Download Survival Analysis

Survival Analysis
Key variable = time until some event
• time from treatment to death
• time for a fracture to heal
• time from surgery to relapse
Censored observations
• subjects removed from data set at
some stage without suffering an event
[lost to follow-up or died from unrelated
event]
• study period ends with some
subjects not suffering an event
Example
Patient
Time at
entry
(months)
Time at
death/
censoring
Dead or
censored
Survival
time
1
0.0
11.8
D
11.8
2
0.0
12.5
C
12.5 *
3
0.4
18.0
C
17.6*
4
1.2
6.6
D
5.4
5
3.0
18.0
C
15.0*
Survival analysis uses information
about subjects who suffer an event
and subjects who do not suffer an
event
Life Table
• Shows pattern of survival for a group of
subjects
• Assesses number of subjects at risk at each
time point and estimates the probability of
survival at each point
Motion sickness data
N=21 subjects placed in a cabin and subjected
to vertical motion
Endpoint = time to vomit
Motion sickness data
• 14 survived 2 hours without vomiting
• 5 subjects vomited at 30, 50, 51, 82 and 92
minutes respectively
• 2 subjects requested an early stop to the
experiment at 50 and 66 minutes
respectively
Life table
Subject
1
2
3
Survival time
(min)
30
50
50 *
Survival
proportion
0.952
0.905
4
5
6
7
8 – 21
51
66*
82
92
120*
0.855
0.801
0.748
Calculation of survival probabilities
pk = pk-1 x (rk – fk)/ rk
where p = probability of surviving to time k
r = number of subjects still at risk
f = number of events (eg. death) at
time k
Calculation of survival probabilities
Time 30 mins : (21 – 1)/21 = 0.952
Time 50 mins : 0.952 x (20 – 1)/20 = 0.905
Time 51 mins : 0.905 x (18 – 1)/18 = 0.854
Kaplan-Meier survival curve
• Graph of the proportion of subjects
surviving against time
• Drawn as a step function (the proportion
surviving remains unchanged between
events)
Survival Curve
1.0
Survival probability
.8
.6
.4
.2
0.0
0
30
60
TIME (mins)
90
120
Kaplan-Meier survival curve
• times of censored observations
indicated by ticks
• numbers at risk shown at regular
time intervals
Summary statistics
1. Median survival time
2. Proportion surviving at a specific
time point
Survival Curve
1.0
Survival probability
.8
.6
.4
.2
0.0
0
30
60
TIME (mins)
90
120
Comparison of survival in two groups
Log rank test
Nonparametric – similar to chi-square test
SPSS Commands
• Analyse – Survival – Kaplan-Meier
• Time = length of time up to event or last
follow-up
• Status = variable indicating whether event
has occurred
• Options – plots - survival
SPSS Commands
(more than one group)
• Factor = categorical variable showing
grouping
• Compare factor – choose log rank test
Example
RCT of 23 cancer patients
11 received chemotherapy
Main outcome = time to relapse
Proportion relapse-free
Chemotherapy example
1.0
.8
.6
.4
Chemotherapy
Yes
.2
Yes-censored
No
0.0
No-censored
0
20
40
60
80
100
120
Time (weeks)
140
160
180
Chemotherapy example
No chemotherapy
Median relapse-free time = 23 weeks
Proportion surviving to 28 weeks = 0.39
Chemotherapy
Median relapse-free time = 31 weeks
Proportion surviving to 28 weeks = 0.61
The Cox model
Proportional hazards regression analysis
Generalisation of simple survival analysis to
allow for multiple independent variables
which can be binary, categorical and
continuous
The Cox Model
Dependent variable = hazard
Hazard = probability of dying at a point in
time, conditional on surviving up to that
point in time
= “instantaneous failure rate”
The Cox Model
Log [hi(t)] =
log[h0(t)] + ß1x1 + ß2x2 + …….. ßkxk
where [h0(t)] = baseline hazard
and x1 ,x2 , …xk are covariates associated
with subject i
The Cox Model
hi(t) =
h0(t) exp [ß1x1 + ß2x2 + …….. ßkxk]
where [h0(t)] = baseline hazard
and x1 ,x2 , …xk are covariates associated
with subject i
The Cox Model
Interpretation of binary predictor variable defining
groups A and B:
Exponential of regression coefficient, b,
= hazard ratio (or relative risk)
= ratio of event rate in group A and event rate in
group B
= relative risk of the event (death) in group A
compared to group B
The Cox Model
Interpretation of continuous predictor
variable:
Exponential of regression coefficient, b,
refers to the increase in hazard (or relative
risk) for a unit increase in the variable
The Cox Model
Model fitting:
• Similar to that for linear or logistic
regression analysis
• Can use stepwise procedures such as
‘Forward Wald’ to obtain the ‘best’ subset
of predictors
The Cox model
Proportional hazards regression analysis
Assumption:
Effects of the different variables on event
occurrence are constant over time
[ie. the hazard ratio remains constant over
time]
SPSS Commands
• Analyse – Survival – Cox regression
• Time = length of time up to event or last follow-up
• Status = variable indicating whether event has
occurred
• Covariates = predictors (continuous and
categorical)
• Options – plots and 95% CI for exp(b)
The Cox model
Check of assumption of proportional hazards (for
categorical covariate):
• Survival curves
• Hazard functions
• Complementary log-log curves
For each, the curves for each group should not cross
and should be approximately parallel