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

Abstract

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...

Estimating treatment effects under untestable assumptions with nonignorable missing data

“…Our findings add to previous evidence on the performance of Heckman‐based selection models by showing that this approach requires not only a valid, but also a moderate or strong exclusion restriction variable to work well in practice. Our simulations also corroborate previous studies that Heckman‐based approaches may be less sensitive to departures from the assumed distributional assumptions compared to full‐likelihood approaches, provided that the exclusion restriction is not weak. In addition, a common criticism of MI is that MI is only valid (and hence useful) under the MAR assumption.…”

“…Second, strong exclusion restriction variables are rare in practice, and hence understanding the implications of a invalid and/or weak exclusion restriction to different selection model approaches and its impact on inferences is required. Third, other aspects of selection models such as model specification and distributional assumptions have received wider attention in the last few years …”

“…The development of flexible, user‐friendly software tools for implementing selection models to handle nonnormal outcomes with nonignorable missing data is warranted. For example, Gomes et al are exploring the flexibility of copula‐based approaches in this context, which provides an encouraging starting point.…”

“…However, the extent to which this Heckman‐based MI approach relies on the exclusion restriction, and more specifically the relevance assumption is unclear. We hypothesize that this imputation approach may be less reliant on the exclusion restriction compared to the original two‐step Heckman, while being more robust to departures from bivariate normality compared to the full maximum likelihood approach. We now investigate the implications of the alternative exclusion restriction assumptions, for the bias and efficiency in the estimates of treatment effectiveness following the alternative selection approaches.…”

A Bayesian Parametric Approach to Handle Missing Longitudinal Outcome Data in Trial-based Health Economic Evaluations

“…A full Baysian non-parametric specification could also be used to handle sparse data (Linero and Daniels, 2015). Gomes et al (2019) have recently proposed a non-ignorable missingness approach for bivariate outcomes which combines multiple-imputation methods with a copula selection model to allow for non-normal outcomes while simultaneously imputing missing outcome data under MNAR. However, this approach relies on parametric assumptions about the joint distribution of the observed and missing data, which are typically difficult to check and do not allow us to introduce sensitivity parameters (Daniels and Hogan, 2008).…”