On the shape of the cross-ratio function in bivariate survival models induced by truncated and folded normal frailty distributions

Abstract

In shared frailty models for bivariate survival data the frailty is identifiable through the cross-ratio function (CRF), which provides a convenient measure of association for correlated survival variables. The CRF may be used to compare patterns of dependence across models and data sets. We explore the shape of the CRF for the families of one-sided truncated normal and folded normal frailty distributions.

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References

  1. Aalen OO, Borgan Ø, Gjessing HK (2008) Survival and event history analysis: a process point of view. Springer, New York

    Book  MATH  Google Scholar 

  2. Anderson JE, Louis TA, Holm NV, Harvald B (1992) Time-dependent association measures for bivariate survival distributions. J Am Stat Assoc 87:641–650

    MathSciNet  Article  MATH  Google Scholar 

  3. Azzalini A, Capitanio A (2014) The skew-normal and related families. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  4. Azzalini A, Kotz S (2003) Log-skew-normal and log-skew-\(t\) distributions as models for family income data. J Income Distrib 11:12–20

    Google Scholar 

  5. Callegaro A, Iacobelli S (2012) The cox shared frailty model with log-skew normal frailties. Stat Model 12:399–418

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  7. Clayton DG, Cuzick J (1985) Multivariate generalizations of the proportional hazards model. J R Stat Soci Ser A 148:82–117

    MathSciNet  Article  MATH  Google Scholar 

  8. Duchateau L, Janssen P (2008) The frailty model. Springer, New York

    MATH  Google Scholar 

  9. Farrington CP, Unkel S, Anaya-Izquierdo K (2012) The relative frailty variance and shared frailty models. J R Stat Soc Ser B 74:673–696

    MathSciNet  Article  Google Scholar 

  10. Hougaard P (2000) Analysis of multivariate survival data. Springer, New York

    Book  MATH  Google Scholar 

  11. Hougaard P (2014) Frailty models. In: Klein JP, van Houwelingen HC, Ibrahim JG, Scheike TH (eds) Handbook of survival analysis. Chapman and Hall, London, pp 457–474

    Google Scholar 

  12. Oakes D (1989) Bivariate survival models induced by frailities. J Am Stat Assoc 84:487–493

    Article  MATH  Google Scholar 

  13. Paik MC, Tsai W-Y, Ottman R (1994) Multivariate survival analysis using piecewise gamma frailty. Biometrics 50:975–988

    Article  MATH  Google Scholar 

  14. R Core Team (2015) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. https://www.R-project.org

  15. Sahu S K, Dey D K (2004) On a Bayesian multivariate survival model with a skewed frailty. In: Genton M G (ed) Skew-elliptical distributions and their applications: a journey beyond normality. CRC Press, Boca Raton, pp 321–338

    Google Scholar 

  16. Viswanathan B, Manatunga AK (2001) Diagnostic plots for assessing the frailty distribution in multivariate survival data. Lifetime Data Anal 7:143–155

    MathSciNet  Article  MATH  Google Scholar 

  17. Wienke A (2011) Frailty models in survival analysis. CRC Press, Boca Raton

    Google Scholar 

Download references

Acknowledgements

The author gratefully acknowledges David Ellenberger for proofreading the manuscript. One reviewer made valuable comments and suggestions on the first draft of this paper.

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Correspondence to Steffen Unkel.

Appendix

Appendix

One-sided truncated normal distribution

$$\begin{aligned} \mathcal {K}'(-s/\mu )= & {} \frac{\sigma ^2 s}{\mu ^2} -\frac{\xi }{\mu } - \frac{\sigma \phi \left( \frac{\sigma s}{\mu }-\frac{\xi }{\sigma }\right) }{\mu \left( 1-\Phi \left( \frac{\sigma s}{\mu }-\frac{\xi }{\sigma } \right) \right) } \end{aligned}$$
(14)
$$\begin{aligned} \mathcal {K}''(-s/\mu )= & {} \frac{\sigma ^2}{\mu ^2} - \frac{\sigma ^2 \phi \left( \frac{\sigma s}{\mu } - \frac{\xi }{\sigma } \right) ^2}{\mu ^2\left( 1-\Phi \left( \frac{\sigma s}{\mu } - \frac{\xi }{\sigma } \right) \right) ^2 } - \frac{\sigma ^2 \phi ' \left( \frac{\sigma s}{\mu } - \frac{\xi }{\sigma } \right) }{\mu ^2\left( 1-\Phi \left( \frac{\sigma s}{\mu } - \frac{\xi }{\sigma } \right) \right) } \end{aligned}$$
(15)

Folded normal distribution

$$\begin{aligned}&\mathcal {K}'(-s/\mu ) \nonumber \\&\quad = \frac{2\xi \exp (2\xi s/\mu )\left( 1-\Phi \left( \frac{\sigma s}{\mu } +\frac{\xi }{\sigma } \right) \right) - \sigma \phi \left( \frac{\sigma s}{\mu } - \frac{\xi }{\sigma } \right) - \sigma \exp (2\xi s /\mu )\phi \left( \frac{\sigma s}{\mu } + \frac{\xi }{\sigma } \right) }{\mu \left\{ 1-\Phi \left( \frac{\sigma s}{\mu } -\frac{\xi }{\sigma } \right) + \exp (2\xi s/\mu )\left[ 1- \Phi \left( \frac{\sigma s}{\mu } + \frac{\xi }{\sigma } \right) \right] \right\} } + \frac{\sigma ^2 s}{\mu ^2} - \frac{\xi }{\mu } \end{aligned}$$
(16)
$$\begin{aligned}&\mathcal {K}''(-s/\mu )\nonumber \\&\quad = \frac{1}{\mu ^2}\left\{ \frac{4 \xi ^2 \exp (2\xi s/\mu )\left( 1-\Phi \left( \frac{\sigma s}{\mu } + \frac{\xi }{\sigma } \right) \right) - \sigma ^2 \phi '\left( \frac{\sigma s}{\mu } -\frac{\xi }{\sigma } \right) - \sigma ^2\exp (2\xi s/\mu ) \phi '\left( \frac{\sigma s}{\mu } +\frac{\xi }{\sigma } \right) - 4 \xi \sigma \exp (2\xi s/\mu )\phi \left( \frac{\sigma s}{\mu } +\frac{\xi }{\sigma } \right) }{1-\Phi \left( \frac{\sigma s}{\mu } - \frac{\xi }{\sigma }\right) +\exp (2\xi s/\mu )\left( 1-\Phi \left( \frac{\sigma s}{\mu } + \frac{\xi }{\sigma } \right) \right) } \right. \nonumber \\&\quad \quad - \left. \frac{\left( 2\xi \exp (2\xi s/\mu )\left( 1-\Phi \left( \frac{\sigma s}{\mu } +\frac{\xi }{\sigma } \right) \right) - \sigma \phi \left( \frac{\sigma s}{\mu } - \frac{\xi }{\sigma } \right) - \sigma \exp (2\xi s /\mu )\phi \left( \frac{\sigma s}{\mu } + \frac{\xi }{\sigma } \right) \right) ^2 }{\left( 1-\Phi \left( \frac{\sigma s}{\mu } - \frac{\xi }{\sigma }\right) +\exp (2\xi s/\mu )\left( 1-\Phi \left( \frac{\sigma s}{\mu } + \frac{\xi }{\sigma } \right) \right) \right) ^2} + \sigma ^2 \right\} \end{aligned}$$
(17)

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Unkel, S. On the shape of the cross-ratio function in bivariate survival models induced by truncated and folded normal frailty distributions. Metrika 80, 351–362 (2017). https://doi.org/10.1007/s00184-016-0608-6

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Keywords

  • Cross-ratio function
  • Frailty
  • Heterogeneity
  • Student-t distributions
  • Survival data
  • Truncation