Tracking Cross-Validated Estimates of Prediction Error as Studies Accumulate

Abstract

In recent years, “reproducibility” has emerged as a key factor in evaluating x applications of statistics to the biomedical sciences, for example, learning predictors of disease phenotypes from high-throughput “omics” data. In particular, “validation” is undermined when error rates on newly acquired data are sharply higher than those originally reported. More precisely, when data are collected from m “studies” representing possibly different subphenotypes, more generally different mixtures of subphenotypes, the error rates in cross-study validation (CSV) are observed to be larger than those obtained in ordinary randomized cross-validation (RCV), although the “gap” seems to close as m increases. Whereas these findings are hardly surprising for a heterogenous underlying population, this discrepancy is then seen as a barrier to translational research. We provide a statistical formulation in the large-sample limit: studies themselves are modeled as components of a mixture and all error rates are optimal (Bayes) for a two-class problem. Our results cohere with the trends observed in practice and suggest what is likely to be observed with large samples and consistent density estimators, namely, that the CSV error rate exceeds the RCV error rates for any m, the latter (appropriately averaged) increases with m, and both converge to the optimal rate for the whole population.

• Bayes’ rule
• Classification
• Mixture model

Additional information

Notes on contributors

Lo-Bin Chang

Lo-Bin Chang is Assistant Professor, Department of Statistics, Ohio State University, Columbus, OH 43210 (E-mail: [email protected]). Donald Geman is Professor, Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218 (E-mail: [email protected]). The authors gratefully acknowledge the support of the Defense Advanced Research Projects Agency under contract FA8650-11-1-7151, and the National Science Council under grant 100-2115-M-009-007-MY2, partial support of the Center of Mathematical Modeling & Scientific Computing and the National Center for Theoretical Science, Hsinchu, Taiwan.

Donald Geman

Lo-Bin Chang is Assistant Professor, Department of Statistics, Ohio State University, Columbus, OH 43210 (E-mail: [email protected]). Donald Geman is Professor, Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218 (E-mail: [email protected]). The authors gratefully acknowledge the support of the Defense Advanced Research Projects Agency under contract FA8650-11-1-7151, and the National Science Council under grant 100-2115-M-009-007-MY2, partial support of the Center of Mathematical Modeling & Scientific Computing and the National Center for Theoretical Science, Hsinchu, Taiwan.