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.

Appendix

1.1 One-sided truncated normal distribution

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

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

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1.2 Folded normal distribution

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

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

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