impson's paradox reminds people that the statistical inference in a low-dimensional space probably distorts the reality in a high one seriously.To study the paradox with respect to Yule's measure,this paper discusses simple collapsibility,strong collapsibility and consecutive collapsibility,and presents necessary and sufficient conditions of them.In fact,these conditions are of great importance for observational and experimental designs,eliminating confounding bias,categorizing discrete variables and so on.
The primary goal of a phase I clinical trial is to find the maximum tolerable dose of a treatment. In this paper, we propose a new stepwise method based on confidence bound and information incorporation to determine the maximum tolerable dose among given dose levels. On the one hand, in order to avoid severe even fatal toxicity to occur and reduce the experimental subjects, the new method is executed from the lowest dose level, and then goes on in a stepwise fashion. On the other hand,in order to improve the accuracy of the recommendation, the final recommendation of the maximum tolerable dose is accomplished through the information incorporation of an additional experimental cohort at the same dose level. Furthermore, empirical simulation results show that the new method has some real advantages in comparison with the modified continual reassessment method.
