Transcript Survival Analysis-1

Survival Analysis-1



In Survival Analysis the outcome of interest is
time to an event
The event does not necessarily have to be death
It can be HRT and first thrombotic episode
OR
Exclusive breast feeding and another pregnancy
OR
Exercise and time to maximum tolerance.
Survival Analysis-2
Studying time to an event poses two difficulties:
Time interval can vary from one subject to another.
Also at the end of the study the event may not have
occurred for many subjects.
So time to event is not Normally distributed.
In a long period of follow-up many subjects drop
out. The only information we have about them is
until last follow-up. These are Censored
Observations.
Assumptions in Survival Analysis




Survival prospects are assumed to stay the same
throughout the study.
Subjects lost to follow-up have the same
prognosis as those who continue with the study.
The probability that an individual is censored is
unrelated to the probability that the individual
suffers an event.
Anything which affects the hazard does so by
the same proportion at all time.
Calculating Kaplan –Meier Survival Curve
At the start of the study (t0), proportion surviving = 1.
Order survival time by increasing duration starting with the
shortest.
At each time interval from 1 to n survival time is calculated by
S (ti) = ( ri – di / ri ) x S (ti -1)
Where di = number of events at time ti
For censored observations di = 0.
ri = number alive just before it.
For censored observations survival curve remains unchanged. At the
next event the number ‘at risk’ is reduced by the number censored
Hazard Curve
H0 is the baseline hazard at time t i.e. h0(t)
For any individual subject the hazard at time t is hi(t).
Hi(t) is linked to the baseline hazard h0(t) by
loge {hi(t)} = loge{h0(t)} + β1X1 + β2X2 +……..+ βpXp
where X1, X2 and Xp are variables associated with the subject.
Then hi(t) = h0(t) e β1X1 + β2X2 +….+ βpXp
H0(t) represents the hazard for an individual where X1=X2=Xp=0.
h(t) / h0(t) depends on the predictor variables and not on t
h(t) / h0(t) is the hazard ratio
Data Analysis

The study to be analysed is about comparing
survival time of 49 patients with colorectal
cancer after random allocation to γ linoleic acid
and controls
Results - 1
Standard
Mean(MTTF)
Error
23.7425
3.0506
IQR =
95.0% Normal CI
Lower
Upper
17.7634
29.7216
Median =
23.0000 Q1 =
24.0000
13.0000
Q3 =
36.0000
Kaplan-Meier Estimates
Time
1.000
5.0000
9.0000
10.0000
12.0000
13.0000
Number
at Risk
25
24
21
20
17
12
Number
Failed
1
1
1
1
1
1
Survival
Probability
0.9600
0.9200
0.8762
0.8324
0.7834
0.7181
Standard
Error
0.0392
0.0543
0.0671
0.0767
0.0864
0.1009
95.0% Normal CI
Lower
Upper
0.8832
1.0000
0.8137
1.0000
0.7447
1.0000
0.6821
0.9827
0.6140
0.9528
0.5204
0.9159
Mean Time to Failure is the measure of the average survival time
The Median gives the survival time for 50% of the subjects followed by survival probability
table.
Results - 2
Nonparametric Survival Plot for Time linoleic-Time Cntrol
Empirical Hazard Function
Censoring Column in Death-Censored
Censoring Column in Death-Censored
Time linoleic
Time Cntrol
1.0
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0
5
10
15
20
25
30
35
40
45
0
Time to Failure
5
10
15
20
25
Time to Failure
Comparison of Survival Curves
Test Statistics
Method
Chi-Square
Log-Rank
0.06986
Time linoleic
Time Cntrol
1.0
0.9
Rate
Probability
Nonparametric Hazard Plot for Time linoleic-Time Cntrol
Kaplan-Meier Method
DF
1
P-Value
0.7915
30
35
40
45