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One-sided tests in shared frailty models

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Abstract

Tests for the presence of heterogeneity in frailty models use an alternative hypothesis in which the heterogeneity parameter is subject to an inequality constraint. As a result, the classical likelihood ratio asymptotic chi-square distribution theory is no longer valid. Our main result states the limiting distribution of the likelihood ratio and score statistic for the one-sided testing problem. The resulting distribution is a mixture of chi-square distributed random variables. The results are shown for gamma and positive stable frailty distributions, and hold when covariate information is present. A data example illustrates the tests. We also assess, in a simulation study, the performance of the tests regarding the significance level and power.

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References

  • Bjarnason H, Hougaard P (2000) Fisher information for two gamma frailty bivariate Weibull models. Lifetime Data Anal 6:59–71

    Article  MATH  Google Scholar 

  • Chernoff H (1954) On the distribution of the likelihood ratio. Ann Math Stat 25:573–578

    Article  MathSciNet  Google Scholar 

  • Claeskens G (2004) Restricted likelihood ratio lack-of-fit tests using mixed spline models. J Roy Stat Soc Ser B 66(4):909–926

    Article  MATH  MathSciNet  Google Scholar 

  • Clayton D, Cuzick J (1985) Multivariate generalizations of the proportional hazards model. J Roy Stat Soc Ser A 148(2):82–117

    Article  MATH  MathSciNet  Google Scholar 

  • Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65(1):141–151

    Article  MATH  MathSciNet  Google Scholar 

  • Commenges D, Andersen PK (1995) Score test of homogeneity for survival data (with discussions). Lifetime Data Anal 1(2):145–159

    Article  MATH  MathSciNet  Google Scholar 

  • Duchateau L, Janssen P, Lindsey P, Legrand C, Nguti R, Sylvester R (2002) The shared frailty model and the power for heterogeneity tests in multicenter trials. Comput Stat Data Anal 40(3):603–620

    Article  MATH  MathSciNet  Google Scholar 

  • Feller W (1971) An introduction to probability theory and its applications, vol II, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Ferguson TS (1996) A course in large sample theory. Texts in statistical science series. Chapman and Hall, London

    MATH  Google Scholar 

  • Geyer CJ (1994) On the asymptotics of constrained M-estimation. Ann Stat 22(4):1993–2010

    Article  MATH  MathSciNet  Google Scholar 

  • Hougaard P (2000) Analysis of multivariate survival data. Statistics for biology and health. Springer, New York

    MATH  Google Scholar 

  • Huster WJ, Brookmeyer R, Self SG (1989) Modelling paired survival data with covariates. Biometrics 45(1):145–156

    Article  MATH  MathSciNet  Google Scholar 

  • Maller RA, Zhou X (2003) Testing for individual heterogeneity in parametric models for event-history data. Math Methods Stat 12(3):276–304

    MathSciNet  Google Scholar 

  • Morgan BJT (1992) Analysis of quantal response data. Chapman and Hall, London

    Google Scholar 

  • Murphy SA (1995) Asymptotic theory for the frailty model. Ann Stat 23(1):182–198

    Article  MATH  MathSciNet  Google Scholar 

  • Murphy SA, van der Vaart AW (1997) Semiparametric likelihood ratio inference. Ann Stat 25(4):1471–1509

    Article  MATH  Google Scholar 

  • Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley series in probability and mathematical statistics: Probability and mathematical statistics. Wiley, Chichester

    MATH  Google Scholar 

  • Self SG, Liang K-Y (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J Am Stat Assoc 82(398):605–610

    Article  MATH  MathSciNet  Google Scholar 

  • Sen PK, Silvapulle MJ (2002) An appraisal of some aspects of statistical inference under inequality constraints. J. Stat Plan Inference 107(1–2):3–43

    Article  MATH  MathSciNet  Google Scholar 

  • Silvapulle MJ, Silvapulle P (1995) A score test against one-sided alternatives. J Am Stat Assoc 90(429):342–349

    Article  MATH  MathSciNet  Google Scholar 

  • Stram DO, Lee JW (1994) Variance components testing in the longitudinal mixed effects models. Biometrics 50:1171–1177

    Article  MATH  Google Scholar 

  • Tawn JA (1988) Bivariate extreme value theory: models and estimation. Biometrika 75(3):397–415

    Article  MATH  MathSciNet  Google Scholar 

  • Therneau TM, Grambsch PM (2000) Modeling survival data: Extending the Cox model. Statistics for biology and health. Springer, New York

    MATH  Google Scholar 

  • Vaida F, Xu R (2000) Proportional hazards model with random effects. Stat Med 19:3309–3324

    Article  Google Scholar 

  • Verbeke G, Molenberghs G (2003) The use of score tests for inference on variance components. Biometrics 59(2):254–262

    Article  MathSciNet  Google Scholar 

  • Vu HTV, Knuiman MW (2002) A hybrid ML-EM algorithm for calculation of maximum likelihood estimates in semiparametric shared frailty models. Comput Stat Data Anal 40(1):173–187

    Article  MATH  MathSciNet  Google Scholar 

  • Vu HTV, Zhou S (1997) Generalization of likelihood ratio tests under nonstandard conditions. Ann Stat 25(2):897–916

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. ORSTAT and University Center for Statistics, Katholieke Universiteit Leuven, Naamsestraat 69, 3000, Leuven, Belgium

    Gerda Claeskens

  2. Department of Mathematics, University of Nairobi, Nairobi, Kenya

    Rosemary Nguti

  3. Center for Statistics, Universiteit Hasselt, Hasselt, Belgium

    Paul Janssen

Authors
  1. Gerda Claeskens
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  2. Rosemary Nguti
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  3. Paul Janssen
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Correspondence to Gerda Claeskens.

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Claeskens, G., Nguti, R. & Janssen, P. One-sided tests in shared frailty models. TEST 17, 69–82 (2008). https://doi.org/10.1007/s11749-006-0023-9

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  • DOI: https://doi.org/10.1007/s11749-006-0023-9

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