Skip to main content

Using Differential Geometry for Sparse High-Dimensional Risk Regression Models

  • Conference paper
  • First Online:
Models for Data Analysis (SIS 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 402))

Included in the following conference series:

  • 292 Accesses

Abstract

With the introduction of high-throughput technologies in clinical and epidemiological studies, the need for inferential tools that are able to deal with fat data-structures, i.e., relatively small number of observations compared to the number of features, is becoming more prominent. In this paper we propose an extension of the dgLARS method to high-dimensional risk regression models. The main idea of the proposed method is to use the differential geometric structure of the partial likelihood function in order to select the optimal subset of covariates.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Augugliaro, L., Mineo, A.M., Wit, E.C.: Differential geometric least angle regression: a differential geometric approach to sparse generalized linear models. J. R. Stat. Soc. Ser. B 75(3), 471–498 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Augugliaro, L., Mineo, A.M., Wit, E.C.: dglars: an R package to estimate sparse generalized linear models. J. Stat. Softw. 59(8), 1–40 (2014)

    Article  Google Scholar 

  3. Augugliaro, L., Mineo, A.M., Wit, E.C.: A differential geometric approach to generalized linear models with grouped predictors. Biometrika 103(3), 563–577 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bao, L., Kimzey, A., Sauter, G., Sowadski, J.M., Lu, K.P., Wang, D.G.: Prevalent overexpression of prolyl isomerase Pin1 in human cancers. Am. J. Pathol. 164(5), 1727–1737 (2004)

    Article  Google Scholar 

  5. Boldrini, L., Pistolesi, S., Gisfredi, S., Ursino, S., Ali, G., Pieracci, N., Basolo, F., Parenti, G., Fontanini, G.: Telomerase activity and hTERT mRNA expression in glial tumors. Int. J. Oncol. 28(6), 1555–1560 (2006)

    Google Scholar 

  6. Cox, D.R.: Regression models and life-tables. J. R. Stat. Soc. Ser. B 34(2), 187–220 (1972)

    MathSciNet  MATH  Google Scholar 

  7. Cox, D.R.: Partial likelihood. Biometrika 62(2), 269–276 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cox DR (1981) Discussion of paper by D. Oakse entitled “survival times: aspects of partial likelihood”. Int. Stat. Rev. 49(3), 258

    Google Scholar 

  9. Cox, D.R., Oakes, D.: Analysis of Survival Data. Monographs on Statistics and Applied Probability. Chapman and Hall, London (1984)

    Google Scholar 

  10. Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96(456), 1348–1360 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fan, Y., Tang, C.Y.: Tuning parameter selection in high dimensional penalized likelihood. J R. Stat. Soc.: Ser. B 75(3), 531–552 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gillet, J.P., Calcagno, A.M., Varma, S., Davidson, B., Elstrand, M.B., Ganapathi, R., Kamat, A.A., Sood, A.K., Ambudkar, S.V., Seiden, M.V., Rueda, B.R., Gottesman, M.M.: Multidrug resistance-linked gene signature predicts overall survival of patients with primary ovarian serous carcinoma. Clin. Cancer Res. 18(11), 3197–3206 (2012)

    Article  Google Scholar 

  14. Goeman, J.J.: L1 penalized estimation in the Cox proportional hazards model. Biometr. J. 52(1), 70–84 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Gui, J., Li, H.: Penalized Cox regression analysis in the high-dimensional and low-sample size settings, with applications to microarray gene expression data. Bioinformatics 21(13), 3001–3008 (2005)

    Article  Google Scholar 

  16. Heagerty, P.J., Lumley, T., Pepe, M.S.: Time-dependent roc curves for censored survival data and a diagnostic marker. Biometrics 56(2), 337–344 (2000)

    Article  MATH  Google Scholar 

  17. Jönsson, G., Busch, C., Knappskog, S., Geisler, J., Miletic, H., Ringnér, Lillehaug JR., Borg, A., Lønning, P.E.: Gene expression profiling-based identification of molecular subtypes in stage IV melanomas with different clinical outcome. Clin. Cancer Res. 16(13), 3356–67 (2010)

    Article  Google Scholar 

  18. Konishi, S., Kitagawa, G.: Generalised information criteria in model selection. Biometrika 83(4), 875–890 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Loboda, A., Nebozhyn, M.V., Watters, J.W., Buser, C.A., Shaw, P.M., Huang, P.S., Van’t Veer, L.R.A.T., Jackson, D.B, Agrawal, D., Dai, H., Yeatman, T.J.: EMT is the dominant program in human colon cancer. BMC Medical Genomics, pp. 4–9 (2011)

    Google Scholar 

  20. McCullagh, P., Nelder, J.A.: Generalized Linear Models, 2nd edn. Chapman & Hall, London (1989)

    Book  MATH  Google Scholar 

  21. Moolgavkar, S.H., Venzon, D.J.: Confidence regions in curved exponential families: application to matched case-control and survival studies with general relative risk function. Ann. Stat. 15(1), 346–359 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nagel, G., Bjørge, T., Stocks, T., Manjer, J., Hallmans, G., Edlinger, M., Häggström, C., Engeland, A., Johansen, D., Kleiner, A., Selmer, R., Ulmer, H., Tretli, S., Jonsson, H., Concin, H., Stattin, P., Lukanova, A.: Metabolic risk factors and skin cancer in the metabolic syndrome and cancer project (Me-Can). Brit. J. Dermatol. 167(1), 59–67 (2012)

    Article  Google Scholar 

  23. Oakes, D.: Survival times: aspects of partial likelihood. Int. Stat. Rev. 49(3), 235–252 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  24. Park, M.Y., Hastie, T.: L1-regularization path algorithm for generalized linear models. J. R. Stat. Soc.: Ser. B 69(4), 659–677 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pazira, H., Augugliaro, L., Wit, E.C.: Extended differential geometric lars for high-dimensional glms with general dispersion parameter. Stat. Comput. 28(4), 753–774 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Peto, R., Peto, J.: Asymptotically efficient rank invariant test procedures. J. R. Stat. Soc. Ser. A 135(2), 185–207 (1972)

    Article  MATH  Google Scholar 

  27. Prentice, R.L., Mason, M.W.: On the application of linear relative risk regression models. Biometrics 42(1), 109–120 (1996)

    Article  Google Scholar 

  28. Prentice, R.L., Yoshimoto, Y., Mason, M.: Relationship of cigarette smoking and radiation exposure to cancer mortality in Hiroshima and Nagasaki. J. Nat. Cancer Inst. 70(4), 611–622 (1983)

    Google Scholar 

  29. Rao, C.R.: On the distance between two populations. Sankhyā 9, 246–248 (1949)

    MathSciNet  Google Scholar 

  30. Rippe, R.C.A., Meulman, J.J., Eilers, P.H.C.: Visualization of genomic changes by segmented smoothing using an \(L_0\) penalty. PLoS One 7(6), e38230 (2012)

    Article  Google Scholar 

  31. Ross, R.W., Galsky, M.D., Scher, H.I., Magidson, J., Wassmann, K., Lee, G.S.M., Katz, L., Subudhi, S.K., Anand, A., Fleisher, M., Kantoff, P.W., Oh, W.K.: A whole-blood RNA transcript-based prognostic model in men with castration-resistant prostate cancer: a prospective study. Lancet Oncol 13(11), 1105–13 (2012)

    Article  Google Scholar 

  32. Simon, N., Friedman, J.H., Hastie, T., Tibshirani, R.: Regularization paths for Cox’s proportional hazards model via coordinate descent. J. Stat. Softw. 39(5), 1–13 (2011)

    Article  Google Scholar 

  33. Sohn, I., Kim, J., Jung, S.H., Park, C.: Gradient lasso for Cox proportional hazards model. Bioinformatics 25(14), 1775–1781 (2009)

    Article  Google Scholar 

  34. Thomas, D.C.: Addendum to the paper by Liddell, McDonald, Thomas and Cunliffe. J. R. Stat. Soc. Ser. A 140(4), 483–485 (1977)

    Google Scholar 

  35. Thomas, D.C.: General relative-risk models for survival time and matched case-control analysis. Biometrics 37(4), 673–686 (1981)

    Article  Google Scholar 

  36. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58(1), 267–288 (1996)

    MathSciNet  MATH  Google Scholar 

  37. Tibshirani, R.: The lasso method for variable selection in the Cox model. Stat. Med. 16, 385–395 (1997)

    Article  Google Scholar 

  38. Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B 67(2), 301–320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Luigi Augugliaro .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Augugliaro, L., Wit, E.C., Pazira, H., González, J., Abegaz, F., Mineo, A.M. (2023). Using Differential Geometry for Sparse High-Dimensional Risk Regression Models. In: Brentari, E., Chiodi, M., Wit, EJ.C. (eds) Models for Data Analysis. SIS 2018. Springer Proceedings in Mathematics & Statistics, vol 402. Springer, Cham. https://doi.org/10.1007/978-3-031-15885-8_2

Download citation

Publish with us

Policies and ethics