Maximum likelihood methods for nonignorable missing responses and covariates in random effects models

This article analyzes quality of life (QOL) data from an Eastern Cooperative Oncology Group (ECOG) melanoma trial that compared treatment with ganglioside vaccination to treatment with high-dose interferon. The analysis of this data set is challenging due to several difficulties, namely, nonignorable missing longitudinal responses and baseline covariates. Hence, we propose a selection model for estimating parameters in the normal random effects model with nonignorable missing responses and covariates. Parameters are estimated via maximum likelihood using the Gibbs sampler and a Monte Carlo expectation maximization (EM) algorithm. Standard errors are calculated using the bootstrap. The method allows for nonmonotone patterns of missing data in both the response variable and the covariates. We model the missing data mechanism and the missing covariate distribution via a sequence of one-dimensional conditional distributions, allowing the missing covariates to be either categorical or continuous, as well as time-varying. We apply the proposed approach to the ECOG quality-of-life data and conduct a small simulation study evaluating the performance of the maximum likelihood estimates. Our results indicate that a patient treated with the vaccine has a higher QOL score on average at a given time point than a patient treated with high-dose interferon.     

Frailty models with missing covariates

We present a method for estimating the parameters in random effects models for survival data when covariates are subject to missingness. Our method is more general than the usual frailty model as it accommodates a wide range of distributions for the random effects, which are included as an offset in the linear predictor in a manner analogous to that used in generalized linear mixed models. We propose using a Monte Carlo EM algorithm along with the Gibbs sampler to obtain parameter estimates. This method is useful in reducing the bias that may be incurred using complete-case methods in this setting. The methodology is applied to data from Eastern Cooperative Oncology Group melanoma clinical trials in which observations were believed to be clustered and several tumor characteristics were not always observed.     

Maximum likelihood estimation in random effects cure rate models with nonignorable missing covariates

We introduce a method of parameter estimation for a random effects cure rate model. We also propose a methodology that allows us to account for nonignorable missing covariates in this class of models. The proposed method corrects for possible bias introduced by complete case analysis when missing data are not missing completely at random and is motivated by data from a pair of melanoma studies conducted by the Eastern Cooperative Oncology Group in which clustering by cohort or time of study entry was suspected. In addition, these models allow estimation of cure rates, which is desirable when we do not wish to assume that all subjects remain at risk of death or relapse from disease after sufficient follow-up. We develop an EM algorithm for the model and provide an efficient Gibbs sampling scheme for carrying out the E-step of the algorithm.     

Zero‐inflated Poisson factor model with application to microbiome read counts

Dimension reduction of high‐dimensional microbiome data facilitates subsequent analysis such as regression and clustering. Most existing reduction methods cannot fully accommodate the special features of the data such as count‐valued and excessive zero reads. We propose a zero‐inflated Poisson factor analysis model in this paper. The model assumes that microbiome read counts follow zero‐inflated Poisson distributions with library size as offset and Poisson rates negatively related to the inflated zero occurrences. The latent parameters of the model form a low‐rank matrix consisting of interpretable loadings and low‐dimensional scores that can be used for further analyses. We develop an efficient and robust expectation‐maximization algorithm for parameter estimation. We demonstrate the efficacy of the proposed method using comprehensive simulation studies. The application to the Oral Infections, Glucose Intolerance, and Insulin Resistance Study provides valuable insights into the relation between subgingival microbiome and periodontal disease.     

Joint model selection of marginal mean regression and correlation structure for longitudinal data with missing outcome and covariates

This work develops a joint model selection criterion for simultaneously selecting the marginal mean regression and the correlation/covariance structure in longitudinal data analysis where both the outcome and the covariate variables may be subject to general intermittent patterns of missingness under the missing at random mechanism. The new proposal, termed “joint longitudinal information criterion” (JLIC), is based on the expected quadratic error for assessing model adequacy, and the second‐order weighted generalized estimating equation (WGEE) estimation for mean and covariance models. Simulation results reveal that JLIC outperforms existing methods performing model selection for the mean regression and the correlation structure in a two stage and hence separate manner. We apply the proposal to a longitudinal study to identify factors associated with life satisfaction in the elderly of Taiwan.     

Score test for missing at random or not under logistic missingness models

Missing data are frequently encountered in various disciplines and can be divided into three categories: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). Valid statistical approaches to missing data depend crucially on correct identification of the underlying missingness mechanism. Although the problem of testing whether this mechanism is MCAR or MAR has been extensively studied, there has been very little research on testing MAR versus MNAR. A critical challenge that is faced when dealing with this problem is the issue of model identification under MNAR. In this paper, under a logistic model for the missing probability, we develop two score tests for the problem of whether the missingness mechanism is MAR or MNAR under a parametric model and a semiparametric location model on the regression function. The implementation of the score tests circumvents the identification issue as it requires only parameter estimation under the null MAR assumption. Our simulations and analysis of human immunodeficiency virus data show that the score tests have well-controlled type I errors and desirable powers.     

Conditional and unconditional categorical regression models with missing covariates

We consider methods for analyzing categorical regression models when some covariates (Z) are completely observed but other covariates (X) are missing for some subjects. When data on X are missing at random (i.e., when the probability that X is observed does not depend on the value of X itself), we present a likelihood approach for the observed data that allows the same nuisance parameters to be eliminated in a conditional analysis as when data are complete. An example of a matched case-control study is used to demonstrate our approach.     

A new multivariate zero‐adjusted Poisson model with applications to biomedicine

Recently, although advances were made on modeling multivariate count data, existing models really has several limitations: (i) The multivariate Poisson log‐normal model (Aitchison and Ho, 1989) cannot be used to fit multivariate count data with excess zero‐vectors; (ii) The multivariate zero‐inflated Poisson (ZIP) distribution (Li et al., 1999) cannot be used to model zero‐truncated/deflated count data and it is difficult to apply to high‐dimensional cases; (iii) The Type I multivariate zero‐adjusted Poisson (ZAP) distribution (Tian et al., 2017) could only model multivariate count data with a special correlation structure for random components that are all positive or negative. In this paper, we first introduce a new multivariate ZAP distribution, based on a multivariate Poisson distribution, which allows the correlations between components with a more flexible dependency structure, that is some of the correlation coefficients could be positive while others could be negative. We then develop its important distributional properties, and provide efficient statistical inference methods for multivariate ZAP model with or without covariates. Two real data examples in biomedicine are used to illustrate the proposed methods.     

A new mixed‐effects regression model for the analysis of zero‐modified hierarchical count data

Count data sets are traditionally analyzed using the ordinary Poisson distribution. However, such a model has its applicability limited as it can be somewhat restrictive to handle specific data structures. In this case, it arises the need for obtaining alternative models that accommodate, for example, (a) zero‐modification (inflation or deflation at the frequency of zeros), (b) overdispersion, and (c) individual heterogeneity arising from clustering or repeated (correlated) measurements made on the same subject. Cases (a)–(b) and (b)–(c) are often treated together in the statistical literature with several practical applications, but models supporting all at once are less common. Hence, this paper's primary goal was to jointly address these issues by deriving a mixed‐effects regression model based on the hurdle version of the Poisson–Lindley distribution. In this framework, the zero‐modification is incorporated by assuming that a binary probability model determines which outcomes are zero‐valued, and a zero‐truncated process is responsible for generating positive observations. Approximate posterior inferences for the model parameters were obtained from a fully Bayesian approach based on the Adaptive Metropolis algorithm. Intensive Monte Carlo simulation studies were performed to assess the empirical properties of the Bayesian estimators. The proposed model was considered for the analysis of a real data set, and its competitiveness regarding some well‐established mixed‐effects models for count data was evaluated. A sensitivity analysis to detect observations that may impact parameter estimates was performed based on standard divergence measures. The Bayesian ‐value and the randomized quantile residuals were considered for model diagnostics.     

Bayesian analysis for generalized linear models with nonignorably missing covariates

We propose Bayesian methods for estimating parameters in generalized linear models (GLMs) with nonignorably missing covariate data. We show that when improper uniform priors are used for the regression coefficients, phi, of the multinomial selection model for the missing data mechanism, the resulting joint posterior will always be improper if (i) all missing covariates are discrete and an intercept is included in the selection model for the missing data mechanism, or (ii) at least one of the covariates is continuous and unbounded. This impropriety will result regardless of whether proper or improper priors are specified for the regression parameters, beta, of the GLM or the parameters, alpha, of the covariate distribution. To overcome this problem, we propose a novel class of proper priors for the regression coefficients, phi, in the selection model for the missing data mechanism. These priors are robust and computationally attractive in the sense that inferences about beta are not sensitive to the choice of the hyperparameters of the prior for phi and they facilitate a Gibbs sampling scheme that leads to accelerated convergence. In addition, we extend the model assessment criterion of Chen, Dey, and Ibrahim (2004a, Biometrika 91, 45-63), called the weighted L measure, to GLMs and missing data problems as well as extend the deviance information criterion (DIC) of Spiegelhalter et al. (2002, Journal of the Royal Statistical Society B 64, 583-639) for assessing whether the missing data mechanism is ignorable or nonignorable. A novel Markov chain Monte Carlo sampling algorithm is also developed for carrying out posterior computation. Several simulations are given to investigate the performance of the proposed Bayesian criteria as well as the sensitivity of the prior specification. Real datasets from a melanoma cancer clinical trial and a liver cancer study are presented to further illustrate the proposed methods.     

Simultaneous inference for semiparametric nonlinear mixed-effects models with covariate measurement errors and missing responses

Semiparametric nonlinear mixed-effects (NLME) models are flexible for modeling complex longitudinal data. Covariates are usually introduced in the models to partially explain interindividual variations. Some covariates, however, may be measured with substantial errors. Moreover, the responses may be missing and the missingness may be nonignorable. We propose two approximate likelihood methods for semiparametric NLME models with covariate measurement errors and nonignorable missing responses. The methods are illustrated in a real data example. Simulation results show that both methods perform well and are much better than the commonly used naive method.     

Maximum likelihood methods for cure rate models with missing covariates

We propose maximum likelihood methods for parameter estimation for a novel class of semiparametric survival models with a cure fraction, in which the covariates are allowed to be missing. We allow the covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one-dimensional conditional distributions. We propose a novel EM algorithm for maximum likelihood estimation and derive standard errors by using Louis's formula (Louis, 1982, Journal of the Royal Statistical Society, Series B 44, 226-233). Computational techniques using the Monte Carlo EM algorithm are discussed and implemented. A real data set involving a melanoma cancer clinical trial is examined in detail to demonstrate the methodology.     

Semiparametric models for missing covariate and response data in regression models

We consider a class of semiparametric models for the covariate distribution and missing data mechanism for missing covariate and/or response data for general classes of regression models including generalized linear models and generalized linear mixed models. Ignorable and nonignorable missing covariate and/or response data are considered. The proposed semiparametric model can be viewed as a sensitivity analysis for model misspecification of the missing covariate distribution and/or missing data mechanism. The semiparametric model consists of a generalized additive model (GAM) for the covariate distribution and/or missing data mechanism. Penalized regression splines are used to express the GAMs as a generalized linear mixed effects model, in which the variance of the corresponding random effects provides an intuitive index for choosing between the semiparametric and parametric model. Maximum likelihood estimates are then obtained via the EM algorithm. Simulations are given to demonstrate the methodology, and a real data set from a melanoma cancer clinical trial is analyzed using the proposed methods.     

Analysis of Zero‐Inflated Poisson Data Incorporating Extent of Exposure

When analyzing Poisson count data sometimes a high frequency of extra zeros is observed. The Zero‐Inflated Poisson (ZIP) model is a popular approach to handle zero‐inflation. In this paper we generalize the ZIP model and its regression counterpart to accommodate the extent of individual exposure. Empirical evidence drawn from an occupational injury data set confirms that the incorporation of exposure information can exert a substantial impact on the model fit. Tests for zero‐inflation are also considered. Their finite sample properties are examined in a Monte Carlo study.     

Semiparametric frailty models for zero‐inflated event count data in the presence of informative dropout

Recurrent events data are commonly encountered in medical studies. In many applications, only the number of events during the follow‐up period rather than the recurrent event times is available. Two important challenges arise in such studies: (a) a substantial portion of subjects may not experience the event, and (b) we may not observe the event count for the entire study period due to informative dropout. To address the first challenge, we assume that underlying population consists of two subpopulations: a subpopulation nonsusceptible to the event of interest and a subpopulation susceptible to the event of interest. In the susceptible subpopulation, the event count is assumed to follow a Poisson distribution given the follow‐up time and the subject‐specific characteristics. We then introduce a frailty to account for informative dropout. The proposed semiparametric frailty models consist of three submodels: (a) a logistic regression model for the probability such that a subject belongs to the nonsusceptible subpopulation; (b) a nonhomogeneous Poisson process model with an unspecified baseline rate function; and (c) a Cox model for the informative dropout time. We develop likelihood‐based estimation and inference procedures. The maximum likelihood estimators are shown to be consistent. Additionally, the proposed estimators of the finite‐dimensional parameters are asymptotically normal and the covariance matrix attains the semiparametric efficiency bound. Simulation studies demonstrate that the proposed methodologies perform well in practical situations. We apply the proposed methods to a clinical trial on patients with myelodysplastic syndromes.     

A Bayesian nonparametric model for zero‐inflated outcomes: Prediction,clustering, and causal estimation

Researchers are often interested in predicting outcomes, detecting distinct subgroups of their data, or estimating causal treatment effects. Pathological data distributions that exhibit skewness and zero‐inflation complicate these tasks—requiring highly flexible, data‐adaptive modeling. In this paper, we present a multipurpose Bayesian nonparametric model for continuous, zero‐inflated outcomes that simultaneously predicts structural zeros, captures skewness, and clusters patients with similar joint data distributions. The flexibility of our approach yields predictions that capture the joint data distribution better than commonly used zero‐inflated methods. Moreover, we demonstrate that our model can be coherently incorporated into a standardization procedure for computing causal effect estimates that are robust to such data pathologies. Uncertainty at all levels of this model flow through to the causal effect estimates of interest—allowing easy point estimation, interval estimation, and posterior predictive checks verifying positivity, a required causal identification assumption. Our simulation results show point estimates to have low bias and interval estimates to have close to nominal coverage under complicated data settings. Under simpler settings, these results hold while incurring lower efficiency loss than comparator methods. We use our proposed method to analyze zero‐inflated inpatient medical costs among endometrial cancer patients receiving either chemotherapy or radiation therapy in the SEER‐Medicare database.