Healthcare
Helpm31990
9781284108217_CH11_SLID.pptChapter 11
Survival Analysis
Learning Objectives (1 of 2)
- 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 (2 of 2)
- 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
<Insert Figure 11-2>
Is the Following Different?
Survival Curve – Survival Function
Survival Curve with 95% CI
Estimating the Survival Function
- There are many parametric approaches (which make certain assumptions about survival times).
- We focus on two nonparametric approaches.
- Actuarial or life-table approach
- Kaplan–Meier approach
- Participants are 65 years and older, followed for up to 24 years until they die, the study ends, or they drop out.
- n = 20 participants are enrolled over a 5-year period.
Example 11.2.
Estimating the Survival Function
(1 of 2)
Example 11.2.
Estimating the Survival Function
(2 of 2)
- 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 Two 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.
RCT to Compare Two Treatments for Advanced Gastric Cancer
Example 11.3.
Log-Rank Test (1 of 2)
H0: Two survival curves are identical
H1: Two survival curves are not identical
Test statistic:
Example 11.3.
Log-Rank Test (2 of 2)
- Reject H0 if c2 ≥ 3.84.
- Reject H0 because 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 (1 of 2)
- 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 (2 of 2)
- Model
ln(h(t)/h0(t)) = b1X1 + b2X2 + … + bpXp
- exp(bi) = hazard ratios
Example 11.5.
Cox Proportional Hazards Regression
(1 of 3)
- Framingham Study
- Outcome = all-cause mortality
- N = 5180 participants ≥ 45 years
- 10-year follow-up
- Analysis with Cox proportional hazards regression
Example 11.5.
Cox Proportional Hazards Regression
(2 of 3)
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
(3 of 3)
- 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)
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