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Our Voyage So Far…

Where we have been:

- Continuous Outcome Data
- T-tests, ANOVA, Linear Regression
- Binary Outcome Data
- Odds ratios, Logistic Regression
- Count Outcome Data
- Poisson Regression (briefly!)

Where we are now:

- Binary Outcome (yes/no) with Follow-up time
- Survival Analysis

Creating Rational

- A randomized clinical trial compared radiation vs. surgery + radiation for treatment of cancers that had spread to the brain. (N Engl J Med 1990;322:494-500). After two years, nearly all the patients had died:
- Chi-square probability 0.1868 => No association between death and type of treatment. Is this a reasonable conclusion?
- What happens to the rates if we follow the patients for 100 years?

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 (censored)
- 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 (other types of censoring exist, but this is most common).

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

Example of Censoring

Begin time

C

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

0 5 10

Years Since Marriage

Plot of time to event data

- Suppose we hatch a large number of mosquitoes in a container, and measure how long they live in days.

Credits: AP Photo/Jim Newman, University of Florida/IFAS

Survival Function EstimationS(t)=P(T>t)

- 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

Estimating Survival Curves

Kaplan-Meier Method

- Sometimes called the product-limit method
- 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;

Calculating Kaplan-Meier estimates

Consider Survival Data: 6, 6, 6, 6+, 7, 10, 11+, 13, 13, 15, 17+, 18, 20, 21, 22, 22, 23+ , 24, 24+, 25, 28

- n=number at risk
- d=number of events
- d/n=chance of an event
- 1-d/n = chance of no event
- S(t) = Kaplan Meier Estimate

Calculating Kaplan-Meier estimates

Consider Survival Data: 6, 6, 6, 6+, 7, 10, 11+, 13, 13, 15, 17+, 18, 20, 21, 22, 22, 23+ , 24, 24+, 25, 28

Number at risk

- SAS calculates these automatically

0.8571

x 0.9412

x 0.9375

x 0.8571

Survival Analysis

- Describe the rate (probability) of the event over time
- Called the survival function S(t)
- Compare survival function among groups
- log rank, Wilcoxon significance tests
- Examine risk factors for having the event taking into consideration the time of the event
- Cox proportional hazard regression

Survival After Diagnosis of Lung Cancer

S (t) is the probability of surviving to at least t

S (200) = 0.37

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 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;

Comparing 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

(log-rank assumes constant risk – called hazard ratio)-long term

- Wilcoxon more powerful for detecting short term risk

Patient died 72 days after diagnosis

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 alive after 100 days but status after that time is unknown (100+)

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

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

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

Product-Limit Survival Estimates

Survival

Standard Number Number

SurvTime 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

………..more…..

Survtime: time t

Survival: Kaplan-Meier Estimate of the survivor function S(t)

Failure: The Kaplan –Meier Estimate of cumulative mortality (1-S(t))

Survival Standard Error: The point-wise standard error of the estimate S(t)

Number Failed: the total number of events and censored observations, NOT the number of events.

Number Left: the number still under observation and at risk for an event.

Product-Limit Survival Estimates

Survival

Standard Number Number

SurvTime 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

X-Y points for life table graph

First death after 12 days

Product-Limit Survival Estimates

Survival

Standard Number Number

SurvTime 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) = 1 (100%)

S(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

Product-Limit Survival Estimates

Survival

Standard Number Number

SurvTime 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 comparison 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
- If they cross, use alternative method to compare (beyond the scope of this class)

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
- Survival curves
- log rank, Wilcoxon significance tests
- Log-rank test more powerful when relative risk is constant
- Wilcoxon more powerful for detecting short term risk
- Cox proportional hazards regression
- Relate covariates to survival

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