The one-step-late PXEM algorithm

The EM algorithm is a popular method for computing maximum likelihood estimates or posterior modes in models that can be formulated in terms of missing data or latent structure. Although easy implementation and stable convergence help to explain the popularity of the algorithm, its convergence is sometimes notoriously slow. In recent years, however, various adaptations have significantly improved the speed of EM while maintaining its stability and simplicity. One especially successful method for maximum likelihood is known as the parameter expanded EM or PXEM algorithm. Unfortunately, PXEM does not generally have a closed form M-step when computing posterior modes, even when the corresponding EM algorithm is in closed form. In this paper we confront this problem by adapting the one-step-late EM algorithm to PXEM to establish a fast closed form algorithm that improves on the one-step-late EM algorithm by insuring monotone convergence. We use this algorithm to fit a probit regression model and a variety of dynamic linear models, showing computational savings of as much as 99.9%, with the biggest savings occurring when the EM algorithm is the slowest to converge.     

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.     

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.     

Robust methods for generalized linear models with nonignorable missing covariates

The EM algorithm is often used for finding the maximum likelihood estimates in generalized linear models with incomplete data. In this article, the author presents a robust method in the framework of the maximum likelihood estimation for fitting generalized linear models when nonignorable covariates are missing. His robust approach is useful for downweighting any influential observations when estimating the model parameters. To avoid computational problems involving irreducibly high‐dimensional integrals, he adopts a Metropolis‐Hastings algorithm based on a Markov chain sampling method. He carries out simulations to investigate the behaviour of the robust estimates in the presence of outliers and missing covariates; furthermore, he compares these estimates to the classical maximum likelihood estimates. Finally, he illustrates his approach using data on the occurrence of delirium in patients operated on for abdominal aortic aneurysm.     

Forecasting time series with missing data using Holt's model

This paper deals with the prediction of time series with missing data using an alternative formulation for Holt's model with additive errors. This formulation simplifies both the calculus of maximum likelihood estimators of all the unknowns in the model and the calculus of point forecasts. In the presence of missing data, the EM algorithm is used to obtain maximum likelihood estimates and point forecasts. Based on this application we propose a leave-one-out algorithm for the data transformation selection problem which allows us to analyse Holt's model with multiplicative errors. Some numerical results show the performance of these procedures for obtaining robust forecasts.     

Full information maximum likelihood estimation in factor analysis with a large number of missing values

We consider the problem of full information maximum likelihood (FIML) estimation in factor analysis when a majority of the data values are missing. The expectation–maximization (EM) algorithm is often used to find the FIML estimates, in which the missing values on manifest variables are included in complete data. However, the ordinary EM algorithm has an extremely high computational cost. In this paper, we propose a new algorithm that is based on the EM algorithm but that efficiently computes the FIML estimates. A significant improvement in the computational speed is realized by not treating the missing values on manifest variables as a part of complete data. When there are many missing data values, it is not clear if the FIML procedure can achieve good estimation accuracy. In order to investigate this, we conduct Monte Carlo simulations under a wide variety of sample sizes.     

A general maximum likelihood analysis of overdispersion in generalized linear models

This paper presents an EM algorithm for maximum likelihood estimation in generalized linear models with overdispersion. The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully non-parametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters may be sensitive to the specification of a parametric form for the mixing distribution. A listing of a GLIM4 algorithm for fitting the overdispersed binomial logit model is given in an appendix.A simple method is given for obtaining correct standard errors for parameter estimates when using the EM algorithm.Several examples are discussed.     

On the use of the selection matrix in the maximum likelihood estimation of normal distribution models with missing data

In this article, by using the constant and random selection matrices, several properties of the maximum likelihood (ML) estimates and the ML estimator of a normal distribution with missing data are derived. The constant selection matrix allows us to obtain an explicit form of the ML estimates and the exact relationship between the EM algorithm and the score function. The random selection matrix allows us to clarify how the missing-data mechanism works in the proof of the consistency of the ML estimator, to derive the asymptotic properties of the sequence by the EM algorithm, and to derive the information matrix.     

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.     

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.     

Use of Orthogonal Polynomial Approximations for Inference in Exponential Distribution Based on K-Sample Doubly Type-II Censored Data

In this article, an EM algorithm approach to obtain the maximum likelihood estimates of parameters for analyzing bivariate skew normal data with non monotone missing values is presented. A simulation study is implemented to investigate the performance of the presented algorithm. Results of an application are also reported where a Bootstrap approach is used to find the variances of the parameter estimates.     

Estimating a Finite Mixed Exponential Distribution under Progressively Type-II Censored Data

The Type-II progressive censoring scheme has become very popular for analyzing lifetime data in reliability and survival analysis. However, no published papers address parameter estimation under progressive Type-II censoring for the mixed exponential distribution (MED), which is an important model for reliability and survival analysis. This is the problem that we address in this paper. It is noted that maximum likelihood estimation of unknown parameters cannot be obtained in closed form due to the complicated log-likelihood function. We solve this problem by using the EM algorithm. Finally, we obtain closed form estimates of the model. The proposed methods are illustrated by both some simulations and a case analysis.     

Maximum likelihood estimates in the multivariate normal with patterned mean and covariance via the em algorithm

The maximum likelihood equations for a multivariate normal model with structured mean and structured covariance matrix may not have an explicit solution. In some cases the model's error term may be decomposed as the sum of two independent error terms, each having a patterned covariance matrix, such that if one of the unobservable error terms is artificially treated as "missing data", the EM algorithm can be used to compute the maximum likelihood estimates for the original problem. Some decompositions produce likelihood equations which do not have an explicit solution at each iteration of the EM algorithm, but within-iteration explicit solutions are shown for two general classes of models including covariance component models used for analysis of longitudinal data.     

Inference based on progressive Type I interval censored data from log-normal distribution

This article considers inference for the log-normal distribution based on progressive Type I interval censored data by both frequentist and Bayesian methods. First, the maximum likelihood estimates (MLEs) of the unknown model parameters are computed by expectation-maximization (EM) algorithm. The asymptotic standard errors (ASEs) of the MLEs are obtained by applying the missing information principle. Next, the Bayes’ estimates of the model parameters are obtained by Gibbs sampling method under both symmetric and asymmetric loss functions. The Gibbs sampling scheme is facilitated by adopting a similar data augmentation scheme as in EM algorithm. The performance of the MLEs and various Bayesian point estimates is judged via a simulation study. A real dataset is analyzed for the purpose of illustration.     

Inference and optimal design of multiple constant-stress testing for generalized half-normal distribution under type-II progressive censoring

The generalized half-normal (GHN) distribution and progressive type-II censoring are considered in this article for studying some statistical inferences of constant-stress accelerated life testing. The EM algorithm is considered to calculate the maximum likelihood estimates. Fisher information matrix is formed depending on the missing information law and it is utilized for structuring the asymptomatic confidence intervals. Further, interval estimation is discussed through bootstrap intervals. The Tierney and Kadane method, importance sampling procedure and Metropolis-Hastings algorithm are utilized to compute Bayesian estimates. Furthermore, predictive estimates for censored data and the related prediction intervals are obtained. We consider three optimality criteria to find out the optimal stress level. A real data set is used to illustrate the importance of GHN distribution as an alternative lifetime model for well-known distributions. Finally, a simulation study is provided with discussion.     

The EM algorithm with gradient function update for discrete mixtures with known (fixed) number of components

The paper is focussing on some recent developments in nonparametric mixture distributions. It discusses nonparametric maximum likelihood estimation of the mixing distribution and will emphasize gradient type results, especially in terms of global results and global convergence of algorithms such as vertex direction or vertex exchange method. However, the NPMLE (or the algorithms constructing it) provides also an estimate of the number of components of the mixing distribution which might be not desirable for theoretical reasons or might be not allowed from the physical interpretation of the mixture model. When the number of components is fixed in advance, the before mentioned algorithms can not be used and globally convergent algorithms do not exist up to now. Instead, the EM algorithm is often used to find maximum likelihood estimates. However, in this case multiple maxima are often occuring. An example from a meta-analyis of vitamin A and childhood mortality is used to illustrate the considerable, inferential importance of identifying the correct global likelihood. To improve the behavior of the EM algorithm we suggest a combination of gradient function steps and EM steps to achieve global convergence leading to the EM algorithm with gradient function update (EMGFU). This algorithms retains the number of components to be exactly k and typically converges to the global maximum. The behavior of the algorithm is highlighted at hand of several examples.     

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.     

AN ALTERNATIVE PARAMETRIC APPROACH FOR DISCRETE MISSING DATA PROBLEMS

We propose an iterative method of estimation for discrete missing data problems that is conceptually different from the Expectation–Maximization (EM) algorithm and that does not in general yield the observed data maximum likelihood estimate (MLE). The proposed approach is based conceptually upon weighting the set of possible complete-data MLEs. Its implementation avoids the expectation step of EM, which can sometimes be problematic. In the simple case of Bernoulli trials missing completely at random, the iterations of the proposed algorithm are equivalent to the EM iterations. For a familiar genetics-oriented multinomial problem with missing count data and for the motivating example with epidemiologic applications that involves a mixture of a left censored normal distribution with a point mass at zero, we investigate the finite sample performance of the proposed estimator and find it to be competitive with that of the MLE. We give some intuitive justification for the method, and we explore an interesting connection between our algorithm and multiple imputation in order to suggest an approach for estimating standard errors.     

Maximum likelihood analysis of masked series system lifetime data

Component lifetime parameters of a series system are estimated from system lifetimes and masked cause of failure observations. The time and cause of system failures are assumed to follow a competing risks model. The masking probabilities of the minimum random subsets are not subjected to the symmetry assumption. Sufficient regularity conditions are provided, justifying the maximum likelihood analysis. Maximum likelihood estimates of both the lifetime parameters and masking probabilities are generically computed via an EM algorithm. An appropriate set of asymptotically pivotal quantities are also derived. Such maximum likelihood based estimates are further refined by bootstrap. The developed techniques are illustrated by numerical examples of independent Weibull component lifetimes with distinct scale and shape parameters.     

The competing risks analysis for parallel and series systems using Type-II progressive censoring

Abstract

Recently, the study of the lifetime of systems in reliability and survival analysis in the presence of several causes of failure (competing risks) has attracted attention in the literature. In this paper, series and parallel systems with exponential lifetime for each item of the system are considered. Several causes of failure independently affect lifetime distributions and observations of failure times of the systems are considered under progressive Type-II censored scheme. For series systems, the maximum likelihood estimates of parameters are computed and confidence intervals for parameters of the model are obtained using Fisher information matrix. For parallel systems, the generalized EM algorithm which uses the Newton-Raphson algorithm inside the EM algorithm is used to compute the maximum likelihood estimates of parameters. Also, the standard errors of the maximum likelihood estimates are computed by using the supplemented EM algorithm. The simulation study confirms the good performance of the introduced approach.