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Register a new userMissing at Random or Not: A Semiparametric Testing Approach
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Practical problems with missing data are common, and statistical methods have been developed concerning the validity and/or efficiency of statistical procedures. On a central focus, there have been longstanding interests on the mechanism governing data missingness, and correctly deciding the appropriate mechanism is crucially relevant for conducting proper practical investigations. The conventional notions include the three common potential classes -- missing completely at random, missing at random, and missing not at random. In this paper, we present a new hypothesis testing approach for deciding between missing at random and missing not at random. Since the potential alternatives of missing at random are broad, we focus our investigation on a general class of models with instrumental variables for data missing not at random. Our setting is broadly applicable, thanks to that the model concerning the missing data is nonparametric, requiring no explicit model specification for the data missingness. The foundational idea is to develop appropriate discrepancy measures between estimators whose properties significantly differ only when missing at random does not hold. We show that our new hypothesis testing approach achieves an objective data oriented choice between missing at random or not. We demonstrate the feasibility, validity, and efficacy of the new test by theoretical analysis, simulation studies, and a real data analysis.
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Missing data occur frequently in empirical studies in health and social sciences, often compromising our ability to make accurate inferences. An outcome is said to be missing not at random (MNAR) if, conditional on the observed variables, the missing data mechanism still depends on the unobserved outcome. In such settings, identification is generally not possible without imposing additional assumptions. Identification is sometimes possible, however, if an instrumental variable (IV) is observed for all subjects which satisfies the exclusion restriction that the IV affects the missingness process without directly influencing the outcome. In this paper, we provide necessary and sufficient conditions for nonparametric identification of the full data distribution under MNAR with the aid of an IV. In addition, we give sufficient identification conditions that are more straightforward to verify in practice. For inference, we focus on estimation of a population outcome mean, for which we develop a suite of semiparametric estimators that extend methods previously developed for data missing at random. Specifically, we propose inverse probability weighted estimation, outcome regression-based estimation and doubly robust estimation of the mean of an outcome subject to MNAR. For illustration, the methods are used to account for selection bias induced by HIV testing refusal in the evaluation of HIV seroprevalence in Mochudi, Botswana, using interviewer characteristics such as gender, age and years of experience as IVs.
We study the identification and estimation of statistical functionals of multivariate data missing non-monotonically and not-at-random, taking a semiparametric approach. Specifically, we assume that the missingness mechanism satisfies what has been previously called no self-censoring or itemwise conditionally independent nonresponse, which roughly corresponds to the assumption that no partially-observed variable directly determines its own missingness status. We show that this assumption, combined with an odds ratio parameterization of the joint density, enables identification of functionals of interest, and we establish the semiparametric efficiency bound for the nonparametric model satisfying this assumption. We propose a practical augmented inverse probability weighted estimator, and in the setting with a (possibly high-dimensional) always-observed subset of covariates, our proposed estimator enjoys a certain double-robustness property. We explore the performance of our estimator with simulation experiments and on a previously-studied data set of HIV-positive mothers in Botswana.
Two-sample hypothesis testing for random graphs arises naturally in neuroscience, social networks, and machine learning. In this paper, we consider a semiparametric problem of two-sample hypothesis testing for a class of latent position random graphs. We formulate a notion of consistency in this context and propose a valid test for the hypothesis that two finite-dimensional random dot product graphs on a common vertex set have the same generating latent positions or have generating latent positions that are scaled or diagonal transformations of one another. Our test statistic is a function of a spectral decomposition of the adjacency matrix for each graph and our test procedure is consistent across a broad range of alternatives. We apply our test procedure to real biological data: in a test-retest data set of neural connectome graphs, we are able to distinguish between scans from different subjects; and in the {em C.elegans} connectome, we are able to distinguish between chemical and electrical networks. The latter example is a concrete demonstration that our test can have power even for small sample sizes. We conclude by discussing the relationship between our test procedure and generalized likelihood ratio tests.
We offer a natural and extensible measure-theoretic treatment of missingness at random. Within the standard missing data framework, we give a novel characterisation of the observed data as a stopping-set sigma algebra. We demonstrate that the usual missingness at random conditions are equivalent to requiring particular stochastic processes to be adapted to a set-indexed filtration of the complete data: measurability conditions that suffice to ensure the likelihood factorisation necessary for ignorability. Our rigorous statement of the missing at random conditions also clarifies a common confusion: what is fixed, and what is random?
Classical semiparametric inference with missing outcome data is not robust to contamination of the observed data and a single observation can have arbitrarily large influence on estimation of a parameter of interest. This sensitivity is exacerbated when inverse probability weighting methods are used, which may overweight contaminated observations. We introduce inverse probability weighted, double robust and outcome regression estimators of location and scale parameters, which are robust to contamination in the sense that their influence function is bounded. We give asymptotic properties and study finite sample behaviour. Our simulated experiments show that contamination can be more serious a threat to the quality of inference than model misspecification. An interesting aspect of our results is that the auxiliary outcome model used to adjust for ignorable missingness by some of the estimators, is also useful to protect against contamination. We also illustrate through a case study how both adjustment to ignorable missingness and protection against contamination are achieved through weighting schemes, which can be contrasted to gain further insights.
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