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April 18. Intro to survival analysis Le 11.1 – 11.2 Not covered in C & S. 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 - PowerPoint PPT Presentation
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April 18
• Intro to survival analysis
• Le 11.1 – 11.2
• Not covered in C & S
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
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
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
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
Comparing Two Survival Curves
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
Kaplan-Meier Life Curves
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)
Examples
• Death after diagnosis of cancer– 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
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
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
Patient no longer “at risk” for study purposes
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
Types of censoring
• Uninformative– “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
Example of Follow-up Times
Years Since Marriage
C
O
U
P
E
S
D
C
C
0 5 10
Has been married 10 years at time of analyses
Divorced after 6 years
One spouse dies after 3 yrs
C No contact with couple after 5 years
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
200
10 D (5 each from 2002 and 2003 marriages)
190
95 C
95
8 D
87 S (1) = 190/200 = .95
S (2) = S (1) * S (2| S>1)
= .95 * .92 = .870
Year 1 of follow-upYear 2 of follow-up
100 couples married in 2002 followed 2 years
100 couples married in 2003 followed 1 year
Follow-up through 2004
Life Table Calculation
Note: S (1) is estimated with more precision than S(2)
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;
S(t) (1 di /ni) (1 d1 /n1)(1 d2 /n2)...
Calculating Kaplan-Meier estimates
ti ni di 1-di/ni S(ti)
6 21 3 0.8571 0.8571
7 17 1 0.9412 0.8067
10 16 1 0.9375 0.7563
13 14 2 0.8571 0.6483
• SAS calculates these automatically
0.8571 x 0.9412 x 0.9375 x 0.8571
Number at risk
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
Comparing Two Survival Curves
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
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
Estimating survival curves
• Survival curve estimates less precise over time
• SAS can produce confidence intervals for the survival curve
• 95% CI of form;
S(t)exp(1.96SE(S(t))
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
Obs Age Cell death SurVTime
1 69 squamous 1 72
2 64 squamous 1 411
10 70 squamous 0 100
11 81 squamous 1 42
12 63 squamous 1 8
13 63 squamous 1 144
14 52 squamous 0 25
15 48 squamous 1 11
23 41 large 1 200
24 66 large 1 156
25 62 large 0 182
26 60 large 1 143
Patient died 72 days after diagnosis
Patient alive after 100 days but status after that time is unknown
USING SAS
PROC LIFETEST PLOTS = (s); WHERE cell in('squamous','large'); TIME survtime*death(0); STRATA cell;
Tells SAS that values of 0 are censored observations
Tells SAS to compute life table estimates separately for each cell type
Tells SAS to draw life table plot
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.pngFTP over to PC and insert file into word
insert/ picture/ from file
PROC LIFETEST OUTPUT
Summary of the Number of Censored and Uncensored Values
Percent
Stratum Cell Total Failed Censored Censored
1 large 27 26 1 3.70
2 squamous 35 31 4 11.43
---------------------------------------------------------------
Total 62 57 5 8.06
Test of Equality over Strata
Pr >
Test Chi-Square DF Chi-Square
Log-Rank 0.8226 1 0.3644
Wilcoxon 0.0520 1 0.8197
-2Log(LR) 1.0218 1 0.3121
Tests equality of 2 survival functions
Stratum 1: Cell = large
Product-Limit Survival Estimates
Survival Standard Number NumberSurvTime Survival Failure Error Failed Left
0.000 1.0000 0 0 0 27 12.000 0.9630 0.0370 0.0363 1 26 15.000 0.9259 0.0741 0.0504 2 25 19.000 0.8889 0.1111 0.0605 3 24 43.000 0.8519 0.1481 0.0684 4 23 49.000 0.8148 0.1852 0.0748 5 22 52.000 0.7778 0.2222 0.0800 6 21 53.000 0.7407 0.2593 0.0843 7 20 100.000 0.7037 0.2963 0.0879 8 19 103.000 0.6667 0.3333 0.0907 9 18 105.000 0.6296 0.3704 0.0929 10 17 111.000 0.5926 0.4074 0.0946 11 16 133.000 0.5556 0.4444 0.0956 12 15 143.000 0.5185 0.4815 0.0962 13 14 156.000 0.4815 0.5185 0.0962 14 13 162.000 0.4444 0.5556 0.0956 15 12 164.000 0.4074 0.5926 0.0946 16 11 177.000 0.3704 0.6296 0.0929 17 10 182.000* . . . 17 9 200.000 0.3292 0.6708 0.0913 18 8
First death after 12 days
X-Y points for life table graph
Stratum 1: Cell = large
Product-Limit Survival Estimates
Survival Standard Number NumberSurvTime Survival Failure Error Failed Left
0.000 1.0000 0 0 0 27 12.000 0.9630 0.0370 0.0363 1 26 15.000 0.9259 0.0741 0.0504 2 25 19.000 0.8889 0.1111 0.0605 3 24 S(0) = 1S(12) = .9630 (26/27)
S(15) = .9259 (25/27) which is also 26/27 * 25/26
S(19) = .8889 (24/27)
What is S(17) ?
Estimated survival function is a step function
Stratum 2: Cell = squamous
Product-Limit Survival Estimates
Survival Standard Number NumberSurvTime Survival Failure Error Failed Left
0.000 1.0000 0 0 0 35 1.000 . . . 1 34 1.000 0.9429 0.0571 0.0392 2 33 8.000 0.9143 0.0857 0.0473 3 32 10.000 0.8857 0.1143 0.0538 4 31 11.000 0.8571 0.1429 0.0591 5 30 15.000 0.8286 0.1714 0.0637 6 29 25.000 0.8000 0.2000 0.0676 7 28 25.000* . . . 7 27 30.000 0.7704 0.2296 0.0713 8 26
2 patients died after 1 day
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
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
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