Confidence Intervals for the Risk Difference in Combined Unilateral and Bilateral Data Incorporating a Distribution-Based Approach

Technical Summary: Confidence Intervals for the Risk Difference in Combined Unilateral and Bilateral Data

Problem Statement
In comparative clinical studies involving paired organs (e.g., eyes, ears), binary outcomes are frequently correlated within subjects. Practical datasets often comprise a mixture of unilateral (one organ observed) and bilateral (both organs observed) data due to missing measurements, prior surgeries, or congenital absence. While existing literature has extensively addressed hypothesis testing and relative measures (odds ratios, relative risks) for such data, the risk difference ($\delta = \pi_2 - \pi_1$) remains a critical absolute measure for clinical interpretation.

Current methods for constructing confidence intervals (CIs) for the risk difference in this setting rely primarily on asymptotic normality (Wald, likelihood ratio, and score tests) or the Method of Variance Estimates Recovery (MOVER). These approaches may fail to adequately reflect finite-sample distributional features, particularly skewness, leading to potential inaccuracies in coverage probabilities and interval widths when sample sizes are small or baseline probabilities are extreme.

Methodology
The authors propose a distribution-based approach for constructing confidence intervals for the risk difference under Donner's constant correlation model ($\rho$), which parameterizes within-subject dependence.

1. Distribution-Based Approach:

   • Characteristic Function: The method derives the characteristic function of the risk difference estimator ($\tilde{\delta}$) by conditioning on the total counts of unilateral and bilateral observations.
   • Density Recovery: Using the inverse Fourier transform, the probability density function (PDF) of the estimator is recovered numerically via the Fastest Fourier Transform in the West (FFTW) algorithm. This avoids reliance on large-sample normal approximations.
   • Interval Construction: Confidence intervals are constructed using a Neyman-type inversion. The acceptance region is defined by the highest density region (HDR) of the recovered PDF, analogous to a Highest Posterior Density (HPD) region but within a frequentist framework. An adaptive backtracking search algorithm determines the upper and lower bounds of $\delta$ by identifying values where the observed estimator falls within the acceptance region defined by the nuisance parameters (profile Maximum Likelihood Estimates).

2. Competing Methods:
   The study compares the proposed method against:

   • Three standard asymptotic methods: Wald-type, Likelihood Ratio, and Score tests.
   • A modified MOVER procedure: The original MOVER is adjusted to account for intra-subject correlation ($\rho$) by introducing an effect factor that modifies the variance estimates of the individual proportions. Two variants are tested using Wilson score and Agresti-Coull intervals for the component proportions.

Key Contributions

• Novel Framework: The paper introduces a distribution-based inference framework that reconstructs the sampling distribution of the risk difference estimator directly from its characteristic function, bypassing asymptotic normality assumptions.
• Handling Skewness: By utilizing the exact (numerically approximated) finite-sample distribution, the method captures skewness and boundary effects that asymptotic methods often miss.
• Adjusted MOVER: The authors provide a specific adjustment to the MOVER procedure to incorporate intra-subject correlation, a feature often overlooked in standard implementations for paired data.
• Practical Tool: An online calculator is provided to facilitate the application of these six methods (three asymptotic, two MOVER, two distribution-based) for researchers.

Results
Extensive simulation studies ($N=10,000$ replications) and analyses of two real-world datasets (otitis media and myopia treatment) were conducted.

• Coverage Probability (ECP): Across a broad range of parameter settings, all methods achieved coverage probabilities close to the nominal 95% level as sample sizes increased. In small sample settings ($m=n=20$) with low baseline probabilities or high correlation, some methods exhibited conservative or liberal behavior. The distribution-based method (specifically PDF1 using true nuisance parameters) tended to be conservative, while PDF2 (using profile MLEs) showed variability but remained within acceptable bounds.
• Interval Width (MIW): The distribution-based procedures generally produced shorter confidence intervals compared to asymptotic and MOVER methods, particularly when intra-subject correlation was low to moderate. As sample sizes increased ($m=n=100$), the differences in interval widths among methods diminished.
• Skewness and Asymmetry (RMNCP): The Ratio of Mesial to Distal Non-Coverage Probability (RMNCP) analysis revealed that asymptotic and MOVER methods tended to produce symmetric tail allocations (RMNCP $\approx$ 0.5). In contrast, the distribution-based approach demonstrated the ability to accommodate asymmetry in the sampling distribution, resulting in non-central coverage patterns in small samples where the estimator's distribution was skewed.
• Real-World Applications: In both the otitis media and myopia datasets, all methods yielded similar point estimates and confidence intervals that included zero, leading to consistent inferential conclusions (no significant difference between treatment groups).

Significance and Claims
The paper claims that the proposed distribution-based approach provides a viable alternative framework for interval estimation of the risk difference in studies with combined unilateral and bilateral binary outcomes. Its primary significance lies in its ability to reflect finite-sample distributional features, specifically skewness, which are not captured by methods relying on asymptotic normality.

The authors modestly note that while the proposed approach offers a more realistic representation of the sampling distribution, it does not uniformly outperform competing methods in terms of coverage accuracy across all small-sample scenarios. They attribute potential minor deviations in coverage to the numerical evaluation of the PDF when the estimator possesses a highly discrete distribution in very small samples. The study concludes that the method is a useful alternative, particularly for capturing distributional nuances in small samples, and suggests future extensions to other dependence models and effect measures (risk ratio, odds ratio).