Cox regression and time scale Does X (hazard) variable in Cox proportional hazard regression analysis always have to be time? If not, could you provide an example, please?
Can age of cancer patient be a hazard variable? If so, can it be interpreted as the risk of getting cancer at a certain age? Would Cox regression be a legitimate analysis to study the association between gene expression and age?
 A: Usually, age at baseline is used as a covariate (because it is often associated to disease/death), but it can be used as your time scale as well (I think it is used in some longitudinal studies, because you need to have enough people at risk along the time scale, but I can't remember actually -- just found these slides about Analysing cohort studies assuming a continuous time scale which talk about cohort studies). In the interpretation, you should replace event time by age, and you might include age at diagnosis as a covariate. This would make sense when you study age-specific mortality of a particular disease (as illustrated in these slides).
Maybe this article is interesting since it contrasts the two approaches, time-on-study vs. chronological age: Time Scales in Cox Model: Effect of Variability Among Entry Ages on Coefficient Estimates. Here is another paper:
Cheung, YB, Gao, F, and Khoo, KS (2003). Age at diagnosis and the choice of survival analysis methods in cancer epidemiology. Journal of Clinical Epidemiology, 56(1), 38-43.
But there are certainly better papers.
A: No, it doesn't always have to be time.  Many censored responses can be modeled with survival analysis techniques.  In his book Nondetects and Data Analysis, Dennis Helsel advocates using the negative of a concentration in place of time (in order to cope with nondetects, which when negated become right-censored values).  A synopsis is available on the Web (pdf format) and an R package, NADA, implements this.
A: On the age-scale vs. time-scale issue, chl has some good references and captures the essentials -- in particular, the requirement that the at-risk set contain sufficient subjects from all ages as would arise in a longitudinal study.  
I would only note that there is no general consensus around this yet, but there is some literature to suggest that age should be preferred as the time scale in certain cases.  In particular, if you have a situation where time doesn't accumulate in the same way for all subjects, for example due to exposure to some toxic material, then age may be more appropriate.  
On the other hand, you can handle that specific example on a time-scale Cox PH model by using age as a time varying covariate -- rather than a fixed covariate at start time.  You need to think about the mechanism behind your object of study to figure out which time scale is more appropriate.  Sometimes it's worth fitting both models to existing data to see if discrepancies arise and how they might be explained before designing your new study.  
Finally, the obvious difference in analyzing the two is that on an age-scale, the interpretation of survival is with respect to an absolute scale (age), whereas on a time-scale, it's relative to the start/entry date of the study.
A: Per the OP's request, heres another application I have seen survival analysis used in a spatial context (although obviously different than measuring environmental substances mentioned by whuber) is modeling the distance between events in space. Heres one example in criminology and here is one in epidemiology.
The reasoning behind using survival analysis to measure the distance between events is not per say an issue of censoring (although censoring can definately occur in a spatial context), it is more so because of the similar distributions between time to event characteristics and distance between events characteristics (i.e. they both have similar types of error structures (frequently distance decay) that violate OLS and so the non-parametric solutions are ideal for both).

Because of my poor citation practices I had to spend and hour finding the correct link/reference to the link above.
For the example in criminology, 
Kikuchi, George, Mamoru Amemiya, Tomonori Saito, Takahito Shimada & Yutaka Harada. 2010. A Spatio-Temporal analysis of near repeat victimization in Japan. 8th National Crime Mapping Conference. Jill Dando Institute of Crime Science. PDF currently available at referenced webpage.
In epidemiology,
Reader, Steven. 2000. Using survival analysis to study spatial point patterns in geographical epidemiology. Social Science & Medicine 50(7-8): 985-1000.
