Threshold regression inverse Gaussian model Aalen and Xxxxxxxx advocated an alternative way of approaching survival analysis, through the hazard rate (Aalen and Gjessing 2001). First hitting time or threshold regression (TR) models refer to a statistical model for time to event data in which the time to event is defined as the first hitting time of an absorbing boundary by an underlying stochastic process. In terms of the general structure, it is assumed that underlying the observed data there is a (latent) parent stochastic process, 𝑋(𝑡) 𝑤here 𝑋 denotes the level of a patient’s status at time 𝑡, measured from randomisation. The event occurs at the random time, when 𝑋(𝑡) first reaches a certain boundary (threshold). Several TR models following this concept are reviewed by Xxx and Xxxxxxxx (Xxx and Xxxxxxxx 2006). Previous work that has also considered regression structures for first hitting time models includes (Xxxxxxxx 1983), (Xxxxxxxx, Xxxxxxx et al. 1998), (Xxx, XxXxxxxxxx et al. 2000) and (Xxx, Xxxxxxxx et al. 2004). TR survival procedures construct survival functions based on events and non-events, ignoring the data collected on continuous measurements. TR can be modelled for various distributions, depending on the nature of this underlying process. The inverse Gaussian (IG) distribution arises from an underlying Weiner process. The advantages of the IG distribution as a model for failure times was set out by (Xxxxxxx 1957): (1) analogous to the Weibull and gamma families it accommodates different shapes of hazard; (2) it has the structure of an exponential family, and many convenient properties for the associated sampling distribution, which are often analogous to the normal distribution; (3) “Its derivation from a plausible stochastics formulation of the failure process often provides a physical support to its empirical fit” . The parametrisation of the IG when it is derived as a TR model has the appealing feature of two parameters; 𝑐 a starting level for the underlying Weiner process which can be seen as distance from the threshold and, 𝜇 the drift velocity towards the threshold; both 𝑐 and 𝜇 can depend on covariates. Consider the application of the IG model to a trial with randomised treatment and covariates measures prior to randomisation, baseline, and during treatment. 𝑐 is the distance in the score from the event threshold at the time of randomisation, and could be linked to most covariates measured prior to randomisation. 𝜇 is the rate of symptom im...
