This paper reviews recent advances in missing data research using graphical models to represent multivariate dependencies. We first examine the limitations of traditional frameworks from three different perspectives: . We then show how procedures based on graphical models can overcome these limitations and provide meaningful performance guarantees even when data are Missing Not At Random (MNAR). In particular, we identify conditions that guarantee consistent estimation in broad categories of missing data problems, and derive procedures for implementing this estimation. Finally we derive testable implications for missing data models in both MAR (Missing At Random) and MNAR categories.
The handling of MNAR data is more or less limited to performing sensitivity analysis . Methods of performing sensitivity analysis has been suggested in research publications such as @cite_29 @cite_30 and @cite_5 . Special handling of MNAR problems based on use of selection models ( @cite_38 ) and pattern mixture models is discussed in @cite_33 . Missing data discussed so far is a special case of , namely data that contains observations made in the power set rather than the sample space of variables of interest @cite_16 . The notion of coarsening at random (CAR) was introduced in @cite_16 and identifies the condition under which coarsening mechanism can be ignored while drawing inferences on the distribution of variables of interest ( @cite_43 ). The notion of sequential CAR has been discussed in @cite_17 . Detailed discussions on coarsened data are available in @cite_21 .
@cite_30 @cite_38 @cite_33 @cite_29 @cite_21 @cite_43 @cite_5 @cite_16 @cite_17
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The notion of coarsening at random (CAR) was introduced by Heitjan and Rubin (1991) to describe the most general form of randomly grouped, censored, or missing data, for which the coarsening mechanism can be ignored when making likelihood-based inference about the parameters of the distribution of the variable of interest. The CAR assumption is popular, and applications abound. However the full implications of the assumption have not been realized. Moreover a satisfactory theory of CAR for continuously distributed data—which is needed in many applications, particularly in survival analysis—hardly exists as yet. This paper gives a detailed study of CAR. We show that grouped data from a finite sample space always fit a CAR model: a nonparametric model for the variable of interest together with the assumption of an arbitrary CAR mechanism puts no restriction at all on the distribution of the observed data. In a slogan, CAR is everything. We describe what would seem to be the most general way CAR data could occur in practice, a sequential procedure called randomized monotone coarsening. We show that CAR mechanisms exist which are not of this type. Such a coarsening mechanism uses information about the underlying data which is not revealed to the observer, without this affecting the observer’s conclusions. In a second slogan, CAR is more than it seems. This implies that if the analyst can argue from subject-matter considerations that coarsened data is CAR, he or she has knowledge about the structure of the coarsening mechanism which can be put to good use in non-likelihood-based inference procedures. We argue that this is a valuable option in multivariate survival analysis. We give a new definition of CAR in general sample spaces, criticising earlier proposals, and we establish parallel results to the discrete case. The new definition focusses on the distribution rather than the density of the data. It allows us to generalise the theory of CAR to the important situation where coarsening variables (e.g., censoring times) are partially observed as well as the variables of interest. The past decade has seen a noticeable shift in missing data handling techniques that assume a missing at random (MAR) mechanism, where the propensity for missing data on an outcome is related to other analysis variables. Although MAR is often reasonable, there are situations where this assumption is unlikely to hold, leading to biased parameter estimates. One such example is a longitudinal study of substance use where participants with the highest frequency of use also have the highest likelihood of attrition, even after controlling for other correlates of missingness. There is a large body of literature on missing not at random (MNAR) analysis models for longitudinal data, particularly in the field of biostatistics. Because these methods allow for a relationship between the outcome variable and the propensity for missing data, they require a weaker assumption about the missing data mechanism. This article describes 2 classic MNAR modeling approaches for longitudinal data: the selection model and the pattern mixture model. To date, these models have been slow to migrate to the social sciences, in part because they required complicated custom computer programs. These models are now quite easy to estimate in popular structural equation modeling programs, particularly Mplus. The purpose of this article is to describe these MNAR modeling frameworks and to illustrate their application on a real data set. Despite their potential advantages, MNAR-based analyses are not without problems and also rely on untestable assumptions. This article offers practical advice for implementing and choosing among different longitudinal models. In a companion paper we described what intuitively would seem to be the most general possible way to generate Coarsening at Random mechanisms, a sequential procedure called randomized monotone coarsening. Counter-examples showed that CAR mechanisms exist which cannot be represented in this way. Here, we further develop these results in two directions. Firstly, we consider what happens when data is coarsened at random in two or more phases. We show that the resulting coarsening mechanism is not CAR anymore, but under suitable assumptions is identified and can provide interesting alternative analysis of data under a non-CAR model. Secondly, we look at sequential mechanisms for generating MAR data, missing components of a multivariate random vector. Randomised monotone missingness schemes, in which one variable at a time is observed and depending on its value, another variable is chosen or the procedure is terminated, supply in our opinion the broadest class of physically interpretable MAR mechanisms. We show that every randomised monotone missingness scheme can be represented by a Markov monotone missingness scheme, in which the choice of which variable to observe next only depends on the set of previously observed variables and their values, not on the sequence in which they were measured. We also show that MAR mechanisms exist which cannot be represented sequentially. We propose in this paper a new approach to train the Generative Adversarial Nets (GANs) with a mixture of generators to overcome the mode collapsing problem. The main intuition is to employ multiple generators, instead of using a single one as in the original GAN. The idea is simple, yet proven to be extremely effective at covering diverse data modes, easily overcoming the mode collapsing problem and delivering state-of-the-art results. A minimax formulation was able to establish among a classifier, a discriminator, and a set of generators in a similar spirit with GAN. Generators create samples that are intended to come from the same distribution as the training data, whilst the discriminator determines whether samples are true data or generated by generators, and the classifier specifies which generator a sample comes from. The distinguishing feature is that internal samples are created from multiple generators, and then one of them will be randomly selected as final output similar to the mechanism of a probabilistic mixture model. We term our method Mixture Generative Adversarial Nets (MGAN). We develop theoretical analysis to prove that, at the equilibrium, the Jensen-Shannon divergence (JSD) between the mixture of generators’ distributions and the empirical data distribution is minimal, whilst the JSD among generators’ distributions is maximal, hence effectively avoiding the mode collapsing problem. By utilizing parameter sharing, our proposed model adds minimal computational cost to the standard GAN, and thus can also efficiently scale to large-scale datasets. We conduct extensive experiments on synthetic 2D data and natural image databases (CIFAR-10, STL-10 and ImageNet) to demonstrate the superior performance of our MGAN in achieving state-of-the-art Inception scores over latest baselines, generating diverse and appealing recognizable objects at different resolutions, and specializing in capturing different types of objects by the generators.