Approaches to regression analysis with multiple measurements from individual sampling units

Application of ordinary least-squares regression to data sets which contain multiple measurements from individual sampling units produces an unbiased estimator of the parameters but a biased estimator of the covariance matrix of the parameter estimates. The present work considers a random coefficient, linear model to deal with such data sets: this model permits many senses in which multiple measurements are taken from a sampling unit, not just when it is measured at several times. Three procedures to estimate the covariance matrix of the error term of the model are considered. Given these, three procedures to estimate the parameters of the model and their covariance matrix are considered; these are ordinary least-squares, generalized least-squares, and an adjusted ordinary least-squares procedure which produces an unbiased estimator of the covariance matrix of the parameters with small samples. These various procedures are compared in simulation studies using three examples from the biological literature. The possibility of testing hypotheses about the vector of parameters is also considered. It is found that all three procedures for regression estimation produce estimators of the parameters with bias of no practical consequence, Both generalized least-squares and adjusted ordinary least-squares generally produce estimators of the covariance matrix of the parameter estimates with bias of no practical consequence, while ordinary least-squares produces a negatively biased estimator. Neither ordinary nor generalized least-squares provide satisfactory hypothesis tests of the vector of parameter estimates. It is concluded that adjusted ordinary least-squares, when applied with either of two of the procedures used to estimate the error coveriance matrix, shows promise for practical application with data sets of the nature considered here.     

Using entropy distance measures for testing odds ratio parameters under logistic regression models based on case–control data

We propose two retrospective test statistics for testing the vector of odds ratio parameters under the logistic regression model based on case–control data by exploiting the density ratio structure under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed test statistics are based on Kullback–Leibler entropy distance and are particularly relevant to the case–control sampling plan. These two test statistics have identical asymptotic chi-squared distributions under the null hypothesis and identical asymptotic noncentral chi-squared distributions under local alternatives to the null hypothesis. Moreover, the proposed test statistics require computation of the maximum semiparametric likelihood estimators of the underlying parameters, but are otherwise easily computed. We present some results on simulation and on the analysis of two real data sets.     

Semiparametric log-linear regression for longitudinal measurements subject to outcome-dependent follow-up

A common problem for longitudinal data analyses is that subjects follow-up is irregular, often related to the past outcome or other factors associated with the outcome measure that are not included in the regression model. Analyses unadjusted for outcome-dependent follow-up yield biased estimates. We propose a longitudinal data analysis that can provide consistent estimates in regression models that are subject to outcome-dependent follow-up. We focus on semiparametric marginal log-link regression with arbitrary unspecified baseline function. Based on estimating equations, the proposed class of estimators are root n consistent and asymptotically normal. We present simulation studies that assess the performance of the estimators under finite samples. We illustrate our approach using data from a health services research study.     

Multiple imputation of a randomly censored covariate improves logistic regression analysis

Randomly censored covariates arise frequently in epidemiologic studies. The most commonly used methods, including complete case and single imputation or substitution, suffer from inefficiency and bias. They make strong parametric assumptions or they consider limit of detection censoring only. We employ multiple imputation, in conjunction with semi-parametric modeling of the censored covariate, to overcome these shortcomings and to facilitate robust estimation. We develop a multiple imputation approach for randomly censored covariates within the framework of a logistic regression model. We use the non-parametric estimate of the covariate distribution or the semi-parametric Cox model estimate in the presence of additional covariates in the model. We evaluate this procedure in simulations, and compare its operating characteristics to those from the complete case analysis and a survival regression approach. We apply the procedures to an Alzheimer's study of the association between amyloid positivity and maternal age of onset of dementia. Multiple imputation achieves lower standard errors and higher power than the complete case approach under heavy and moderate censoring and is comparable under light censoring. The survival regression approach achieves the highest power among all procedures, but does not produce interpretable estimates of association. Multiple imputation offers a favorable alternative to complete case analysis and ad hoc substitution methods in the presence of randomly censored covariates within the framework of logistic regression.     

Bias-corrected maximum semiparametric likelihood estimation under logistic regression models based on case–control data

We consider the corrective approach (Theoretical Statistics, Chapman & Hall, London, 1974, p. 310) and preventive approach (Biometrica 80 (1993) 27) to bias reduction of maximum likelihood estimators under the logistic regression model based on case–control data. The proposed bias-corrected maximum likelihood estimators are based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the prospective and retrospective analyses on the basis of the corrective approach to bias reduction produce identical bias-corrected maximum likelihood estimators of the odds ratio parameter, but this does not hold when using the preventive approach unless the case and control sample sizes are identical. We present some results on simulation and on the analysis of two real data sets.     

Efficient inverse probability weighting method for quantile regression with nonignorable missing data

Quantitle regression (QR) is a popular approach to estimate functional relations between variables for all portions of a probability distribution. Parameter estimation in QR with missing data is one of the most challenging issues in statistics. Regression quantiles can be substantially biased when observations are subject to missingness. We study several inverse probability weighting (IPW) estimators for parameters in QR when covariates or responses are subject to missing not at random. Maximum likelihood and semiparametric likelihood methods are employed to estimate the respondent probability function. To achieve nice efficiency properties, we develop an empirical likelihood (EL) approach to QR with the auxiliary information from the calibration constraints. The proposed methods are less sensitive to misspecified missing mechanisms. Asymptotic properties of the proposed IPW estimators are shown under general settings. The efficiency gain of EL-based IPW estimator is quantified theoretically. Simulation studies and a data set on the work limitation of injured workers from Canada are used to illustrated our proposed methodologies.     

Fitting logistic regression models with contaminated case–control data

Errors in measurement frequently occur in observing responses. If case–control data are based on certain reported responses, which may not be the true responses, then we have contaminated case–control data. In this paper, we first show that the ordinary logistic regression analysis based on contaminated case–control data can lead to very serious biased conclusions. This can be concluded from the results of a theoretical argument, one example, and two simulation studies. We next derive the semiparametric maximum likelihood estimate (MLE) of the risk parameter of a logistic regression model when there is a validation subsample. The asymptotic normality of the semiparametric MLE will be shown along with consistent estimate of asymptotic variance. Our example and two simulation studies show these estimates to have reasonable performance under finite sample situations.     

Using logistic regression for semiparametric comparison of population means and variances

Abstract

We propose to compare population means and variances under a semiparametric density ratio model. The proposed method is easy to implement by employing logistic regression procedures in many statistical software, and it often works very well when data are not normal. In this paper, we construct semiparametric estimators of the differences of two population means and variances, and derive their asymptotic distributions. We prove that the proposed semiparametric estimators are asymptotically more efficient than the corresponding non parametric ones. In addition, a simulation study and the analysis of two real data sets are presented. Finally, a short discussion is provided.     

Semiparametric Analysis of Isotonic Errors-in-Variables Regression Models with Missing Response

This article is concerned with the estimation problem in the semiparametric isotonic regression model when the covariates are measured with additive errors and the response is missing at random. An inverse marginal probability weighted imputation approach is developed to estimate the regression parameters and a least-square approach under monotone constraint is employed to estimate the functional component. We show that the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the isotonic estimator of the functional component, at a fixed point, is cubic root-n consistent. A simulation study is conducted to examine the finite-sample properties of the proposed estimators. A data set is used to demonstrate the proposed approach.     

Parameter estimation approaches to tackling measurement error and multicollinearity in ordinal probit models

Abstract

The regression model with ordinal outcome has been widely used in a lot of fields because of its significant effect. Moreover, predictors measured with error and multicollinearity are long-standing problems and often occur in regression analysis. However there are not many studies on dealing with measurement error models with generally ordinal response, even fewer when they suffer from multicollinearity. The purpose of this article is to estimate parameters of ordinal probit models with measurement error and multicollinearity. First, we propose to use regression calibration and refined regression calibration to estimate parameters in ordinal probit models with measurement error. Second, we develop new methods to obtain estimators of parameters in the presence of multicollinearity and measurement error in ordinal probit model. Furthermore we also extend all the methods to quadratic ordinal probit models and talk about the situation in ordinal logistic models. These estimators are consistent and asymptotically normally distributed under general conditions. They are easy to compute, perform well and are robust against the normality assumption for the predictor variables in our simulation studies. The proposed methods are applied to some real datasets.     

M-estimation Under a Two-Sample Semiparametric Model

We consider M -estimation under a two-sample semiparametric model in which the log ratio of two unknown density functions has a known parametric form. This two-sample semiparametric model, arising naturally from case-control studies and logistic discriminant analysis, can be regarded as a biased sampling model. A new class of M -estimators are constructed on the basis of the maximum semiparametric likelihood estimator of the underlying distribution function. It is shown that the proposed M -estimators are consistent and asymptotically normally distributed. A simulation study is presented to demonstrate the performance of the proposed M -estimators.     

Bayesian semiparametric models for nonignorable missing mechanisms in generalized linear models

Semiparametric models provide a more flexible form for modeling the relationship between the response and the explanatory variables. On the other hand in the literature of modeling for the missing variables, canonical form of the probability of the variable being missing (p) is modeled taking a fully parametric approach. Here we consider a regression spline based semiparametric approach to model the missingness mechanism of nonignorably missing covariates. In this model the relationship between the suitable canonical form of p (e.g. probit p) and the missing covariate is modeled through several splines. A Bayesian procedure is developed to efficiently estimate the parameters. A computationally advantageous prior construction is proposed for the parameters of the semiparametric part. A WinBUGS code is constructed to apply Gibbs sampling to obtain the posterior distributions. We show through an extensive Monte Carlo simulation experiment that response model coefficent estimators maintain better (when the true missingness mechanism is nonlinear) or equivalent (when the true missingness mechanism is linear) bias and efficiency properties with the use of proposed semiparametric missingness model compared to the conventional model.     

Pseudo likelihood-based estimation and testing of missingness mechanism function in nonignorable missing data problems

In nonignorable missing response problems, we study a semiparametric model with unspecified missingness mechanism model and a exponential family model for response conditional density. Even though existing methods are available to estimate the parameters in exponential family, estimation or testing of the missingness mechanism model nonparametrically remains to be an open problem. By defining a “synthesis" density involving the unknown missingness mechanism model and the known baseline “carrier" density in the exponential family model, we treat this “synthesis" density as a legitimate one with biased sampling version. We develop maximum pseudo likelihood estimation procedures and the resultant estimators are consistent and asymptotically normal. Since the “synthesis" cumulative distribution is a functional of the missingness mechanism model and the known carrier density, proposed method can be used to test the correctness of the missingness mechanism model nonparametrically andindirectly. Simulation studies and real example demonstrate the proposed methods perform very well.     

On the Breslow–Holubkov estimator

Breslow and Holubkov (J Roy Stat Soc B 59:447–461 1997a) developed semiparametric maximum likelihood estimation for two-phase studies with a case–control first phase under a logistic regression model and noted that, apart for the overall intercept term, it was the same as the semiparametric estimator for two-phase studies with a prospective first phase developed in Scott and Wild (Biometrica 84:57–71 1997). In this paper we extend the Breslow–Holubkov result to general binary regression models and show that it has a very simple relationship with its prospective first-phase counterpart. We also explore why the design of the first phase only affects the intercept of a logistic model, simplify the calculation of standard errors, establish the semiparametric efficiency of the Breslow–Holubkov estimator and derive its asymptotic distribution in the general case.     

Semiparametric multiple kernel estimators and model diagnostics for count regression functions

Abstract

This study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estimators. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivariate regression model used as start regression function belongs to a neighborhood of the true regression model.     

Conditional maximum likelihood estimation in semiparametric transformation model with LTRC data

Left-truncated data often arise in epidemiology and individual follow-up studies due to a biased sampling plan since subjects with shorter survival times tend to be excluded from the sample. Moreover, the survival time of recruited subjects are often subject to right censoring. In this article, a general class of semiparametric transformation models that include proportional hazards model and proportional odds model as special cases is studied for the analysis of left-truncated and right-censored data. We propose a conditional likelihood approach and develop the conditional maximum likelihood estimators (cMLE) for the regression parameters and cumulative hazard function of these models. The derived score equations for regression parameter and infinite-dimensional function suggest an iterative algorithm for cMLE. The cMLE is shown to be consistent and asymptotically normal. The limiting variances for the estimators can be consistently estimated using the inverse of negative Hessian matrix. Intensive simulation studies are conducted to investigate the performance of the cMLE. An application to the Channing House data is given to illustrate the methodology.     

Empirical likelihood for nonlinear regression models with nonignorable missing responses

This article develops three empirical likelihood (EL) approaches to estimate parameters in nonlinear regression models in the presence of nonignorable missing responses. These are based on the inverse probability weighted (IPW) method, the augmented IPW (AIPW) method and the imputation technique. A logistic regression model is adopted to specify the propensity score. Maximum likelihood estimation is used to estimate parameters in the propensity score by combining the idea of importance sampling and imputing estimating equations. Under some regularity conditions, we obtain the asymptotic properties of the maximum EL estimators of these unknown parameters. Simulation studies are conducted to investigate the finite sample performance of our proposed estimation procedures. Empirical results provide evidence that the AIPW procedure exhibits better performance than the other two procedures. Data from a survey conducted in 2002 are used to illustrate the proposed estimation procedure. The Canadian Journal of Statistics 48: 386–416; 2020 © 2020 Statistical Society of Canada     

Semiparametric ROC surface estimation for continuous diagnostic tests via polytomous logistic regression procedures

In this paper, we propose a semiparametric method of estimating receiver operating characteristic (ROC) surfaces for continuous diagnostic tests under density ratio models. Implementation of our method is easy since the usual polytomous logistic regression procedures in many statistical software packages can be employed. A simulated example is provided to facilitate the implementation of our method. Simulation results show that the proposed semiparametric ROC surface estimator is more efficient than the nonparametric counterpart and the parametric counterpart whether the normality assumption of data holds or not. Moreover, some simulation results on the underlying semiparametric distribution function estimators are also reported. In addition, some discussions on the proposed method as well as analysis of a real data set are provided.     

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

Generalized case–cohort sampling

A class of cohort sampling designs, including nested case–control, case–cohort and classical case–control designs involving survival data, is studied through a unified approach using Cox's proportional hazards model. By finding an optimal sample reuse method via local averaging, a closed form estimating function is obtained, leading directly to the estimators of the regression parameters that are relatively easy to compute and are more efficient than some commonly used estimators in case–cohort and nested case–control studies. A semiparametric efficient estimator can also be found with some further computation. In addition, the class of sampling designs in this study provides a variety of sampling options and relaxes the restrictions of sampling schemes that are currently available.