In survival analysis, why do we use semi-parametric models (Cox proportional hazards) instead of fully parametric models? I've been studying the Cox Proportional Hazards model, and  this question is glossed over in most texts.  
Cox proposed fitting the coefficients of the Hazard function using a partial likelihood method, but why not just fit the coefficients of a parametric Survival function using the maximum likelihood method and a linear model?  
In any cases where you have censored data, you could just find the area under the curve.  For example, if your estimate is 380 with standard deviation of 80, and a sample is censored >300, then there is an 84% probability for that sample in the likelihood calculation assuming normal error.
 A: If you know the parametric distribution that your data follows then using a maximum likelihood approach and the distribution makes sense.  The real advantage of Cox Proportional Hazards regression is that you can still fit survival models without knowing (or assuming) the distribution.  You give an example using the normal distribution, but most survival times (and other types of data that Cox PH regression is used for) do not come close to following a normal distribution.  Some may follow a log-normal, or a Weibull, or other parametric distribution, and if you are willing to make that assumption then the maximum likelihood parametric approach is great.  But in many real world cases we do not know what the appropriate distribution is (or even a close enough approximation).  With censoring and covariates we cannot do a simple histogram and say "that looks like a ... distribution to me".  So it is very useful to have a technique that works well without needing a specific distribution.
Why use the hazard instead of the distribution function?  Consider the following statement:  "People in group A are twice as likely to die at age 80 as people in group B".  Now that could be true because people in group B tend to live longer than those in group A, or it could be because people in group B tend to live shorter lives and most of them are dead long before age 80, giving a very small probability of them dying at 80 while enough people in group A live to 80 that a fair number of them will die at that age giving a much higher probability of death at that age.  So the same statement could mean being in group A is better or worse than being in group B.  What makes more sense is to say, of those people (in each group) that lived to 80, what proportion will die before they turn 81.  That is the hazard (and the hazard is a function of the distribution function/survival function/etc.).  The hazard is easier to work with in the semi-parametric model and can then give you information about the distribution.
A: "We" don't necessarily. The range of survival analysis tools ranges from the fully non-parametric, like the Kaplan-Meier method, to fully parametric models where you specify the distribution of the underlying hazard. Each has their advantages and disadvantages.
Semi-parametric methods, like the Cox proportional hazards model, let you get away with not specifying the underlying hazard function. This can be helpful, as we don't always know the underlying hazard function and in many cases also don't care. For example, many epidemiology studies want to know "Does exposure X decrease the time until event Y?" That they care about is the difference in patients who have X and who do not have X. In that case, the underlying hazard doesn't really matter, and the risk of misspecifying it is worse than the consequences of not knowing it.
There are times however when this also isn't true. I've done work with fully parametric models because the underlying hazard was of interest.
