Survival and Hazard Functions for Progressive Diseases Using Saddlepoint Approximations

Abstract

Saddlepoint approximations for the computation of survival and hazard functions are introduced in the context of parametric survival analysis. Although these approximations are computationally fast, accurate, and relatively straightforward to implement, their use in survival analysis has been lacking. We approximate survival functions using the Lugannani and Rice saddlepoint approximation to the distribution function or by numerically integrating the saddlepoint density approximation. The hazard function is approximated using the saddlepoint density and distribution functions. The approximations are especially useful for consideration of survival and hazard functions for waiting times in complicated models. Examples include total or partial waiting times for a disease that progresses through various stages (convolutions of distributions).

The empirical saddlepoint method applied to testing for serial correlation in panel time series data

“…The approximation in (5) can have instability for u close to the expected value ofŪ as noted by, for example, Reid (1996) and illustrated in Huzurbazar and Huzurbazar (1999). However we are typically interested in larger values of u and so this should not be an issue when approximating p-values.…”

Analysis of censored and incomplete survival data using flowgraph models

“…A saddlepoint approximation to the CDF was derived by Lugannani and Rice [15]. However, this is numerically unstable near the mean of the distribution [16] and cannot be used when dealing with convolutions, ÿnite mixtures of distributions, or convolutions of ÿnite mixtures of distributions as occur commonly in owgraph models. Alternatively, we use the numerically integrated saddlepoint density approximation to approximate the CDF.…”

GEARSHIFT: Guaranteeing availability requirements in SLAs using hybrid fault tolerance

“…The convolution of gamma distributed downtimes is obtained directly by using the Laplace transform. For the case of Weibull distributed uptimes, P (N (τ ) = n) can be obtained by expanding the Weibull function as a Taylor series as presented in [18], and the convolution of Weibull distributed downtimes may be approximated using the Saddle Point Approximation [15]. Finally, several numerical methods can be used to approximate convolution operations like the ones shown in [17] or the Fast Fourier Transform.…”