Non-parametric statistical methods are commonly used to plan studies and analyze biomedical data. A key feature of these methods is the use of asymptotic theory to derive the approximate permutational distribution of the relevant test statistic. Inference based on such asympototic approximations is not always satisfactory. Many data sets do not satisfy the large sample criteria that are needed for the asymptotic results to hold. It is thus desirable to have the option to base inference on the exact permutational distribution of the test statistic. In the past, this has not been a practical option because of the formidable computational difficulties inherent in such an approach. Recent spectacular advances in computer technology, combined with the availability of efficient numberical algorithms for exact permutational inference, can change the situation. The main goal of this research project is to develop such numerical algorithms for four commonly encountered biomedical problems; linear rank tests, logistic regression, group sequential design, and contingency tables. An interdisciplinary approach will be taken. Powerful techniques of network optimization, recursion, MonteCarlo sampling with variance reduction, and hashing, from the fields of operations research and computer science, will play a prominent role in all the algorithms. Many new areas for methodological research will be stimulated by the development of these algorithms. Some examples are, importance sampling as a new way to quantify asymptotic rates of convergence, comparison of exact and asymptotic methods of inference, the impact of time trends on twosample linear rank tests, extending exact regression to ordered categorical or censored survival data, and exact power calculations for the Wilcoxon test with tied data. All these new directions will be investigated. The results of this research will be useful to practicing statisticians for planning and analyzing cancer trials, to applied epidemiologists conducting casecontrol studies, and to theoretical statisticians needing exact benchmarks against which to compare their asymptotic results.