1. April 18 Intro to survival analysis Le 11.1 – 11.2
Not covered in C & S

2. Intro to Survival Data Our voyage so far… Continuous outcome data
T-tests, linear regression, ANOVA
Categorical data
Odds ratios, relative risk, chi-square tests, logistic regression
New scenario; time to event data
Categorical outcome (yes/no)
Follow-up time

3. Rational
Want to take into account not just whether a patient has an event of interest but the amount of time from some starting point until the event.
Patient who dies 2 weeks after diagnosis of cancer should be considered differently than a patient who dies 2 years after diagnosis

4. Goals Describe the rate (probability) of the event over time
Called the survival function
Compare survival function among groups
Examine risk factors for having the event taking into consideration the time of the event

5. Kaplan-Meier survival curve Survival After Diagnosis of Lung Cancer
S (t) is the probability of surviving to at least t
S (200) = 0.37

6. Comparing Two Survival Curves

7. Time To ? Death after diagnosis of cancer
CVD event after enrolled in a study
Re-arrest after release from prison
Divorce after marriage
Survival analyses better described as “Time to Event” analyses
Note: The event does not have to inevitable

8. Kaplan-Meier Life Curves

9. Nature of Data Definitive starting point (become “at risk”)
Definitive ending point
If had event then date of event
If did not have event then date last know not to have had the event
Analyses based on two factors:
Had event or did not have event (0/1 variable)
Length of time followed (ending – starting date)

10. Examples Death after diagnosis of cancer Divorce after marriage
Starting point: date of diagnosis
Ending point: date of death or date last know to be alive
Divorce after marriage
Starting point: date of marriage
Ending point: date of divorce or date last know to be still married

11. Censoring
After a certain period of time the patient does not have the event but it is unknown as to whether the patient had the event after this time.
Called right censoring

12. Reasons for Censoring
Patient no longer followed (thus event status not know after a certain date)
Patient has a different event that make the primary event not possible
Primary event: death from cancer but patient dies from CHD
Primary event: divorce but one spouse dies
Study could end or patient becomes lost
Patient no longer “at risk” for study purposes

13. Censoring example Follow-up for study is 365 days
Patient survives 245 days then is lost
At that point, we KNOW that they survived 245 days but we do NOT KNOW whether they survived between days 246 and 365
If we exclude them from any end-point calculations we ignore 245 days worth of information

14. Types of censoring Uninformative Informative
“lost” status not related to outcome
Those lost similar to those not lost (usually not true)
Informative
“lost” status is related to outcome
Those who are lost are more likely to be dead than those not lost
Most methods assume we have uninformative censoring
Could be true, say an entire clinical center closes

15. Example of Follow-up TimesC O U P E S
Divorced after 6 years D C
Has been married 10 years at time of analyses C
One spouse dies after 3 yrs C
No contact with couple after 5 years
Years Since Marriage

16. Survival Function Estimation
Patients are followed for different length of time
Like to use all the data to estimate the survival function
Patients followed 1-year can help estimate survival function in first year
Patients followed 2-years can help estimate survival function in first 2-years

17. Life Table Calculation
100 couples married in followed 2 years
100 couples married in followed 1 year
Follow-up through 2004
Year 1 of follow-up
Year 2 of follow-up
10 D (5 each from 2002 and 2003 marriages)
200
8 D
95 C
190 95
S (1) = 190/200 = .95 87
S (2) = S (1) * S (2| S>1)
= .95 * .92 = .870
Note: S (1) is estimated with more precision than S(2)

18. Estimating Survival Curves
Kaplan-Meier Method
Also called Product-Limit or Life-table curve
For each time where 1 or more events occur, calculate number who die at that point over number who survived to that point (di/ni)
Multiply all these quantities;

19. Calculating Kaplan-Meier estimatesni di
1-di/ni
S(ti) 6 21 3
0.8571 7 17 1
0.9412
0.8067 10 16
0.9375
0.7563 13 14 2
0.6483
Number at risk
SAS calculates these automatically
0.8571 x x x

20. Questions
What is the survival rate over time for persons diagnosed with lung cancer?
Is the survival rate over time different for different types of cancer?
Are patient characteristics related to survival

21. Comparing Two Survival Curves

22. How do we describe this data?
Logistic regression?
Model risk of death
Would ignore the amount of follow-up time
Linear regression?
Model survival time
How do you handle those who died vs. those who survived?
Survival times not normally distributed (all >0)
Need new methods that incorporate follow-up time information
Survival or time-to-event analyses

23. Comparing survival curves
For any time point, can see probability of survival for either group
Median survival time; point where probability surviving = 50%
Rank Tests – Compare entire curves

24. Estimating survival curves
Survival curve estimates less precise over time
SAS can produce confidence intervals for the survival curve
95% CI of form;

25. Testing survival curves
Formal statistical tests exist
Log-rank test and Wilcoxon test
Both assess whether survival distributions are equal
Null hypothesis: survival distributions (curves) are equal
Alternative hypothesis: survival distributions (curves) are not equal; one greater/less than other
Each compares survival distributions in a slightly different way
Log-rank test more powerful when relative risk is constant
Wilcoxon more powerful for detecting short term risk

26. USING SAS Patient died 72 days after diagnosis
Obs Age Cell death SurVTime
squamous
squamous
squamous
squamous
squamous
squamous
squamous
squamous
large
large
large
large
Patient alive after 100 days but status after that time is unknown

27. PROC LIFETEST PLOTS = (s); WHERE cell in('squamous','large');
TIME survtime*death(0);
STRATA cell;
Tells SAS to draw life table plot
Tells SAS that values of 0 are censored observations
Tells SAS to compute life table estimates separately for each cell type

28. RUNNING ON SATURN (UNIX)
GOPTIONS DEVICE = png htext=0.8 htitle=1
ftext=swissb gsfmode=replace
PROC LIFETEST PLOTS = (s);
WHERE cell in('squamous','large');
TIME survtime*death(0);
STRATA cell;
Creates a file called sasgraph.png
FTP over to PC and insert file into word
insert/ picture/ from file

29. PROC LIFETEST OUTPUT
Summary of the Number of Censored and Uncensored Values
Percent
Stratum Cell Total Failed Censored Censored
1 large
2 squamous
Total

30. Test of Equality over Strata
Pr >
Test Chi-Square DF Chi-Square
Log-Rank
Wilcoxon
-2Log(LR)
Tests equality of 2 survival functions

31. X-Y points for life table graph
Stratum 1: Cell = large
Product-Limit Survival Estimates
Survival
Standard Number Number
SurvTime Survival Failure Error Failed Left *
X-Y points for life table graph
First death after 12 days

32. Stratum 1: Cell = large
Product-Limit Survival Estimates
Survival
Standard Number Number
SurvTime Survival Failure Error Failed Left
S(0) = 1
S(12) = (26/27)
S(15) = (25/27) which is also 26/27 * 25/26
S(19) = (24/27)
What is S(17) ?
Estimated survival function is a step function

33. 2 patients died after 1 day
Stratum 2: Cell = squamous
Product-Limit Survival Estimates
Survival
Standard Number Number
SurvTime Survival Failure Error Failed Left
25.000*
2 patients died after 1 day

34. Crossing Survival curves
Validity of tests require risk in one group always greater than risk in other group
When survival curves cross, terms used in calculating test statistic cancel out
Give test statistic value near zero
P-value is larger than it should be
Graph survival curves to check for crossing
Use alternative method

35. Censoring vs. missing data
Censoring is a special case of having missing data
Missing; don’t know whether or not person had outcome
Censoring; don’t know whether or not person had outcome, but know they didn’t have outcome after being followed for some time

36. Statistical Techniques for censored data
Kaplan-Meier (life table analysis)
Survival curves
log rank, wilcoxon significance tests
Tests to compare survival curves
Cox proportional hazards regression
Relate covariates to survival