A robust imputation method for missing responses and covariates in sample selection models

Missing not at random (MNAR) data pose key challenges for statistical inference because the substantive model of interest is typically not identifiable without imposing further (eg, distributional) assumptions. Selection models have been routinely used for handling MNAR by jointly modeling the outcome and selection variables and typically assuming that these follow a bivariate normal distribution. Recent studies have advocated parametric selection approaches, for example, estimated by multiple imputation and maximum likelihood, that are more robust to departures from the normality assumption compared with those assuming that nonresponse and outcome are jointly normally distributed.However, the proposed methods have been mostly restricted to a specific joint distribution (eg, bivariate t-distribution). This paper discusses a flexible copula-based selection approach (which accommodates a wide range of non-Gaussian outcome distributions and offers great flexibility in the choice of functional form specifications for both the outcome and selection equations) and proposes a flexible imputation procedure that generates plausible imputed values from the copula selection model. A simulation study characterizes the relative performance of the copula model compared with the most commonly used selection models for estimating average treatment effects with MNAR data.We illustrate the methods in the REFLUX study, which evaluates the effect of laparoscopic surgery on long-term quality of life in patients with reflux disease.We provide software code for implementing the proposed copula framework using the R package GJRM. In such cases, the data are said to be missing not at random (MNAR), and nonresponse must be modeled together with the substantive model for the observed data.Selection models have been commonly used to handle MNAR data in clinical and epidemiological research, 4-6 by jointly modeling the outcome and missingness models and typically assuming bivariate normality. Heckman 7 was one of the first to propose such model (Heckman selection model) using a simultaneous equation approach, where the error terms were assumed to follow a bivariate Gaussian. At that time, to circumvent some of the difficulties associated with direct likelihood maximization, Heckman proposed a two-stage least-squares estimation procedure. 8 This involved combining a probit model for the probability of observing the outcome (first stage) with a linear regression model for the outcome (second stage), which was a function of the estimates obtained in the first stage. Although this approach (method of moments estimator) is somehow robust to deviations from normality, it relies crucially on the availability of exclusion restrictions, ie, variables that predict missingness but are unrelated to the outcome of interest. 9 An alternative approach to estimating selection models is the single-step full-information maximum likelihood (FIML), which jointly estimates the outcome and missing data equations. For example, Diggle and Kenward 10 combined a marg...

Section: Discussionmentioning

confidence: 98%

Section: Copula Selection Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Copula selection models for non‐Gaussian outcomes that are missing not at random

Gomes

Radice

Brenes

et al. 2018

“…21 The information from covariates can be used to derive two predictive scores to identify imputing sets for missing observations, following the NN-based multiple imputation idea in the case of MAR. 7,8 This will allow us to compare the proposed approach with the approaches that directly uses Heckman's selection model to perform imputation 14,15 and will be explored in the future study.…”

Section: Discussionmentioning

confidence: 99%

“…Specifically, parametric multiple imputation approaches based on the Heckman's selection model and its extensions have been proposed. 14,15 Another strategy uses a selection model to first induce MNAR and then develop dual imputation procedures to iteratively impute both missing indicator and missing variable. 16 These approaches involve a direct estimation of parameters of the selection model.…”

Section: Multiple Imputation-based Sensitivity Analysismentioning

confidence: 99%

A multiple imputation‐based sensitivity analysis approach for data subject to missing not at random

Hsu

et al. 2020

Missingness mechanism is in theory unverifiable based only on observed data. If there is a suspicion of missing not at random, researchers often perform a sensitivity analysis to evaluate the impact of various missingness mechanisms. In general, sensitivity analysis approaches require a full specification of the relationship between missing values and missingness probabilities. Such relationship can be specified based on a selection model, a pattern-mixture model or a shared parameter model. Under the selection modeling framework, we propose a sensitivity analysis approach using a nonparametric multiple imputation strategy. The proposed approach only requires specifying the correlation coefficient between missing values and selection (response) probabilities under a selection model. The correlation coefficient is a standardized measure and can be used as a natural sensitivity analysis parameter. The sensitivity analysis involves multiple imputations of missing values, yet the sensitivity parameter is only used to select imputing/donor sets. Hence, the proposed approach might be more robust against misspecifications of the sensitivity parameter. For illustration, the proposed approach is applied to incomplete measurements of level of preoperative Hemoglobin A1c, for patients who had high-grade carotid artery stenosisa and were scheduled for surgery. A simulation study is conducted to evaluate the performance of the proposed approach.

Multiple imputation of incomplete multilevel data using Heckman selection models

Muñoz,

Efthimiou,

Audigier

et al. 2023

Missing data is a common problem in medical research, and is commonly addressed using multiple imputation. Although traditional imputation methods allow for valid statistical inference when data are missing at random (MAR), their implementation is problematic when the presence of missingness depends on unobserved variables, that is, the data are missing not at random (MNAR). Unfortunately, this MNAR situation is rather common, in observational studies, registries and other sources of real‐world data. While several imputation methods have been proposed for addressing individual studies when data are MNAR, their application and validity in large datasets with multilevel structure remains unclear. We therefore explored the consequence of MNAR data in hierarchical data in‐depth, and proposed a novel multilevel imputation method for common missing patterns in clustered datasets. This method is based on the principles of Heckman selection models and adopts a two‐stage meta‐analysis approach to impute binary and continuous variables that may be outcomes or predictors and that are systematically or sporadically missing. After evaluating the proposed imputation model in simulated scenarios, we illustrate it use in a cross‐sectional community survey to estimate the prevalence of malaria parasitemia in children aged 2‐10 years in five regions in Uganda.