The Gini concentration test for survival data Article

First Online: 02 September 2009 Received: 25 January 2009 Accepted: 07 August 2009 DOI :
10.1007/s10985-009-9125-5

Cite this article as: Bonetti, M., Gigliarano, C. & Muliere, P. Lifetime Data Anal (2009) 15: 493. doi:10.1007/s10985-009-9125-5
Abstract We apply the well known Gini index to the measurement of concentration in survival times within groups of patients, and as a way to compare the distribution of survival times across groups of patients in clinical studies. In particular, we propose an estimator of a restricted version of the index from right censored data. We derive the asymptotic distribution of the resulting Gini statistic, and construct an estimator for its asymptotic variance. We use these results to propose a novel test for differences in the heterogeneity of survival distributions, which may suggest the presence of a differential treatment effect for some groups of patients. We focus in particular on traditional and generalized cure rate models, i.e., mixture models with a distribution of the lifetimes of the cured patients that is either degenerate at infinity or has a density. Results from a simulation study suggest that the Gini index is useful in some situations, and that it should be considered together with existing tests (in particular, the Log-rank, Wilcoxon, and Gray–Tsiatis tests). Use of the test is illustrated on the classic data arising from the Eastern Cooperative Oncology Group melanoma clinical trial E1690.

Keywords Cure rate model Gini index Linear rank test Melanoma Right censored survival data

References Aghevli BB, Mehran F (1981) Optimal grouping of income distribution data. J Am Stat Assoc 76: 22–26

CrossRef MathSciNet Andersen PK, Borgan O, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New York

MATH Atkinson T (1970) On the measurement of inequality. J Econ Theory 2: 244–263

CrossRef Chen M, Harrington DP, Ibrahim JG (2002) Bayesian cure rate models for malignant melanoma: a case-study of eastern cooperative oncology group trial E1690. Appl Stat 51(2): 135–150

MATH MathSciNet Esteban J, Gradín C, Ray D (2007) An extension of a measure of polarization, with an application to the income distribution of five OECD countries. J Econ Inequal 5: 1–19

CrossRef Ewell M, Ibrahim JG (1997) The Large Sample distribution of the weighted Log rank statistic under general local alternatives. Lifetime Data Anal 3: 5–12

MATH CrossRef Gail MH, Gastwirth JL (1978) A scale-free goodness-of-fit test for the exponential distribution based on the Gini statistic. J Roy Stat Soc Ser B 40(3): 350–357

MATH MathSciNet Gastwirth JL (1971) A general definition of the Lorenz curve. Econometrica 31(6): 1037–1039

CrossRef Gill RD (1980) Censoring and stochastic integrals. Mathematisch Centrum, Amsterdam

MATH Gini C (1912) Variabilità e mutabilità. Contributo allo studio delle distribuzioni e relazioni statistiche. Studi Economico-Giuridici dell’Università di Cagliari III

Gini C (1914) Sulla misura della concentrazione e della Variabilità dei caratteri. Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti LXXIII(part 2): 1203–1248

Gray RJ, Tsiatis AA (1989) A linear rank test for use when the main interest is in differences in cure rates. Biometrics 45: 899–904

MATH CrossRef Halpern J, Brown BW (1987) Cure rate models: power of the logrank and generalized Wilcoxon tests. Stat Med 6: 483–489

CrossRef Hanada K (1983) A formula of Gini’s concentration ratio and its applications to life tables. J Jpn Stat Soc 19: 293–325

Harrington DP, Fleming TR (1982) A class of rank test procedures for censored survival data. Biometrika 69(3): 553–566

MATH CrossRef MathSciNet Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19(3): 293–325

MATH CrossRef MathSciNet Ibrahim JG, Chen M, Sinha D (2001) Bayesian survival analysis. Springer, New York

MATH Kakwani NC (1980) Income inequality and poverty: methods of estimation and policy applications. Oxford University Press, Oxford

Kaplan EM, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53: 457–481

MATH CrossRef MathSciNet Kendall M, Stuart A (1977) The advanced theory of statistics. Mac Millan Publishing, New York

MATH Kirkwood JM, Ibrahim JG, Sondak VK, Richards J, Flaherty LE, Ernstoff MS, Smith TJ, Rao U, Steele M, Blum RH (2000) High- and low-dose interferon alfa-2b in high-risk melanoma: first analysis of intergroup trial E1690/S9111/C9190. J Clin Oncol 18(12): 2444–2458

Laska EM, Meisner MJ (1992) Nonparametric estimation and testing in a cure rate model. Biometrics 48: 1223–1234

CrossRef Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, New York

MATH Lorenz MO (1905) Methods of measuring the concentration of wealth. Publ Am Stat Assoc 9(70): 209–219

Michetti B, Dall′Aglio G (1957) La differenza semplice media. Statistica 7(2): 159–255

MathSciNet Muliere P, Petrone S (1992) Generalized Lorenz curve and Monotone dependence orderings. METRON-Int J Stat L 3–4: 19–38

MathSciNet Muliere P, Scarsini M (1989) A note on stochastic dominance and inequality measures. J Econ Theory 49: 314–323

MATH CrossRef MathSciNet Niewiadomska-Bugaj M, Kowalczyk T, Ouda H (2006) A new test of association and other tests based on the Gini mean difference. METRON-Int J Stat LXIV(3): 399–409

MathSciNet Nikitin YaYu, Tchirina AV (1996) Bahadur efficiency and local optimality of a test for the exponential distribution based on the Gini statistic. J Ital Stat Soc 5(1): 163–175

MATH CrossRef Nygard F, Sandröm A (1981) Measuring income inequality. Almqvist and Wilsell International, Stockholm

Pietra G (1915) Delle relazioni tra gli indici di Variabilità, I, II. Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti LXXIV(II): 775–804

Rao JS, Goria MN (2004) A test of goodness-of-fit based on Gini’s index of spacings. Stat Prob Lett 68: 177–187

MATH CrossRef Shaked M, Shanthikumar JG (1994) Stochastic orders and their applications. Academic Press Inc, Boston

MATH Shkolnikov VM, Andreev EE, Begun AZ (2003) Gini coefficient as a life table function: computation from discrete data, decomposition of differences and empirical examples. Demogr Res 8: 305–358

CrossRef Sposto R, Sather HN, Baker SA (1992) A comparison of tests of the difference in the proportion of patients who are cured. Biometrics 48: 87–99

CrossRef Tarone RE, Ware J (1977) On distribution-free tests for equality of survival distributions. Biometrika 64(1): 156–160

MATH CrossRef MathSciNet Tse SM (2006) Lorenz curve for truncated and censored data. Ann Inst Stat Math 58: 675–686

MATH CrossRef MathSciNet van der Vaart AW, Wellner LA (1996) Weak convergence and empirical processes: with applications to statistics. Springer-Verlag, New York

MATH Wang R, Lagakos SW, Ware JH, Hunter DJ, Drazen JM (2007) Statistics in medicine-reporting of subgroups analyses in clinical trials. New England J Med 357: 2189–2194

CrossRef Yin G, Ibrahim JG (2005) A general class of bayesian survival models with zero and nonzero cure fractions. Biometrics 61: 403–412

MATH CrossRef MathSciNet © Springer Science+Business Media, LLC 2009

Authors and Affiliations 1. Department of Decision Sciences Bocconi University Milan Italy 2. Department of Economics Università Politecnica delle Marche Ancona Italy