Quadratic Inference Function Research Articles

Compare the marginal effects for environmental exposure and biomonitoring data with repeated measurements and values below the limit of detection.

Environmental exposure and biomonitoring data with repeated measurements from environmental and occupational studies are commonly right-skewed and in the presence of limits of detection (LOD). However, existing model has not been discussed for small-sample properties and highly skewed data with non-detects and repeated measurements. Marginal modeling provides an alternative to analyzing longitudinal and cluster data, in which the parameter interpretations are with respect to marginal or population-averaged means. We outlined the theories of three marginal models, i.e., generalized estimating equations (GEE), quadratic inference functions (QIF), and generalized method of moments (GMM). With these approaches, we proposed to incorporate the fill-in methods, including single and multiple value imputation techniques, such that any measurements less than the limit of detection are assigned values. We demonstrated that the GEE method works well in terms of estimating the regression parameters in small sample sizes, while the QIF and GMM outperform in large-sample settings, as parameter estimates are consistent and have relatively smaller mean squared error. No specific fill-in method can be deemed superior as each has its own merits. Marginal modeling is firstly employed to analyze repeated measures data with non-detects, in which only the mean structure needs to be correctly provided to obtain consistent parameter estimates. After replacing non-detects through substitution methods and utilizing small-sample bias corrections, in a simulation study we found that the estimating approaches used in the marginal models have corresponding advantages under a wide range of sample sizes. We also applied the models to longitudinal and cluster working examples.

Estimation and inference for multi-kink expectile regression with nonignorable dropout

In this paper, we consider parameter estimation, kink points testing and statistical inference for a longitudinal multi-kink expectile regression model with nonignorable dropout. In order to accommodate both within-subject correlations and nonignorable dropout, the bias-corrected generalized estimating equations are constructed by combining the inverse probability weighting and quadratic inference function approaches. The estimators for the kink locations and regression coefficients are obtained by using the generalized method of moments. A selection procedure based on a modified BIC is applied to estimate the number of kink points. We theoretically demonstrate the number selection consistency of kink points and the asymptotic normality of all estimators. A weighted cumulative sum type statistic is proposed to test the existence of kink effects at a given expectile, and its limiting distributions are derived under both the null and the local alternative hypotheses. Simulation studies show that the proposed estimators and test have desirable finite sample performance in both homoscedastic and heteroscedastic errors. An application to the Nation Growth, Lung and Health Study dataset is also presented.

The effect of the working correlation on fitting models to longitudinal data

AbstractWe present a detailed discussion of the theoretical properties of quadratic inference function estimators of the parameters in marginal linear regression models. We consider the effect of the choice of working correlation on fundamental questions including the existence of quadratic inference function estimators, their relationship with generalized estimating equations estimators, and the robustness and asymptotic relative efficiency of quadratic inference function and generalized estimating equations estimators. We show that the quadratic inference function estimators do not always exist and propose a way to handle this. We then show that they have unbounded influence functions and can be more or less asymptotically efficient than generalized estimating equations estimators. We also present empirical evidence to demonstrate these results. We conclude that the choice of working correlation can have surprisingly large effects.

Orthogonality-based bias-corrected empirical likelihood inference for partial linear varying coefficient EV models with longitudinal data

In this paper, we study the model estimation for the partial linear varying coefficient errors-in-variables (EV) models with longitudinal data. Based on the empirical likelihood and quadratic inference functions, we propose an orthogonality-based bias-corrected empirical likelihood estimation method using the QR decomposition method of matrix. The proposed method can handle the measurement errors of covariates and unknown within-subject correlation simultaneously, estimate the parametric and nonparametric components separately. Under some regularity conditions, the resulting estimator of the parametric component is asymptotic normal and the estimators of the nonparametric varying coefficients achieve the optimal convergence rate. Furthermore, we conduct simulation studies and a real data analysis to compare the performance of the proposed method with existing methods in finite samples. The results show that the feasibility of the proposed method is obvious.

Analysing cluster randomised controlled trials using GLMM, GEE1, GEE2, and QIF: results from four case studies

BackgroundUsing four case studies, we aim to provide practical guidance and recommendations for the analysis of cluster randomised controlled trials.MethodsFour modelling approaches (Generalized Linear Mixed Models with parameters estimated by maximum likelihood/restricted maximum likelihood; Generalized Linear Models with parameters estimated by Generalized Estimating Equations (1st order or second order) and Quadratic Inference Function, for analysing correlated individual participant level outcomes in cluster randomised controlled trials were identified after we reviewed the literature. We systematically searched the online bibliography databases of MEDLINE, EMBASE, PsycINFO (via OVID), CINAHL (via EBSCO), and SCOPUS. We identified the above-mentioned four statistical analytical approaches and applied them to four case studies of cluster randomised controlled trials with the number of clusters ranging from 10 to 100, and individual participants ranging from 748 to 9,207. Results were obtained for both continuous and binary outcomes using R and SAS statistical packages.ResultsThe intracluster correlation coefficient (ICC) estimates for the case studies were less than 0.05 and are consistent with the observed ICC values commonly reported in primary care and community-based cluster randomised controlled trials. In most cases, the four methods produced similar results. However, in a few analyses, quadratic inference function produced different results compared to the generalized linear mixed model, first-order generalized estimating equations, and second-order generalized estimating equations, especially in trials with small to moderate numbers of clusters.ConclusionThis paper demonstrates the analysis of cluster randomised controlled trials with four modelling approaches. The results obtained were similar in most cases, however, for trials with few clusters we do recommend that the quadratic inference function should be used with caution, and where possible a small sample correction should be used. The generalisability of our results is limited to studies with similar features to our case studies, for example, studies with a similar-sized ICC. It is important to conduct simulation studies to comprehensively evaluate the performance of the four modelling approaches.

Fused mean structure learning in data integration with dependence

AbstractMotivated by image‐on‐scalar regression with data aggregated across multiple sites, we consider a setting in which multiple independent studies each collect multiple dependent vector outcomes, with potential mean model parameter homogeneity between studies and outcome vectors. To determine the validity of a joint analysis of these data sources, we must learn which of them share mean model parameters. We propose a new model fusion approach that delivers improved flexibility and statistical performance over existing methods. Our proposed approach specifies a quadratic inference function within each data source and fuses mean model parameter vectors in their entirety based on a new formulation of a pairwise fusion penalty. We establish theoretical properties of our estimator and propose an asymptotically equivalent weighted oracle meta‐estimator that is more computationally efficient. Simulations and an application to the ABIDE neuroimaging consortium highlight the flexibility of the proposed approach. An R package is provided for ease of implementation.

Estimation and variable selection of quantile partially linear additive models for correlated data

Longitudinal data arise frequently in many economic studies and epidemiological research. In this paper, we investigate the partially linear additive model for longitudinal data in the framework of quantile regression. To incorporate the within-subject correlation, we develop an estimation procedure using quadratic inference function (QIF) and polynomial spline approximation for unknown nonparametric functions. The theoretical properties of the resulting estimators are established, where the nonparametric functions achieve the optimal convergence rate and the parametric components are asymptotically normal even when the number of parameters in the linear part is diverging. We also propose a variable selection procedure based on penalization. Since the objective function is discontinuous, a practical estimation procedure is proposed using induced smoothing and we prove that the smoothed estimator is asymptotically equivalent to the original estimator. The proposed methods are evaluated via simulation studies and a real data application.

Modeling Factors Associated with Dialysis Adequacy Using Longitudinal Data Analysis: Generalized Estimating Equation Versus Quadratic Inference Function.

In hemodialysis patients, changes in dialysis adequacy (DA) are examined longitudinally. The aim of this study was to determine factors affecting DA using the generalized estimating equation (GEE) and to compare them with the quadratic inference function (QIF). A longitudinal study. This longitudinal study examined the records of 153 end-stage renal disease (ESRD) patients. The longitudinal data on the DA and baseline demographic and clinical characteristics were obtained from patients' files. The GEE1, GEE2, and QIF models were fitted with different correlation structures, and then the best correlation structure was selected using the quasi-likelihood information criterion (QIC), Akaike information criterion (AIC), and Bayes information criterion (BIC) fitting criteria. The majority of patients (59.5%) had unfavorable DA (KT/V<1.2). Women and patients<60 years had more favorable DA. In the GEE model, the coefficients of female gender (β=0.079, 95% confidence interval [CI]: 0.032, 0.062), age at starting dialysis (β=-0.002, 95% CI: -0.004, -0.0001), hypertension (HTN, β=-0.055, 95% CI: -0.007, -0.103), diabetes (β=-0.088,95% CI: -0.021, -0.155), dialysis duration (β=0.132, 95% CI: 0.085, 0.178), and weight (β=-0.004, 95% CI: -0.006, -0.003) demonstrated a significant relationship with DA. The three models resulted in a similar estimate of regression coefficients. The relative efficiencies of QIF versus GEE1, QIF versus GEE2, and GEE2 versus GEE1 were 1.175, 1.056, and 1.113, respectively. DA is not optimal in most hemodialysis patients, and gender, age at the start of dialysis, HTN, diabetes, dialysis duration, and weight had a significant association with DA. The three different models yielded quite similar coefficient estimates, but the QIF model resulted more efficient than GEE1 and GEE2.

Improving marginal hazard ratio estimation using quadratic inference functions.

Clustered and multivariate failure time data are commonly encountered in biomedical studies and a marginal regression approach is often employed to identify the potential risk factors of a failure. We consider a semiparametric marginal Cox proportional hazards model for right-censored survival data with potential correlation. We propose to use a quadratic inference function method based on the generalized method of moments to obtain the optimal hazard ratio estimators. The inverse of the working correlation matrix is represented by the linear combination of basis matrices in the context of the estimating equation. We investigate the asymptotic properties of the regression estimators from the proposed method. The optimality of the hazard ratio estimators is discussed. Our simulation study shows that the estimator from the quadratic inference approach is more efficient than those from existing estimating equation methods whether the working correlation structure is correctly specified or not. Finally, we apply the model and the proposed estimation method to analyze a study of tooth loss and have uncovered new insights that were previously inaccessible using existing methods.

Semiparametric penalized quadratic inference functions for longitudinal data in ultra-high dimensions

In many biomedical and health studies, multivariate data arise from repeated measurements on a sample of subjects over time. In order to analyze such longitudinal data, we need to consider the correlations from the same subject, and it is inappropriate to use a simple multivariate model assuming independence structure. Motivated by a large scale longitudinal public health study that requires longitudinal data analysis with correlated multivariate discrete responses from repeated measurements and very high dimensional covariates, we adopt a flexible semiparametric approach for simultaneous variable selection and estimation without the requirement of specifying the full likelihood. Specifically, we propose generalized partially linear single-index models using penalized quadratic inference functions for longitudinal data in ultra-high dimensions. A key feature is that we allow the number of single-index covariates in the nonparametric term to diverge and even to be in ultra-high dimensions. The penalized quadratic inference functions easily incorporate within-subject correlation and pursue efficient estimation, and the single-index models can incorporate nonlinearity and some interactions while avoiding the curse of dimensionality. In this challenging setting, we contribute both an efficient algorithm and new asymptotic theory for our proposed approach for diverging and even ultra-dimensional covariates and a multivariate correlated response in longitudinal data. We apply our method to investigate diabetes status within a continuing longitudinal public health study with very high-dimensional genetic variables and phenotype variables.

Estimation of A Partial Linear Model with Instrumental Variable for the Longitudinal Data

A partial linear model with instrumental variables was developed for longitudinal data. In the partially linear model, the explanatory variable is an endogenous variable, which is correlated with the error term. The endogenous variable was expressed by an instrumental variable and an error item. The endogenous variable was estimated by the instrumental variable through the least square method. B-spline regression combined with QR decomposition was used to approximate the nonparametric function. For the estimation of parametric, the Quadratic inference function and Secant method were applied. Under some conditions, the estimator was consistent and asymptotic normality. Some simulation was conducted to prove the finite sample behavior of the estimator.

Statistical inference for streamed longitudinal data

Summary Modern longitudinal data, for example from wearable devices, may consist of measurements of biological signals on a fixed set of participants at a diverging number of time-points. Traditional statistical methods are not equipped to handle the computational burden of repeatedly analysing the cumulatively growing dataset each time new data are collected. We propose a new estimation and inference framework for dynamic updating of point estimates and their standard errors along sequentially collected datasets with dependence, both within and between the datasets. The key technique is a decomposition of the extended inference function vector of the quadratic inference function constructed over the cumulative longitudinal data into a sum of summary statistics over data batches. We show how this sum can be recursively updated without the need to access the whole dataset, resulting in a computationally efficient streaming procedure with minimal loss of statistical efficiency. We prove consistency and asymptotic normality of our streaming estimator as the number of data batches diverges, even as the number of independent participants remains fixed. Simulations demonstrate the advantages of our approach over traditional statistical methods that assume independence between data batches. Finally, we investigate the relationship between physical activity and several diseases through analysis of accelerometry data from the National Health and Nutrition Examination Survey.

Joint integrative analysis of multiple data sources with correlated vector outcomes

We propose a distributed quadratic inference function framework to jointly estimate regression parameters from multiple potentially heterogeneous data sources with correlated vector outcomes. The primary goal of this joint integrative analysis is to estimate covariate effects on all outcomes through a marginal regression model in a statistically and computationally efficient way. We develop a data integration procedure for statistical estimation and inference of regression parameters that is implemented in a fully distributed and parallelized computational scheme. To overcome computational and modeling challenges arising from the high-dimensional likelihood of the correlated vector outcomes, we propose to analyze each data source using Qu, Lindsay and Li’s (Biometrika 87 (2000) 823–836) quadratic inference functions and then to jointly reestimate parameters from each data source by accounting for correlation between data sources using a combined meta-estimator in a similar spirit to the generalized method of moments put forward by Hansen (Econometrica 50 (1982) 1029–1054). We show both theoretically and numerically that the proposed method yields efficiency improvements and is computationally fast. We illustrate the proposed methodology with the joint integrative analysis of the association between smoking and metabolites in a large multicohort study and provide an R package for ease of implementation.

Robust statistical inference for longitudinal data with nonignorable dropouts

In this paper, we propose robust statistical inference and variable selection method for generalized linear models that accommodate the outliers, nonignorable dropouts and within-subject correlations. The purpose of our study is threefold. First, we construct the robust and bias-corrected generalized estimating equations (GEEs) by combining the Mallows-type weights, Huber's score function and inverse probability weighting approaches to against the influence of outliers and account for nonignorable dropouts. Subsequently, the generalized method of moments is utilized to estimate the parameters in the nonignorable dropout propensity based on sufficient instrumental estimating equations. Second, in order to incorporate the within-subject correlations under an informative working correlation structure, we borrow the idea of quadratic inference function and hybrid-GEE to obtain the improved empirical likelihood procedures. The asymptotic properties of the proposed estimators and their confidence regions are derived. Third, the robust variable selection and algorithm are investigated. We evaluate the performance of proposed estimators through simulation and illustrate our method in an application to HIV-CD4 data.

Estimation in regret-regression using quadratic inference functions with ridge estimator.

In this paper, we propose a new estimation method in estimating optimal dynamic treatment regimes. The quadratic inference functions in myopic regret-regression (QIF-MRr) can be used to estimate the parameters of the mean response at each visit, conditional on previous states and actions. Singularity issues may arise during computation when estimating the parameters in ODTR using QIF-MRr due to multicollinearity. Hence, the ridge penalty was introduced in rQIF-MRr to tackle the issues. A simulation study and an application to anticoagulation dataset were conducted to investigate the model’s performance in parameter estimation. The results show that estimations using rQIF-MRr are more efficient than the QIF-MRr.

Sparse group variable selection for gene-environment interactions in the longitudinal study.

Penalized variable selection for high-dimensional longitudinal data has received much attention as it can account for the correlation among repeated measurements while providing additional and essential information for improved identification and prediction performance. Despite the success, in longitudinal studies, the potential of penalization methods is far from fully understood for accommodating structured sparsity. In this article, we develop a sparse group penalization method to conduct the bi-level gene-environment (G E) interaction study under the repeatedly measured phenotype. Within the quadratic inference function framework, the proposed method can achieve simultaneous identification of main and interaction effects on both the group and individual levels. Simulation studies have shown that the proposed method outperforms major competitors. In the case study of asthma data from the Childhood Asthma Management Program, we conduct G E study by using high-dimensional single nucleotide polymorphism data as genetic factors and the longitudinal trait, forced expiratory volume in 1 s, as the phenotype. Our method leads to improved prediction and identification of main and interaction effects with important implications.

Improving estimation efficiency for multivariate failure time data with auxiliary covariates

For the analysis of multivariate failure time data when the primary covariates are incomplete but the auxiliary covariates for them are available for the whole cohort subjects, an estimated quadratic inference function method is developed to improve the estimation efficiency under the marginal hazard model with common baseline hazard function. Both the auxiliary information and the intra-cluster correlation between the failure times are incorporated in the estimation procedure. The proposed estimator is shown to be consistent and asymptotically normal. Simulation studies demonstrate that when the intra-cluster correlation is strong or moderate, the proposed method gains noticeable efficiency compared to other existing methods.

Real-Time Regression Analysis of Streaming Clustered Data With Possible Abnormal Data Batches

This article develops an incremental learning algorithm based on quadratic inference function (QIF) to analyze streaming datasets with correlated outcomes such as longitudinal data and clustered data. We propose a renewable QIF (RenewQIF) method within a paradigm of renewable estimation and incremental inference, in which parameter estimates are recursively renewed with current data and summary statistics of historical data, but with no use of any historical subject-level raw data. We compare our renewable estimation method with both offline QIF and offline generalized estimating equations (GEE) approach that process the entire cumulative subject-level data all together, and show theoretically and numerically that our renewable procedure enjoys statistical and computational efficiency. We also propose an approach to diagnose the homogeneity assumption of regression coefficients via a sequential goodness-of-fit test as a screening procedure on occurrences of abnormal data batches. We implement the proposed methodology by expanding existing Spark’s Lambda architecture for the operation of statistical inference and data quality diagnosis. We illustrate the proposed methodology by extensive simulation studies and an analysis of streaming car crash datasets from the National Automotive Sampling System-Crashworthiness Data System (NASS CDS). Supplementary materials for this article are available online.

Estimated quadratic inference function for correlated failure time data.

An estimated quadratic inference function method is proposed for correlated failure time data with auxiliary covariates. The proposed method makes efficient use of the auxiliary information for the incomplete exposure covariates and preserves the property of the quadratic inference function method that requires the covariates to be completely observed. It can improve the estimation efficiency and easily deal with the situation when the cluster size is large. The proposed estimator which minimizes the estimated quadratic inference function is shown to be consistent and asymptotically normal. A chi-squared test based on the estimated quadratic inference function is proposed to test hypotheses about the regression parameters. The small-sample performance of the proposed method is investigated through extensive simulation studies. The proposed method is then applied to analyze the Study of Left Ventricular Dysfunction (SOLVD) data as anillustration.

Smoothed quantile regression with nonignorable dropouts

In this paper, we adopt a three-stage estimation procedure and statistical inference methods for quantile regression (QR) based on empirical likelihood (EL) approach with nonignorable dropouts. In the first stage, we consider a parametric model on the dropout propensity of response and handle the parameter identifiability issue by using nonresponse instrument. With the estimated dropout propensity, in the second stage the inverse probability weighting and kernel smoothing methods are applied to construct the bias-corrected and smoothed generalized estimating equations for nonignorable dropouts. In the third stage, borrowing the matrix expansion idea of quadratic inference function, we obtain the proposed estimators that can accommodate the within-subject correlations and improve the estimation efficiency simultaneously. A class of improved estimators and their confidence regions for QR coefficient are derived. Further, the penalized EL method and algorithm for variable selection are investigated. Simulation studies and a real example on HIV-CD4 data set are also provided to show the performance of the proposed estimators.