Cellwise and Casewise Robust Multivariate Regression with Inference

Technical Summary: Cellwise and Casewise Robust Multivariate Regression with Inference

Problem Statement
Multivariate linear regression is a fundamental statistical tool for modeling relationships between $p$-dimensional predictors and $q$-dimensional responses. However, classical estimators like Ordinary Least Squares (OLS) are highly sensitive to outliers. Traditional robust statistics have primarily addressed casewise outliers (entire observations that deviate from the majority), requiring at least 50% of the data to be clean. However, modern high-dimensional datasets frequently suffer from cellwise outliers, where individual entries in the predictor or response matrix are contaminated. In high dimensions, even a small proportion of contaminated cells can render a large fraction of cases "bad" (e.g., with $p=100$ and $q=2$, 1% cell contamination affects 64% of cases), causing casewise robust methods to fail. Furthermore, contemporary applications often involve missing values and dimensions where $p \geq n$, leading to instability in classical estimation. Finally, while robust estimation is established, providing valid statistical inference (e.g., confidence intervals) under simultaneous cellwise and casewise contamination, particularly with missing data, remains an open challenge.

Methodology
The authors propose cellMR (cellwise Multivariate Regression) and cellBoot (cellwise Bootstrap) to address these challenges.

1. The cellMR Estimator:

   • Foundation: The method builds upon a cellwise robust covariance estimator (cellCov) developed by Centofanti et al. (2025).
   • Mechanism: It utilizes a robust Principal Component Analysis (PCA) approach called cellPCA to handle the combined matrix of predictors and responses $[X; Y]$. This method models the data as a low-rank structure plus noise, employing valid $\rho$-functions (specifically a hyperbolic tangent function) to bound the influence of both cellwise and casewise outliers.
   • Regularization: To ensure numerical stability in high-dimensional settings ($p \geq n$), the regression coefficients are estimated using ridge regularization. The regression parameters are derived by plugging the robust location ($\hat{\mu}$) and scatter ($\hat{\Sigma}$) estimates from cellCov into the standard ridge regression formulas.
   • Missing Data: The framework naturally accommodates missing values through the cellPCA mechanism, which uses missingness indicators to weight contributions during estimation.
   • Prediction: For out-of-sample prediction, the method employs an imputation strategy where suspicious cells in new data are cleaned and missing cells are filled in based on the robust structure learned from the training data.

2. The cellBoot Inference Procedure:

   • Challenge: Direct bootstrap inference on the cellMR estimator is problematic because the estimator is not guaranteed to be consistent due to its cellwise construction.
   • Solution: The authors introduce cellBoot, a bootstrap procedure based on Indirect Inference (II).
   • Auxiliary Estimator: A computationally efficient, one-step approximation of cellCov, termed FastCellCov, is used as the auxiliary estimator. While FastCellCov is inconsistent, it captures the robustness properties of the full estimator.
   • Bias Correction: The II framework constructs a consistent estimator by matching the auxiliary estimator computed on the observed data with the average of the auxiliary estimator computed on simulated datasets generated from a parametric model.
   • Implementation: The algorithm generates bootstrap samples, computes FastCellCov estimates, applies the II bias correction to obtain consistent estimates of the location and scatter, and finally derives regression coefficients. Confidence intervals are constructed using the empirical quantiles of these corrected bootstrap estimates.

Key Contributions

• Unified Framework: To the authors' knowledge, this is the first multivariate regression method capable of simultaneously handling cellwise outliers, casewise outliers, missing data, and high dimensionality ($p \geq n$).
• Theoretical Properties:
  • The authors derive the influence functions (IF) for the cellMR regression estimators under both casewise and cellwise contamination models, proving that the estimators are bounded and thus robust.
  • They prove the asymptotic consistency of the cellBoot procedure.
  • They derive the influence functions for the center and length of the resulting confidence intervals, demonstrating their robustness.
• Inference Validity: The paper establishes that the cellBoot confidence intervals are asymptotically exact, providing valid uncertainty quantification in settings where standard bootstrap or robust bootstrap methods (like FRB) fail.

Results

• Simulations: Extensive Monte Carlo experiments demonstrate that cellMR achieves superior predictive performance (lower Mean Squared Error) compared to competing methods (including RIDGE, SEST, PENSE, CRM, REGCELL, and SHOOT) across various contamination scenarios (cellwise, casewise, and mixed) and dimensions. Competing methods either fail in high dimensions or perform poorly under cellwise contamination.
• Inference Performance: The cellBoot method maintains nominal coverage probabilities (e.g., 90%) even under significant contamination and missing data. In contrast, OLS-based intervals suffer from severe undercoverage, and the Fast and Robust Bootstrap (FRB) fails in high-dimensional settings.
• Real Data Application: The methods were applied to a genomics dataset involving 59 cancer cell lines, 50 genes, and 3 proteins. The analysis successfully identified robust associations between gene and protein expression levels, visualized through outlier maps and forest plots, while handling potential outliers and missing values effectively.

Significance and Claims
The paper claims to provide a unified methodology that addresses the "triple threat" of modern data analysis: cellwise contamination, casewise contamination, and missingness in high dimensions. It asserts that cellMR is the first approach to jointly tackle these issues within a multivariate regression framework. Furthermore, the introduction of cellBoot is claimed to be the first inference framework specifically designed to provide valid confidence intervals in the presence of simultaneous cellwise and casewise contamination combined with missing values. The authors emphasize that their approach moves beyond prediction to enable reliable scientific interpretation through robust uncertainty quantification.