A Permutation Test for a Modified Spearman’s Rank Correlation Using Martingale-Residual Ranks Under Right Censoring

Technical Summary: A Permutation Test for a Modified Spearman's Rank Correlation Using Martingale-Residual Ranks Under Right Censoring

Problem Statement
Rank-based measures of association, particularly Spearman's rank correlation coefficient ($\rho_s$), are fundamental tools in nonparametric statistics for capturing monotone relationships. However, their application to right-censored survival data presents significant challenges. While recent literature has proposed estimators of "Spearman-like" correlations that reduce to the classical coefficient in the absence of censoring, these methods generally fail to provide a valid hypothesis test for $H_0: \rho_s = 0$ with correct Type I error control under censoring. Existing alternatives, such as Kendall-type concordance measures adapted for censoring, offer independence tests but do not target $\rho_s$ and are often conservative. Consequently, a gap exists for a method that (i) achieves asymptotically correct Type I error control under right censoring, (ii) applies when one or both margins are censored, and (iii) reduces exactly to the classical Spearman permutation test when censoring is absent. To the best of the authors' knowledge, no existing method satisfies all three criteria simultaneously.

Methodology
The authors propose a robust permutation testing framework based on Univariate Martingale Residuals (UMRs). The methodology proceeds through the following steps:

1. Data Transformation: For paired failure times $(S_i, T_i)$ subject to independent right-censoring $(C_i, D_i)$, the observed data are $(X_i, \delta_{Si})$ and $(Y_i, \delta_{Ti})$. The method constructs UMRs using the Kaplan–Meier cumulative hazard estimators ($\hat{H}_S$ and $\hat{H}_T$):
   $$m_{Si} = \delta_{Si} - \hat{H}_S(X_i), \quad m_{Ti} = \delta_{Ti} - \hat{H}_T(Y_i)$$
   These residuals serve as a rank-generating transformation. When no censoring is present, they reduce to strictly monotone transformations of the original data. Under censoring, they incorporate the survival information via the Kaplan–Meier estimator while preserving the ordering of the observed times with high probability.

2. Rank Calculation: The UMRs are ranked to obtain paired ranks $(a_i, b_i)$.

3. Test Statistic: A modified Spearman rank correlation coefficient, $r_s(m_S, m_T)$, is calculated based on these ranks. To facilitate the permutation test, a studentized statistic $R_s = r_s / \hat{\tau}_n$ is constructed, where $\hat{\tau}_n$ is a variance estimator derived from the moments of the ranks.

4. Permutation Framework: The null hypothesis $H_0: \rho_s = 0$ is tested by permuting the ranks of the second variable ($b_i$) while keeping the first fixed. The $p$-value is derived from the permutation distribution of the studentized statistic.

Theoretical Foundations
The validity of the test rests on several regularity conditions standard in survival analysis (i.i.d. failure times, independent censoring, continuous distributions, and uniform strong consistency of Kaplan–Meier estimators).

• Asymptotic Exchangeability: Under $H_0$, the normalized ranks of the UMRs converge almost surely to independent uniform random variables. This establishes that the joint distribution of the ranked pairs is asymptotically invariant under permutations.
• Validity of Permutation Distribution: By leveraging the Hoeffding decomposition of the statistic and the studentization process, the authors demonstrate that the permutation distribution consistently estimates the sampling distribution of the test statistic.
• Asymptotic Type I Error Control: The procedure guarantees that the rejection probability converges to the nominal significance level $\alpha$ as the sample size increases.
• Reduction Property: In the absence of censoring, the UMRs become strictly monotone transformations of the original data, ensuring the test reduces exactly to the classical studentized permutation test for Spearman's $\rho_s$.

Results
The authors evaluated the method through simulation studies and real-data applications:

• Simulation Studies: Two scenarios were examined with varying degrees of censoring (moderate $\approx$ 50% and heavy $\approx$ 67%).
  • Type I Error: The proposed test demonstrated strong control of Type I error across all sample sizes and significance levels ($\alpha = 0.01, 0.05, 0.10$).
  • Power: The test exhibited competitive power compared to the classical Pearson correlation test applied to fully observed data, even in the presence of censoring.
  • Comparison: The proposed method substantially outperformed the modified Kendall concordance statistic (Oakes, 1987), which was found to be highly conservative (Type I error $< 0.002$) and possessed significantly lower power.
• Real Data Examples:
  1. Skin Graft Survival: Analysis of paired survival times for burn patients ($n=11$) revealed a significant positive association ($p=0.044$).
  2. Kidney Infection Recurrence: Analysis of recurrence times for kidney patients using dialysis ($n=38$) with mixed censoring patterns showed a statistically significant positive association ($p=0.031$).

Significance and Claims
The paper claims to address a critical methodological gap by providing a hypothesis test for $H_0: \rho_s = 0$ under right censoring that satisfies three specific criteria: asymptotic correctness, applicability to singly or doubly censored margins, and exact reduction to the classical test when censoring is absent. To the best of the authors' knowledge, no existing method satisfies all three criteria. The authors emphasize that the method is "straightforward and robust," offering a reliable tool for biomedical applications where paired survival outcomes are common. The availability of R software implementations is highlighted as a key factor in facilitating the adoption of this methodology. The authors position this work as a foundation for future extensions, such as exact tests of independence and covariate-adjusted rank correlations, but do not claim immediate solutions for these broader problems.