Multiple imputation (MI) is a popular method for dealing with missing values. However, the suitable way for applying clustering after MI remains unclear: how to pool partitions? How to assess the clustering instability when data are incomplete? By answering both questions, this paper proposed a complete view of clustering with missing data using MI. The problem of partitions pooling is here addressed using consensus clustering while, based on the bootstrap theory, we explain how to assess the instability related to observed and missing data. The new rules for pooling partitions and instability assessment are theoretically argued and extensively studied by simulation. Partitions pooling improves accuracy, while measuring instability with missing data enlarges the data analysis possibilities: it allows assessment of the dependence of the clustering to the imputation model, as well as a convenient way for choosing the number of clusters when data are incomplete, as illustrated on a real data set.
In this paper, we expand the methodology presented in Mertens et. al (2020, Biometrical Journal) to the study of life-time (survival) outcome which is subject to censoring and when imputation is used to account for missing values. We consider the problem where missing values can occur in both the calibration data as well as newly - to-be-predicted - observations (validation). We focus on the Cox model. Methods are described to combine imputation with predictive calibration in survival modeling subject to censoring. Application to cross-validation is discussed. We demonstrate how conclusions br…
Incomplete observability of data generates an identification problem. There is no panacea for missing data. What one can learn about a population parameter depends on the assumptions one finds credible to maintain. The credibility of assumptions varies with the empirical setting. No specific assumptions can provide a realistic general solution to the problem of inference with missing data. Yet Rubin has promoted random multiple imputation (RMI) as a general way to deal with missing values in public-use data. This recommendation has been influential to empirical researchers who seek a simple fi…
Two major ideas in the analysis of missing data are (a) the EM algorithm [Dempster, Laird and Rubin, J. Roy. Statist. Soc. Ser. B 39 (1977) 1--38] for maximum likelihood (ML) estimation, and (b) the formulation of models for the joint distribution of the data ${Z}$ and missing data indicators ${M}$, and associated "missing at random"; (MAR) condition under which a model for ${M}$ is unnecessary [Rubin, Biometrika 63 (1976) 581--592]. Most previous work has treated ${Z}$ and ${M}$ as single blocks, yielding selection or pattern-mixture models depending on how their joint distribution is factori…
Università degli Studi di Perugia（佩鲁贾大学）；University College Dublin（都柏林大学学院）
Gaussian Mixture models (GMMs) are a powerful tool for clustering, classification and density estimation when clustering structures are embedded in the data. The presence of missing values can largely impact the GMMs estimation process, thus handling missing data turns out to be a crucial point in clustering, classification and density estimation. Several techniques have been developed to impute the missing values before model estimation. Among these, multiple imputation is a simple and useful general approach to handle missing data. In this paper we propose two different methods to fit Gaussi…
The $k$-means algorithm is often used in clustering applications but its usage requires a complete data matrix. Missing data, however, is 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 …
Large-sample properties of multiple imputation estimators for parameters of logistic regression with covariates missing at random separately or simultaneously
研究两组分类协变量分别或同时 MAR 时 logistic 回归 MI 估计量的渐近性质，提出修正方差估计，并通过模拟和逢甲夜市调查数据验证其优势。
We are concerned in clustering continuous data sets subject to non-ignorable missingness. We perform clustering with a specific semi-parametric mixture, under the assumption of conditional independence given the component. The mixture model isused for clustering and not for estimating the density of the full variables (observed and unobserved), thus we do not need other assumptions on the component distribution neither to specify the missingness mechanism. Estimation is performed by maximizing an extension of smoothed likelihood allowing missingness. This optimization is achieved by a Majoriza…
Robust clustering from incomplete data is an important topic because, in many practical situations, real data sets are heavy-tailed, asymmetric, and/or have arbitrary patterns of missing observations. Flexible methods and algorithms for model-based clustering are presented via mixture of the generalized hyperbolic distributions and its limiting case, the mixture of multivariate skew-t distributions. An analytically feasible EM algorithm is formulated for parameter estimation and imputation of missing values for mixture models employing missing at random mechanisms. The proposed methodologies a…
Multiple imputation (MI) is a popular method for dealing with missing values. One main advantage of MI is to separate the imputation phase and the analysis one. However, both are related since they are based on distribution assumptions that have to be consistent. This point is well known as congeniality. In this paper, we discuss congeniality for clustering on continuous data. First, we theoretically highlight how two joint modeling (JM) MI methods (JM-GL and JM-DP) are congenial with various clustering methods. Then, we propose a new fully conditional specification (FCS) MI method with the sa…
Partially recorded data are frequently encountered in many applications and usually clustered by first removing incomplete cases or features with missing values, or by imputing missing values, followed by application of a clustering algorithm to the resulting altered dataset. Here, we develop clustering methodology through a model-based approach using the marginal density for the observed values, assuming a finite mixture model of multivariate $t$ distributions. We compare our approximate algorithm to the corresponding full expectation-maximization (EM) approach that considers the missing valu…
We consider the problem of model-based clustering in the presence of many correlated, mixed continuous and discrete variables, some of which may have missing values. Discrete variables are treated with a latent continuous variable approach and the Dirichlet process is used to construct a mixture model with an unknown number of components. Variable selection is also performed to identify the variables that are most influential for determining cluster membership. The work is motivated by the need to cluster patients thought to potentially have autism spectrum disorder (ASD) on the basis of many …
Multiple imputation is widely used to handle missing data. Although Rubin's combining rule is simple, it is not clear whether or not the standard multiple imputation inference is consistent when coupled with the commonly-used full sample estimators. This article establishes a unified martingale representation of multiple imputation for a wide class of asymptotically linear full sample estimators. This representation invokes the wild bootstrap inference to provide consistent variance estimation under the correct specification of the imputation models. As a motivating application, we illustrate …
Inverse probability of treatment weighting (IPTW) is a popular propensity score (PS)-based approach to estimate causal effects in observational studies at risk of confounding bias. A major issue when estimating the PS is the presence of partially observed covariates. Multiple imputation (MI) is a natural approach to handle missing data on covariates, but its use in the PS context raises three important questions: (i) should we apply Rubin's rules to the IPTW treatment effect estimates or to the PS estimates themselves? (ii) does the outcome have to be included in the imputation model? (iii) ho…
A Unified Framework for Variable Selection in Model-Based Clustering with Missing Not at Random
Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam（越南科技大学数学与计算机科学学院）；Vietnam National University Ho Chi Minh City, Vietnam（越南胡志明市国家大学）；School of Information and Communications Technology, Hanoi University of Science and Technology, Ha Noi, Vietnam（河内科学技术大学信息与通信技术学院）；ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems（细胞系统数学分析卓越中心）；School of Mathematical Sciences, Queensland University of Technology, Brisbane City, Australia（昆士兰科技大学数学科学学院）
Eli Lilly and Company（礼来公司）；Iowa State University（爱荷华州立大学）
The $k$-means algorithm is arguably the most popular nonparametric clustering method but cannot generally be applied to datasets with incomplete records. The usual practice then is to either impute missing values under an assumed missing-completely-at-random mechanism or to ignore the incomplete records, and apply the algorithm on the resulting dataset. We develop an efficient version of the $k$-means algorithm that allows for clustering in the presence of incomplete records. Our extension is called $k_m$-means and reduces to the $k$-means algorithm when all records are complete. We also provi…
In some contexts, mixture models can fit certain variables well at the expense of others in ways beyond the analyst's control. For example, when the data include some variables with non-trivial amounts of missing values, the mixture model may fit the marginal distributions of the nearly and fully complete variables at the expense of the variables with high fractions of missing data. Motivated by this setting, we present a mixture model for mixed ordinal and nominal data that splits variables into two groups, focus variables and remainder variables. The model allows the analyst to specify a ric…
Reference based multiple imputation methods have become popular for handling missing data in randomised clinical trials. Rubin's variance estimator is well known to be biased compared to the reference based imputation estimator's true repeated sampling variance. Somewhat surprisingly given the increasingly popularity of these methods, there has been relatively little debate in the literature as to whether Rubin's variance estimator or alternative (smaller) variance estimators targeting the repeated sampling variance are more appropriate. We review the arguments made on both sides of this debat…
In many applied fields incomplete covariate vectors are commonly encountered. It is well known that this can be problematic when making inference on model parameters, but its impact on prediction performance is less understood. We develop a method based on covariate dependent partition models that seamlessly handles missing covariates while completely avoiding any type of imputation. The method we develop allows in-sample predictions as well as out-of-sample prediction, even if the missing pattern in the new subjects' incomplete covariate vector was not seen in the training data. Any data type…
Clinical trials with longitudinal outcomes typically include missing data due to missed assessments or structural missingness of outcomes after intercurrent events handled with a hypothetical strategy. Approaches based on Bayesian random multiple imputation and Rubin's rule for pooling results across multiple imputed datasets are increasingly used in order to align the analysis of these trials with the targeted estimand. We propose and justify deterministic conditional mean imputation combined with the jackknife for inference as an alternative approach. The method is applicable to imputations …
We propose a copula based method to handle missing values in multivariate data of mixed types in multilevel data sets. Building upon the extended rank likelihood of \cite{hoff2007extending} and the multinomial probit model, our model is a latent variable model which is able to capture the relationship among variables of different types as well as accounting for the clustering structure. We fit the model by approximating the posterior distribution of the parameters and the missing values through a Gibbs sampling scheme. We use the multiple imputation procedure to incorporate the uncertainty due…
Focusing on regression based analysis of extremes in a presence of systematically missing covariates, this work presents a data-driven spatio-temporal regression based clustering of threshold excesses. It is shown that in a presence of systematically missing covariates the behavior of threshold excesses becomes nonstationary and nonhomogenous. The presented approach describes this complex behavior by a set of local stationary Generalized Pareto Distribution (GPD) models, where the parameters are expressed as regression models, and a latent spatio-temporal switching process. The spatio-temporal…
Missing data is inevitable in longitudinal clinical trials. Conventionally, the missing at random assumption is assumed to handle missingness, which however is unverifiable empirically. Thus, sensitivity analysis is critically important to assess the robustness of the study conclusions against untestable assumptions. Toward this end, regulatory agencies often request using imputation models such as return-to-baseline, control-based, and washout imputation. Multiple imputation is popular in sensitivity analysis; however, it may be inefficient and result in an unsatisfying interval estimation by…
We investigate the problem of calibration and assessment of predictive rules in prognostic designs when missing values are present in the predictors. Our paper has two key objectives which are entwined. The first is to investigate how the calibration of the prediction rule can be combined with the use of multiple imputation to account for missing predictor observations. The second objective is to propose such methods that can be implemented with current multiple imputation software, while allowing for unbiased predictive assessment through validation on new observations for which outcome is no…
When data are incomplete, a random vector Y for the data process together with a binary random vector R for the process that causes missing data, are modelled jointly. We review conditions under which R can be ignored for drawing likelihood inferences about the distribution for Y. The standard approach of Rubin (1976) and Seaman et. al. (2013) Statist. Sci., 28:2 pp. 257--268 emulates complete-data methods exactly, and directs an investigator to choose a full model in which missing at random (MAR) and distinct of parameters holds if the goal is not to use a full model. Another interpretation o…
In problems with large amounts of missing data one must model two distinct data generating processes: the outcome process which generates the response and the missing data mechanism which determines the data we observe. Under the ignorability condition of Rubin (1976), however, likelihood-based inference for the outcome process does not depend on the missing data mechanism so that only the former needs to be estimated; partially because of this simplification, ignorability is often used as a baseline assumption. We study the implications of Bayesian ignorability in the presence of high-dimensi…
Prior work has shown that combining bootstrap imputation with tree-based machine learning variable selection methods can provide good performances achievable on fully observed data when covariate and outcome data are missing at random (MAR). This approach however is computationally expensive, especially on large-scale datasets. We propose an inference-based method, called RR-BART, which leverages the likelihood-based Bayesian machine learning technique, Bayesian additive regression trees, and uses Rubin's rule to combine the estimates and variances of the variable importance measures on multip…
Causally interpretable meta-analysis combines information from a collection of randomized controlled trials to estimate treatment effects in a target population in which experimentation may not be possible but covariate information can be collected from a simple random sample. In such analyses, a key practical challenge is systematically missing data when some baseline covariates are not collected in all trials. Here, we provide identification results for potential (counterfactual) outcome means and average treatment effects in the target population when covariate data are systematically missi…
The self-organizing map is an unsupervised neural network which is widely used for data visualisation and clustering in the field of chemometrics. The classical Kohonen algorithm that computes self-organizing maps is suitable only for complete data without any missing values. However, in many applications, partially observed data are the norm. In this paper, we propose an extension of self-organizing maps to incomplete data via a new criterion that also defines estimators of the missing values. In addition, an adaptation of the Kohonen algorithm, named missSOM, is provided to compute these sel…
This paper tackles the problem of robust covariance matrix estimation when the data is incomplete. Classical statistical estimation methodologies are usually built upon the Gaussian assumption, whereas existing robust estimation ones assume unstructured signal models. The former can be inaccurate in real-world data sets in which heterogeneity causes heavy-tail distributions, while the latter does not profit from the usual low-rank structure of the signal. Taking advantage of both worlds, a covariance matrix estimation procedure is designed on a robust (mixture of scaled Gaussian) low-rank mode…
