Testing the Equivalence of Survival Distributions using PP- and PPP-Plots


  • Trevor F. Cox Cancer Research UK Liverpool Cancer Trials Unit, University of Liverpool, Liverpool, UK




Crossing survival curves, Hazard ratio, Kaplan-Meier, Log-rank test, PP-plot, Wilcoxon test


This paper discusses the use of PP-plots for survival distributions where for a pair of survival distributions, one is plotted against the other. This is another way of visualizing the nature of the relationship between the two survival distributions along with typical Kaplan-Meier plots. For three survival distributions, the PPP-plot is introduced where the survival distributions are plotted against each other in three-dimensions. At the population level, measures of divergence between distributions are introduced based on areas and lengths associated with the PP- and PPP- plots. At the sample level, two test statistics are defined, based on these areas and lengths, to test the null hypothesis of equivalent survival curves. A simulation exercise showed that, overall, the new tests are worthy competitors to the log-rank and Wilcoxon tests and also to a Levine-type test and a Kolmogorov-Smirnov type test for the case of crossing survival curves. The paper also shows how the PP-plot can be used to estimate the hazard ratio and to assess the ratio of hazard functions if proportional hazards are not appropriate. Finally, the methods introduced are illustrated on two cancer data sets

Author Biography

Trevor F. Cox, Cancer Research UK Liverpool Cancer Trials Unit, University of Liverpool, Liverpool, UK

Cancer Research UK Liverpool Cancer Trials Unit


Wilk MB, Gnanadesikan R. Probability plotting methods for the analysis of data. Biometrika 1968; 55; 1-17. DOI: https://doi.org/10.1093/biomet/55.1.1

Michael JR. The stabilized probability plot. Biometrika 1983; 70; 11-17. http://dx.doi.org/10.1093/biomet/70.1.11 DOI: https://doi.org/10.1093/biomet/70.1.11

Zhou XH, Obuchowski NA, McClish DK. Statistical Methods in Diagnostic Medicine. 2nd ed. Hoboken: Wiley 2011. DOI: https://doi.org/10.1002/9780470906514

Krzanowski WJ, Hand DJ. ROC curves for continuous data. Boca Raton: Chapman and Hall/CRC 2009. DOI: https://doi.org/10.1201/9781439800225

Yang H, Carlin D. ROC surface: a generalisation of ROC curve analysis. J Biopharm Stat 2000; 10; 183-96. http://dx.doi.org/10.1081/BIP-100101021 DOI: https://doi.org/10.1081/BIP-100101021

Kullback S, Leibler RA. On information and sufficiency. Ann Math Stat 1951; 22; 79-86. http://dx.doi.org/10.1214/aoms/1177729694 DOI: https://doi.org/10.1214/aoms/1177729694

Kullback S. The Kullback-Leibler distance. Am Stat 1987; 41: 341-42.

Struik DJ. Lectures on Classical Differential Geometry. 2nd ed. London: Dover 1961.

O'Neill B. Elementary Differential Geometry. Revised 3rd ed. Burlington: Academic Press 2006.

Collett D. Modelling Survival Data in Medical Research. 2nd ed. Boca Raton: Chapman and Hall/CRC 2003.

Mantel N. Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemotherapy Report 1966; 50; 163-70.

Peto R. Contribution to the discussion of a paper by D.R. Cox. JRSS A 1972; 34; 205-207.

Gehan EA. A generalized Wilcoxon test for comparing arbitrarily singly censored samples. Biometrika 1965; 52; 203-23. http://dx.doi.org/10.1093/biomet/52.1-2.203 DOI: https://doi.org/10.2307/2333825

Tarone RE, Ware J. On distribution free tests for equality of survival distributions. Biometrika 1977; 64; 156-60. http://dx.doi.org/10.1093/biomet/64.1.156 DOI: https://doi.org/10.1093/biomet/64.1.156

Harrington DP, Fleming TR. A class of rank test procedures for censored survival data. Biometrika 1982; 69; 553-66. http://dx.doi.org/10.1093/biomet/69.3.553 DOI: https://doi.org/10.1093/biomet/69.3.553

Jones MP, Crowley J. A general class of nonparametric tests for survival analysis. Biometrics 1989; 45; 157-70. http://dx.doi.org/10.2307/2532042 DOI: https://doi.org/10.2307/2532042

Stablein DM, Carter WH Jr, Novak JW. Analysis of survival data with nonproportional hazard functions. Control Clin Trials 1981; 2: 149-59. http://dx.doi.org/10.1016/0197-2456(81)90005-2 DOI: https://doi.org/10.1016/0197-2456(81)90005-2

Mantel N, Stablein DM. The crossing hazard function problem. Statistician 1988; 37: 59-64. http://dx.doi.org/10.2307/2348379 DOI: https://doi.org/10.2307/2348379

Liu K, Qiu P, Sheng J. Comparing two crossing hazard rates by Cox proportional hazards modelling. Stat Med 2007; 26: 375-91. http://dx.doi.org/10.1002/sim.2544 DOI: https://doi.org/10.1002/sim.2544

Bouliotis G, Billingham L. Crossing survival curves: alternatives to the log-rank test. Trials 2011; 12(Suppl. 1): A137. http://dx.doi.org/10.1186/1745-6215-12-S1-A137 DOI: https://doi.org/10.1186/1745-6215-12-S1-A137

Logan BR, Klein JP, Zhang MJ. Comparing treatments in the presence of crossing survival curves: an application to bone marrow transplantation. Biometrics 2008; 64: 733-40. http://dx.doi.org/10.1111/j.1541-0420.2007.00975.x DOI: https://doi.org/10.1111/j.1541-0420.2007.00975.x

Putter H, Sasako M, Hartgrink HH, van de Velde CJ H, van Houwelingen JC. Long-term survival with non-proportional hazards: results from the Dutch gastric cancer trial. Stat Med 2005; 24: 2807-21. http://dx.doi.org/10.1002/sim.2143 DOI: https://doi.org/10.1002/sim.2143

Le CT. Statistical methods for the comparison of crossing survival curves. In: Balakrishnan N, Rao CR, Eds. Handbook of Statistics, Amsterdam: Elsevier 2004; vol. 23: pp. 277-289. DOI: https://doi.org/10.1016/S0169-7161(03)23015-7

Yang S, Prentice R. Semiparametric analysis of short-term and long-term hazard ratios with two-sample survival data. Biometrika 2005; 92: 1-17. http://dx.doi.org/10.1093/biomet/92.1.1 DOI: https://doi.org/10.1093/biomet/92.1.1

Yang S, Prentice R. Improved logrank-type tests for survival data using adaptive weights. Biometrics 2010; 66: 30-38. http://dx.doi.org/10.1111/j.1541-0420.2009.01243.x DOI: https://doi.org/10.1111/j.1541-0420.2009.01243.x

Fleming T, O'Fallon JR, O'Brien PC. Modified Kolmogorov-Smirnov test procedures with application to arbitrarily right-censored data. Biometrics 1980; 36: 607-625. http://dx.doi.org/10.2307/2556114 DOI: https://doi.org/10.2307/2556114

Lin X, Xu Q. A new method for the comparison of survival distributions. Pharmaceut Statist 2010; 9: 67-76. http://dx.doi.org/10.1002/pst.376 DOI: https://doi.org/10.1002/pst.376

Van Besien K, Loberiza F, Bajorunaite, R, et al. Comparison of autologous and allogenetic hematopoietic stem cell transplantation for follicular lymphoma. Blood 2003; 102: 3521-29. http://dx.doi.org/10.1182/blood-2003-04-1205 DOI: https://doi.org/10.1182/blood-2003-04-1205




How to Cite

Cox, T. F. (2014). Testing the Equivalence of Survival Distributions using PP- and PPP-Plots. International Journal of Statistics in Medical Research, 3(2), 161–173. https://doi.org/10.6000/1929-6029.2014.03.02.10



General Articles