Survival Analysis:
From Square One to
Square Two
Yin Bun Cheung, Ph.D.
Paul Yip, Ph.D.
Readings
Lecture structure
• Basic concepts
• Kaplan-Meier analysis
• Cox regression
• Computer practice
What’s in a name?
• time-to-event data
• failure-time data
• censored data
(unobserved outcome)
Types of censoring
– loss to follow-up
during the study period
– study closure
Examples of survival analysis
1. Marital status & mortality
2. Medical treatments & tumor
recurrence & mortality in
cancer patients
3. Size at birth & developmental
milestones in infants
Why survival analysis ?
• Censoring (time of
event not observed)
• Unequal follow-up time
What is time?
What is the origin of time?
In epidemiology:
•Age (birth as time 0) ?
•Calendar time since a
baseline survey ?
What is the origin of time?
In clinical trials:
• Since randomisation ?
• Since treatment begins ?
• Since onset of exposure ?
The choice
of origin of time
• Onset of continuous exposure
• Randomisation to treatment
• Strongest effect on the hazard
Types of survival analysis
1. Non-parametric method
Kaplan-Meier analysis
2. Semi-parametric method
Cox regression
3. Parametric method
Square 1 to square 2
This lecture focuses on two
commonly used methods
• Kaplan-Meier method
• Cox regression model
KM survival curve
Day Death / At
(t) Cens. risk
1
4
2
1d
4
3
1c
3
4
1d
2
5
1c
1
Pt(d) Pt(surv)
S(t)
0.00
0.25
0.00
0.50
0.00
1.00
0.75
0.75
0.38
0.38
1.00
0.75
1.00
0.50
1.00
5
* d=death, c=censored, surv=survival
KM survival curve
1.00
S(t)
0.75
0.50
0.25
0.00
0
1
2
Day
3
4
5
No. of expected deaths
Expected death in group A at time
i, assuming equality in survival:
EAi =no. at risk in group A i  death i
total no. at risk i
Total expected death in group A:
EA =  EAi
Log rank test
•A comparison of the number
of expected and observed
deaths.
•The larger the discrepancy,
the less plausible the null
hypothesis of equality.
An approximation
The log rank test statistic is
often approximated by
X2 = (OA-EA)2/EA+ (OB-EB)2/EB,
where OA & EA are the
observed & expected number
of deaths in group A, etc.
1
1
.8
.8
.6
.6
S(t)
S(t)
Proportional hazard assumption
.4
.4
.2
.2
0
0
0
5
10
Time
15
20
Log rank test
preferred (PH true )
0
5
10
Time
15
20
Breslow test
preferred (non-PH)
Proportion
Risk, conditional risk, hazard
.5
.4
.3
.2
.1
0
0
1
2
Month
3
Hazard
Hazard
Another look of PH
0
5
10 15 20
Time
0
5
10 15 20
Time
Log rank test
Breslow test
preferred (PH true ) preferred (non-PH)
Cox regression model
• Handles 1 exposure variables.
• Covariate effects given as
Hazard Ratios.
• Semi-parametric: only assumes
proportional hazard.
Cox model in the case
of a single variable
1. hi(t) = hB(t)  exp(BXi)
2. hj(t) = hB(t)  exp(BXj)
3. hi(t)/hj(t) =exp[B(Xi-Xj)]
exp(B) is a Hazard Ratio
Test of proportional
hazard assumption
• Scaled Schoenfeld residuals
• Grambsch-Therneau test
• Test for treatmentperiod
interaction
• Example: mortality of widows
Computer practice
A clinical trial of
stage I bladder tumor
Thiotepa vs Control
Data from StatLib
Data structure
Two most important variables:
• Time to recurrence (>0)
• Indicator of failure/censoring
(0=censored; 1=recurrence)
(coding depends on software)
KM estimates
1.00
S(t)
0.75
Thiotepa
0.50
Control
0.25
0.00
0
20
40
Months
60
Log rank test
Recurrence
Group
Observed Expected
Control
29
24.9
Thiotepa
18
22.1
chi2(1) = 1.52
Pr>chi2 = 0.22
Cox regression models
Model I Model II
HR
HR
Thiotepa
0.70
0.60
(vs Control) (P=0.23) (P=0.11)
Number of
1.26
tumor
(P<0.01)
Test of PH assumption
Grambsch-Therneau test
for PH in model II
• Thiotepa
P=0.55
• Number of tumor P=0.60
Major References
(Examples)
Ex 1. Cheung. Int J Epidemiol
2000;29:93-99.
Ex 2. Sauerbrei et al. J Clin Oncol
2000;18:94-101.
Ex 3. Cheung et al. Int J Epidemiol
2001;30:66-74.
Major References
(General)
• Allison. Survival Analysis using
the SAS® System.
• Collett. Modelling Survival
Data in Medical Research.
• Fisher, van Belle. Biostatistics:
A Methodology for the Health
Sciences.