Statistics in Health Care Management week 7
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week7.pdf
Chapter 11
Survival Analysis
Learning Objectives
• Identify applications with time to event
outcomes
• Construct a life table using the actuarial
approach
• Construct a life table using the Kaplan-Meier
approach
Learning Objectives
• Perform and interpret the log-rank test
• Compute and interpret a hazard ratio
• Interpret regression coefficients in a Cox
proportional hazards regression analysis
Survival Analysis
• Outcome is time to event
– Time to heart attack, cancer remission, death
• Measure whether person has event or not
(Yes/No) and Time to event
• Estimate “survival time”
• Determine factors associated with longer
survival
Issues with Time to Event Data
• Times are positive (often skewed)
• Incomplete follow-up information
– Some participants enroll late
– Some participants drop-out
– Study ends
• Censoring
– Measure follow-up time and not time to event
– We know survival time > follow-up time
Experiences of n=10 Participants
Experiences of Same n=10 Participants, Time
Projected to Zero
Is the Following Different?
Survival Curve – Survival Function
Survival Curve with 95% CI
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 5 10 15 20 25
Time, Years
S ur
vi va
l P
ro ba
bi li
ty
Estimating the Survival Function
• There are many parametric approaches (which
make certain assumptions about survival
times)
• We focus on two non-parametric approaches
– Actuarial or life table approach
– Kaplan-Meier approach
Example 11.2.
Estimating the Survival Function • Participants are 65 years and older, followed
for up to 24 years until the die, until the study
ends or until they drop out.
• n=20 participants are enrolled over a 5 year
period.
Example 11.2.
Estimating the Survival Function Year of Death or Year of Last Contact
• Years of Death: 3, 14, 1, 23, 5, 17
• Years of Last Contact: 24, 11, 19, 24, 13, 2,
18, 17, 24, 21, 12, 10, 6, 9
Notation
Nt = number of participants who are event free and
considered at risk during interval
Dt = number who suffer event during interval
Ct = number censored during interval
qt = proportion suffering event during interval
pt = proportion surviving interval
St = proportion surviving past interval
Example 11.2. Life Table
Example 11.2.
Life Table – Actuarial Approach
Example 11.2. Life Table – Kaplan-Meier Approach
Example 11.2 Survival Function
Comparing Survival Curves
• Log rank test to compare survival in two or
more independent groups
• Chi-square test that compares the observed
numbers of events to what would be expected
if the groups had equal survival
Example 11.3.
Comparing Survival
• Clinical trial to compare two treatments for advanced
gastric cancer
• n=20 participants with stage IV cancer are randomly
assigned to receive chemotherapy before surgery or
chemotherapy after surgery
• Primary outcome is death
• Participants are followed for up to 48 months
following enrollment
RCT to Compare 2 Treatments for
Advanced Gastric Cancer
Log Rank Test
H0: Two survival curves are identical
H1: Two survival curves are not identical
Test statistic:
Reject H0 if c2 > c2,df where df=k-1 and
k=number of comparison groups
jt
2
jtjt2
E
)EO( χ
RCT to Compare 2 Treatments for
Advanced Gastric Cancer
Example 11.3
Log Rank Test
H0: Two survival curves are identical
H1: Two survival curves are not identical
Test statistic:
151.6 380.6
)380.63(
620.2
)620.26(
E
)EO( χ
22
jt
2
jtjt2
Example 11.3.
Log Rank Test
Reject H0 if c2 > 3.84.
Reject H0 since 6.151> 3.84. We have statistical
evidence that two survival curves are not
identical.
Comparing Survival Curves
H0: Two survival curves are equal
c2 Test with df=1. Reject H0 if c2 > 3.84
c2 = 6.151. Reject H0.
Cox Proportional Hazards Regression
• Model
h(t) = h0(t) exp (b1X1 + b2X2 + … + bpXp)
• Where h(t) = hazard at time t (risk of
failure at time t),
h0(t)= baseline hazard,
Xi are predictors,
bi are regression coefficients
Cox Proportional Hazards Regression
• Model
ln(h(t)/h0(t)) = b1X1 + b2X2 + … + bpXp
• exp(bi) = hazard ratios
Example 11.5.
• Framingham Study
– Outcome = all-cause mortality
– N=5,180 participants > 45 years
– 10 year follow-up
– Analysis with Cox Proportional Hazards
Regression
Example 11.5.
Cox Proportional Hazards Regression
bi p HR
Age 0.11149 0.0001 1.118
Male Sex 0.67958 0.0001 1.973
Example 11.5.
Cox Proportional Hazards Regression
Multivariable Model
bi p HR (95% CI)
Age 0.11691 0.0001 1.12 (1.11-1.14)
Male Sex 0.40359 0.0001 1.50 (1.22-1.85)
SBP 0.11691 0.0001 1.02 (1.01-1.02)
Current
Smoker 0.40359 0.0001 2.16 (1.76-2.64)
Total Chol 0.40359 0.0001 1.00 (0.99-1.00)
Diabetes 0.40359 0.0001 0.82 (0.62-1.08)
