Graphical Procedures for Multiple Comparisons Under General Dependence

It has been more than half a century since Tukey first introduced graphical displays that relate non-overlap of confidence intervals to statistically significant differences between parameter estimates. In this paper, we show how Tukey’s graphical overlap procedure can be modified to accommodate general forms of dependence within and across samples. We also develop a procedure that can be used to more effectively resolve rankings within the tails of the distributions of parameter values, thereby generalizing existing methods for "multiple comparisons with the best." We show that these new procedures retain the simplicity of Tukey’s original procedure while maintaining asymptotic control of the familywise error rate under very general conditions. Simple examples are used throughout to illustrate the procedures