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
1535091708 2099891228 2097147565 1572302037 1573240883 2050127204 1921855215
In this paper, I present a simple characterization of the sample selection bias problem that is also applicable to the conceptually distinct econometric problems that arise from truncated samples and from models with limited dependent variables. The problem of sample selection bias is fit within the conventional specification error framework of Griliches and Theil. A simple estimator is discussed that enables analysts to utilize ordinary regression methods to estimate models free of selection bias. The techniques discussed here are applied to re-estimate and test a model of female labor supply developed by the author. (1974). This paper is in three parts. In the first section, selection bias is presented within the specification error framework. In this section, general distributional assumptions are maintained. In section two, specific results are presented for the case of normal regression disturbances. Simple estimators are proposed and discussed. In the third section, empirical results are presented. 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. Abstract We consider inference about the parameter β* indexing the conditional mean of a vector of correlated outcomes given a vector of explanatory variables when some of the outcomes are missing in a subsample of the study and the probability of response depends on both observed and unobserved data values; that is, nonresponse is nonignorable. We propose a class of augmented inverse probability of response weighted estimators that are consistent and asymptotically normal (CAN) for estimating β* when the response probabilities can be parametrically modeled and a CAN estimator exists. The proposed estimators do not require full specification of a parametric likelihood, and their computation does not require numerical integration. Our estimators can be viewed as an extension of generalized estimating equation estimators that allows for nonignorable nonresponse. We show that our class essentially consists of all CAN estimators of β*. We also show that the asymptotic variance of the optimal estimator in our ... Introduction.- General Methodology.- Monotone Censored Data.- CrossSectional Data and Right Censored Data Combined.- Multivariate RightCensored Multivariate Data.- Unified Approach for Causal Inference andCensored Data. 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. 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.