Accelerated downtime model

In medical statistics, and especially in survival time analysis , the model for accelerated downtimes , also called the accelerated survival time model , is a parametric model that can be linearized by logarithmic transformation and is an alternative to the frequently used proportional hazard model.

Introduction to the problem

It is often said that one dog year equals seven human years. If and the survival functions for humans (M) and dogs (H), and if we continue to assume that dogs age about 7 times faster than humans, then applies ${\displaystyle S_{M}(t)}$ ${\displaystyle S_{H}(t)}$

${\displaystyle S_{H}(t)=S_{M}(7t)}$.

The factor is called the acceleration factor. In general ${\displaystyle \gamma =7}$

${\displaystyle S_{1}(t)=S_{2}(\gamma t)}$or . ${\displaystyle \gamma T_{1}=T_{2}}$

This means an advantage and a disadvantage in terms of service life. ${\displaystyle \gamma >1}$ ${\displaystyle \gamma <1}$

Model representations

Let be the time to the occurrence of an event (e.g. death) and be a vector of time-constant explanatory variables. The accelerated survival model says that the survival function of an individual with the vector of explanatory variables at time is the same as the survival function of an individual with a basic survival function at time , where is a vector of regression parameters . In other words, the accelerated (delayed) survival times model is given by the relationship ${\displaystyle T}$ ${\displaystyle \mathbf {Z} }$ ${\displaystyle \mathbf {Z} }$ ${\displaystyle t}$ ${\displaystyle t\cdot \exp({\boldsymbol {\theta }}^{\top }\mathbf {Z} )}$ ${\displaystyle {\boldsymbol {\theta }}^{\top }=(\theta _{1},\theta _{2},\ldots ,\theta _{k})}$

${\displaystyle S(t\mid \mathbf {Z} )=S_{0}(\exp({\boldsymbol {\theta }}^{\top }\mathbf {Z} )t),\quad \mathrm {f{\ddot {u}}r\;} {\text{alle}}\;t}$.

The factor is called the acceleration factor and provides the user with information about how a change in the explanatory variable changes the time scale in relation to the base time scale. One implication of this model is that the hazard rate for a person with the vector of explanatory variables with a hazard rate is related to the base hazard rate through ${\displaystyle \exp({\boldsymbol {\theta }}^{\top }\mathbf {Z} )}$ ${\displaystyle \mathbf {Z} }$

${\displaystyle h(t\mid \mathbf {Z} )=\exp({\boldsymbol {\theta }}^{\top }\mathbf {Z} )h_{0}(\exp({\boldsymbol {\theta }}^{\top }\mathbf {Z} )t),\quad \mathrm {f{\ddot {u}}r\;} {\text{alle}}\;t}$.

Another implication is that the median survival time with the vector of the explanatory variable is equal to the median base survival time divided by the acceleration factor. ${\displaystyle \mathbf {Z} }$

The second representation of the relationship between the values ​​of the explanatory variables and survival time is the linear relationship between the logarithmic survival time and the values ​​of the explanatory variables. The following loglinear approach is used here:

${\displaystyle Y=\log(T)={\boldsymbol {\mu }}+{\boldsymbol {\gamma }}^{\top }\mathbf {Z} +\sigma W}$,

where is a vector of regression parameters and represents the distribution of the disturbance variable . ${\displaystyle {\boldsymbol {\gamma }}^{\top }=(\gamma _{1},\gamma _{2},\ldots ,\gamma _{k})}$ ${\displaystyle W}$

Individual evidence

1. ↑ International Statistical Institute : Glossary of statistical terms.
2. ↑ Lothar Sachs , Jürgen Hedderich: Applied Statistics: Collection of Methods with R. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2018, ISBN 978-3-662-56657-2 , p. 900
3. ↑ Lothar Sachs , Jürgen Hedderich: Applied Statistics: Collection of Methods with R. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2018, ISBN 978-3-662-56657-2 , p. 899
4. ↑ Dietz, K., et al .: Statistics for Biology and Health. Survival Analysis, Edition Springer 7 (2002), p. 394