Survival Analysis
Mar 29, 2015
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 tableSubject Survival time
(min)Survival proportion
1 30 0.952
2 50 0.905
3 50 *
4 51 0.855
5 66*
6 82 0.801
7 92 0.748
8 – 21 120*
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
TIME (mins)
1209060300
Su
rviv
al p
rob
ab
ility
1.0
.8
.6
.4
.2
0.0
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
TIME (mins)
1209060300
Su
rviv
al p
rob
ab
ility
1.0
.8
.6
.4
.2
0.0
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
Chemotherapy example
Time (weeks)
180160140120100806040200
Pro
po
rtio
n r
ela
pse
-fre
e1.0
.8
.6
.4
.2
0.0
Chemotherapy
Yes
Yes-censored
No
No-censored
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 modelProportional 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 modelProportional 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