Prof. Dr. Helmut Finner
Deutsches Diabetes-Zentrum (DDZ), Leibniz-Zentrum fur Diabetes-Forschung an der Heinrich-Heine-Universitat Dusseldorf
From Closed Multiple Testing via Ranking and Selecting to Partitioning Principles
In this talk I will review the way from closed multiple testing via ranking and selecting to partitioning principles from a personal perspective.
A starting point is Eckart Sonnemann's article 'Allgemeine Losungen multipler Testprobleme' (1982, in: EDV in Medizin und Biologie 13(4)). Among others, this paper contains a formal derivation of the closure principle introduced by Einot and Gabriel in a Biometrika paper in 1976. End of 1983, I moved to University of Trier and Eckart Sonnemann encouraged me to focus on multiple testing problems and to write my thesis on closed multiple range tests for pairwise comparisons. The first thing I learned was that the closure principle may lead to extremely complex systems of hypotheses and test statistics. Moreover, new issues appeared, e.g., (1) monotonicity of critical values and the validity of suitable probability inequalities, (2) directional error control, (3) construction of compatible confidence bounds for stepwise test procedures, and, (4) construction of one sided confidence bounds if a two sided hypothesis is rejected.
A first important contribution with respect to compatible confidence sets based on partitioning the parameter space is due to Stefansson, Kim and Hsu in 1988 (In: Statistical Decision Theory and Related Topics IV, Vol. 1). Around that time, Sonnemann supported my work on some general correspondence theorems between confidence sets, families of confidence sets and multiple tests. In 1987, I met Guido Giani at a symposium entitled 'Multiple Hypotheses Testing' (organized by P. Bauer, G. Hommel and E. Sonnemann) in Gerolstein (small city in Germany) and became involved in some problems in the field of ranking and selection. Among others, based on the closure principle we developed some duality theory between multiple hypotheses testing and the selection approach. Years later (around 1998), Klaus Strassburger mentioned in passing that he was not able to reconstruct a specific selection procedure with our duality method. Finally, this observation lead us to the idea to formulate suitable partitioning principles (a weak and strong version, respectively, cf. Finner and Strassburger (2002), Ann. Stat. 30(4), 1194-1213) in order to improve multiple test procedures based on a formal application of the closure principle.
We take a closer look at all these principles and illustrate how they work in selected examples