Turkiye Klinikleri Journal of Biostatistics

.: ORIGINAL RESEARCH
Kesirli Polinomlar ile Cox Regresyon Modeli: Prostat Kanseri Veri Kümesi Üzerine Bir Uygulama
Cox Regression Model with Fractional Polynomials: An Application on Prostate Cancer Data Set
Hazal DİNÇa, Nihal ATA TUTKUNa
aHacettepe Üniversitesi Fen Fakültesi, İstatistik Bölümü, Ankara, TÜRKİYE
Turkiye Klinikleri J Biostat. 2020;12(1):38-52
doi: 10.5336/biostatic.2019-71879
Article Language: TR
Full Text
ÖZET
 Amaç: Kesirli polinomlu regresyon modelleri, değişkenlerin doğrusal olmayan etkilerini modellemek için önerilmiştir. Değişkenlerin sürekli değişken olarak modele dahil edilmesi ya da kategorik biçime dönüştürülerek modele dahil edilmesi sonuçları değiştirebilir. Sürekli değişkenleri kategorik biçime dönüştürmek bilgi kaybına neden olmaktadır. Bu durumda da kesirli polinomlu modeller uygun bir alternatif olmaktadır. Bu çalışmanın amacı kesirli polinomların Cox regresyon modelinde kullanımını incelemektir. Gereç ve Yöntemler: Prostat kanseri hastalığının 3. ve 4. aşamalarında olan ve 5 yıl boyunca izlenen 475 hastaya ait bir veri kümesine (Byar and Green, 1980) klasik ve kesirli polinomlu Cox regresyon modelleri uygulanmıştır.1 Bulgular: Tedavi türü ve evre değişkenleri kategorik ve yaş değişkeni sürekli kabul edilerek klasik Cox regresyon çözümlemesi yapılmış, yaş değişkenin ve evre değişkenin istatisitksel olarak anlamlı olduğu ancak orantılı tehlikeler varsayımının sağlanmadığı görülmüştür. Yaş değişkeni kategorik olarak ele alındığında ise 81 yaş ve üzeri yaş grubunun ve evre değişkenin anlamlı çıktığı görülmüştür. Bununla birlikte, orantılı tehlikeler varsayımı sağlanmadığından klasik Cox regresyon modelinin kullanılması da uygun değildir. Prostat kanseri veri kümesi için kesirli polinomlu Cox regresyon modeli incelendiğinde ise yaş değişkeninin yaşam süresi ile doğrusal olmayan bir ilişkiye sahip olduğu belirlenmiştir. Yaş ve evre değişkenlerinin istatistiksel olarak anlamlı olduğu ve kesirli polinomlu Cox regresyon modeli için orantılı tehlikeler varsayımının da sağlandığı görülmüştür. Sonuç: Kesirli polinomlu Cox regresyon modelinin prostat kanseri veri kümesi için daha uygun olduğu sonucuna ulaşılmıştır.

Anahtar Kelimeler: Cox regresyon modeli; doğrusal olmama; kesirli polinomlar; orantılı tehlikeler varsayımı
ABSTRACT
 Objective: The use of regression models with fractional polynomial was proposed to model nonlinear effects of covariates. The inclusion of covariates into the model as continuous or categorical can change the results. However, converting continuous covariates into categorical format causes a loss of information. In this case, models with fractional polynomial become an appropriate alternative. The aim of this study is examining the use of fractional polynomials in Cox regression model. Material and Methods: Classical Cox regression model and Cox regression model with fractional polynomial were applied to the data set of 475 patients (Byar and Green, 1980) who were in the 3rd and 4th stages of prostate cancer disease and followed for 5 years.1 Results: Classical Cox regression analysis was performed by including type of treatment and stage of disease as categorical and age as continuous and it was found that age and stage of disease were statistically significant, but the assumption of proportional hazards was violated. When age was taken as a categorical covariate, the 81 and more level of age and the stage of disease was significant. However, the use of classical Cox regression model is not correct since the proportional hazards assumption was violated. When the Cox regression model with fractional polynomials was run for prostate cancer data set, it was found that the age variable had a nonlinear relationship with the survival time. Age and stage of disease were statistically significant and the assumption of proportional hazards was provided. Conclusion: It was concluded that the Cox regression model with fractional polynomials is more suitable for the prostate cancer dataset.

Keywords: Cox regression model; nonlinearity; fractional polynomials; proportional hazards assumption
REFERENCES:
  1. Byar DP, Green SB. The choice of treatment for cancer patients based on covariate information. Bull Cancer. 1980;67(4):477-90.
  2. Austin PC, Park-Wyllie LY, Juurlink DN. Using fractional polynomials to model the effect of cumulative duration of exposure on outcomes: applications to cohort and nested case, control designs. Pharmacoepidemiol Drug Saf. 2014;23(8):819-29. [Crossref]  [PubMed]   [PMC]  
  3. Royston P, Altman DG. Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling. Applied Statistics. 1994;43(3):429- 67. [Crossref] 
  4. Royston P, Ambler G, Sauerbrei W. The use of fractional polynomials to model continuous risk variables in epidemiology. Int J Epidemiol. 1999;28(5):964-74. [Crossref]  [PubMed] 
  5. Berger, Ursula, Gerein, Pia, Ulm, Kurt, Schäfer, Juliane. On the use of Fractional Polynomials in Dynamic Cox Models. Collaborative Research Center 386, Discussion Paper 207; 2000 ( [Crossref] 
  6. Berger U, Schäfer J, Ulm K. Dynamic Cox modelling based on fractional polynomials: time-variations in gastric cancer prognosis. Stat Med. 2003;22(7):1163-80. [Crossref]  [PubMed] 
  7. Bellera CA, MacGrogan G, Debled M, de Lara CT, Brouste V, Mathoulin Pélissier S. Variables with time-varying effects and the Cox model: some statistical concepts illustrated with a prognostic factor study in breast cancer. BMC Med Res Methodol. 2010;10(20):1471-2288. [Crossref]  [PubMed]  [PMC] 
  8. Buchholz A, Sauerbrei W. Comparison of procedures to assess non-linear and time-varying effects in multivariable models for survival data. Biom J. 2011;53(2):308-31. [Crossref]  [PubMed] 
  9. Zhang Z, Chen K, Ni H, Fan H. Predictive value of lactate in unselected critically ill patients: an analysis using fractional polynomials. J Thorac Dis. 2014;6(7):995- 1003.
  10. Buchholz A, Sauerbrei W, Royston P. A measure for assessing functions of time-varying effects in survival analysis. Open J Stat. 2014;4(11):977-98. [Crossref] 
  11. Sauerbrei W, Royston P, Meier-Hirmer C, Benner A. Multivariable regression model building by using fractional polynomials: description of SAS, STATA, R programs. Comput Stat Data Anal. 2006;50(12):3464-85. [Crossref] 
  12. Miller AJ. Least-squares computations. In: Isham V, Keiding N, Louis T, Reid N, Tibshirani R, Tong H, eds. Subset Selection in Regression. 2nd ed. New York: Chapman & Hall; 1990. p.11-34. [Crossref]  [PMC] 
  13. Altman DG, Lausen B, Sauerbrei W, Schumacher M. The dangers of using "optimal" cutpoints in the evaluation of prognostic factors. J Natl Cancer Inst. 1994;86(11):829-35. [Crossref]  [PubMed] 
  14. Miller R, Siegmund D. Maximally selected chi-square statistics. Biometrics. 1982;38(4):1011-6. [Crossref] 
  15. Lausen B, Schumacher M. Evaluating the effect of optimized cut-off values in the assessment of prognostic factors. Comput Stat Data Anal. 1996;21(3):307-26. [Crossref] 
  16. Ata Tutkun N. [Kesirli polinomlu lojistik regresyon modeli]. Journal of Statisticians: Statistics and Actuarial Sciences. 2015;8(2):27-35.
  17. Royston P, Sauerbrei W. Fractional polynomials for one variable, MFP: multivariable model-building with fractional polynomials. In: Balding DJ, Cressie NAC, Fitzmaurice GM, Johnstone IM, Molenberghs G, Scott DW, et al, eds. Multivariable Model-Building. 1st ed. New York: John Wiley & Sons, Inc; 2008. p.72-4.
  18. Ambler G, Royston P. Fractional polynomial model selection procedures: investigation of type I error rate. J Stat Comput Simul. 2001;69(1):89-108. [Crossref] 
  19. Royston P, Sauerbrei W. Fractional polynomials for one variable, MFP: multivariable model-building with fractional polynomials. In: Balding DJ, Cressie NAC, Fitzmaurice GM, Johnstone IM, Molenberghs G, Scott DW, et al, eds. Multivariable Model-Building. 1st ed. New York: John Wiley & Sons, Inc; 2008. p.84-5.
  20. Royston P, Sauerbrei W. Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials. Journal of the Royal Statistical Society Series A. 1999;162(1):71-94. [Crossref] 
  21. Royston P, Sauerbrei W. Fractional polynomials for one variable, MFP: multivariable model-building with fractional polynomials. In: Balding DJ, Cressie NAC, Fitzmaurice GM, Johnstone IM, Molenberghs G, Scott DW, et al, eds. Multivariable Model-Building. 1st ed. New York: John Wiley & Sons, Inc; 2008. p.116-8.
  22. Kay R. Treatment effects in competing-risks analysis of prostate cancerdata. Biometrics. 1986;42(1):203-11. [Crossref]  [PubMed] 
  23. Schemper M, Heinze G. Probability imputation revisited for prognostic factor studies. Stat Med. 1997;16(1-3):73-80. [Crossref] 
  24. Cheng SC, Fine JP, Wei LJ. Prediction of cumulative incidence function under the proportional hazards model. Biometrics. 1998;54(1):219-28. [Crossref]  [PubMed] 
  25. Hunt L, Jorgensen M. Mixture model clustering using the multimix program. Australia and New Zealand Journal of Statistics. 1999;41(2):154-71. [Crossref] 

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