Document Type


Date of Award


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Ji Meng Loh

Second Advisor

Wenge Guo

Third Advisor

Sundarraman Subramanian

Fourth Advisor

Antai Wang

Fifth Advisor

Zhi Wei


With increasing complexity of research objectives in clinical trials, a variety of relatively complex and less intuitive multiple testing procedures (MTPs) have been developed and applied in clinical data analysis. In order to make testing strategies more explicit and intuitive to communicate with non-statisticians, several flexible and powerful graphical approaches have recently been introduced in the literature for developing and visualizing newer MTPs. Nevertheless, some theoretical as well as methodological issues still remain to be fully addressed. This dissertation addresses several important issues arising in graphical approaches and related selective inference problems. It consists of three parts.

In the first part of this dissertation, a generalized graphical approach is introduced, which allows one to reject more than one hypothesis at each step. This overcomes a main drawback of existing graphical approaches in which only one rejection is allowed at each step. Through some clinical examples, the proposed approach is illustrated to be more flexible and computationally efficient than existing graphical approaches. Theoretically, it is shown that the generalized graphical approach strongly controls the FWER under arbitrary dependence. To show the FWER control of the proposed method, as a by-product, a new concept of a multivariate critical value function is introduced and based on this function, the sequential rejection principle (Goeman and Solari, 2010) is generalized from the case of univariate critical value function to that of multivariate.

In the second part of this dissertation, a new graphical approach for general logically related multiple hypotheses testing is developed. By re-assigning critical values between testable and non-testable hypotheses, all local critical values can be made fully used. Theoretically, it is shown that the proposed graphical approach strongly controls the FWER at level a under arbitrary dependence, by employing the generalized sequential rejection principle developed in the first part of this dissertation. Through some clinical examples, it demonstrates that the proposed graphical approach is more flexible and computationally efficient than entangled graphical approach for testing general logically related hypotheses (Maurer and Bretz, 2013).

In the third part of this dissertation, several powerful MTPs based on the very recently introduced ideas and methods of selective inference are proposed, which can be applied in large scale data analysis, such as microarray study, genomewise association study (GWAS), etc. By further developing the idea of independent filtering (Bourgon et al., 2010; Dai et al., 2012; Du and Zhang, 2014; Ignatiadis et al., 2016), where hypotheses are splitted into two blocks by selection process, three two-stage MTPs, adaptive two-stage Bonferroni procedure, selective parallel gatekeeping procedure and data-driven weighted selective procedure, are proposed. The proposed MTPs can not only exploit information of selected hypotheses more explicitly by estimating the true null proportion, but also deal with non-selected hypotheses. In order to exploit information of each null hypothesis more explicitly, the proposed procedures are further generalized from two blocks to multiple blocks. Theoretically, it is shown that the proposed MTPs strongly control the FWER at level a. Under independence, the proposed procedures are evaluated through extensive simulation studies.



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