Turkiye Klinikleri Journal of Biostatistics

.: ORIGINAL RESEARCH
Comparison of the Statistical Tests for Two Crossing Survival Functions
Kesişen İki Yaşam Fonksiyonu İçin Kullanılan İstatistiksel Testlerin Karşılaştırılması
Hülya ÖZENa, Cengiz BALa, Ertuğrul ÇOLAKa
aDepartment of Biostatistics, Eskişehir Osmangazi University Faculty of Medicine, Eskişehir, TURKEY
Turkiye Klinikleri J Biostat. 2021;13(1):13-27
doi: 10.5336/biostatic.2020-80208
Article Language: EN
Full Text
ABSTRACT
 Objective: Testing the equality of two survival functions is frequently used in survival analysis studies. The most preferred method in the literature is the log-rank (LR) test, but it provides misleading results in evaluating the crossing survival functions in which the proportional hazards assumption is violated. Therefore, different comparison tests have been proposed. In this study, currently proposed partitioned LR tests and weighted LinWang tests were compared with LR and weighted LR tests. Material and Methods: Type I error rates and powers of comparison tests were compared in different sample sizes and censoring rates using the Monte Carlo simulation technique. The weaknesses and strengths of the comparison tests were examined in various scenarios. In addition to the simulation study, comparison tests were evaluated using open source real-life data set. Results: Simulation results showed that all tests provided reasonable Type I error rates. The most successful results in the scenarios of crossing survival functions belonged to partitioned LR tests. It was observed that the location of the crossing point affected the performance of the tests adversely. Conclusion: Proportional hazards assumption should be tested before comparing two survival functions. It is recommended to use the LR test, when the proportional hazards assumption is provided and the use of partitioned LR tests in the comparison of crossing survival functions.

Keywords: Log-rank test; partitioned log-rank tests; survival analysis; weighted lin-wang tests
ÖZET
 Amaç: İki yaşam fonksiyonunun benzerliğinin araştırılması yaşam analizi çalışmalarında sıklıkla kullanılmaktadır. Literatürde, en çok tercih edilen yöntem log-rank (LR) testidir ancak bu yöntem orantılı hazardlar varsayımının ihlal edildiği kesişen yaşam fonksiyonlarını değerlendirmede yanıltıcı sonuçlar sunmaktadır. Bu nedenle kesişen yaşam fonksiyonlarını değerlendirmek için literatürde farklı karşılaştırma testleri önerilmiştir. Bu çalışmada, literatürde önerilen güncel yöntemlerden parçalanmış LR testleri ve ağırlıklandırılmış Lin-Wang testlerinin, LR ve ağırlıklandırılmış LR testleri ile karşılaştırılması amaçlanmaktadır. Gereç ve Yöntemler: Monte Carlo simülasyon tekniği kullanılarak, karşılaştırma testlerinin Tip I hata oranları ve güçleri, farklı örneklem büyüklükleri ve sansür oranlarında karşılaştırıldı. Karşılaştırma testlerinin zayıf ve güçlü yönleri çeşitli senaryolarda incelendi. Simülasyon çalışmasına ek olarak, karşılaştırma testleri gerçek hayattaki bir veri seti üzerinde değerlendirildi. Bulgular: Simülasyon sonuçları, tüm testlerin makul Tip I hata oranları sağladığını gösterdi. Kesişen yaşam fonksiyonlarına ait senaryolarda en başarılı sonuçlar parçalanmış LR testlerine aitti. Kesişme noktasının konumunun, karşılaştırma testlerinin performansını olumsuz yönde etkilediği gözlendi. Sonuç: Orantılı hazardlar varsayımı, 2 yaşam fonksiyonu karşılaştırılmadan önce test edilmelidir. Orantılı hazardlar varsayımı sağlandığında LR testinin, kesişen yaşam fonksiyonlarının karşılaştırılmasında ise parçalanmış LR testlerinin kullanılması önerilmektedir.

Anahtar Kelimeler: Log-rank testi; parçalanmış log-rank testleri; yaşam analizi; ağırlıklandırılmış lin-wang testleri
REFERENCES:
  1. Mantel N. Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemother Rep. 1966;50(3):163-70. [PubMed] 
  2. Liu Y, Yin G. Partitioned log-rank tests for the overall homogeneity of hazard rate functions. Lifetime Data Anal. 2017;23(3):400-25. [Crossref]  [PubMed] 
  3. Lee CT. Statistical Methods for the Comparison of Crossing Survival Curves. In: Balakrishnan N, Rao CR, eds. Advances in survival analysis. Vol 23. 1st ed. Amsterdam: Elsevier; 2004. p. 277-89. [Crossref] 
  4. Krishnan A, Pasquini MC, Logan B, Stadtmauer EA, Vesole DH, Alyea E 3rd, et al; Blood Marrow Transplant Clinical Trials Network (BMT CTN). Autologous haemopoietic stem-cell transplantation followed by allogeneic or autologous haemopoietic stem-cell transplantation in patients with multiple myeloma (BMT CTN 0102): a phase 3 biological assignment trial. Lancet Oncol. 2011;12(13):1195-203. [Crossref]  [PubMed]  [PMC] 
  5. Park JH, Rivière I, Gonen M, Wang X, Sénéchal B, Curran KJ, et al. Long-Term Follow-up of CD19 CAR Therapy in Acute Lymphoblastic Leukemia. N Engl J Med. 2018;378(5):449-59. [Crossref]  [PubMed]  [PMC] 
  6. Weeda VB, Murawski M, McCabe AJ, Maibach R, Brugières L, Roebuck D, et al. Fibrolamellar variant of hepatocellular carcinoma does not have a better survival than conventional hepatocellular carcinoma--results and treatment recommendations from the Childhood Liver Tumour Strategy Group (SIOPEL) experience. Eur J Cancer. 2013;49(12):2698-704. [Crossref]  [PubMed] 
  7. Kristiansen IS. PRM39 Survival curve convergences and crossing: a threat to validity of meta-analysis? Value in health. 2012;15(7):A652 [Crossref] 
  8. Suciu GP, Lemeshow S, Moeschberger M. Statistical Tests of the Equality of Survival Curves: Reconsidering the Options. In: Balakrishnan N, Rao CR, eds. Advances in survival analysis. Vol 23. 1st ed. Amsterdam: Elsevier; 2004. p. 251-61. [Crossref] 
  9. Koziol JA. Two sample cramér-von mises test for randomly censored data. Biometrical Journal. 1978;20(6):603-8. [Crossref] 
  10. Schumacher M. Two-Sample Tests of Cramér-von Mises and Kolmogorov-Smirnov-Type for Randomly Censored Data. Int Stat Rev. 1984;263-81. [Crossref] 
  11. Gill RD. Censoring and stochastic integrals. Stat Neerl. 1980;34(2):124. [Crossref] 
  12. Fleming TR, O'Fallon JR, O'Brien PC, Harrington DP. Modified Kolmogorov-Smirnov test procedures with application to arbitrarily right-censored data. Biometrics 1980;(36)4:607-625. [Crossref] 
  13. Pepe MS, Fleming TR. Weighted Kaplan-Meier statistics: a class of distance tests for censored survival data. Biometrics. 1989;45(2):497-507. [Crossref]  [PubMed] 
  14. Lin X, Wang H. A new testing approach for comparing the overall homogeneity of survival curves. Biometrical Journal. 2004;46(5):489-96. [Crossref] 
  15. Qiu P, Sheng J. A two‐stage procedure for comparing hazard rate functions. J R Stat Soc Series B Stat Methodol. 2008;70(1):191-208. [Link] 
  16. Kraus D. Adaptive Neyman's smooth tests of homogeneity of two samples of survival data. J Stat Plan Infer. 2009;139(10):3559-69. [Crossref] 
  17. Lin X, Xu Q. A new method for the comparison of survival distributions. Pharm Stat. 2010;9(1):67-76. [Crossref]  [PubMed] 
  18. Koziol JA, Jia Z. Weighted Lin-Wang tests for crossing hazards. Comput Math Methods Med. 2014;2014:643457. [Crossref]  [PubMed]  [PMC] 
  19. Tubert-Bitter P, Kramar A, Chale JJ, Moreau T. Linear rank tests for comparing survival in two groups with crossing hazards. Comput Stat Data Anal. 1994;18(5):547-59. [Crossref] 
  20. Li H, Han D, Hou Y, Chen H, Chen Z. Statistical inference methods for two crossing survival curves: a comparison of methods. PLoS One. 2015;10(1):e0116774. [Crossref]  [PubMed]  [PMC] 
  21. Liu K, Qiu P, Sheng J. Comparing two crossing hazard rates by Cox proportional hazards modelling. Stat Med. 2007;26(2):375-91. [Crossref]  [PubMed] 
  22. Gehan EA. A Generalized wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika. 1965;52:203-23. [Crossref]  [PubMed] 
  23. Tarone RE, Ware J. On distribution-free tests for equality of survival distributions. Biometrika. 1977;64(1):156-160. [Crossref] 
  24. Peto R, Peto J. Asymptotically efficient rank invariant test procedures. J R Stat Soc Ser A Stat Soc. 1972;135(2):185-207. [Crossref] 
  25. Andersen PK, Borgan O, Gill RD, Keiding N. Statistical models based on counting processes: Nonparametric Hypothesis Testing. 1st ed. New York: Springer-Verlag; 1993. p. 393-5. [Crossref] 
  26. Andersen PK, Gill RD. Cox's regression model for counting processes: a large sample study Ann Stat. 1982;10(4):1100-20. [Crossref] 
  27. Harrington DP, Fleming TR. A class of rank test procedures for censored survival data. Biometrika. 1982;69(3):553-66. [Crossref] 
  28. R Studio Team. R Studio: Integrated Development for R. RStudio. PBC, Boston, MA. 2019. [Link] 
  29. R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. 2019. [Link] 
  30. Kassambara A, Kosinski M, Biecek P, Fabian S. Package 'survminer'. 2017. [Link] 
  31. Therneau TM, Lumley T. Package 'survival'. 2014. [Link] 
  32. Wickham H, Wickham MH. Package 'dplyr'. 2013. [Link] 
  33. Dardis C, Dardis MC. Package 'survMisc'. 2018. [Link] 
  34. A comparison of combination chemotherapy and combined modality therapy for locally advanced gastric carcinoma. Gastrointestinal Tumor Study Group. Cancer. 1982;49(9):1771-7. [Crossref]  [PubMed] 
  35. Stablein DM, Koutrouvelis IA. A two-sample test sensitive to crossing hazards in uncensored and singly censored data. Biometrics. 1985;41(3):643-52. [Crossref]  [PubMed] 
  36. Klein JP, Moeschberger ML. Hypothesis Testing. Survival analysis: techniques for censored and truncated data. 2nd ed. New York: Springer-Verlag; 2003. p. 224-6. [Crossref] 
  37. Grambsch PM, Therneau TM. Proportional hazards tests and diagnostics based on weighted residuals. Biometrika 1994;81(3):515-26. [Crossref] 
  38. Hsieh JJ, Chen HY. A testing strategy for two crossing survival curves. Commun Stat Simul Comput. 2017;46(8):6685-96. [Crossref] 

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