survival notes

survival.Rmd

---
title: Notes on survival analysis
author: Gibran Hemani
date: "`r Sys.Date()`"
output: pdf_document
bibliography: survival.bib
---

These notes are based on [http://data.princeton.edu/wws509/notes](http://data.princeton.edu/wws509/notes).

## The survival function

$T$ is a continuous random variable with pdf $f(t)$ and cdf $F(t) = P(T < t)$, giving the probability that the event has occurred by duration T. An example is the Gompertz-Makeham law, which has pdf

$$
(\alpha e^{\beta x} + \lambda)\cdot \exp(-\lambda x-\frac{\alpha}{\beta}(e^{\beta x}-1))
$$

and CDF

$$
1-\exp(-\lambda x-\frac{\alpha}{\beta}(e^{\beta x}-1))
$$

These distributions look like:

```{r }

gm_pdf <- function(x, a, b, lambda)
{
	(a * exp(b*x) + lambda) * exp(-lambda * x - a / b * (exp(b*x) - 1))
}

gm_cdf <- function(x, a, b, lambda)
{
	1 - exp( -lambda * x - a / b * (exp(b*x) - 1) )
}

age <- 1:100
a <- 7.478359e-05
b <- 8.604875e-02
lambda <- -1.846973e-03
par(mfrow=c(2,1))
plot(
	x=age, y=gm_pdf(age, a, b, lambda), 
	main="PDF of Gompertz-Makeham"
)
plot(
	x=age, y=gm_cdf(age, a, b, lambda), 
	main="CDF of Gompertz-Makeham"
)

```

The survival function gives the probability of being alive before duration $t$, e.g. the event has not occurred by duration $t$. e.g.

$$
S(t) = 1 - F(t) = \int_{t}^{\infty} f(x)dx
$$

So $S(x) \leq S(t)$ for all $x > t$. The hazard function is 

$$
\lambda(t) = \lim_{dt \to 0} \frac{P(t \leq T < t+dt \mid T \geq t)}{dt}
$$

which is basically just the rate of occurrance per unit time. For the Gompertz-Makeham this looks like:


## The hazard function

The hazard function 

$$
\lambda(t) = \frac{f(t)}{S(t)}
$$

e.g.

```{r }

gm_hz <- function(x, a, b, lambda)
{
	a * exp(b*x) + lambda
}

plot(
	x=age, y=gm_hz(age, a, b, lambda), 
	main="Hazard function of Gompertz-Makeham"
)

```

relates to the cdf and pdf as

```{r }

plot(
	x = gm_hz(age, a, b, lambda), 
	y = gm_pdf(age, a, b, lambda) / (1 - gm_cdf(age, a, b, lambda))
)

```

So the rate of occurrance of the event at duration $t$ equals the density of events at $t$ divided by the probability of surviving to that duration without the event happening.

Because we know that $S(t) = 1 - F(t) = \int_{t}^{\infty} f(x)dx$, we know that the derivative of $S(t)$ is $-f(t)$. So, we can rewrite the hazard function as 

$$
\lambda(t) = - \frac{d}{dt}logS(t)
$$

This allows us to rewrite $S(t) as a function of surviving all hazards up to $t$

$$
S(t) = exp( - \int_{0}^{t} \lambda(x)dx)
$$

The integral part of this is known as the cumulative hazard ($\Delta(t)$).

## Explanatory variables

Introducing explanatory variables is understood in the context of $T_i$ being the time to event for individual $i$, and modelled as

$$
\log(T_i) = \mathbf{{x_i}'\beta} + \epsilon_i
$$

where $\epsilon_i$ is a suitable error term, or indeed the baseline value for $T_i$ when there are no explanatory variables. e.g. The explanatory variable shifts the standard baseline value. This can be exponentiated

$$
T_i = exp(\mathbf{x_i'\beta})T_{0i}
$$

where $T_{0i}$ is the exponentiated error term. We can use $\gamma$ as shorthand for the multiplicative effect $exp(\mathbf{x_i'\beta})$ of the covariates.

Interpretation of this is then straightforward - if an explanatory variable $x$ is binary (smokers vs non-smokers) then if smokers live half as long as non-smokers then $\gamma = 0.5$, or $\beta = \log(0.5) =$ `r log(0.5)`. Relating this to the survivor function, the model is interpreted as *life acceleration*. If $S_1(t)$ are smokers and $S_0(t)$ are non-smokers then 

$$
S_1(t) = S_0(t/\gamma)
$$

An alternative approach is proportional hazards [@Cox1972] which focuses on the hazard function directly.

## References