Gaussian Scale Mixture Models for Robust Linear Multivariate Regression with Missing Data

We present an algorithm for multivariate robust Bayesian linear regression with missing data. The iterative algorithm computes an approximative posterior for the model parameters based on the variational Bayes (VB) method. Compared to the EM algorithm, the VB method has the advantage that the variance for the model parameters is also computed directly by the algorithm. We consider three families of Gaussian scale mixture models for the measurements, which include as special cases the multivariate t distribution, the multivariate Laplace distribution, and the contaminated normal model. The observations can contain missing values, assuming that the missing data mechanism can be ignored. A Matlab/Octave implementation of the algorithm is presented and applied to solve three reference examples from the literature.     

Tukey's gh Distribution for Multiple Imputation

Tukey proposed a class of distributions, the g-and-h family (gh family), based on a transformation of a standard normal variable to accommodate different skewness and elongation in the distribution of variables arising in practical applications. It is easy to draw values from this distribution even though it is hard to explicitly state the probability density function. Given this flexibility, the gh family may be extremely useful in creating multiple imputations for missing data. This article demonstrates how this family, as well as its generalizations, can be used in the multiple imputation analysis of incomplete data. The focus of this article is on a scalar variable with missing values. In the absence of any additional information, data are missing completely at random, and hence the correct analysis is the complete-case analysis. Thus, the application of the gh multiple imputation to the scalar cases affords comparison with the correct analysis and with other model-based multiple imputation methods. Comparisons are made using simulated datasets and the data from a survey of adolescents ascertaining driving after drinking alcohol.     

Bayesian analysis of multivariate t linear mixed models with missing responses at random

The multivariate t linear mixed model (MtLMM) has been recently proposed as a robust tool for analysing multivariate longitudinal data with atypical observations. Missing outcomes frequently occur in longitudinal research even in well controlled situations. As a powerful alternative to the traditional expectation maximization based algorithm employing single imputation, we consider a Bayesian analysis of the MtLMM to account for the uncertainties of model parameters and missing outcomes through multiple imputation. An inverse Bayes formulas sampler coupled with Metropolis-within-Gibbs scheme is used to effectively draw the posterior distributions of latent data and model parameters. The techniques for multiple imputation of missing values, estimation of random effects, prediction of future responses, and diagnostics of potential outliers are investigated as well. The proposed methodology is illustrated through a simulation study and an application to AIDS/HIV data.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

Synthetic data method to incorporate external information into a current study

We consider the situation where there is a known regression model that can be used to predict an outcome, Y, from a set of predictor variables X . A new variable B is expected to enhance the prediction of Y. A dataset of size n containing Y, X and B is available, and the challenge is to build an improved model for Y| X ,B that uses both the available individual level data and some summary information obtained from the known model for Y| X . We propose a synthetic data approach, which consists of creating m additional synthetic data observations, and then analyzing the combined dataset of size n + m to estimate the parameters of the Y| X ,B model. This combined dataset of size n + m now has missing values of B for m of the observations, and is analyzed using methods that can handle missing data (e.g., multiple imputation). We present simulation studies and illustrate the method using data from the Prostate Cancer Prevention Trial. Though the synthetic data method is applicable to a general regression context, to provide some justification, we show in two special cases that the asymptotic variances of the parameter estimates in the Y| X ,B model are identical to those from an alternative constrained maximum likelihood estimation approach. This correspondence in special cases and the method's broad applicability makes it appealing for use across diverse scenarios. The Canadian Journal of Statistics 47: 580–603; 2019 © 2019 Statistical Society of Canada     

M-Estimators Based on Inverse Probability Weighted Estimating Equations with Response Missing at Random

Asymptotic properties of M-estimators with complete data are investigated extensively. In the presence of missing data, however, the standard inference procedures for complete data cannot be applied directly. In this article, the inverse probability weighted method is applied to missing response problem to define M-estimators. The existence of M-estimators is established under very general regularity conditions. Consistency and asymptotic normality of the M-estimators are proved, respectively. An iterative algorithm is applied to calculating the M-estimators. It is shown that one step iteration suffices and the resulting one-step M-estimate has the same limit distribution as in the fully iterated M-estimators.     

Application of a predictive distribution formula to Bayesian computation for incomplete data models

We consider exact and approximate Bayesian computation in the presence of latent variables or missing data. Specifically we explore the application of a posterior predictive distribution formula derived in Sweeting And Kharroubi (2003), which is a particular form of Laplace approximation, both as an importance function and a proposal distribution. We show that this formula provides a stable importance function for use within poor man’s data augmentation schemes and that it can also be used as a proposal distribution within a Metropolis-Hastings algorithm for models that are not analytically tractable. We illustrate both uses in the case of a censored regression model and a normal hierarchical model, with both normal and Student t distributed random effects. Although the predictive distribution formula is motivated by regular asymptotic theory, it is not necessary that the likelihood has a closed form or that it possesses a local maximum.     

Handling of missing data in long‐term clinical trials: a case study

Missing data in clinical trials is a well‐known problem, and the classical statistical methods used can be overly simple. This case study shows how well‐established missing data theory can be applied to efficacy data collected in a long‐term open‐label trial with a discontinuation rate of almost 50%. Satisfaction with treatment in chronically constipated patients was the efficacy measure assessed at baseline and every 3 months postbaseline. The improvement in treatment satisfaction from baseline was originally analyzed with a paired t‐test ignoring missing data and discarding the correlation structure of the longitudinal data. As the original analysis started from missing completely at random assumptions regarding the missing data process, the satisfaction data were re‐examined, and several missing at random (MAR) and missing not at random (MNAR) techniques resulted in adjusted estimate for the improvement in satisfaction over 12 months. Throughout the different sensitivity analyses, the effect sizes remained significant and clinically relevant. Thus, even for an open‐label trial design, sensitivity analysis, with different assumptions for the nature of dropouts (MAR or MNAR) and with different classes of models (selection, pattern‐mixture, or multiple imputation models), has been found useful and provides evidence towards the robustness of the original analyses; additional sensitivity analyses could be undertaken to further qualify robustness. Copyright © 2012 John Wiley & Sons, Ltd.     

Logistic regression with a partially observed covariate

We present results of a Monte Carlo study comparing four methods of estimating the parameters of the logistic model logit (pr (Y = 1 | X, Z)) = α₀ + α ₁ X + α ₂ Z where X and Z are continuous covariates and X is always observed but Z is sometimes missing. The four methods examined are 1) logistic regression using complete cases, 2) logistic regression with filled-in values of Z obtained from the regression of Z on X and Y, 3) logistic regression with filled-in values of Z and random error added, and 4) maximum likelihood estimation assuming the distribution of Z given X and Y is normal. Effects of different percent missing for Z and different missing value mechanisms on the bias and mean absolute deviation of the estimators are examined for data sets of N = 200 and N = 400.     

k-POD: A Method for k-Means Clustering of Missing Data

The k-means algorithm is often used in clustering applications but its usage requires a complete data matrix. Missing data, however, are common in many applications. Mainstream approaches to clustering missing data reduce the missing data problem to a complete data formulation through either deletion or imputation but these solutions may incur significant costs. Our k-POD method presents a simple extension of k-means clustering for missing data that works even when the missingness mechanism is unknown, when external information is unavailable, and when there is significant missingness in the data.

[Received November 2014. Revised August 2015.]     

How to Make Model‐free Feature Screening Approaches for Full Data Applicable to the Case of Missing Response?

It is quite a challenge to develop model‐free feature screening approaches for missing response problems because the existing standard missing data analysis methods cannot be applied directly to high dimensional case. This paper develops some novel methods by borrowing information of missingness indicators such that any feature screening procedures for ultrahigh‐dimensional covariates with full data can be applied to missing response case. The first method is the so‐called missing indicator imputation screening, which is developed by proving that the set of the active predictors of interest for the response is a subset of the active predictors for the product of the response and missingness indicator under some mild conditions. As an alternative, another method called Venn diagram‐based approach is also developed. The sure screening property is proven for both methods. It is shown that the complete case analysis can also keep the sure screening property of any feature screening approach with sure screening property.     

Numerical Bayesian inference with arbitrary prior

The purpose of the paper, is to explain how recent advances in Markov Chain Monte Carlo integration can facilitate the routine Bayesian analysis of the linear model when the prior distribution is completely user dependent. The method is based on a Metropolis-Hastings algorithm with a Student-t source distribution that can generate posterior moments as well as marginal posterior densities for model parameters. The method is illustrated with numerical examples where the combination of prior and likelihood information leads to multimodal posteriors due to prior-likelihood conflicts, and to cases where prior information can be summarized by symmetric stable Paretian distributions.     

Bayesian adaptive Lasso for quantile regression models with nonignorably missing response data

Abstract

Handling data with the nonignorably missing mechanism is still a challenging problem in statistics. In this paper, we develop a fully Bayesian adaptive Lasso approach for quantile regression models with nonignorably missing response data, where the nonignorable missingness mechanism is specified by a logistic regression model. The proposed method extends the Bayesian Lasso by allowing different penalization parameters for different regression coefficients. Furthermore, a hybrid algorithm that combined the Gibbs sampler and Metropolis-Hastings algorithm is implemented to simulate the parameters from posterior distributions, mainly including regression coefficients, shrinkage coefficients, parameters in the non-ignorable missing models. Finally, some simulation studies and a real example are used to illustrate the proposed methodology.     

A skew-normal random effects model for longitudinal ordinal categorical responses with missing data

Missing values are common in longitudinal data studies. The missing data mechanism is termed non-ignorable (NI) if the probability of missingness depends on the non-response (missing) observations. This paper presents a model for the ordinal categorical longitudinal data with NI non-monotone missing values. We assumed two separate models for the response and missing procedure. The response is modeled as ordinal logistic, whereas the logistic binary model is considered for the missing process. We employ these models in the context of so-called shared-parameter models, where the outcome and missing data models are connected by a common set of random effects. It is commonly assumed that the random effect follows the normal distribution in longitudinal data with or without missing data. This can be extremely restrictive in practice, and it may result in misleading statistical inferences. In this paper, we instead adopt a more flexible alternative distribution which is called the skew-normal distribution. The methodology is illustrated through an application to Schizophrenia Collaborative Study data [19 D. Hedeker, Generalized linear mixed models, in Encyclopedia of Statistics in Behavioral Science, B. Everitt and D. Howell, eds., John Wiley, London, 2005, pp. 729–738. [Google Scholar]] and a simulation.     

Objective Bayesian Analysis for Linear Degradation Models

We propose an objective Bayesian approach to analyze degradation models. For the linear degradation models, two reference priors are derived, and based on this we show the posterior distributions are proper. Since the lifetime of the product is of interest in practice, a transformation is introduced to obtain the reference priors of the medium lifetime. In the posterior analysis, we explore two sampling procedures: Monte Carlo (MC) procedure and Monte Carlo Markov Chain (MCMC) procedure. A real data from Takeda and Suzuki (1983 Takeda , E. , Suzuki , N. ( 1983 ). An empirical model for device degradation due to hot-carrier injection . IEEE Electron Dev. Lett. 4 : 111 – 113 .[Crossref], [Web of Science ®] , [Google Scholar]) is analyzed, and we find the results obtained by both procedures are close to the given literature.     

A note on dealing with missing standard errors in meta-analyses of continuous outcome measures in WinBUGS

A meta-analysis of a continuous outcome measure may involve missing standard errors. This is not a problem depending on assumptions made about the population standard deviation. Multiple imputation can be used to impute missing values while allowing for uncertainty in the imputation. Markov chain Monte Carlo simulation is a multiple imputation technique for generating posterior predictive distributions for missing data. We present an example of imputing missing variances using WinBUGS. The example highlights the importance of checking model assumptions, whether for missing or observed data.     

Particle Learning for Fat-Tailed Distributions

It is well known that parameter estimates and forecasts are sensitive to assumptions about the tail behavior of the error distribution. In this article, we develop an approach to sequential inference that also simultaneously estimates the tail of the accompanying error distribution. Our simulation-based approach models errors with a t_ν-distribution and, as new data arrives, we sequentially compute the marginal posterior distribution of the tail thickness. Our method naturally incorporates fat-tailed error distributions and can be extended to other data features such as stochastic volatility. We show that the sequential Bayes factor provides an optimal test of fat-tails versus normality. We provide an empirical and theoretical analysis of the rate of learning of tail thickness under a default Jeffreys prior. We illustrate our sequential methodology on the British pound/U.S. dollar daily exchange rate data and on data from the 2008–2009 credit crisis using daily S&P500 returns. Our method naturally extends to multivariate and dynamic panel data.     

Bayesian Inference in Multivariate Regression With Missing Observations on the Response Variables

We discuss the case of the multivariate linear model Y = XB + E with Y an (n × p) matrix, and so on, when there are missing observations in the Y matrix in a so-called nested pattern. We propose an analysis that arises by incorporating the predictive density of the missing observations in determining the posterior distribution of B, and its mean and variance matrix. This involves us with matric-T variables. The resulting analysis is illustrated with some Canadian economic data.     

A Study for Missing Values in PINAR(1)T Processes

In this paper, we propose several approaches to estimate the parameters of the periodic first-order integer-valued autoregressive process with period T (PINAR(1)_T) in the presence of missing data. By using incomplete data, we propose two approaches that are based on the conditional expectation and conditional likelihood to estimate the parameters of interest. Then we study three kinds of imputation methods for the missing data. The performances of these approaches are compared via simulations.     

Data-driven sensitivity analysis to detect missing data mechanism with applications to structural equation modelling

Missing data are a common problem in almost all areas of empirical research. Ignoring the missing data mechanism, especially when data are missing not at random (MNAR), can result in biased and/or inefficient inference. Because MNAR mechanism is not verifiable based on the observed data, sensitivity analysis is often used to assess it. Current sensitivity analysis methods primarily assume a model for the response mechanism in conjunction with a measurement model and examine sensitivity to missing data mechanism via the parameters of the response model. Recently, Jamshidian and Mata (Post-modelling sensitivity analysis to detect the effect of missing data mechanism, Multivariate Behav. Res. 43 (2008), pp. 432–452) introduced a new method of sensitivity analysis that does not require the difficult task of modelling the missing data mechanism. In this method, a single measurement model is fitted to all of the data and to a sub-sample of the data. Discrepancy in the parameter estimates obtained from the the two data sets is used as a measure of sensitivity to missing data mechanism. Jamshidian and Mata describe their method mainly in the context of detecting data that are missing completely at random (MCAR). They used a bootstrap type method, that relies on heuristic input from the researcher, to test for the discrepancy of the parameter estimates. Instead of using bootstrap, the current article obtains confidence interval for parameter differences on two samples based on an asymptotic approximation. Because it does not use bootstrap, the developed procedure avoids likely convergence problems with the bootstrap methods. It does not require heuristic input from the researcher and can be readily implemented in statistical software. The article also discusses methods of obtaining sub-samples that may be used to test missing at random in addition to MCAR. An application of the developed procedure to a real data set, from the first wave of an ongoing longitudinal study on aging, is presented. Simulation studies are performed as well, using two methods of missing data generation, which show promise for the proposed sensitivity method. One method of missing data generation is also new and interesting in its own right.