Comparing Survival Functions

1
Comparing Survival Functions
1.00
0.75
High
Survival Distribution Function
0.50
Low
0.25
Medium
0.00
0
10
20
30
40
50
60
Time
2
Log-Rank Test

• The log-rank test
• tests whether the survival functions are
  statistically equivalent
• is a large-sample chi-square test that uses the
  observed and expected cell counts across the
  event times
• has maximum power when the ratio of hazards is
  constant over time.

3
Wilcoxon Test

• The Wilcoxon test
• weights the observed number of events minus the
  expected number of events by the number at risk
  across the event times
• can be biased if the pattern of censoring is
  different between the groups.

4
Uninformative Censoring

• Censoring is uninformative if it
• occurs when the reasons for termination are
  unrelated to the risk of the event
• assumes that subjects who are censored at time X
  should be representative of all those subjects
  with the same values of the predictor variables
  who survive to time X
• does not bias the parameter estimates and
  statistical inference.

5
Informative Censoring

• Censoring is informative if it
• occurs when the reasons for termination of the
  observation are related to the risk of the event
• results in biased parameter estimates and
  inaccurate statistical inference about the
  survival experience.

6
Time Origin Recommendations

• Choose a time origin that marks the onset of
  continuous exposure to the risk of the event.
• Choose the time of randomization to treatment as
  the time origin in experimental studies.
• If there are several time origins available,
  consider controlling for the other time origins
  by including them as covariates.

7
Log-rank versus Wilcoxon Test

• Log-rank test
• is more sensitive than the Wilcoxon test to
  differences between groups in later points in
  time.
• Wilcoxon test
• is more sensitive than the log-rank test to
  differences between groups that occur in early
  points in time.

8
LIFETEST Procedure
PROC LIFETEST DATASAS-data-set ltoptionsgt TIME
variable ltcensor(list)gt STRATA variable
lt(list)gt lt...variable lt(list)gtgt TEST
variables RUN
9
Life Table Method

• The life table method
• is useful when there are a large number of
  observations
• groups the event times into intervals
• can produce estimates and plots of the hazard
  function.

10
Differences between KM and Life Table Methods

• In the Kaplan-Meier method,
• time interval boundaries are determined by the
  event times themselves
• censored observations are assumed to be at risk
  for the whole event time period.
• In the life table method,
• time interval boundaries are determined by the
  user
• censored observations are censored at the
  midpoint of the time interval.

11
Cox Proportional Hazards Model
12
Objectives

• Explain the concepts behind the Cox proportional
  hazards model.
• Explain the concept of partial likelihood.
• Explain the methods for handling ties.
• Fit a proportional hazards model in the PHREG
  procedure.

13
Survival Models

• Models in survival analysis
• are written in terms of the hazard function
• assess the relationship of predictor variables to
  survival time
• can be parametric or nonparametric models.

14
Parametric versus Nonparametric Models

• Parametric models require that
• the distribution of survival time is known
• the hazard function is completely specified
  except for the values of the unknown parameters.
• Examples include the Weibull model, the
  exponential model, and the log-normal model.

15
Parametric versus Nonparametric Models

• Properties of nonparametric models are
• the distribution of survival time is unknown
• the hazard function is unspecified.
• An example is the Cox proportional hazards model.

16
Cox Proportional Hazards Model
...
17
Popularity of the Cox Model

• The Cox proportional hazards model
• provides the primary information desired from a
  survival analysis, hazard ratios and adjusted
  survival curves, with a minimum number of
  assumptions
• is a robust model where the regression
  coefficients closely approximate the results from
  the correct parametric model.

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Measure of Effect
hazard in group Ahazard in group B
Hazard ratio
19
Properties of the Hazard Ratio
No Association
Group B Higher Hazard
Group A Higher Hazard
0 1
20
Proportional Hazards Assumption
Females
Log h(t)
Males
Time
21
Nonproportional Hazards
Males
Log h(t)
Females
Time
22
Shortcomings of the Cox Model

• The Cox model
• has no estimated intercept term
• does not provide an equation that can be used to
  predict survival time
• does not provide group-specific hazard rates.

23
Maximum Likelihood Estimation
Log-likelihood
24
Partial Likelihood

• Partial likelihood differs from maximum
  likelihood because
• it does not use the likelihoods for all subjects
• it only considers likelihoods for subjects that
  experience the event
• it considers subjects as part of the risk set
  until they are censored.

25
Partial Likelihood
Subject
Survival Time
Status
C
2.0
1
B
3.0
1
A
4.0
0
D
5.0
1
E
6.0
0
26
Partial Likelihood
27
Partial Likelihood
28
Tied Event Times

• The exact method
• assumes that ties are due to the lack of
  precision in measuring survival time
• computes all possible orderings of the tied event
  times
• is very CPU intensive with large data sets that
  contain many ties.

29
Tied Event Times

• The discrete method
• assumes events occurred at exactly the same time
• computes probabilities that the events occurred
  to a set of subjects with tied event times
• is very CPU intensive, but not as much as the
  exact method.

30
Tied Event Times

• The Breslow method
• is an approximation of the exact method
• works well when the number of ties are relatively
  few
• yields coefficients biased towards 0 when the
  number of ties is large
• is the default in PROC PHREG
• uses less CPU time for large data sets compared
  to the exact and discrete methods.

31
Tied Event Times

• The Efron method
• is also an approximation to the exact method
• yields coefficients that are closer to the exact
  method compared to coefficients obtained from the
  Breslow method
• also yields coefficients biased towards 0 when
  the number of ties is large
• uses approximately the same CPU time as the
  Breslow method.

32
Convergence Problems
Number Censored
Number of Events
Yes
0(0)
0(2)
0(2)
0(6)
Treatment
No
2(1)
1(0)
1(1)
2(2)
1
2
3
4
Time
33
Convergence Problems
Variable
SE
Treatment
18.58
3751
Test
Chi-Square
P-value
Likelihood Ratio
10.38
0.0013
Score
8.29
0.0041
Wald
0.00
0.9960