evolution of the coronavirus. Using two different types of time-varying coefficient models, local polynomial regression models and piecewise linear regression models, we analyze the province-level data in Canada as well as country-level data using cumulative counts. We use out-of-sample prediction to evaluate the model performance. Based on our data analyses, both time-varying coefficient modeling strategies work well. Local polynomial regression models generally work better than piecewise linear regression models, especially when the pattern of the relationship between the two time series of counts gets more complicated (e.g., more segments are needed to portray the pattern). Our proposed methods can be easily and quickly implemented via existing R packages.

Bayesian hierarchical quantile regression with application to characterizing the immune architecture of lung cancer

The successful development and implementation of precision immuno-oncology therapies requires a deeper understanding of the immune architecture at a patient level. T-cell receptor (TCR) repertoire sequencing is a relatively new technology that enables monitoring of T-cells, a subset of immune cells that play a central role in modulating immune response. These immunologic relationships are complex and are governed by various distributional aspects of an individual patient's tumor profile. We propose Bayesian QUANTIle regression for hierarchical COvariates (QUANTICO) that allows simultaneous modeling of hierarchical relationships between multilevel covariates, conducts explicit variable selection, estimates quantile and patient-specific coefficient effects, to induce individualized inference. We show QUANTICO outperforms existing approaches in multiple simulation scenarios. We demonstrate the utility of QUANTICO to investigate the effect of TCR variables on immune response in a cohort of lung cancer patients. At population level, our analyses reveal the mechanistic role of T-cell proportion on the immune cell abundance, with tumor mutation burden as an important factor modulating this relationship. At a patient level, we find several outlier patients based on their quantile-specific coefficient functions, who have higher mutational rates and different smoking history.

Individualized risk assessment of preoperative opioid use by interpretable neural network regression

Preoperative opioid use has been reported to be associated with higher preoperative opioid demand, worse postoperative outcomes, and increased postoperative healthcare utilization and expenditures. Understanding the risk of preoperative opioid use helps establish patient-centered pain management. In the field of machine learning, deep neural network (DNN) has emerged as a powerful means for risk assessment because of its superb prediction power; however, the blackbox algorithms may make the results less interpretable than statistical models. Bridging the gap between the statistical and machine learning fields, we propose a novel Interpretable Neural Network Regression (INNER), which combines the strengths of statistical and DNN models. We use the proposed INNER to conduct individualized risk assessment of preoperative opioid use. Intensive simulations and an analysis of 34,186 patients expecting surgery in the Analgesic Outcomes Study (AOS) show that the proposed INNER not only can accurately predict the preoperative opioid use using preoperative characteristics as DNN, but also can estimate the patient-specific odds of opioid use without pain and the odds ratio of opioid use for a unit increase in the reported overall body pain, leading to more straight-forward interpretations of the tendency to use opioids than DNN. Our results identify the patient characteristics that are strongly associated with opioid use and is largely consistent with the previous findings, providing evidence that INNER is a useful tool for individualized risk assessment of preoperative opioid use.

Decision tree boosted varying coefficient models

Varying coefficient models are a flexible extension of generic parametric models whose coefficients are functions of a set of effect-modifying covariates instead of fitted constants. They are capable of achieving higher model complexity while preserving the structure of the underlying parametric models, hence generating interpretable predictions. In this paper we study the use of gradient boosted decision trees as those coefficient-deciding functions in varying coefficient models with linearly structured outputs. In contrast to the traditional choices of splines or kernel smoothers, boosted trees are more flexible since they require no structural assumptions in the effect modifier space. We introduce our proposed method from the perspective of a localized version of gradient descent, prove its theoretical consistency under mild assumptions commonly adapted by decision tree research, and empirically demonstrate that the proposed tree boosted varying coefficient models achieve high performance qualified by their training speed, prediction accuracy and intelligibility as compared to several benchmark algorithms.

Heterogeneity in general multinomial choice models

Different voters behave differently at the polls, different students make different university choices, or different countries choose different health care systems. Many research questions important to social scientists concern choice behavior, which involves dealing with nominal dependent variables. Drawing on the principle of maximum random utility, we propose applying a flexible and general heterogeneous multinomial logit model to study differences in choice behavior. The model systematically accounts for heterogeneity that classical models do not capture, indicates the strength of heterogeneity, and permits examining which explanatory variables cause heterogeneity. As the proposed approach allows incorporating theoretical expectations about heterogeneity into the analysis of nominal dependent variables, it can be applied to a wide range of research problems. Our empirical example uses individual-level survey data to demonstrate the benefits of the model in studying heterogeneity in electoral decisions.

Sparse regression for low-dimensional time-dynamic varying coefficient models with application to air quality data

Time dynamic varying coefficient models play an important role in applications of biology, medicine, environment, finance, etc. Traditional methods, such as kernel smoothing and spline smoothing, are popular. But explicit expressions are unavailable using these methods, and the convergence rate of coefficient function estimators is slow. To address these problems, we expand the varying component with appropriate basis functions. And then we solve a sparse regression problem via a sequential thresholded least-squares estimator. The "parameterization" leads to explicit expressions and fast computation speed. Convergence of the sequential thresholded least squares algorithm is guaranteed. The asymptotic distribution of the coefficient function estimator is derived under certain assumptions. Our simulation shows the proposed method has higher precision and computing speed. Finally, our proposed method is applied to the study of PM 2.5 concentration in Beijing. We analyze the relationship between PM 2.5 and other impact factors.

Model estimation and selection for partial linear varying coefficient EV models with longitudinal data

In this paper, we consider the estimation and model selection for longitudinal partial linear varying coefficient errors-in-variables (EV) models when the covariates are measured with some additive errors. Bias-corrected penalized quadratic inference functions method is proposed based on quadratic inference functions with two penalty function terms. The proposed method can not only handle the measurement errors of covariates and within-subject correlations but also estimate and select significant non-zero parametric and nonparametric components simultaneously. With some regularization conditions, the resulting estimators of parameters are asymptotically normal and the estimators of nonparametric varying coefficient achieves the optimal convergence rate. Furthermore, we present simulation studies and a real example analysis to evaluate the finite sample performance of the proposed method.

Backfitting tests in generalized structured models

We introduce bootstrap tests for semiparametric generalized structured models. These can be used for testing different kinds of model specifications like separability, functional forms and homogeneity of effects, or for performing variable selection in a large class of semiparametric models. The test statistics are based on the comparison of non- and semiparametric alternatives in which both the null hypothesis and the alternative are non- or semiparametric. All estimators are obtained by smooth backfitting. Simulation studies show excellent performance of the test procedures.

Interactions between ambient air pollutants and temperature on emergency department visits: Analysis of varying-coefficient model in Guangzhou, China

BACKGROUND

At present, there are few studies on the effect of short-term interactions between ambient air pollutants and temperature on cause-specific emergency department visits in China. This study aimed to explore their short-term interactions on cause-specific emergency department visits using data collected from a total of 65 public hospitals in Guangzhou city, south China.

MATERIAL AND METHODS

We included a total of 226,443 emergency department visits which were diagnosed as neurological, respiratory and circulatory disease in Guangzhou from January 1, 2014 to December 31, 2017. Average daily concentrations of air pollutants including carbon monoxide (CO), particulate matter having a median diameter of 2.5 μm or less (PM_2.5), sulfur dioxide (SO₂), nitrogen dioxide (NO₂) and ozone (O₃) were collected from the Guangzhou Environmental Protection Bureau. We employed quasi-Poisson varying coefficient regression models to assess the interaction effects between air pollutants and daily temperature levels (DTLs) on emergency department visits for neurological, respiratory and circulatory diseases, respectively.

RESULTS

Average number of emergency department visits for neurological, respiratory and circulatory diseases were 92, 26 and 38, respectively. After controlling for other pollutants, meteorological factors and other time-varying confounders, we found the interactions between NO₂ and the 1st DTL (3.4-17.1 °C), NO₂ and the 2nd DTL (17.1-23.5 °C) for neurological emergency department visits were statistically significant, displaying a nonlinear relationship. Additionally, we found that the interactions between SO₂ and the 4th DTL (27.4-31.1 °C) also had a significantly adverse effect on respiratory emergency department visits.

CONCLUSIONS

Our findings provide novel evidence on SO₂-by-temperature interactions, and NO₂-by-temperature interactions for emergency department visits of cause-specific diseases.

A Review on Variable Selection in Regression Analysis

In this paper, we investigate several variable selection procedures to give an overview of the existing literature for practitioners. “Let the data speak for themselves” has become the motto of many applied researchers since the number of data has significantly grown. Automatic model selection has been promoted to search for data-driven theories for quite a long time now. However, while great extensions have been made on the theoretical side, basic procedures are still used in most empirical work, e.g., stepwise regression. Here, we provide a review of main methods and state-of-the art extensions as well as a topology of them over a wide range of model structures (linear, grouped, additive, partially linear and non-parametric) and available software resources for implemented methods so that practitioners can easily access them. We provide explanations for which methods to use for different model purposes and their key differences. We also review two methods for improving variable selection in the general sense.

How to analyze the Visual Analogue Scale: Myths, truths and clinical relevance

BACKGROUND AND AIMS

The Visual Analogue Scale (VAS) is a popular tool for the measurement of pain. A variety of statistical methods are employed for its analysis as an outcome measure, not all of them optimal or appropriate. An issue which has attracted much discussion in the literature is whether VAS is at a ratio or ordinal level of measurement. This decision has an influence on the appropriate method of analysis. The aim of this article is to provide an overview of current practice in the analysis of VAS scores, to propose a method of analysis which avoids the shortcomings of more traditional approaches, and to provide best practice recommendations for the analysis of VAS scores.

METHODS

We report on the current usage of statistical methods, which fall broadly into two categories: those that assume a probability distribution for VAS, and those that do not. We give an overview of these methods, and propose continuous ordinal regression, an extension of current ordinal regression methodology, which is appropriate for VAS at an ordinal level of measurement. We demonstrate the analysis of a published data set using a variety of methods, and use simulation to compare the power of the various methods to detect treatment differences, in differing pain situations.

RESULTS

We demonstrate that continuous ordinal regression provides the most powerful statistical analysis under a variety of conditions.

CONCLUSIONS AND IMPLICATIONS

We recommend that in the situation in which no covariates besides treatment group are included in the analysis, distribution-free methods (Wilcoxon, Mann-Whitney) be used, as their power is indistinguishable from that of the proposed method. In the situation in which there are covariates which affect VAS, the proposed method is optimal. However, in this case, if the VAS scores are not concentrated around either extreme of the scale, normal-distribution methods (t-test, linear regression) are almost as powerful, and are recommended as a pragmatic choice. In the case of small sample size and VAS skewed to either extreme of the scale, the proposed method has vastly superior power to other methods.

A penalized robust semiparametric approach for gene-environment interactions

In genetic and genomic studies, gene-environment (G×E) interactions have important implications. Some of the existing G×E interaction methods are limited by analyzing a small number of G factors at a time, by assuming linear effects of E factors, by assuming no data contamination, and by adopting ineffective selection techniques. In this study, we propose a new approach for identifying important G×E interactions. It jointly models the effects of all E and G factors and their interactions. A partially linear varying coefficient model is adopted to accommodate possible nonlinear effects of E factors. A rank-based loss function is used to accommodate possible data contamination. Penalization, which has been extensively used with high-dimensional data, is adopted for selection. The proposed penalized estimation approach can automatically determine if a G factor has an interaction with an E factor, main effect but not interaction, or no effect at all. The proposed approach can be effectively realized using a coordinate descent algorithm. Simulation shows that it has satisfactory performance and outperforms several competing alternatives. The proposed approach is used to analyze a lung cancer study with gene expression measurements and clinical variables. Copyright © 2015 John Wiley & Sons, Ltd.