Background: Synthetic data has been proposed as a solution for sharing anonymized versions of sensitive biomedical datasets. Ideally, synthetic data should preserve the structure and statistical properties of the original data, while protecting the privacy of the individual subjects. Differential privacy (DP) is currently considered the gold standard approach for balancing this trade-off. Objectives: The aim of this study is to evaluate the Mann-Whitney U test on DP-synthetic biomedical data in terms of Type I and Type II errors, in order to establish whether statistical hypothesis testing performed on privacy preserving synthetic data is likely to lead to loss of test's validity or decreased power. Methods: We evaluate the Mann-Whitney U test on DP-synthetic data generated from real-world data, including a prostate cancer dataset (n=500) and a cardiovascular dataset (n=70 000), as well as on data drawn from two Gaussian distributions. Five different DP-synthetic data generation methods are evaluated, including two basic DP histogram release methods and MWEM, Private-PGM, and DP GAN algorithms. Conclusion: Most of the tested DP-synthetic data generation methods showed inflated Type I error, especially at privacy budget levels of $\epsilon\leq 1$. This result calls for caution when releasing and analyzing DP-synthetic data: low p-values may be obtained in statistical tests simply as a byproduct of the noise added to protect privacy. A DP smoothed histogram-based synthetic data generation method was shown to produce valid Type I error for all privacy levels tested but required a large original dataset size and a modest privacy budget ($\epsilon\geq 5$) in order to have reasonable Type II error levels.

How many different binary classification problems a single learning algorithm can solve on a fixed data with exactly zero or at most a given number of cross-validation errors? While the number in the former case is known to be limited by the no-free-lunch theorem, we show that the exact answers are given by the theory of error detecting codes. As a case study, we focus on the AUC performance measure and leave-pair-out cross-validation (LPOCV), in which every possible pair of data with different class labels is held out at a time. We show that the maximal number of classification problems with fixed class proportion, for which a learning algorithm can achieve zero LPOCV error, equals the maximal number of code words in a constant weight code (CWC), with certain technical properties. We then generalize CWCs by introducing light CWCs, and prove an analogous result for nonzero LPOCV errors and light CWCs. Moreover, we prove both upper and lower bounds on the maximal numbers of code words in light CWCs. Finally, as an immediate practical application, we develop new LPOCV based randomization tests for learning algorithms that generalize the classical Wilcoxon-Mann-Whitney U test.