Controlling the false discovery rate via knockoffs (1404.5609v3)

Abstract: In many fields of science, we observe a response variable together with a large number of potential explanatory variables, and would like to be able to discover which variables are truly associated with the response. At the same time, we need to know that the false discovery rate (FDR) - the expected fraction of false discoveries among all discoveries - is not too high, in order to assure the scientist that most of the discoveries are indeed true and replicable. This paper introduces the knockoff filter, a new variable selection procedure controlling the FDR in the statistical linear model whenever there are at least as many observations as variables. This method achieves exact FDR control in finite sample settings no matter the design or covariates, the number of variables in the model, or the amplitudes of the unknown regression coefficients, and does not require any knowledge of the noise level. As the name suggests, the method operates by manufacturing knockoff variables that are cheap - their construction does not require any new data - and are designed to mimic the correlation structure found within the existing variables, in a way that allows for accurate FDR control, beyond what is possible with permutation-based methods. The method of knockoffs is very general and flexible, and can work with a broad class of test statistics. We test the method in combination with statistics from the Lasso for sparse regression, and obtain empirical results showing that the resulting method has far more power than existing selection rules when the proportion of null variables is high.

• The paper introduces the knockoff filter, a novel approach that creates synthetic variables to precisely control the false discovery rate in linear models.
• It details a methodology combining knockoff construction, Lasso-based importance statistics, and data-dependent thresholding for effective variable selection.
• Empirical results using genetic data validate its superior performance over traditional FDR methods, highlighting its practical utility in high-dimensional research.

Controlling the False Discovery Rate via Knockoffs

The paper "Controlling the false discovery rate via knockoffs" by Rina Foygel Barber and Emmanuel J. Candés presents a novel methodology for variable selection in linear models, focusing on controlling the false discovery rate (FDR). This technique addresses a fundamental concern in high-dimensional statistics: identifying truly significant variables while minimizing the chances of false discoveries.

Overview

The authors introduce the knockoff filter, a procedural innovation designed specifically to manage FDR in statistical linear models. This approach operates effectively when the number of observations is at least as great as the number of explanatory variables. The primary contribution lies in the generation of knockoff variables—synthetic variables that replicate the correlation structure of the original variables. These knockoffs do not require new data for their construction, thereby maintaining the integrity of the original dataset.

Methodology

The paper meticulously outlines the steps involved in utilizing the knockoff filter:

1. Knockoff Construction: For each variable in the dataset, a corresponding knockoff variable is generated. These knockoff variables imitate the covariance properties of the real variables while also ensuring the FDR can be precisely controlled. The construction relies on computing a specialized Gram matrix that conforms to specific criteria, ensuring that the knockoffs and their original counterparts exhibit desired properties.
2. Statistic Calculation: Utilizing methods such as the Lasso for sparse regression, the authors generate statistics that compare the importance of each variable relative to its knockoff. The method determines which variables enter the model and at what penalization levels, hence indicating their significance in an informed manner.
3. Thresholding for Selection: A data-dependent threshold is calculated to determine which variables should be included in the final model. This threshold ensures that the expected FDR does not exceed a predefined level, satisfying a robustness criterion across various designs and noise levels.

Theoretical Insights

The methodology ensures exact FDR control irrespective of factors such as design complexity, number of variables, or unknown noise levels. Through simulations and theoretical proofs, the authors demonstrate that the knockoff filter can outperform conventional methods such as the Benjamini-Hochberg procedure, particularly in scenarios with complex correlation structures among explanatory variables.

Empirical Validation

The robustness of the knockoff filter is validated through empirical studies, including a detailed analysis of genetic data related to HIV drug resistance. By comparing selected genetic mutations against known panels, the authors show that the knockoff method can effectively identify important variables with minimal false discoveries, outperforming alternative methods.

Implications and Future Work

Practically, this research provides a powerful tool for statisticians and researchers working with high-dimensional data. The method's ability to control FDR without prior knowledge of noise levels or the number of true signals positions it as a versatile option for exploratory data analysis and high-stakes decision-making.

Theoretically, the innovation invites further exploration into extensions, including scenarios where the number of variables exceeds the number of observations, and the integration of more complex dependencies. The potential adaptation of this method into non-linear or generalized linear models could markedly extend its applicability.

Conclusion

The knockoff filter represents a significant stride in the control of false discovery in high-dimensional regression problems. Its blend of theoretical rigor and practical performance offers a compelling alternative to traditional methods, making it a noteworthy advancement in statistical science. As the field of high-dimensional data continues to expand, techniques like the knockoff filter are crucial in maintaining the balance between discovery and reliability.