Introduction to Survival Analysis October 13

1
Introduction to Survival AnalysisOctober 13
20, 2009

• Brian F. Gage, MD, MSc
• with thanks to Bing Ho, MD, MPH
• Dept. of Medicine
• Washington University in St. Louis

2
Goal Conceptual and Graphical Understanding of
Survival Analyses

• What is survival analysis
• When to use it?
• How it compares to alternative statistics
• Univariate method Kaplan-Meier curves
• Multivariate methods Cox-proportional hazards
  model
• Assessment of adequacy of model

3
Sample Kidney Transplant (Tx) Data
PID Years Donor Tx Fails 1
6.1790 Cadaveric 1 2 10.1604
Living-related 1 3 0.0260
Cadaveric 1 4 4.2967
Living-related 1 5 3.8560
Cadaveric 1 6 2.3644
Living-related 1 7 0.8420
Cadaveric 1 8 2.8048
Living-related 1 9 2.7940
Cadaveric 1 10 5.4670
Living-related 1 11 5.1450
Cadaveric 1 12 3.7554
Living-related 1
4
Univariate analysis of Tx survival in recipients
of cadaveric kidney
Mean 1.9 years Median1.3 years
5
Univariate analysis of Tx survival in recipients
of living-related kidney
Mean 3.0 years Median2.15 years
6
How Would You Analyze Those Data?

• All 2000 simulated pts. were followed until time
  of rejection or Tx failure.
• ltwrite your data analysis plan heregt

7
Univariate analysis of logarithm (Tx survival) in
recipients of living-related kidney
8
Univariate analysis of logarithm (Tx survival) in
recipients of cadaveric kidney
9
Comparisons of Log (Tx Survival)

• Variable Method Variances DF
  Pr gt t
• LnYears Pooled Equal 1998
  lt.0001
• LnYears Satterthwaite Unequal 1988
  lt.0001
• Variable Method Two-Sided Pr gt
  Z
• LnYears Wilcoxon/Mann-Whitney Two-Sample Test
  lt.0001

10
Suppose You Only Have Time/Money to Follow
Participants for 4.5 Years or that some Patients
Enrolled Late
PID Years Donor Tx Fails 1
4.5 Cadaveric 0 2 4.5
Living-related 0 3 0.0260
Cadaveric 1 4 4.2967
Living-related 1 5 3.8560
Cadaveric 1 6 2.3644
Living-related 1 7 0.8420
Cadaveric 1 8 2.8048
Living-related 1 9 2.7940
Cadaveric 1 10 4.5
Living-related 0 11 4.5
Cadaveric 0 12 3.7554
Living-related 1
11
Univariate analysis of Tx survival in recipients
of cadaveric kidney
Data censored at 4.5 years
12
Univariate analysis of Tx survival in recipients
of living-related kidney
Data censored at 4.5 years
13
Now, Survival Times are Censored

• A t-test is no longer appropriate
• We dont know how long patients will survive past
  the observation window
• We cant compute the mean (or SD) of survival
  time between the 2 cohorts
• although may be able to observe medians

14
To Analyze Censored Data, We Need to Use
Time-to-Event Analysis, Such as St

• Survivor function, S(t) defines the probability
  of surviving longer than time t
• Known as Kaplan-Meier curves or product-limit
• Does not account for other covariates
• Model time to failure or time to event
• Survival analysis has a dichotomous (binary)
  outcome
• Unlike logistic regression, survival analysis
  analyzes the time to an event
• Able to account for censoring
• But not covariates
• When is this OK?
• Can compare survival between 2 groups

15
Kaplan-Meier Plots of Kidney Tx
St
P lt .0001
Median survival
Living-Related Donor
Cadaveric Donor
16
How to Compare Kaplan-Meier Curves?

• Hypothesis test (test of significance)
• H0 the curves are statistically the same
• HA the curves are statistically different
• Compares observed to expected cell counts
• Test statistic is compared to ?2
• Do you weigh each failure equally?
• Yes gt Log-Rank (Mantel-Haenszel) Test
• or do you penalize early failure more?
• Yes gt Generalized Wilcoxon (Breslow) Test.

17
Time to Cardiovascular Adverse Event in VIGOR
Trial
P lt .001
1-S(t)
18
Censoring is Variable

• Subject does not experience event of interest
• Incomplete follow-up
• Lost to follow-up
• Withdraws from study
• Death (if not an endpoint)

Death
Death
Death
19
Importance of censored data

• Why are censored data important?
• In a Cox model, what is the key assumption of
  censoring?

20
When to use Survival Analysis

• When one suspects that 1 explanatory variable(s)
  explains the differences in time to an event
• Examples
• Time to death or clinical endpoint
• Time in remission after treatment of disease
• Recidivism rate after addiction treatment
• Especially when follow-up is incomplete or
  variable

21
P .0001
Gage B et al. Adverse outcomes and predictors of
underuse of antithrombotic therapy in Medicare
beneficiaries with chronic atrial fibrillation.
Stroke 200031822-7.
22
Limitation of Kaplan-Meier curves

• What happens when you have several covariates
  that you believe contribute to survival?
• Example
• Smoking, hyperlipidemia, diabetes, hypertension,
  contribute to time to myocardial infarct or
  stroke.
• Can use stratified K-M curves, but only for 2 or
  maybe 3 categorical covariates.
• Need another approach Cox proportional hazards
  model is most common for many covariates, esp.
  continuous ones

23
Multivariate method Cox proportional hazards

• Can assess the effect of multiple covariates on
  survival
• Cox-proportional hazards is the most commonly
  used multivariate survival method
• Easy to implement in SPSS, Stata, JMP, SAS, or R
• Parametric approaches are an alternative, but
  they require stronger assumptions about h(t)
• They yield a closed eqn. for S(t) and H(t)

24
Cox model Proportional hazard assumption

• Hazard Ratio (HR) exp(B) is a multiplicative
  riskthis is the proportional hazard assumption
• Can handle both continuous and categorical
  predictor variables
• Can stratify results using a categorical variable
• Cox models distinguish individual contributions
  of covariates on survival.

25
Hazard Rate h(t)

• of pts. dying per unit time in the interval
• of pts. alive at t
• h(t) is called the hazard rate, hazard
  function, conditional failure rate, or
  instantaneous failure rate.

ht
26
The Hazard Rate h(t)
ht lim ?(1-St)/ ?t / St
?t ? 0
?(1-St)
? t
27
Cox proportional hazard model

• Separates baseline hazard function (ho(t), which
  can be any shape) from covariates
• Baseline hazard function over time
• h(t) ho(t)exp(B1XBo)
• Covariates are not usually time independent
• But they can be
• B1 is used to calculate the hazard ratio, which
  is similar to the relative risk
• semiparametric

28
Time to Cardiovascular Adverse Event in VIGOR
Trial Should be Summarized w/ a Single HR,
Instead of
RR 2.6
RR 2.4
RR 1.9
RR 1.9
RR 1.2
29
Use These 2 Eqns. to Show How the Hazard Ratio
Changes when Binary Factor B1 Is Present (X1)
Rather than Absent (X0)

• ht hot exp(B1XBo)
• Hazard ratio (HR) htX1 / htX0
• Hint exp (a) / exp (b) exp (a-b)
• Relative risk reduction, RRR, 1-HR

30
Cox proportional hazards models

• Hazard Ratio (HR) exp(B) is a multiplicative
  riskthis is the proportional hazard assumption
• Sometimes can be compensated for by using an
  interaction term
• Can handle both continuous and categorical
  predictor variables
• can stratify results using a categorical variable
• no distribution assumption is required in that
  case

31
Output of Cox Proportional Hazard Model From
Simulated Kidney Tx Data
Analysis of Maximum Likelihood
Estimates Parameter
Standard Parameter DF Estimate Error
Chi-Square Pr gt ChiSq Donor 1
0.474 0.0493 92.3 lt.0001
Hazard 95 Hazard Ratio Parameter
Ratio Confidence Limits Donor
1.61 1.46 1.77
Thus, cadaveric Tx were 61 more likely to fail.
32
Limitations of Cox PH model

• Normally, does not include variables that change
  over time
• Luckily most variables (e.g. gender, ethnicity,
  and congenital condition, birth year) are constant

33
Example Tumor Extent

• 3000 patients derived from SEER cancer registry
  and Medicare billing information
• Explore the relationship between tumor extent and
  survival
• Hypothesis is that more extensive tumor
  involvement is related to poorer survival

34
Log-Rank ?2 269 p lt.0001
35
Example Tumor Extent

• Tumor stage may not be the only covariate that
  affects survival
• Medical comorbidities poor functional status
  may be associated with poorer outcome
• Ethnicity and gender may contribute
• Tumor grade and genotype may contribute
• Etc.
• Cox proportional hazards model could quantify
  these relationships

36
Summary of Kaplan-Meier Curves

• Model time to failure or time to event
• Survival analysis has a dichotomous (binary)
  outcome
• Unlike logistic regression, survival analysis
  analyzes the time to an event
• Able to account for censoring
• Can compare survival between 2 groups

37
Summary of time-to-event analyses

• Quantifies time to a single, dichotomous event
• Handles censored data well
• Cox models distinguish individual contributions
  of covariates on survival, provided certain
  assumptions are met
• Cox models are used commonly in outcomes
  research. To learn more, take a full-course in
  survival analysis
• E.g. Math 434 - Survival Analysis or
  http//k30.im.wustl.edu/program/interm20biostats
  20syllabus.doc
• BST.520 Survival Data Analysis at SLU