**4. Discussion**

In cancer research, prognosis modeling with omics measurements plays an essential role. The existing studies mostly conduct analysis on one single type of cancer and often suffer from a lack of sufficient information. Integrative analysis represents an emerging trend in recent biomedical studies, among which the most common is the integrative analysis of multiple types of omics data, including gene expressions, copy number variations, and some others, and has led to interesting findings beyond single type omics data-based analysis. In this study, we have taken a different perspective and conducted integrative analysis on multiple cancer types to facilitate across-cancer information borrowing. Similarity across cancer types has been extensively studied in the literature, which provides a solid biological ground for our integrative analysis. Both marginal and joint analysis have been developed with two types of similarity-based penalty, which have intuitive formulations and solid statistical basis. We have analyzed mRNA gene expression data on nine TCGA cancer types

with censored survival outcomes. Biologically sensible findings different from the benchmark analysis have been made.

The proposed analysis can be directly applied to other types of omics data and other cancer types. In this study, we have focused on prognosis data and the AFT model. A continuous outcome can be regarded as a special case of prognosis outcome without censoring, and thus the proposed analysis can be applied directly. It can also be extended to accommodate categorical outcomes using, for example, generalized linear models. With the availability of multiple types of omics data on multiple cancer types, it can be of interest to conduct the two types of integration simultaneously. More functional examination of the data analysis results will be needed to confirm the findings.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2073-4425/10/8/604/s1: Detailed results referred to in Section 3 are available in the Supplementary Excel file. Table S1: Detailed estimation and identification results.

**Author Contributions:** All authors contributed to conceptualization, methodology, and writing. S.W. conducted data analysis.

**Funding:** This research was partly funded by the National Institutes of Health [CA216017, CA204120]; National Natural Science Foundation of China [91546202, 71331006]; Bureau of Statistics of China [2018LD02]; "Chenguang Program" supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission [18CG42]; and Program for Innovative Research Team of Shanghai University of Finance and Economics.

**Acknowledgments:** We are very grateful to the reviewers for their careful review and insightful comments, which have led to a significant improvement of this article.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
