Time To Failure From Survival Analysis Techniques I am interested to know if/how certain survival analysis techniques can be used to also infer expected time to event or estimate probability density functions over time to event.
As  I understand it, given a time to event random variable T > 0, the survival function denotes the probability of surviving beyond time t i.e.:
$$
S(t \vert x) = P(T > t \vert  x) = 1 - \int_{0}^{t}  f(t \vert x) dt 
$$
With covariates x, and where time to event random variable T has a probability density function given by: $$f(t \vert x)$$
Then the expected time to event i.e. expectation of the random variable is given as:
$$ E[T-t \vert T >t] = \int_{z=0}^{z=\infty}  zf(z \vert x) dz  $$
Two questions:

*

*For non-parametric (e.g. Kaplan-Meier/Cox-Regression) techniques that estimate the survival function, is there a way to also obtain an estimate of the probability density function (|) and expectation of that random variable i.e. expected time to event?

*I have read that non-parametric survival analysis techniques like Cox's regression are not suitable for the task of absolute time-to-event prediction because they optimise the relative risks in the population and not the absolute time to event prediction - is anyone able to explain this?

 A: Point 1: in non-parametric (Kaplan-Meier) or semi-parametric (Cox) analyses, you have a problem if the last observation is a censoring rather than an event time, as is often the case. If that observation is a censoring time, you have no information about the survival function beyond that time, so you can't integrate all the way out to infinite time. You will sometimes see "restricted mean survival" values reported in those instances, in which the upper limit of integration in your presentation is some finite value rather than infinite. If mean survival time out to infinite time is of interest, you would have to resort to parametric modeling in such a situation.
Point 2: That might be a bit overstated, but it gets to an important issue. The baseline survival function in a Cox model is empirically derived from the data, based on the underlying assumptions of the Cox model (linearity in predictors, proportional hazards, etc). Absolute time-to-event predictions are always difficult in survival modeling, but related predictions of things like survival probabilities (with error estimates) up to particular times of interest are quite possible with Cox models. How well those predictions hold up in the underlying population depends on how representative the data sample is and how well the  modeling assumptions are met. In principle, that's not much different from the situation with parametric survival modeling. You can estimate how well the modeling approach might apply to predictions in the population by resampling, for example developing models on multiple bootstrap samples of the data and testing performance of all those models on the full data set.
