A review of Scharfstein, Rotnitzky, and Robins (1999) on handling nonignorable missing data and conducting sensitivity analysis using semiparametric methods.
The Data Generating Process: The observed data for a subject consists of: \(O = (Q, \Delta, \Delta Y, \bar{V}(Q))\)
Variable Definitions: * $Q$: Time to drop-out. If a subject completes the study, $Q=T$.
The Goal: Estimation and inference regarding the unconditional expectation (mean) of the outcome variable, denoted as $\mu_0 = \mathbb{E}[Y]$.
The Challenge: The drop-out indicator $\Delta$ is a function of $Q$. If $Q$ is strictly independent of $Y$, estimation is straightforward because the observed data represents a completely random subsample. However, in most clinical or real-world settings, drop-out is informative. For example, patients with severe disease progression may be more likely to drop out, while healthier patients remain. Because $\Delta$ often depends on the potentially unobserved outcome $Y$, analyzing only the observed cases will lead to biased estimation.
To address the estimability of $\mu_0$, the authors assume a specific underlying structure and propose a stratified Cox proportional hazards model.
Model Mechanics: In this framework, the risk of dropping out depends on both the observed history of covariate variables ($\bar{V}$) and the potentially unobserved final outcome ($Y$).
The Bias Parameter: The degree to which the drop-out hazard relies on the unobserved outcome is governed by an unknown selection bias parameter, denoted as $a_0$.
Theoretical Limitations (Theorem 1): Without parametric assumptions on the joint distribution of $Y$ and $\bar{V}$ or $Q$, the paper establishes two critical facts:
Implications of Theorem 1: Because the observed data distribution $F_O$ can be easily estimated, the data itself can never reject a hypothesized value of $a_0$. In short: $a_0$ cannot be estimated from the data alone. However, if a value for $a_0$ is fixed, the outcome mean $\mu_0$ becomes estimable.
To overcome the inherent non-identifiability of the model, the authors propose a structured sensitivity analysis. By treating the selection bias parameter $a_0$ as a known, fixed variable, researchers can estimate the outcome’s mean and compute valid confidence intervals across a plausible range of bias parameters. This allows analysts to evaluate how robust their conclusions are to varying assumptions about the missing data mechanism.
Context: A clinical trial comparing four treatments for HIV.
Outcome Variable: CD4 count measured at the end of the study period.
Initial Findings: Based purely on the mean of the observed outcomes, Treatment 2 appears to be the most effective (as shown in Table 3 of the paper).
Sensitivity Analysis Application: The authors apply their methodology by systematically varying the values of $a_0$ for each treatment arm. By plotting the estimated treatment effects across this range, researchers can visualize the area of uncertainty and determine if Treatment 2 remains superior even under pessimistic assumptions about nonignorable drop-out.