Paper:
Consistent Estimation of Functions of Data Missing Non-Monotonically and Not at Random
Missing records are a perennial problem in analysis of complex data of all types, when the target of inference is some function of the full data law. In simple cases, where data is missing at random or completely at random [15], well-known adjustments exist that result in consistent estimators of target quantities. Assumptions underlying these estimators are generally not realistic in practical missing data problems. Unfortunately, consistent estimators in more complex cases where data is missing not at random, and where no ordering on variables induces monotonicity of missingness status are not known in general, with some notable exceptions [13,18,16]. In this paper, we propose a general class of consistent estimators for cases where data is missing not at random, and missingness status is non-monotonic. Our estimators, which are generalized inverse probability weighting estimators, make no assumptions on the underlying full data law, but instead place independence restrictions, and certain other fairly mild assumptions, on the distribution of missingness status conditional on the data. The assumptions we place on the distribution of missingness status conditional on the data can be viewed as a version of a conditional Markov random field (MRF) corresponding to a chain graph. Assumptions embedded in our model permit identification from the observed data law, and admit a natural fitting procedure based on the pseudo likelihood approach of [2]. We illustrate our approach with a simple simulation study, and an analysis of risk of premature birth in women in Botswana exposed to highly active anti-retroviral therapy.

Review:
This paper is concerned with the problem of deriving consistent estimators for datasets which have fields that are missing non-monotonically and not at random (MNAR). The authors frame this problem using the language of graphical models by introducing a chain graph depiction of the problem, propose a solution using inverse probability weighting and suggest an estimation technique which uses pseudo-likelihood estimation.  A small set of synthetic experiments on a simple synthetic dataset are carried out to evaluate the proposed technique empirically.
I found the paper to be well written and fairly easy to follow overall. However, I have two concerns regarding this work:  1. The lack of discussion around related work makes it difficult to fully judge the contributions of the paper to the causal inference community. For example, the non-monotone MNAR  setting has been studied, and a solution has been proposed using inverse propensity weighting, by Robins et al. (1999) and Vansteelandt et al. (2007). A nice summarization of these approaches is given by Li et al. (2011). A simple discussion of the estimation technique proposed in section 5 to the previously mentioned work would be very helpful. The proposed work seems unique in that the authors explicitly frame the problem within the context of graphical models, and use propose a pseudo-likelihood approach for estimation. However, it is difficult to judge the novelty of the proposed contributions without additional context.  2. The authors use a very simple synthetic domain for the experimental evaluation. I would have liked to see a larger array of functional forms and dependence structures. It is also not clear from the results that the proposed method achieves a substantial improvement over taking no action at all. In many cases, the adjusted values provide a worse estimate than performing no adjustment at all. It may be the case that more substantial improvement is seen in larger sample sizes but this should be shown within the main text of the paper.  There is a small typo in footnote one where the independence symbol is erroneously defined as the conditional independence. The clarification is also not likely to be necessary given the intended audience.   Overall, this paper presents an interesting idea for an important problem but the lack of framing and a thorough evaluation make it difficult for me to suggest acceptance.   * Robins JM, Rotnitzky A, and Scharfstein D. "Sensitivity analysis for selection bias and unmeasured confounding in missing data and causal inference models." Statistical models in epidemiology: the environment and clinical trials. 1999 * Vansteelandt S, Rotnitzky A and Robins J. "Estimation of regression models for the mean of repeated outcomes under nonignorable nonmonotone nonresponse." Biometrika. 2007 * Li, Lingling, et al. "On weighting approaches for missing data." Statistical methods in medical research. 2011