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Testing for Proportional Hazards with Unrestricted Univariate Unobserved Heterogeneity

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  • Bhattacharjee, Arnab

Abstract

We develop tests of the proportional hazards assumption, with respect to a continuous covariate, in the presence of unobserved heterogeneity with unknown distribution at the individual observation level. The proposed tests are specially powerful against ordered alternatives useful for modeling non-proportional hazards situations. By contrast to the case when the heterogeneity distribution is known up to …nite dimensional parameters, the null hypothesis for the current problem is similar to a test for absence of covariate dependence. However, the two testing problems di¤er in the nature of relevant alternative hypotheses. We develop tests for both the problems against ordered alternatives. Small sample performance and an application to real data highlight the usefulness of the framework and methodology.

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File URL: http://repo.sire.ac.uk/handle/10943/114
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Paper provided by Scottish Institute for Research in Economics (SIRE) in its series SIRE Discussion Papers with number 2009-22.

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Date of creation: 2009
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Handle: RePEc:edn:sirdps:114

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Keywords: Two-sample tests; Increasing hazard ratio; Trend tests; Partial orders; Mixed proportional hazards model; Time varying coe¢cients;

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  1. Arnab Bhattacharjee & Chris Higson & Sean Holly & Paul Kattuman, 2007. "Macroeconomic Conditions and Business Exit: Determinants of Failures and Acquisitions of UK Firms," CDMA Working Paper Series 200713, Centre for Dynamic Macroeconomic Analysis.
  2. Horowitz, J. & Gorgens, T., 1995. "Semiparametric Estimation of a Censored Regression Model with an Unknown Transformation of the Dependent Variable," Working Papers 95-15, University of Iowa, Department of Economics.
  3. Baker, Michael & Melino, Angelo, 2000. "Duration dependence and nonparametric heterogeneity: A Monte Carlo study," Journal of Econometrics, Elsevier, vol. 96(2), pages 357-393, June.
  4. Heckman, James J. & Singer, Burton, 1984. "Econometric duration analysis," Journal of Econometrics, Elsevier, vol. 24(1-2), pages 63-132.
  5. Sherman, Robert P, 1993. "The Limiting Distribution of the Maximum Rank Correlation Estimator," Econometrica, Econometric Society, vol. 61(1), pages 123-37, January.
  6. Torben Martinussen, 2002. "Efficient Estimation of Fixed and Time-varying Covariate Effects in Multiplicative Intensity Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 29(1), pages 57-74.
  7. Jenkins, Stephen P, 1995. "Easy Estimation Methods for Discrete-Time Duration Models," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 57(1), pages 129-38, February.
  8. Han, Aaron K., 1987. "Non-parametric analysis of a generalized regression model : The maximum rank correlation estimator," Journal of Econometrics, Elsevier, vol. 35(2-3), pages 303-316, July.
  9. Joel L. Horowitz, 1999. "Semiparametric Estimation of a Proportional Hazard Model with Unobserved Heterogeneity," Econometrica, Econometric Society, vol. 67(5), pages 1001-1028, September.
  10. McCall, Brian P., 1996. "The Identifiability of the Mixed Proportional Hazards Model with Time-Varying Coefficients," Econometric Theory, Cambridge University Press, vol. 12(04), pages 733-738, October.
  11. Murphy, S. A. & Sen, P. K., 1991. "Time-dependent coefficients in a Cox-type regression model," Stochastic Processes and their Applications, Elsevier, vol. 39(1), pages 153-180, October.
  12. Liu, P. Y. & Wei Yann Tsai, 1999. "A modified logrank test for censored survival data under order restrictions," Statistics & Probability Letters, Elsevier, vol. 41(1), pages 57-63, January.
  13. Arnab Bhattacharjee, 2008. "Partial Orders with Respect to Continuous Covariates and Tests for the Proportional Hazards Model," Discussion Paper Series, Department of Economics 200807, Department of Economics, University of St. Andrews.
  14. Thomas H. Scheike & Torben Martinussen, 2004. "On Estimation and Tests of Time-Varying Effects in the Proportional Hazards Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 31(1), pages 51-62.
  15. Horowitz, Joel L, 1996. "Semiparametric Estimation of a Regression Model with an Unknown Transformation of the Dependent Variable," Econometrica, Econometric Society, vol. 64(1), pages 103-37, January.
  16. Bhattacharjee, A., 2003. "Estimation in Hazard Regression Models under Ordered Departures from Proportionality," Cambridge Working Papers in Economics 0337, Faculty of Economics, University of Cambridge.
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