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