This summarizes some very basic concepts in survival analysis from this amazing tutorial, which helps me a lot.

Chapter 3: Estimiating the Survival or Hazard Function (Parametric)

There are two ways of estimating survival/hazard functions

• parametric model for $\lambda(t)$ based on a particular density function
• empirical estimate of survival function (non-parametric)

If without censoring, the emperical $\hat{S}(t)$ is simply the proportion of people dying no earlier that time $t$. It is censoring patients that complicates this problem.

Some Parametric Survival Distribution (Continuos)

The exponential distribution

This is a constant hazard model, with only one parameter.

• $$f(t) = \lambda e^{-\lambda t} \text{ for } t \geq 0$$
• $$S(t) = e^{-\lambda t}$$
• $$\lambda(t) = \lambda$$
• $$\Lambda(t) = \lambda t$$

The Weibull distribution

It is a generalization of exponential model

• $$S(t) = e^{-\lambda t^\mathcal{k}}$$
• $$f(t) = \mathcal{k} \lambda t^{\mathcal{k} -1} e^{-\lambda t^{\mathcal{k}}}$$
• $$\lambda(t) = \mathcal{k} \lambda t^{\mathcal{k} - 1}$$
• $$\Lambda(t) = \lambda t^\mathcal{k}$$

where $\lambda$ os the scale parameter and $\mathcal{k}$ is the shape parameter.

• $\mathcal{k} = 1$: constant hazard
• $0 < \mathcal{k} < 1$: decreasing hazard
• $\mathcal{k} > 1$: increasing hazard

The Rayleigh distribution

Another 2-parameter generalization of exponential.

$$\lambda(t) = \lambda_0 + \lambda_1 t$$

Other models

Compound exponential, log-normal, log-logistic

Summary of Parametric Model

Good sides of parametric model

• easy to estimate and inference
• simple forms of $f(t)$, $S(t)$, and $\lambda(t)$
• qualitative shape of hazard function

Other tips

• With adequacy of fit to a dataset, one can usually distinguish one-parameter and two-parameter models
• Without a lot of data, hard to distinguish between the fits of 2-par models