# Survival analysis: why use a continuous probability distribution?

Estimating the time to an event (survival analysis) is typically modelled in continuous time, assuming the time to event is exponentially distributed (or gamma, Weibull, log-normal).

However, as data are almost always aggregated over discrete intervals (days, weeks, months), and there are discrete probability analogues of continuous distributions that can account for right interval censoring (geometric and exponential), my questions are:

1) Why are continuous probability distributions preferred over discrete probability distributions?

2) Are there any circumstances where a discrete probability distribution is preferable?

(I'm aware of a similar question on CV, though this does not answer my points exactly)

More detailed notation. After $$D$$ days of observation, the event $$X$$ either takes place $$(X=1)$$, or has not yet occured $$(X=0)$$. In a continuous formulation (exponential), the expected time to $$X$$ is the reciprocal of the rate $$\lambda$$.

$$pr(D|X=1) = \lambda e^{-\lambda D}$$

$$pr(D|X=0) = e^{-\lambda D}$$

In a discrete formulation (geometric), the expected time to event is the reciprocal of the probability $$p$$.

$$pr(D|X=1) = p(1-p)^{D-1}$$

$$pr(D|X=0) = (1-p)^D$$

As I understand, both of these approaches would yield the same result.

If the assumption of the rate being constant over time is not valid, then this would necessitate the use of a different continuous probability distribution (gamma, Weibull, log-normal).

However, the discrete probability approach can also be made more flexible by considering the occurrence of the event on each day $$i$$ as a Bernoulli trial, and $$p$$ can be permitted to vary over periods of time (by fitting as a hierarchical / mixed effect logistic regression model). This would, seemingly, have the advantage of not assuming a particular shape for the probability distribution.