Survival Analysis
• In many medical studies, the primary endpoint is
time until an event occurs (e.g. death, remission)
• Data are typically subject to censoring 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:
–
–
–

T  survival time of a randomly selected individual
t  a specific point in time.
S(t) = P(T > t)  Survival Function
l(t)  instantaneous failure rate at time t aka hazard
function
Kaplan-Meier Estimate of Survival Function
• Case with no censoring during the study (notes
give rules when some individuals leave for other
reasons during study)
–
–
–
–
Identify the observed failure times: t(1)<···<t(k)
Number of individuals at risk before t(i)  ni
Number of individuals with failure time t(i)  di
Estimated hazard function at t(i):
^
di
li 
ni
– Estimated Survival Function at time t
^
S (t )  i|t
^
(i )
(1  l i ) 
t
# with T  t
n
(when no censoring)
Example - Navelbine/Taxol vs Leukemia
• Mice given P388 murine leukemia assigned at random
to one of two regimens of therapy
– Regimen A - Navelbine + Taxol Concurrently
– Regimen B - Navelbine + Taxol 1-hour later
• Under regimen A, 9 of nA=49 mice died on days:
6,8,22,32,32,35,41,46, and 54. Remainder > 60 days
• Under regimen B, 9 of nB=15 mice died on days:
8,10,27,31,34,35,39,47, and 57. Remainder > 60 days
Source: Knick, et al (1995)
Example - Navelbine/Taxol vs Leukemia
Regimen B
Regimen A
i
1
2
3
4
5
6
7
8
t(i)
6
8
22
32
35
41
46
54
^ A
l1
^ A
l2
ni
49
48
47
46
44
43
42
41
di
1
1
1
2
1
1
1
1
li
.020
.021
.021
.043
.023
.023
.024
.024
1

 .020
49
1

 .021
48
S(t(i))
.980
.959
.939
.899
.878
.858
.837
.817
i
1
2
3
4
5
6
7
8
9
t(i)
8
10
27
31
34
35
39
47
57
ni
15
14
13
12
11
10
9
8
7
di
1
1
1
1
1
1
1
1
1
li
.067
.071
.077
.083
.091
.100
.111
.125
.143
^ A
S (6)  1  .020  .980
^ A
S (8)  .980(1  .021)  .959
S(t(i))
.933
.867
.800
.733
.667
.600
.533
.467
.400
Example - Navelbine/Taxol vs Leukemia
Survival Functions
1.1
1.0
.9
.8
.7
REGIMEN
Cum Survival
.6
2
.5
2-censored
.4
1
.3
1-censored
0
DAY
10
20
30
40
50
60
70
Log-Rank Test to Compare 2 Survival
Functions
• Goal: Test whether two groups (treatments) differ
wrt population survival functions. Notation:
–
–
–
–
–
t(i)  Time of the ith failure time (across groups)
d1i  Number of failures for trt 1 at time t(i)
d2i  Number of failures for trt 2 at time t(i)
n1i  Number at risk prior for trt 1 prior to time t(i)
n2i  Number at risk prior for trt 2 prior to time t(i)
• Computations:
d i  d1i  d 2 i
e1i 
ni  n1i  n2 i
n1i d1i
ni
O1  E1 
v1i 
 d
i
1i
n1i n2 i d i ( ni  d i )
ni2 ( ni  1)
 e1i 
V1 
v
i
1i
Log-Rank Test to Compare 2 Survival
Functions
• H0: Two Survival Functions are Identical
• HA: Two Survival Functions Differ
T .S . : TMH
O1  E1

V1
R.R. : | TMH | z / 2
P  val : 2 P ( Z | TMH |)
Some software packages conduct this identically as a chi-square
test, with test statistic (TMH)2 which is distributed c12 under H0
Example - Navelbine/Taxol vs Leukemia (SPSS)
Survival Analysis for DAY
REGIMEN
REGIMEN
1
2
Overall
Total
Number
Events
Number
Censored
Percent
Censored
49
15
9
9
40
6
81.63
40.00
64
18
46
71.88
Test Statistics for Equality of Survival Distributions for REGIMEN
Statistic
Log Rank
10.93
df
1
Significance
.0009
This is conducted as a chi-square test, compare with notes.
Relative Risk Regression - Proportional
Hazards (Cox) Model
• Goal: Compare two or more groups (treatments),
adjusting for other risk factors on survival times
(like Multiple regression)
• p Explanatory variables (including dummy
variables)
• Models Relative Risk of the event as function of
time and covariates:
l (t; x1 ,, x p ) l (t; x1 ,, x p )
RR (t ; x1 ,, x p ) 

l (t;0,,0)
l0 (t )
Relative Risk Regression - Proportional
Hazards (Cox) Model
• Common assumption: Relative Risk is constant
over time. Proportional Hazards
• Log-linear Model:
RR (t; x1 ,, x p )  e
1 x x   p x p
• Test for effect of variable xi, adjusting for all other predictors:
•H0: i = 0
(No association between risk of event and xi)
• HA: i  0
(Association between risk of event and xi)
Relative Risk for Individual Factors
• Relative Risk for increasing predictor xi by 1 unit,
controlling for all other predictors:
RRi  e
i
• 95% CI for Relative Risk for Predictor xi:
• Compute a 95% CI for i :
^
^
 i  1.96  
^
i
• Exponentiate the lower and upper bounds for CI for RRi
Example - Comparing 2 Cancer Regimens
• Subjects: Patients with multiple myeloma
• Treatments (HDM considered less intensive):
– High-dose melphalan (HDM)
– Thiotepa, Busulfan, Cyclophosphamide (TBC)
• Covariates (That were significant in tests):
– Durie-Salmon disease stage III at diagnosis (Yes/No)
– Having received 3+ previous treatments (Yes/No)
• Outcome: Progression-Free Survival Time
• 186 Subjects (97 on TBC, 89 on HDM)
Source: Anagnostopoulos, et al (2004)
Example - Comparing 2 Cancer Regimens
• Variables and Statistical Model:
– x1 = 1 if Patient at Durie-Salmon Stage III, 0 ow
– x2 = 1 if Patient has had  3 previos treatments, 0 ow
– x3 = 1 if Patient received HDM, 0 if TBC
RR (t ; x1 , x2 , x3 )  e 1x1   2 x 2   3 x3
Of primary importance is 3:
• 3 = 0  Adjusting for x1 and x2, no difference in risk for HDM and TBC
• 3 > 0  Adjusting for x1 and x2, risk of progression higher for HDM
• 3 < 0  Adjusting for x1 and x2, risk of progression lower for HDM
Example - Comparing 2 Cancer Regimens
• Results: (RR=Relative Risk aka Hazard Ratio)
Variable
D-S Stage 3 (x1)
3+ Previous Trts (x2)
HDM Trt Regimen (x3)
b (sb)
0.54 (0.21)
0.48 (0.18)
-0.16 (0.20)
RR (95%CI)
1.71 (1.14 – 2.54)
1.61 (1.13 – 2.39)
0.85 (0.57 – 1.28)
P-value
.009
.009
.44
Conclusions (adjusting for all other factors):
• Patients at Durie-Salmon Stage III are at higher risk
• Patients who have had  3 previous treatments at higher risk
• Patients receiving HDM at same risk as patients on TBC