SurvivalAnalysis

10
3B. Comparing Survival Functions
1.00
0.75
Survival Distribution 0.50 Function
0.25
Low
High Medium
0.00 0
10
20
30
40
50
60
Time
11
Log-Rank Test
The log-rank test • tests whether the survival functions are
0
5
10
Survival Time
15
20
9
Figure 1. Plot of survival distribution functions for the NCI and the SCI Groups. The Y-axis is the probability of not declining to GDS 3 or above. The X-axis is the time (in years) to decline. (Barry Reisberg et al., 2010; Alzheimer & Dementia; in press.)
◦ However one should use the customized logistic regression procedures designed to account for 5
Key Assumption: Independent Censoring
Those still at risk at time t in the study are a random sample of the population at risk at time t, for all t
◦ T: survival time of a randomly selected
individual
t

◦ t: a specSif(itc)poPin(tTin tti)mee.xp (u)du
0

3
Hazard Function/Rate
Hazard Function (t): instantaneous
lim 0

S(t) S t
Here f(t) is the probabilitfytd endsiFtyt
function of the survival time Td.tThat is,
where F(t) is thFetcum1uSlativePT t
C uThrevKeaplan–Meier estimator is the
nonparametric maximum likelihood
estimate of S(t). It is a product of the
form
Sˆ(t)

r1

d1

r2

d2
...
ri

di
r1
r2
ri
distribution function of T:
4
2. The Key Word is ‘Censoring’
Because of censoring, many common data analysis procedures can not be adopted directly.
For example, one could use the logistic regression model to model the relationship between survival probability and some relevant covariates
failure rate at time t given that the
subject has survived upto time t. That

t
is
lim 0
P(t

T

t
|T

t)

P(t T t P(T t)
)
S(t) S(t ) 1 f t
Data are typically subject to censoring (e.g. when a study ends before the event occurs)
Survival Function - A function describing the proportion of individuals surviving to or beyond a given time. Notation:
rk
is ttkhe number of subjects alive just
dbkefore time
tk
denotes the number who died at time
7
Kaplan-Meier Curve, Example
Time ti # at risk # events
0
This assumption means that the hazard function, λ(t), can be estimated in a fair/unbiased/valid way
6
3A. Kaplan-Meier (Product-
Limit) Estimator of the Survival
20
0
Sˆ
1.00
5
20
2
[1(2/20)]*1.00=0.90
6
18
0
[1(0/18)]*0.90=0.90

10
15
1
[1(1/15)]*0.90=0.84
13
14
2
(1-
8
Kaplan Meier Curve
Proportion Surviving (95% Confidence) 0.6 0.7 0.8 0.9 1.0
Survival Analysis
A Brief Introduction
2
1. Survival Function, Hazard
Function
In many medical studies, the primary endpoint is time until an event occurs (e.g. death, remission)