Covariate Decomposition Methods for Longitudinal Missing‐at‐Random Data and Predictors Associated with Subject‐Specific Effects

Investigators often gather longitudinal data to assess changes in responses over time within subjects and to relate these changes to within‐subject changes in predictors. Missing data are common in such studies and predictors can be correlated with subject‐specific effects. Maximum likelihood methods for generalized linear mixed models provide consistent estimates when the data are ‘missing at random’ (MAR) but can produce inconsistent estimates in settings where the random effects are correlated with one of the predictors. On the other hand, conditional maximum likelihood methods (and closely related maximum likelihood methods that partition covariates into between‐ and within‐cluster components) provide consistent estimation when random effects are correlated with predictors but can produce inconsistent covariate effect estimates when data are MAR. Using theory, simulation studies, and fits to example data this paper shows that decomposition methods using complete covariate information produce consistent estimates. In some practical cases these methods, that ostensibly require complete covariate information, actually only involve the observed covariates. These results offer an easy‐to‐use approach to simultaneously protect against bias from both cluster‐level confounding and MAR missingness in assessments of change.     

Hybrid pairwise‐likelihood estimation methods for incomplete longitudinal binary data

In longitudinal data, missing observations occur commonly with incomplete responses and covariates. Missing data can have a ‘missing not at random’ mechanism, a non‐monotone missing pattern, and moreover response and covariates can be missing not simultaneously. To avoid complexities in both modelling and computation, a two‐stage estimation method and a pairwise‐likelihood method are proposed. The two‐stage estimation method enjoys simplicities in computation, but incurs more severe efficiency loss. On the other hand, the pairwise approach leads to estimators with better efficiency, but can be cumbersome in computation. In this paper, we develop a compromise method using a hybrid pairwise‐likelihood framework. Our proposed approach has better efficiency than the two‐stage method, but its computational cost is still reasonable compared to the pairwise approach. The performance of the methods is evaluated empirically by means of simulation studies. Our methods are used to analyse longitudinal data obtained from the National Population Health Study.     

Influence diagnostics for censored regression models with autoregressive errors

Observations collected over time are often autocorrelated rather than independent, and sometimes include observations below or above detection limits (i.e. censored values reported as less or more than a level of detection) and/or missing data. Practitioners commonly disregard censored data cases or replace these observations with some function of the limit of detection, which often results in biased estimates. Moreover, parameter estimation can be greatly affected by the presence of influential observations in the data. In this paper we derive local influence diagnostic measures for censored regression models with autoregressive errors of order p (hereafter, AR(p)‐CR models) on the basis of the Q‐function under three useful perturbation schemes. In order to account for censoring in a likelihood‐based estimation procedure for AR(p)‐CR models, we used a stochastic approximation version of the expectation‐maximisation algorithm. The accuracy of the local influence diagnostic measure in detecting influential observations is explored through the analysis of empirical studies. The proposed methods are illustrated using data, from a study of total phosphorus concentration, that contain left‐censored observations. These methods are implemented in the R package ARCensReg.     

A Sample Selection Model with Skew‐normal Distribution

Non‐random sampling is a source of bias in empirical research. It is common for the outcomes of interest (e.g. wage distribution) to be skewed in the source population. Sometimes, the outcomes are further subjected to sample selection, which is a type of missing data, resulting in partial observability. Thus, methods based on complete cases for skew data are inadequate for the analysis of such data and a general sample selection model is required. Heckman proposed a full maximum likelihood estimation method under the normality assumption for sample selection problems, and parametric and non‐parametric extensions have been proposed. We generalize Heckman selection model to allow for underlying skew‐normal distributions. Finite‐sample performance of the maximum likelihood estimator of the model is studied via simulation. Applications illustrate the strength of the model in capturing spurious skewness in bounded scores, and in modelling data where logarithm transformation could not mitigate the effect of inherent skewness in the outcome variable.     

INFLUENCE DIAGNOSTICS FOR THE NORMAL LINEAR MODEL WITH CENSORED DATA

Methods of detecting influential observations for the normal model for censored data are proposed. These methods include one-step deletion methods, deletion of observations and the empirical influence function. Emphasis is placed on assessing the impact that a single observation has on the estimation of coefficients of the model. Functions of the coefficients such as the median lifetime are also considered. Results are compared when applied to two sets of data.     

Existence,uniqueness and consistency of estimation of life characteristics of three-parameter Weibull distribution based on Type-II right censored data

In this paper, we propose a new method of estimation for the parameters and quantiles of the three-parameter Weibull distribution based on Type-II right censored data. The method, based on a data transformation, overcomes the problem of unbounded likelihood. In the proposed method, under mild conditions, the estimates always exist uniquely, and the estimators are also consistent over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method of estimation performs well compared to some prominent methods in terms of bias and root mean squared error in small-sample situations. Finally, two real data sets are used to illustrate the proposed method of estimation.     

Regression Analysis of Length‐biased and Right‐censored Failure Time Data with Missing Covariates

Length‐biased and right‐censored failure time data arise from many fields, and their analysis has recently attracted a great deal of attention. Two examples of the areas that often produce such data are epidemiological studies and cancer screening trials. In this paper, we discuss regression analysis of such data in the presence of missing covariates, for which no established inference procedure seems to exist. For the problem, we consider the data arising from the proportional hazards model and propose two inverse probability weighted estimation procedures. The asymptotic properties of the resulting estimators are established, and the extensive simulation study conducted for the evaluation of the proposed methods suggests that they work well for practical situations.     

A Cox‐Aalen Model for Interval‐censored Data

The Cox‐Aalen model, obtained by replacing the baseline hazard function in the well‐known Cox model with a covariate‐dependent Aalen model, allows for both fixed and dynamic covariate effects. In this paper, we examine maximum likelihood estimation for a Cox‐Aalen model based on interval‐censored failure times with fixed covariates. The resulting estimator globally converges to the truth slower than the parametric rate, but its finite‐dimensional component is asymptotically efficient. Numerical studies show that estimation via a constrained Newton method performs well in terms of both finite sample properties and processing time for moderate‐to‐large samples with few covariates. We conclude with an application of the proposed methods to assess risk factors for disease progression in psoriatic arthritis.     

On the asymptotic non‐equivalence of efficient‐GMM and MEL estimators in models with missing data

The generalized method of moments (GMM) and empirical likelihood (EL) are popular methods for combining sample and auxiliary information. These methods are used in very diverse fields of research, where competing theories often suggest variables satisfying different moment conditions. Results in the literature have shown that the efficient‐GMM (GMM_E) and maximum empirical likelihood (MEL) estimators have the same asymptotic distribution to order n^?1/2 and that both estimators are asymptotically semiparametric efficient. In this paper, we demonstrate that when data are missing at random from the sample, the utilization of some well‐known missing‐data handling approaches proposed in the literature can yield GMM_E and MEL estimators with nonidentical properties; in particular, it is shown that the GMM_E estimator is semiparametric efficient under all the missing‐data handling approaches considered but that the MEL estimator is not always efficient. A thorough examination of the reason for the nonequivalence of the two estimators is presented. A particularly strong feature of our analysis is that we do not assume smoothness in the underlying moment conditions. Our results are thus relevant to situations involving nonsmooth estimating functions, including quantile and rank regressions, robust estimation, the estimation of receiver operating characteristic (ROC) curves, and so on.     

Impact of missing data on type 1 error rates in non‐inferiority trials

In this paper, a simulation study is conducted to systematically investigate the impact of different types of missing data on six different statistical analyses: four different likelihood‐based linear mixed effects models and analysis of covariance (ANCOVA) using two different data sets, in non‐inferiority trial settings for the analysis of longitudinal continuous data. ANCOVA is valid when the missing data are completely at random. Likelihood‐based linear mixed effects model approaches are valid when the missing data are at random. Pattern‐mixture model (PMM) was developed to incorporate non‐random missing mechanism. Our simulations suggest that two linear mixed effects models using unstructured covariance matrix for within‐subject correlation with no random effects or first‐order autoregressive covariance matrix for within‐subject correlation with random coefficient effects provide well control of type 1 error (T1E) rate when the missing data are completely at random or at random. ANCOVA using last observation carried forward imputed data set is the worst method in terms of bias and T1E rate. PMM does not show much improvement on controlling T1E rate compared with other linear mixed effects models when the missing data are not at random but is markedly inferior when the missing data are at random. Copyright © 2009 John Wiley & Sons, Ltd.     

Restricted maximum likelihood estimation of joint mean‐covariance models

The class of joint mean‐covariance models uses the modified Cholesky decomposition of the within subject covariance matrix in order to arrive to an unconstrained, statistically meaningful reparameterisation. The new parameterisation of the covariance matrix has two sets of parameters that separately describe the variances and correlations. Thus, with the mean or regression parameters, these models have three sets of distinct parameters. In order to alleviate the problem of inefficient estimation and downward bias in the variance estimates, inherent in the maximum likelihood estimation procedure, the usual REML estimation procedure adjusts for the degrees of freedom lost due to the estimation of the mean parameters. Because of the parameterisation of the joint mean covariance models, it is possible to adapt the usual REML procedure in order to estimate the variance (correlation) parameters by taking into account the degrees of freedom lost by the estimation of both the mean and correlation (variance) parameters. To this end, here we propose adjustments to the estimation procedures based on the modified and adjusted profile likelihoods. The methods are illustrated by an application to a real data set and simulation studies. The Canadian Journal of Statistics 40: 225–242; 2012 © 2012 Statistical Society of Canada     

Minimum chi-squared estimation of stable distributions parameters: an application to the Warsaw Stock Exchange

This paper derives an application of the minimum chi-squared (MCS) methodology to estimate the parameters of the unimodal symmetric stable distribution. The proposed method is especially suitable for large, both regular and non-standard, data sets. Monte Carlo simulations are performed to compare the efficiency of the MCS estimation with the efficiency of the McCulloch quantile algorithm. In the case of grouped observations, evidence in favour of the MCS method is reported. For the ungrouped data the MCS estimation generally performs better than McCulloch's quantile method for samples larger than 400 observations and for high alphas. The relative advantage of the MCS over the McCulloch estimators increases for larger samples. The empirical example analyses the highly irregular distributions of returns on the selected securities from the Warsaw Stock Exchange. The quantile and maximum likelihood estimates of characteristic exponents are generally smaller than the MCS ones. This reflects the bias in the traditional methods, which is due to a lack of adjustment for censored and clustered observations, and shows the flexibility of the proposed MCS approach.     

Robust estimation in generalized linear mixed models

Generalized linear mixed models (GLMMs) are widely used to analyse non-normal response data with extra-variation, but non-robust estimators are still routinely used. We propose robust methods for maximum quasi-likelihood and residual maximum quasi-likelihood estimation to limit the influence of outlying observations in GLMMs. The estimation procedure parallels the development of robust estimation methods in linear mixed models, but with adjustments in the dependent variable and the variance component. The methods proposed are applied to three data sets and a comparison is made with the nonparametric maximum likelihood approach. When applied to a set of epileptic seizure data, the methods proposed have the desired effect of limiting the influence of outlying observations on the parameter estimates. Simulation shows that one of the residual maximum quasi-likelihood proposals has a smaller bias than those of the other estimation methods. We further discuss the equivalence of two GLMM formulations when the response variable follows an exponential family. Their extensions to robust GLMMs and their comparative advantages in modelling are described. Some possible modifications of the robust GLMM estimation methods are given to provide further flexibility for applying the method.     

Aalen's linear model for doubly censored data

Double censoring often occurs in registry studies when left censoring is present in addition to right censoring. In this work, we examine estimation of Aalen's nonparametric regression coefficients based on doubly censored data. We propose two estimation techniques. The first type of estimators, including ordinary least squared (OLS) estimator and weighted least squared (WLS) estimators, are obtained using martingale arguments. The second type of estimator, the maximum likelihood estimator (MLE), is obtained via expectation-maximization (EM) algorithms that treat the survival times of left censored observations as missing. Asymptotic properties, including the uniform consistency and weak convergence, are established for the MLE. Simulation results demonstrate that the MLE is more efficient than the OLS and WLS estimators.     

Semi‐parametric small‐area estimation by combining time‐series and cross‐sectional data methods

In survey sampling, policymaking regarding the allocation of resources to subgroups (called small areas) or the determination of subgroups with specific properties in a population should be based on reliable estimates. Information, however, is often collected at a different scale than that of these subgroups; hence, the estimation can only be obtained on finer scale data. Parametric mixed models are commonly used in small‐area estimation. The relationship between predictors and response, however, may not be linear in some real situations. Recently, small‐area estimation using a generalised linear mixed model (GLMM) with a penalised spline (P‐spline) regression model, for the fixed part of the model, has been proposed to analyse cross‐sectional responses, both normal and non‐normal. However, there are many situations in which the responses in small areas are serially dependent over time. Such a situation is exemplified by a data set on the annual number of visits to physicians by patients seeking treatment for asthma, in different areas of Manitoba, Canada. In cases where covariates that can possibly predict physician visits by asthma patients (e.g. age and genetic and environmental factors) may not have a linear relationship with the response, new models for analysing such data sets are required. In the current work, using both time‐series and cross‐sectional data methods, we propose P‐spline regression models for small‐area estimation under GLMMs. Our proposed model covers both normal and non‐normal responses. In particular, the empirical best predictors of small‐area parameters and their corresponding prediction intervals are studied with the maximum likelihood estimation approach being used to estimate the model parameters. The performance of the proposed approach is evaluated using some simulations and also by analysing two real data sets (precipitation and asthma).     

Linear Regression With Nested Errors Using Probability‐Linked Data

Probabilistic matching of records is widely used to create linked data sets for use in health science, epidemiological, economic, demographic and sociological research. Clearly, this type of matching can lead to linkage errors, which in turn can lead to bias and increased variability when standard statistical estimation techniques are used with the linked data. In this paper we develop unbiased regression parameter estimates to be used when fitting a linear model with nested errors to probabilistically linked data. Since estimation of variance components is typically an important objective when fitting such a model, we also develop appropriate modifications to standard methods of variance components estimation in order to account for linkage error. In particular, we focus on three widely used methods of variance components estimation: analysis of variance, maximum likelihood and restricted maximum likelihood. Simulation results show that our estimators perform reasonably well when compared to standard estimation methods that ignore linkage errors.     

Estimation and goodness-of-fit for the Cox model with various types of censored data

The currently existing estimation methods and goodness-of-fit tests for the Cox model mainly deal with right censored data, but they do not have direct extension to other complicated types of censored data, such as doubly censored data, interval censored data, partly interval-censored data, bivariate right censored data, etc. In this article, we apply the empirical likelihood approach to the Cox model with complete sample, derive the semiparametric maximum likelihood estimators (SPMLE) for the Cox regression parameter and the baseline distribution function, and establish the asymptotic consistency of the SPMLE. Via the functional plug-in method, these results are extended in a unified approach to doubly censored data, partly interval-censored data, and bivariate data under univariate or bivariate right censoring. For these types of censored data mentioned, the estimation procedures developed here naturally lead to Kolmogorov-Smirnov goodness-of-fit tests for the Cox model. Some simulation results are presented.     

Estimating the Burr XII Parameters in Constant-Stress Partially Accelerated Life Tests Under Multiple Censored Data

In this article, we present the performance of the maximum likelihood estimates of the Burr XII parameters for constant-stress partially accelerated life tests under multiple censored data. Two maximum likelihood estimation methods are considered. One method is based on observed-data likelihood function and the maximum likelihood estimates are obtained by using the quasi-Newton algorithm. The other method is based on complete-data likelihood function and the maximum likelihood estimates are derived by using the expectation-maximization (EM) algorithm. The variance–covariance matrices are derived to construct the confidence intervals of the parameters. The performance of these two algorithms is compared with each other by a simulation study. The simulation results show that the maximum likelihood estimation via the EM algorithm outperforms the quasi-Newton algorithm in terms of the absolute relative bias, the bias, the root mean square error and the coverage rate. Finally, a numerical example is given to illustrate the performance of the proposed methods.     

A model with mixed binary responses and censored observations

In this paper, we study the maximum likelihood estimation of a model with mixed binary responses and censored observations. The model is very general and includes the Tobit model and the binary choice model as special cases. We show that, by using additional binary choice observations, our method is more efficient than the traditional Tobit model. Two iterative procedures are proposed to compute the maximum likelihood estimator (MLE) for the model based on the EM algorithm (Dempster et al, 1977) and the Newton-Raphson method. The uniqueness of the MLE is proved. The simulation results show that the inconsistency and inefficiency can be significant when the Tobit method is applied to the present mixed model. The experiment results also suggest that the EM algorithm is much faster than the Newton-Raphson method for the present mixed model. The method also allows one to combine two data sets, the smaller data set with more detailed observations and the larger data set with less detailed binary choice observations in order to improve the efficiency of estimation. This may entail substantial savings when one conducts surveys.     

Estimation for the three-parameter gamma distribution based on progressively censored data

Some work has been done in the past on the estimation for the three-parameter gamma distribution based on complete and censored samples. In this paper, we develop estimation methods based on progressively Type-II censored samples from a three-parameter gamma distribution. In particular, we develop some iterative methods for the determination of the maximum likelihood estimates (MLEs) of all three parameters. It is shown that the proposed iterative scheme converges to the MLEs. In this context, we propose another method of estimation which is based on missing information principle and moment estimators. Simple alternatives to the above two methods are also suggested. The proposed estimation methods are then illustrated with a numerical example. We also consider the interval estimation based on large-sample theory and examine the actual coverage probabilities of these confidence intervals in case of small samples using a Monte Carlo simulation study.