Longitudinal varying coefficient single-index model with censored covariates

It is of interest to health policy research to estimate the population-averaged longitudinal medical cost trajectory from initial cancer diagnosis to death, and understand how the trajectory curve is affected by patient characteristics. This research question leads to a number of statistical challenges because the longitudinal cost data are often non-normally distributed with skewness, zero-inflation, and heteroscedasticity. The trajectory is nonlinear, and its length and shape depend on survival, which are subject to censoring. Modeling the association between multiple patient characteristics and nonlinear cost trajectory curves of varying lengths should take into consideration parsimony, flexibility, and interpretation. We propose a novel longitudinal varying coefficient single-index model. Multiple patient characteristics are summarized in a single-index, representing a patient's overall propensity for healthcare use. The effects of this index on various segments of the cost trajectory depend on both time and survival, which is flexibly modeled by a bivariate varying coefficient function. The model is estimated by generalized estimating equations with an extended marginal mean structure to accommodate censored survival time as a covariate. We established the pointwise confidence interval of the varying coefficient and a test for the covariate effect. The numerical performance was extensively studied in simulations. We applied the proposed methodology to medical cost data of prostate cancer patients from the Surveillance, Epidemiology, and End Results-Medicare-Linked Database.

Bayesian multiple index models for environmental mixtures

An important goal of environmental health research is to assess the risk posed by mixtures of environmental exposures. Two popular classes of models for mixtures analyses are response-surface methods and exposure-index methods. Response-surface methods estimate high-dimensional surfaces and are thus highly flexible but difficult to interpret. In contrast, exposure-index methods decompose coefficients from a linear model into an overall mixture effect and individual index weights; these models yield easily interpretable effect estimates and efficient inferences when model assumptions hold, but, like most parsimonious models, incur bias when these assumptions do not hold. In this paper, we propose a Bayesian multiple index model framework that combines the strengths of each, allowing for non-linear and non-additive relationships between exposure indices and a health outcome, while reducing the dimensionality of the exposure vector and estimating index weights with variable selection. This framework contains response-surface and exposure-index models as special cases, thereby unifying the two analysis strategies. This unification increases the range of models possible for analysing environmental mixtures and health, allowing one to select an appropriate analysis from a spectrum of models varying in flexibility and interpretability. In an analysis of the association between telomere length and 18 organic pollutants in the National Health and Nutrition Examination Survey (NHANES), the proposed approach fits the data as well as more complex response-surface methods and yields more interpretable results.

Linearized Maximum Rank Correlation Estimation

We propose a linearized maximum rank correlation estimator for the single index model. Unlike the existing maximum rank correlation and other rank-based methods, the proposed estimator has a closed-form expression, making it appealing in theory and computation. The proposed estimator is robust to outliers in the response and its construction does not need the knowledge of the unknown link function or the error distribution. Under mild conditions, it is shown to be consistent and asymptotically normal when the predictors satisfy the linearity of expectation assumption. A more general class of estimators is also studied. Inference procedures based on the plug-in rule or random weighting resampling are employed for variance estimation. The proposed method can be easily modified to accommodate censored data. It can also be extended to deal with high-dimensional data combined with a penalty function. Extensive simulation studies provide strong evidence that the proposed method works well in various practical situations. Its application is illustrated with the Beijing PM 2.5 dataset.

Generalized expectile regression with flexible response function

Expectile regression, in contrast to classical linear regression, allows for heteroscedasticity and omits a parametric specification of the underlying distribution. This model class can be seen as a quantile-like generalization of least squares regression. Similarly as in quantile regression, the whole distribution can be modeled with expectiles, while still offering the same flexibility in the use of semiparametric predictors as modern mean regression. However, even with no parametric assumption for the distribution of the response in expectile regression, the model is still constructed with a linear relationship between the fitted value and the predictor. If the true underlying relationship is nonlinear then severe biases can be observed in the parameter estimates as well as in quantities derived from them such as model predictions. We observed this problem during the analysis of the distribution of a self-reported hearing score with limited range. Classical expectile regression should in theory adhere to these constraints, however, we observed predictions that exceeded the maximum score. We propose to include a response function between the fitted value and the predictor similarly as in generalized linear models. However, including a fixed response function would imply an assumption on the shape of the underlying distribution function. Such assumptions would be counterintuitive in expectile regression. Therefore, we propose to estimate the response function jointly with the covariate effects. We design the response function as a monotonically increasing P-spline, which may also contain constraints on the target set. This results in valid estimates for a self-reported listening effort score through nonlinear estimates of the response function. We observed strong associations with the speech reception threshold.

High-dimensional single-index models with censored responses

In this article, we study the estimation of high-dimensional single index models when the response variable is censored. We hybrid the estimation methods for high-dimensional single-index models (but without censorship) and univariate nonparametric models with randomly censored responses to estimate the index parameters and the link function and apply the proposed methods to analyze a genomic dataset from a study of diffuse large B-cell lymphoma. We evaluate the finite sample performance of the proposed procedures via simulation studies and establish large sample theories for the proposed estimators of the index parameter and the nonparametric link function under certain regularity conditions.

Sparse Single Index Models for Multivariate Responses

Joint models are popular for analyzing data with multivariate responses. We propose a sparse multivariate single index model, where responses and predictors are linked by unspecified smooth functions and multiple matrix level penalties are employed to select predictors and induce low-rank structures across responses. An alternating direction method of multipliers (ADMM) based algorithm is proposed for model estimation. We demonstrate the effectiveness of proposed model in simulation studies and an application to a genetic association study.

Large-sample estimation and inference in multivariate single-index models

By optimizing index functions against different outcomes, we propose a multivariate single-index model (SIM) for development of medical indices that simultaneously work with multiple outcomes. Fitting of a multivariate SIM is not fundamentally different from fitting a univariate SIM, as the former can be written as a sum of multiple univariate SIMs with appropriate indicator functions. What have not been carefully studied are the theoretical properties of the parameter estimators. Because of the lack of asymptotic results, no formal inference procedure has been made available for multivariate SIMs. In this paper, we examine the asymptotic properties of the multivariate SIM parameter estimators. We show that, under mild regularity conditions, estimators for the multivariate SIM parameters are indeed n-consistent and asymptotically normal. We conduct a simulation study to investigate the finite-sample performance of the corresponding estimation and inference procedures. To illustrate its use in practice, we construct an index measure of urine electrolyte markers for assessing the risk of hypertension in individual subjects.

SiAM: A Hybrid of Single Index Models and Additive Models

While popular, single index models and additive models have potential limitations, a fact that leads us to propose SiAM, a novel hybrid combination of these two models. We first address model identifiability under general assumptions. The result is of independent interest. We then develop an estimation procedure by using splines to approximate unknown functions and establish the asymptotic properties of the resulting estimators. Furthermore, we suggest a two-step procedure for establishing confidence bands for the nonparametric additive functions. This procedure enables us to make global inferences. Numerical experiments indicate that SiAM works well with finite sample sizes, and are especially robust to model structures. That is, when the model reduces to either single-index or additive scenario, the estimation and inference results are comparable to those based on the true model, while when the model is misspecified, the superiority of our method can be very great.

A Functional Varying-Coefficient Single-Index Model for Functional Response Data

Motivated by the analysis of imaging data, we propose a novel functional varying-coefficient single index model (FVCSIM) to carry out the regression analysis of functional response data on a set of covariates of interest. FVCSIM represents a new extension of varying-coefficient single index models for scalar responses collected from cross-sectional and longitudinal studies. An efficient estimation procedure is developed to iteratively estimate varying coefficient functions, link functions, index parameter vectors, and the covariance function of individual functions. We systematically examine the asymptotic properties of all estimators including the weak convergence of the estimated varying coefficient functions, the asymptotic distribution of the estimated index parameter vectors, and the uniform convergence rate of the estimated covariance function and their spectrum. Simulation studies are carried out to assess the finite-sample performance of the proposed procedure. We apply FVCSIM to investigating the development of white matter diffusivities along the corpus callosum skeleton obtained from Alzheimer's Disease Neuroimaging Initiative (ADNI) study.