The estimation of direct effects and indirect effects (via one or several mediators) of an exposure on survival is a challenging task, that has received much attention lately. A good overview is given by Aalen et al. (2012). They emphazise the importance of explicitly including time in any discussion of causality and mediation: A cause must precede an effect. And many relevant references are found there.
Relevant, recent material
Tyler VanderWeele has written a lot of interesting stuff on mediation, look under “Selected Publications” and “Methodological” on his homepage. Under “Tools and Tutorials” you’ll find VIDEO OF DAY-LONG WORKSHOP ON CAUSAL MEDIATION ANALYSIS in four parts. Play them!
The difficulties with applying mediation analysis to Cox regression are pointed out by Lange and Hansen (2011), and they suggest a solution. In a “follow-up” paper, Lange, Vansteelandt, and Bekaert (2012) give “a simple unified approach” for estimation.
An alternative to Cox’s Proportional Hazards model (PH) is the Accelerated Failure Time (AFT) model. The advantage in the mediation context is that the AFT model is ordinary linear regression, however with right censored and eventually left truncated data. An example of using this approach is given by Broström and Edvinsson (2013) (pdf), with an application in Edvinsson and Broström (2013).
Aalen, O. O., K. Røysland, J. M. Gran, and B. Ledergerber. 2012. “Causality, Mediation and Time: A Dynamic Viewpoint.” Journal of the Royal Statistical Society Series A 175: 831–61. https://doi.org/10.1111/j.1467-985X.2011.01030.x.
Broström, G., and S. Edvinsson. 2013. “A Parametric Model for Old Age Mortality in Mediation Analysis.”
Edvinsson, S., and G. Broström. 2013. “The Effect of Early-Life and Mid-Life Factors on Old Age Mortality.”
Lange, T., and J. Hansen. 2011. “Direct and Indirect Effects in a Survival Context.” Epidemiology 22: 575–81. https://doi.org/10.1097/EDE.0b013e31821c680c.
Lange, T., S. Vansteelandt, and M. Bekaert. 2012. “A Simple Unified Approach for Estimating Natural Direct and Indirect Effects.” American Journal of Epidemiology 176: 190–95. https://doi.org/10.1093/aje/kwr525.