Partially Linear Single-index Models Research Articles

Empirical likelihood in a partially linear single-index model with censored response data

An empirical likelihood (EL) approach for a partial linear single-index model with censored response data is studied. A bias-corrected EL ratio is proposed, and the asymptotic chi-squared distribution of this ratio is obtained. The result can be directly used to construct the confidence regions of the regression parameters. The estimators of regression parameters and link function are constructed, and their asymptotic distributions are obtained. Also, a confidence band of the link function is constructed. The proposed method has two main features: The first feature is that the EL ratio is calibrated directly from within, instead of multiplying an adjustment factor by an EL ratio, which reflects the nature of EL. The second feature is avoiding undersmoothing of nonparametric functions, thus ensuring that the n-consistency of the parameter estimator. As a byproduct, the EL and estimation of a single-index model with censored response data are studied. The performance of the bias-corrected EL is evaluated by the simulation studies. The proposed method is illustrated with an example of a real data analysis.

Research on Two-stage Estimation of Partially Linear Single-index Model with Longitudinal Data

Partial linear single-index model is a kind of semi-parametric model with wide application. In this paper, we deal with the partial linear single-index model under longitudinal data. A "two-stage estimation method" without iteration by using local polynomial and bias correction generalized estimation equation is proposed. under some regularity conditions, the asymptotic properties of the connection function and unknown parameter estimator are investigated. Numerical simulation shows that the proposed method is robust.

Determinants of phthalate exposures in pregnant women in New York City

Previous studies have provided data on determinants of phthalates in pregnant women, but results were disparate across regions. We aimed to identify the food groups and demographic factors that predict phthalate exposure in an urban contemporary pregnancy cohort in the US. The study included 450 pregnant women from the New York University Children's Health and Environment Study in New York City. Urinary concentrations of 22 phthalate metabolites, including metabolites of di-2-ethylhexylphthalate (DEHP), were determined at three time points across pregnancy by liquid chromatography coupled with tandem mass spectrometry. The Diet History Questionnaire II was completed by pregnant women at mid-pregnancy to assess dietary information. Linear mixed models were fitted to examine determinants of urinary phthalate metabolite concentrations. Using partial-linear single-index (PLSI) models, we assessed the major contributors, among ten food groups, to phthalate exposure. Metabolites of DEHP and its ortho-phthalate replacement, diisononyl phthalate (DiNP), were found in >90% of the samples. The sum of creatinine-adjusted DiNP metabolite concentrations was higher in older and single women and in samples collected in summer. Hispanic and non-Hispanic Black women had lower urinary concentrations of summed metabolites of di-n-octyl phthalate (DnOP), but higher concentrations of low molecular weight phthalates compared with non-Hispanic White women. Each doubling of grain products consumed was associated with a 20.9% increase in ∑DiNP concentrations (95%CI: 4.5, 39.9). PLSI models revealed that intake of dried beans and peas was the main dietary factor contributing to urinary ∑DEHP, ∑DiNP, and ∑DnOP levels, with contribution proportions of 76.3%, 35.8%, and 27.4%, respectively. Urinary metabolite levels of phthalates in pregnant women in NYC varied by age, marital status, seasonality, race/ethnicity, and diet. These results lend insight into the major determinants of phthalates levels, and may be used to identify exposure sources and guide interventions to reduce exposures in susceptible populations.

Robust estimation for partial linear single-index models

In this article, minimum average variance estimation (MAVE) based on local modal regression is proposed for partial linear single-index models, which can be robust to different error distributions or outliers. Asymptotic distributions of the proposed estimators are derived, which have the same convergence rate as the original MAVE based on least squares. A modal EM algorithm is provided to implement our robust estimation. Both simulation studies and a real data example are used to evaluate the finite sample performance of the proposed estimation procedure.

Comparison of Some Semi-parametric Methods in Partial Linear Single-Index Model

The research dealt with a comparative study between some semi-parametric estimation methods to the Partial linear Single Index Model using simulation. There are two approaches to model estimation two-stage procedure and MADE to estimate this model. Simulations were used to study the finite sample performance of estimating methods based on different Single Index models, error variances, and different sample sizes , and the mean average squared errors were used as a comparison criterion between the methods were used. The results showed a preference for the two-stage procedure depending on all the cases that were used

Organophosphate pesticide exposure: Demographic and dietary predictors in an urban pregnancy cohort

Pregnant women are widely exposed to organophosphate (OP) pesticides, which are potentially neurotoxicant for the developing fetus. We aimed to identify principal demographic and dietary predictors of OP pesticide exposure among 450 pregnant women participating in the New York University Children’s Health and Environment Study (enrolled 2016–19). Urinary concentrations of six dialkyl phosphate (DAP) metabolites (3 dimethyl (DM) metabolites and 3 diethyl (DE) metabolites) of OP pesticides were determined at three time points across pregnancy. At mid-gestation, the Diet History Questionnaire II was used to assess women’s dietary intake over the past year. Demographic characteristics were obtained using questionnaires and/or electronic health records. We used linear mixed models to evaluate the associations of demographic and food groups with DAP metabolite levels, and partial-linear single-index (PLSI) models to analyze the contribution proportions of food groups to DAP metabolite concentrations and the dose-response relationships between them. We observed that pregnant women in NYC had lower levels of OP pesticide metabolites than pregnant populations in Europe, Asia, and other regions in the U.S. Having lower pre-pregnancy body mass index and being Asian, employed, and single were associated with higher DAP metabolite concentrations. Fruit and grain intakes were associated with higher ∑DM, ∑DE, and ∑DAP levels. ∑DE concentrations increased 9.0% (95% confidence interval (CI) = 1.2%, 17.4%) per two-fold increase in dairy consumption, whereas ∑DE concentrations decreased 1.8% (95%CI = −3.1%, −0.4%) per two-fold increase in seafood consumption. The PLSI model indicated that among the food mixture, fruit and grains were the main food groups contributed to higher levels of ∑DAP, while meat contributed to lower levels of ∑DAP. The contribution proportions of fruit, grains, and meat were 18.7%, 17.9%, and 39.3%, respectively. Our results suggest that fruit, grains, and meat are major dietary components associated with OP pesticide exposure in urban pregnant women.

Optimal shrinkage estimations in partially linear single-index models for binary longitudinal data

This paper focuses on the optimal estimation strategies of partially linear single-index models (PLSIM) for binary longitudinal data. Fitting model between the response and covariates may cause complexity and the linear terms may not be adequate to represent the relationship. In this situation, the PLSIM containing both linear and nonlinear terms is preferable. The objective of this paper is to develop optimal estimation strategies such as, pretest and shrinkage methods, for the analysis of binary longitudinal data under the PLSIM where some regression parameters are subject to restrictions. We estimate the nonparametric component using kernel estimating equations, and then use profile estimating equations to estimate the unrestricted and restricted estimators. To apply the pretest and shrinkage methods, we fit two models: one includes all covariates and the other restricts the regression parameters based on the auxiliary information. The unrestricted and restricted estimators are then combined optimally to get the pretest and shrinkage estimators. We also derive the asymptotic properties of the estimators in terms of biases and risks. Monte Carlo simulations are also conducted to examine the relative performance of the proposed estimators to the unrestricted estimator. An empirical application is also be used to illustrate the usefulness of our methodology.

Locally efficient semiparametric estimator for zero-inflated Poisson model with error-prone covariates

Overdispersion is a common phenomenon in count or frequency responses in Poisson models. For example, number of car accidents on a highway during a year period. A similar phenomenon is observed in electric power systems, where cascading failures often follows some distribution with inflated zero. When the response contains an excess amount of zeros, zero-inflated Poisson (ZIP) is the most favourable model. However, during the data collection process, some of the covariates cannot be accessed directly or are measured with error among numerous disciplines. To the best of our knowledge, little existing work is available in the literature that tackles the population heterogeneity in the count response while some of the covariates are measured with error. With the increasing popularity of such outcomes in modern studies, it is interesting and timely to study zero-inflated Poisson models in which some of the covariates are subject to measurement error while some are not. We propose a flexible partial linear single index model for the log Poisson mean to correct bias potentially due to the error in covariates or the population heterogeneity. We derive consistent and locally efficient semiparametric estimators and study the large sample properties. We further assess the finite sample performance through simulation studies. Finally, we apply the proposed method to a real data application and compare with existing methods that handle measurement error in covariates.

A family of partial-linear single-index models for analyzing complex environmental exposures with continuous, categorical, time-to-event, and longitudinal health outcomes

BackgroundStatistical methods to study the joint effects of environmental factors are of great importance to understand the impact of correlated exposures that may act synergistically or antagonistically on health outcomes. This study proposes a family of statistical models under a unified partial-linear single-index (PLSI) modeling framework, to assess the joint effects of environmental factors for continuous, categorical, time-to-event, and longitudinal outcomes. All PLSI models consist of a linear combination of exposures into a single index for practical interpretability of relative direction and importance, and a nonparametric link function for modeling flexibility.MethodsWe presented PLSI linear regression and PLSI quantile regression for continuous outcome, PLSI generalized linear regression for categorical outcome, PLSI proportional hazards model for time-to-event outcome, and PLSI mixed-effects model for longitudinal outcome. These models were demonstrated using a dataset of 800 subjects from NHANES 2003–2004 survey including 8 environmental factors. Serum triglyceride concentration was analyzed as a continuous outcome and then dichotomized as a binary outcome. Simulations were conducted to demonstrate the PLSI proportional hazards model and PLSI mixed-effects model. The performance of PLSI models was compared with their counterpart parametric models.ResultsPLSI linear, quantile, and logistic regressions showed similar results that the 8 environmental factors had both positive and negative associations with triglycerides, with a-Tocopherol having the most positive and trans-b-carotene having the most negative association. For the time-to-event and longitudinal settings, simulations showed that PLSI models could correctly identify directions and relative importance for the 8 environmental factors. Compared with parametric models, PLSI models got similar results when the link function was close to linear, but clearly outperformed in simulations with nonlinear effects.ConclusionsWe presented a unified family of PLSI models to assess the joint effects of exposures on four commonly-used types of outcomes in environmental research, and demonstrated their modeling flexibility and effectiveness, especially for studying environmental factors with mixed directional effects and/or nonlinear effects. Our study has expanded the analytical toolbox for investigating the complex effects of environmental factors. A practical contribution also included a coherent algorithm for all proposed PLSI models with R codes available.

Detection of the symmetry of model errors for partial linear single-index models

In this paper, we propose a k-th correlation coefficient estimator between the density function and distribution function of the model errors in single-index models and partial linear single-models. This k-th correlation coefficient estimator is used to test whether the density function of the true model error is symmetric or not. First, we propose a moment based estimator of k-th correlation coefficient and present its asymptotic results. Second, we consider statistical inference of the k-th correlation coefficient estimator by using the empirical likelihood method. The empirical likelihood statistic is shown to be asymptotically distributed a centered chi-squared distribution with degree of freedom one. Simulation studies are conducted to examine the performance of the proposed estimators and test statistics.

Checking Heteroscedasticity in Partially Linear Single-Index Models Using Pairwise Distance

In this article, a new test is proposed for partially linear single-index models (PLSIM) based on the pairwise distances of the sample points, to test heteroscedasticity. The statistic can be formulated as a U statistic and does not have to estimate the conditional variance function by using nonparametric methods, such as kernel, local polynomial, or spline. We derive a computationally feasible approximation to deal with the complexity of the limit zero distribution under the null hypothesis. We prove that the proposed bootstrap procedure is valid approximation to the null distribution of the test. It shows that this statistic has an asymptotically normal distribution. The algorithmic program of this test method is easy to implement and has faster convergence than some existing methods. In addition, convergence rate of the statistic does not depend on the dimensions of the covariates, which greatly reduces the impact of the dimensional curse. Finally, we give the numerical simulations and a real data example.

Estimation and variable selection for a class of quantile regression models with multiple index

Quantile regression (QR) for a groupwise additive multiple-index models and its applications are investigated. We find that quantile regression can be used to recovery the directions of the index parameter vectors, and it does not involve the nonparametric treatment completely. Based on this useful finding, a iterative-free QR estimator for the partial linear single index model and a penalized QR for variable selection in the high dimensional sparse models are proposed respectively. Because of inheriting the superiorities of quantile regression, our methods are robust and comprehensive. Simulation studies and real data analysis are included to illustrate the finite sample performance.

Estimation for biased partial linear single index models

In this paper, we propose a novel method to consistently estimate, at the root-n rate, the coefficient parameters in a biased partial linear single-index model whose error term does not have zero conditional expectation. To achieve this purpose, we first transfer the model to a pro forma linear model and then introduce an artificial variable into a linear bias correction model. Based on the bias correction model, the parameters can then be consistently estimated by the linear least squares method. Both numerical studies and real data analyses are conducted to show the effectiveness of the proposed estimation procedure.

Detection of marginal heteroscedasticity for partial linear single-index models

We consider the detection of marginal heteroscedasticity for partial linear single-index models. We regress absolute values of residuals on each covariate by marginal univariate nonparametric smoothing with a sequence of bandwidths, then a ordinary least squares estimator between the quadratic nonparametric estimate and bandwidths are obtained. We rank least squares estimates and obtain the top ranked covariates according to a ridge-type absolute coefficient ratio. Then, a refinement step is based to the smoothly clipped absolute deviation penalization method for marginal heteroscedastic detection. Simulation studies and a real dataset are conducted to demonstrate the performance of the proposed method.

A novel partial-linear single-index model for time series data

Partial-linear single-index models have been widely studied and applied, but their current applications to time series modeling still need some strong and inappropriate assumptions. A novel method which relaxes those assumptions is proposed. It extends the applicability of partial-linear single-index models to time series modeling, taking both lag variables and autocorrelated errors into consideration. An estimation procedure based on Whittle likelihood is proposed and some asymptotical properties of the corresponding estimators are derived. In addition, some simulation studies are conducted to elaborate that the proposed model is necessary in certain situations. The proposed models are also shown to be useful and reasonable in real data analysis, indicating the feasibility and practicability of the proposed estimation method.

Estimation of the error distribution function for partial linear single-index models

We consider the estimation of the error distribution function of partial linear single-index models. The estimation methods for the error distribution function based on the classical empirical distribution function as well as empirical likelihood method are discussed, the latter method allows for incorporation of additional information on the error distribution function into estimation. We show weak convergence of the corresponding empirical processes to Gaussian processes and compare both approaches with the asymptotic theory and by means of simulation studies.

Statistical inference on asymptotic properties of two estimators for the partially linear single-index models

ABSTRACTThe outer product of gradients (OPG) estimation procedure based on least squares (LS) approach has been presented by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] to estimate the single-index parameter in partially linear single-index models (PLSIM). However, its asymptotic property has not been established yet and the efficiency of LS-based method can be significantly affected by outliers and heavy-tailed distributions. In this paper, we firstly derive the asymptotic property of OPG estimator developed by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] in theory, and a novel robust estimation procedure combining the ideas of OPG and local rank (LR) inference is further developed for PLSIM along with its theoretical property. Then, we theoretically derive the asymptotic relative efficiency (ARE) of the proposed LR-based procedure with respect to LS-based method, which is shown to possess an expression that is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Moreover, we demonstrate that the new proposed estimator has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error. Even in the worst case scenarios, the ARE owns a lower bound equalling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric function respectively, versus the LS-based estimators. Finally, some Monte Carlo simulations and a real data analysis are conducted to illustrate the finite sample performance of the estimators.

Bayesian quantile regression and variable selection for partial linear single-index model: Using free knot spline

ABSTRACTPartial linear single-index model (PLSIM) has both the flexibility of nonparametric treatment and interpretability of linear term, yet existing literatures about it mainly focused on mean regression, and quantile regression analysis is scarce. Based on free knot spline approximation, we apply asymmetric Laplace distribution to implement Bayesian quantile regression, and perform variable selection in linear term and index vector via binary indicators. Our approach is exempt from regularity conditions in frequentist method, and could execute variable selection and quantile regression under mutual posterior correction, which is also the first work to implement them jointly for PLSIM in fully Bayesian framework. The numerical simulation manifests the superiority of our approach to previous methods, which embodied in better efficiency of variable selection, index vector estimates and link function approximation with different error distributions. For illustration of its application, we build a power consumption model of A2/O process in wastewater treatment and emphatically analyze the impact of water quality factors.

Variable selection for the partial linear single-index model

In this paper, we consider the issue of variable selection in partial linear single-index models under the assumption that the vector of regression coefficients is sparse. We apply penalized spline to estimate the nonparametric function and SCAD penalty to achieve sparse estimates of regression parameters in both the linear and single-index parts of the model. Under some mild conditions, it is shown that the penalized estimators have oracle property, in the sense that it is asymptotically normal with the same mean and covariance that they would have if zero coefficients are known in advance. Our model owns a least square representation, therefore standard least square programming algorithms can be implemented without extra programming efforts. In the meantime, parametric estimation, variable selection and nonparametric estimation can be realized in one step, which incredibly increases computational stability. The finite sample performance of the penalized estimators is evaluated through Monte Carlo studies and illustrated with a real data set.

Partial linear single-index models with additive distortion measurement errors

ABSTRACTWe study partial linear single-index models (PLSiMs) when the response and the covariates in the parametric part are measured with additive distortion measurement errors. These distortions are modeled by unknown functions of a commonly observable confounding variable. We use the semiparametric profile least-squares method to estimate the parameters in the PLSiMs based on the residuals obtained from the distorted variables and confounding variable. We also employ the smoothly clipped absolute deviation penalty (SCAD) to select the relevant variables in the PLSiMs. We show that the resulting SCAD estimators are consistent and possess the oracle property. For the non parametric link function, we construct the simultaneous confidence bands and obtain the asymptotic distribution of the maximum absolute deviation between the estimated link function and the true link function. A simulation study is conducted to evaluate the performance of the proposed methods and a real dataset is analyzed for illustration.