Correcting for Missing Data When Evaluating Surrogate Markers in a Clinical Trial

1. Problem Statement

In clinical trials, surrogate markers ($S$) are often used to predict long-term primary outcomes ($Y$) to accelerate drug approval (e.g., via the FDA's Accelerated Approval Program). However, evaluating the validity of a surrogate requires estimating the Proportion of Treatment Effect Explained (PTE), denoted as $R_S$.

Existing frameworks for estimating $R_S$ (both parametric and nonparametric) typically assume complete data. In practice, surrogate data are frequently missing due to dropout, failed assays, or missed visits. Standard practice involves Complete Case (CC) analysis, which discards patients with missing surrogates. This approach suffers from two major flaws:

1. Bias: If data are Missing at Random (MAR) rather than Missing Completely at Random (MCAR), CC analysis yields biased estimates of the treatment effect and the PTE.
2. Inefficiency: Discarding data reduces statistical power and widens confidence intervals, potentially obscuring the validity of a surrogate.

Current literature on missing data in mediation or surrogate analysis is limited, often relying on fully parametric models with restrictive assumptions or fully Bayesian approaches that are computationally intensive. There is a lack of flexible, semi-parametric methods that can handle missing surrogates without strong distributional assumptions.

2. Methodology

The authors propose two distinct classes of methods to correct for missing surrogate data within the PTE framework: Design-based and Model-based approaches.

A. Framework and Notation

• Variables: $Z$ (binary treatment), $S$ (continuous surrogate), $Y$ (continuous primary outcome).
• Target: $R_S = 1 - \frac{\Delta_S}{\Delta}$, where $\Delta$ is the total treatment effect on $Y$, and $\Delta_S$ is the residual treatment effect on $Y$ after accounting for $S$.
• Missingness Indicator: $O=1$ if $S$ is observed, $O=0$ otherwise.

B. Design-Based Approach: Inverse Probability Weighting (IPW)

This approach re-weights the observed complete cases to represent the full sample.

• Mechanism: Estimate the probability of the surrogate being observed, $\pi = P(O=1 | \text{covariates})$, using a model (e.g., logistic regression).
• Implementation:
  • Parametric IPW: Fit a weighted linear regression for $Y \sim Z, S$ using weights $w = 1/\pi$. Calculate weighted means for $S$ in each group.
  • Nonparametric IPW: Apply weights to the Nadaraya-Watson kernel estimator for the conditional mean $E(Y|S)$ and the empirical CDF of $S$.
• Pros/Cons: Robust to model misspecification of the outcome/surrogate relationship but can be inefficient (high variance) if weights are extreme.

C. Model-Based Approach: Semiparametric Maximum Likelihood Estimation (SMLE)

This approach uses an Expectation-Maximization (EM) algorithm to maximize the likelihood while treating the distribution of the surrogate nonparametrically.

• Likelihood Structure:
  • $P(Y|S, Z)$ is modeled parametrically (linear regression).
  • $P(S|Z)$ is modeled nonparametrically (discrete probability mass on observed support points).
  • This avoids assuming a specific distribution (e.g., Normal) for the surrogate $S$.
• EM Algorithm:
  • E-step: Compute the conditional expectation of the missing surrogate indicators given observed $Y$ and $Z$.
  • M-step: Update regression coefficients ($\beta$) and the nonparametric probabilities ($p_{kz}$) to maximize the expected log-likelihood.
• Pros/Cons: Generally more efficient than IPW because it utilizes all available data (including $Y$ for patients with missing $S$) and avoids the variance inflation of extreme weights. However, it is computationally slower.

D. Inference

Standard errors and confidence intervals for all estimators are derived using a bootstrap resampling approach ($D=500$ iterations), stratified by treatment group to preserve randomization ratios.

3. Key Contributions

1. Novel Frameworks: The paper introduces the first IPW and SMLE methods specifically tailored for the PTE framework in the presence of missing surrogate data.
2. Semi-Parametric Flexibility: The SMLE approach offers a unique balance: it retains the efficiency of likelihood-based methods while avoiding restrictive parametric assumptions on the surrogate's distribution ($P(S|Z)$).
3. Comprehensive Comparison: The authors provide a rigorous comparison of CC, IPW, and SMLE across various missingness mechanisms (MCAR, MAR dependent on $Z$, MAR dependent on $Y$, and MAR dependent on interactions).
4. Software Implementation: All methods are implemented in the open-source R package MissSurrogate, making these advanced corrections accessible to practitioners.

4. Results

Simulation Studies

Simulations were conducted with $N=2000$ patients under five settings:

• Setting 1 (MCAR): All methods were unbiased. SMLE showed superior efficiency (narrower CIs) compared to CC and IPW.
• Settings 2 & 3 (MAR):
  • When missingness depended on treatment ($Z$), CC remained unbiased.
  • When missingness depended on the outcome ($Y$), CC analysis became biased (approx. 4% bias) and had poor coverage (85–88%).
  • IPW and SMLE remained unbiased and maintained nominal 95% coverage.
  • SMLE consistently outperformed IPW in efficiency (Relative Efficiency $\approx$ 0.75–0.79 vs. 0.65–0.69 for CC/IPW), yielding tighter confidence intervals.
• Setting 4 (Weight Misspecification): IPW performance degraded significantly if the weight model omitted key variables (e.g., interactions). SMLE did not require a weight model and remained robust.
• Setting 5 (Lack of Overlap): When the support of $S$ differed between groups, nonparametric estimators were biased regardless of missingness correction. This highlights a limitation of nonparametric methods when distributions do not overlap.

Data Application (DCCT Trial)

The methods were applied to the Diabetes Control and Complications Trial (DCCT) to evaluate blood glucose ($S$) as a surrogate for LDL cholesterol ($Y$).

• Data: 14.7% missingness in blood glucose.
• Findings:
  • CC and missingness-corrected estimates were similar, suggesting missingness was likely MCAR or MAR given $Z$ in this specific dataset.
  • SMLE vs. IPW: The SMLE produced a 31% narrower confidence interval than IPW, demonstrating the efficiency gain in real-world data.
  • Conclusion: While blood glucose explained a moderate proportion of the treatment effect, the lower bounds of the confidence intervals for all methods were below 0.5. Thus, glucose could not be validated as a surrogate for LDL in this context.

5. Significance

This paper addresses a critical gap in clinical trial analysis where surrogate validation is often compromised by missing data.

• Statistical Validity: It demonstrates that relying on complete cases can lead to biased conclusions about surrogate validity, particularly when missingness is related to the outcome.
• Efficiency: The proposed SMLE method offers a statistically efficient alternative to IPW, allowing researchers to retain statistical power without discarding valuable data.
• Practical Utility: By providing the MissSurrogate R package, the authors enable the immediate adoption of these robust methods in regulatory and clinical research settings, potentially improving the reliability of accelerated drug approvals.
• Future Directions: The authors note that extending these methods to handle Missing Not At Random (MNAR) scenarios and missing primary outcomes remains an important area for future research.