A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

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.     

Using auxiliary data for parameter estimation with non-ignorably missing outcomes

We propose a method for estimating parameters in generalized linear models when the outcome variable is missing for some subjects and the missing data mechanism is non-ignorable. We assume throughout that the covariates are fully observed. One possible method for estimating the parameters is maximum likelihood with a non-ignorable missing data model. However, caution must be used when fitting non-ignorable missing data models because certain parameters may be inestimable for some models. Instead of fitting a non-ignorable model, we propose the use of auxiliary information in a likelihood approach to reduce the bias, without having to specify a non-ignorable model. The method is applied to a mental health study.     

Cox Regression with Incomplete Covariate Measurements using the EM-algorithm

Ibrahim (1990) used the EM-algorithm to obtain maximum likelihood estimates of the regression parameters in generalized linear models with partially missing covariates. The technique was termed EM by the method of weights. In this paper, we generalize this technique to Cox regression analysis with missing values in the covariates. We specify a full model letting the unobserved covariate values be random and then maximize the observed likelihood. The asymptotic covariance matrix is estimated by the inverse information matrix. The missing data are allowed to be missing at random but also the non-ignorable non-response situation may in principle be considered. Simulation studies indicate that the proposed method is more efficient than the method suggested by Paik & Tsai (1997). We apply the procedure to a clinical trials example with six covariates with three of them having missing values.     

The em algorithm for the quasi-likelihood regression model

The objective of this paper is to present a method which can accommodate certain types of missing data by using the quasi-likelihood function for the complete data. This method can be useful when we can make first and second moment assumptions only; in addition, it can be helpful when the EM algorithm applied to the actual likelihood becomes overly complicated. First we derive a loss function for the observed data using an exponential family density which has the same mean and variance structure of the complete data. This loss function is the counterpart of the quasi-deviance for the observed data. Then the loss function is minimized using the EM algorithm. The use of the EM algorithm guarantees a decrease in the loss function at every iteration. When the observed data can be expressed as a deterministic linear transformation of the complete data, or when data are missing completely at random, the proposed method yields consistent estimators. Examples are given for overdispersed polytomous data, linear random effects models, and linear regression with missing covariates. Simulation results for the linear regression model with missing covariates show that the proposed estimates are more efficient than estimates based on completely observed units, even when outcomes are bimodal or skewed.     

Joint regression modeling for missing categorical covariates in generalized linear models

Missing covariates data is a common issue in generalized linear models (GLMs). A model-based procedure arising from properly specifying joint models for both the partially observed covariates and the corresponding missing indicator variables represents a sound and flexible methodology, which lends itself to maximum likelihood estimation as the likelihood function is available in computable form. In this paper, a novel model-based methodology is proposed for the regression analysis of GLMs when the partially observed covariates are categorical. Pair-copula constructions are used as graphical tools in order to facilitate the specification of the high-dimensional probability distributions of the underlying missingness components. The model parameters are estimated by maximizing the weighted log-likelihood function by using an EM algorithm. In order to compare the performance of the proposed methodology with other well-established approaches, which include complete-cases and multiple imputation, several simulation experiments of Binomial, Poisson and Normal regressions are carried out under both missing at random and non-missing at random mechanisms scenarios. The methods are illustrated by modeling data from a stage III melanoma clinical trial. The results show that the methodology is rather robust and flexible, representing a competitive alternative to traditional techniques.     

Missing covariates in generalized linear models when the missing data mechanism is non-ignorable

We propose a method for estimating parameters in generalized linear models with missing covariates and a non-ignorable missing data mechanism. We use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one-dimensional conditional distributions, with each one-dimensional conditional distribution consisting of a logistic regression. We allow the covariates to be either categorical or continuous. The joint covariate distribution is also modelled via a sequence of one-dimensional conditional distributions, and the response variable is assumed to be completely observed. We derive the E- and M-steps of the EM algorithm with non-ignorable missing covariate data. For categorical covariates, we derive a closed form expression for the E- and M-steps of the EM algorithm for obtaining the maximum likelihood estimates (MLEs). For continuous covariates, we use a Monte Carlo version of the EM algorithm to obtain the MLEs via the Gibbs sampler. Computational techniques for Gibbs sampling are proposed and implemented. The parametric form of the assumed missing data mechanism itself is not `testable' from the data, and thus the non-ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model. Therefore, although a model may have `passed' the tests for a certain missing data mechanism, this does not mean that we have captured, even approximately, the correct missing data mechanism. Hence, model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail. Several simulations are given to demonstrate the methodology. In addition, a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed.     

Constrained inference for generalized linear models with incomplete covariate data

Missing data are common in many experiments, including surveys, clinical trials, epidemiological studies, and environmental studies. Unconstrained likelihood inferences for generalized linear models (GLMs) with nonignorable missing covariates have been studied extensively in the literature. However, parameter orderings or constraints may occur naturally in practice, and thus the efficiency of a statistical method may be improved by incorporating parameter constraints into the likelihood function. In this paper, we consider constrained inference for analysing GLMs with nonignorable missing covariates under linear inequality constraints on the model parameters. Specifically, constrained maximum likelihood (ML) estimation is based on the gradient projection expectation maximization approach. Further, we investigate the asymptotic null distribution of the constrained likelihood ratio test (LRT). Simulations study the empirical properties of the constrained ML estimators and LRTs, which demonstrate improved precision of these constrained techniques. An application to contaminant levels in an environmental study is also presented.     

A cautionary note on generalized linear models for covariance of unbalanced longitudinal data

Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positive-definiteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positive-definiteness constraint and use a generalized linear model setup to jointly model the mean and covariance using covariates (Pourahmadi, 2000). However, this approach may not be directly applicable when the longitudinal data are unbalanced, as coherent regression models for the dependence across all times and subjects may not exist. Within the existing generalized linear model framework, we show how to overcome this and other challenges by embedding the covariance matrix of the observed data for each subject in a larger covariance matrix and employing the familiar EM algorithm to compute the maximum likelihood estimates of the parameters and their standard errors. We illustrate and assess the methodology using real data sets and simulations.     

INCOMPLETE DATA IN GENERALIZED LINEAR MODELS WITH CONTINUOUS COVARIATES

This paper proposes a method for estimating the parameters in a generalized linear model with missing covariates. The missing covariates are assumed to come from a continuous distribution, and are assumed to be missing at random. In particular, Gaussian quadrature methods are used on the E-step of the EM algorithm, leading to an approximate EM algorithm. The parameters are then estimated using the weighted EM procedure given in Ibrahim (1990). This approximate EM procedure leads to approximate maximum likelihood estimates, whose standard errors and asymptotic properties are given. The proposed procedure is illustrated on a data set.     

Using the EM-algorithm for survival data with incomplete categorical covariates

Incomplete covariate data is a common occurrence in many studies in which the outcome is survival time. With generalized linear models, when the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM by the method of weights proposed in Ibrahim (1990). In this article, we extend the EM by the method of weights to survival outcomes whose distributions may not fall in the class of generalized linear models. This method requires the estimation of the parameters of the distribution of the covariates. We present a clinical trials example with five covariates, four of which have some missing values.     

Analysis of ordinal outcomes with longitudinal covariates subject to missingness

We propose a mixture model for data with an ordinal outcome and a longitudinal covariate that is subject to missingness. Data from a tailored telephone delivered, smoking cessation intervention for construction laborers are used to illustrate the method, which considers as an outcome a categorical measure of smoking cessation, and evaluates the effectiveness of the motivational telephone interviews on this outcome. We propose two model structures for the longitudinal covariate, for the case when the missing data are missing at random, and when the missing data mechanism is non-ignorable. A generalized EM algorithm is used to obtain maximum likelihood estimates.     

Measurement error in the generalised linear model

This paper considers the problem of estimating the linear parameters of a Generalised Linear Model (GLM) when the explanatory variable is subject to measurement error. In this situation the induced model for dependence on the approximate explanatory variable is not usually of GLM form. However, when the distribution of measurement error is known or estimated from replicated measurements, application of the GLIM iteratively reweighted least squares algorithm with transformed data and weighting is shown to produce maximum quasi likelihood estimates in many cases. Details of this approach are given for two particular generalized linear models; simulation results illustrate the usefulness of the theory for these models.     

Efficient estimation for case‐cohort studies

The author considers time‐to‐event data from case‐cohort designs. As existing methods are either inefficient or based on restrictive assumptions concerning the censoring mechanism, he proposes a semi‐parametrically efficient estimator under the usual assumptions for Cox regression models. The estimator in question is obtained by a one‐step Newton‐Raphson approximation that solves the efficient score equations with initial value obtained from an existing method. The author proves that the estimator is consistent, asymptotically efficient and normally distributed in the limit. He also resorts to simulations to show that the proposed estimator performs well in finite samples and that it considerably improves the efficiency of existing pseudo‐likelihood estimators when a correlate of the missing covariate is available. Although he focuses on the situation where covariates are discrete, the author also explores how the method can be applied to models with continuous covariates.     

Bayesian Analysis of Nonlinear Reproductive Dispersion Mixed Models for Longitudinal Data with Nonignorable Missing Covariates

This article proposes a Bayesian approach, which can simultaneously obtain the Bayesian estimates of unknown parameters and random effects, to analyze nonlinear reproductive dispersion mixed models (NRDMMs) for longitudinal data with nonignorable missing covariates and responses. The logistic regression model is employed to model the missing data mechanisms for missing covariates and responses. A hybrid sampling procedure combining the Gibber sampler and the Metropolis-Hastings algorithm is presented to draw observations from the conditional distributions. Because missing data mechanism is not testable, we develop the logarithm of the pseudo-marginal likelihood, deviance information criterion, the Bayes factor, and the pseudo-Bayes factor to compare several competing missing data mechanism models in the current considered NRDMMs with nonignorable missing covaraites and responses. Three simulation studies and a real example taken from the paediatric AIDS clinical trial group ACTG are used to illustrate the proposed methodologies. Empirical results show that our proposed methods are effective in selecting missing data mechanism models.     

Local influence for generalized linear mixed models

The authors describe a method for assessing model inadequacy in maximum likelihood estimation of a generalized linear mixed model. They treat the latent random effects in the model as missing data and develop the influence analysis on the basis of a Q‐function which is associated with the conditional expectation of the complete‐data log‐likelihood function in the EM algorithm. They propose a procedure to detect influential observations in six model perturbation schemes. They also illustrate their methodology in a hypothetical situation and in two real cases.     

Sparse cluster analysis of large-scale discrete variables with application to single nucleotide polymorphism data

Currently, extreme large-scale genetic data present significant challenges for cluster analysis. Most of the existing clustering methods are typically built on the Euclidean distance and geared toward analyzing continuous response. They work well for clustering, e.g. microarray gene expression data, but often perform poorly for clustering, e.g. large-scale single nucleotide polymorphism (SNP) data. In this paper, we study the penalized latent class model for clustering extremely large-scale discrete data. The penalized latent class model takes into account the discrete nature of the response using appropriate generalized linear models and adopts the lasso penalized likelihood approach for simultaneous model estimation and selection of important covariates. We develop very efficient numerical algorithms for model estimation based on the iterative coordinate descent approach and further develop the expectation–maximization algorithm to incorporate and model missing values. We use simulation studies and applications to the international HapMap SNP data to illustrate the competitive performance of the penalized latent class model.     

Incomplete covariates data in generalized linear models

We consider regression analysis when part of covariates are incomplete in generalized linear models. The incomplete covariates could be due to measurement error or missing for some study subjects. We assume there exists a validation sample in which the data is complete and is a simple random subsample from the whole sample. Based on the idea of projection-solution method in Heyde (1997, Quasi-Likelihood and its Applications: A General Approach to Optimal Parameter Estimation. Springer, New York), a class of estimating functions is proposed to estimate the regression coefficients through the whole data. This method does not need to specify a correct parametric model for the incomplete covariates to yield a consistent estimate, and avoids the ‘curse of dimensionality’ encountered in the existing semiparametric method. Simulation results shows that the finite sample performance and efficiency property of the proposed estimates are satisfactory. Also this approach is computationally convenient hence can be applied to daily data analysis.     

Weighting Method for a Linear Mixed Model

Maximum likelihood is a widely used estimation method in statistics. This method is model dependent and as such is criticized as being non robust. In this article, we consider using weighted likelihood method to make robust inferences for linear mixed models where weights are determined at both the subject level and the observation level. This approach is appropriate for problems where maximum likelihood is the basic fitting technique, but a subset of data points is discrepant with the model. It allows us to reduce the impact of outliers without complicating the basic linear mixed model with normally distributed random effects and errors. The weighted likelihood estimators are shown to be robust and asymptotically normal. Our simulation study demonstrates that the weighted estimates are much better than the unweighted ones when a subset of data points is far away from the rest. Its application to the analysis of deglutition apnea duration in normal swallows shows that the differences between the weighted and unweighted estimates are due to large amount of outliers in the data set.     

Robust inference for generalized partially linear mixed models that account for censored responses and missing covariates – an application to Arctic data analysis

In this article, we propose a family of bounded influence robust estimates for the parametric and non-parametric components of a generalized partially linear mixed model that are subject to censored responses and missing covariates. The asymptotic properties of the proposed estimates have been looked into. The estimates are obtained by using Monte Carlo expectation–maximization algorithm. An approximate method which reduces the computational time to a great extent is also proposed. A simulation study shows that performances of the two approaches are similar in terms of bias and mean square error. The analysis is illustrated through a study on the effect of environmental factors on the phytoplankton cell count.