Semiparametric Theory

Mendelian randomization (MR) is a popular instrumental variable (IV) approach, in which genetic markers are used as IVs. In order to improve efficiency, multiple markers are routinely used in MR analyses, leading to concerns about bias due to possible violation of IV exclusion restriction of no direct effect of any IV on the outcome other than through the exposure in view. To address this concern, we introduce a new class of Multiply Robust MR (MR2) estimators that are guaranteed to remain consistent for the causal effect of interest provided that at least one genetic marker is a valid IV without necessarily knowing which IVs are invalid. We show that the proposed MR2 estimators are a special case of a more general class of estimators that remain consistent provided that a set of at least k† out of K candidate instrumental variables are valid, for k† ≤ K set by the analyst ex ante, without necessarily knowing which IVs are invalid. We provide formal semiparametric theory supporting our results, and characterize the semiparametric efficiency bound for the exposure causal effect which cannot be improved upon by any regular estimator with our favorable robustness property. We conduct extensive simulation studies and apply our methods to a large-scale analysis of UK Biobank data, demonstrating the superior empirical performance of MR2 compared to competing MR methods.

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Methods Based on Semiparametric Theory for Analysis in the Presence of Missing Data

Annual Review of Statistics and Its Application ◽

10.1146/annurev-statistics-040120-025906 ◽

2021 ◽

Vol 9 (1) ◽

Author(s):

Marie Davidian

Keyword(s):

Statistical Model ◽

Probability Distributions ◽

Parametric Models ◽

Annual Review ◽

Publication Date ◽

Full Data ◽

Dimensional Parameter ◽

Finite Dimensional ◽

Key Steps ◽

Semiparametric Theory

A statistical model is a class of probability distributions assumed to contain the true distribution generating the data. In parametric models, the distributions are indexed by a finite-dimensional parameter characterizing the scientific question of interest. Semiparametric models describe the distributions in terms of a finite-dimensional parameter and an infinite-dimensional component, offering more flexibility. Ordinarily, the statistical model represents distributions for the full data intended to be collected. When elements of these full data are missing, the goal is to make valid inference on the full-data-model parameter using the observed data. In a series of fundamental works, Robins, Rotnitzky, and colleagues derived the class of observed-data estimators under a semiparametric model assuming that the missingness mechanism is at random, which leads to practical, robust methodology for many familiar data-analytic challenges. This article reviews semiparametric theory and the key steps in this derivation. Expected final online publication date for the Annual Review of Statistics, Volume 9 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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A Falsifiability Characterization of Double Robustness Through Logical Operators

Journal of Causal Inference ◽

10.1515/jci-2018-0016 ◽

2019 ◽

Vol 7 (1) ◽

Author(s):

Constantine Frangakis

Keyword(s):

Causal Inference ◽

Robust Estimator ◽

Missing Information ◽

Logical Operators ◽

Double Robustness ◽

Inference Problems ◽

Consistent Estimators ◽

Doubly Robust ◽

Semiparametric Theory

AbstractWe address the characterization of problems in which a consistent estimator exists in a union of two models, also termed as a doubly robust estimator. Such estimators are important in missing information, including causal inference problems. Existing characterizations, based on the semiparametric theory of projections, have seen sufficient progress, but can still leave one’s understanding less than satisfied as to when and especially why such estimation works. We explore here a different, explanatory characterization – an exegesis based on logical operators. We show that double robustness exists if and only if we can produce consistent estimators for each contributing model based on an “AND” estimator, i. e., an estimator whose consistency generally needs both models to be correct. We show how this characterization explains double robustness through falsifiability.

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