# Survival Analysis – Lifetime Estimation Using Weibull vs. Survival Methods

reliabilitysurvivalweibull distribution

Survival analysis is mainly about estimating the lifetime of nearly anything or even anything:

How are they different (in general) and which should be used when?

If the failure times have a distribution $$F(t)$$ then the corresponding survival function is $$S(t)=1-F(t)$$. That's a critical thing to keep in mind.

The Weibull plots are just plots of a transformed $$F(t)=1-S(t)$$ against the log of time. So the Weibull plot is just a particular replotting of the survival curve.

A Weibull model can be written in the form:

$$\log T= \alpha + \sigma W,$$

where $$W$$ has a minimum extreme-value distribution and $$\sigma$$ is a scale factor. The y-axis transformation of $$F(t)$$ in a Weibull plot gives a straight line if the underlying distribution $$W$$ is minimum extreme-value. If there is a well fitting line, then the values of $$\alpha$$ and $$\sigma$$ can be deduced from the plot.

My sense is that the plots were most important before modern computational technology became available. I don't see that there's anything they do that can't be done with more general survival modeling, which doesn't restrict you to a Weibull form. You can always generate a Weibull plot from your survival model if you want to.