Survival Analysis

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Survival Analysis

• Bandit Thinkhamrop, PhD. (Statistics)
• Department of Biostatistics and Demography
• Faculty of Public Health, Khon Kaen University

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Begin at the conclusion
Begin at the conclusion
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Type of the study outcome Key for selecting
appropriate statistical methods

• Study outcome
• Dependent variable or response variable
• Focus on primary study outcome if there are more
• Type of the study outcome
• Continuous
• Categorical (dichotomous, polytomous, ordinal)
• Numerical (Poisson) count
• Event-free duration

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The outcome determine statistics
Mean Median
Proportion (Prevalence Or Risk)
Rate per space
Median survival Risk of events at T(t)
Linear Reg.
Logistic Reg.
Poisson Reg.
Cox Reg.
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Statistics quantify errors for judgments
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Back to the conclusion
Appropriate statistical methods
Mean Median
Proportion (Prevalence or Risk)
Rate per space
Median survival Risk of events at T(t)
Magnitude of effect 95 CI P-value
Answer the research question based on lower or
upper limit of the CI
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Study outcome

• Survival outcome event-free duration
• Event (1Yes 0Censor)
• Duration or length of time between
• Start date ()
• End date ()
• At the start, no one had event (event 0) at
  time t(0)
• At any point since the start, event could occur,
  hence, failure (event 1) at time t(t)
• At the end of the study period, if event did not
  occur, hence, censored (event 0)
• Thus, the duration could be either
  time-to-event or time-to-censoring

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Censoring

• Censored data incomplete time to event data
• In the present of censoring, the time to event
  is not known
• The duration indicates there has been no event
  occurred since the start date up to last date
  assessed or observed, a.k.a., the end date.
• The end date could be
• End of the study
• Last observed prior to the end of the study due
  to
• Lost to follow-up
• Withdrawn consent
• Competing events occurred, prohibiting
  progression to the event under observation
• Explanatory variables changed, irrelevance to
  occurrence of event under observation

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• Magnitude of effects
• Median survival
• Survival probability
• Hazard ratio

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SURVIVAL ANALYSIS

• Study aims
• Median survival
• Median survival of liver cancer
• Survival probability
• Five-year survival of liver cancer
• Five-year survival rate of liver cancer
• Hazard ratio
• Factors affecting liver cancer survival
• Effect of chemotherapy on liver cancer survival

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SURVIVAL ANALYSIS
Event
Dead, infection, relapsed, etc
Negative
Cured, improved, conception, discharged, etc
Positive
Neutral
Smoking cessation, ect
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Natural History of Cancer
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Accrual, Follow-up, and Event
ID
Begin the study
End of the study
1
2
3
Dead
Dead
4
5
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Start of accrual
End of accrual
End of follow-up
Recruitment period
Follow-up period
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Time since the beginning of the study
0 1
2 3
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ID
48 months
1
22 months
2
14 months
3
Dead
40 months
Dead
4
26 months
5
13 months
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The data gt48 gt22 14
40 gt26 gt13
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DATA
ID SURVIVAL TIME OUTCOME AT
THE END EVENT (Months) OF THE STUDY

1 48 Still alive at the end of the
study Censored 2 22 Dead due to
accident Censored 3 14 Dead caused by the
disease under investigation Dead 4 40 Dead
caused by the disease under investigation
Dead 5 26 Still alive at the end of the
study Censored 6 13 Lost to
follow-up Censored
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DATA
ID TIME EVENT
ID TIME EVENT
1 48 Censored 2 22 Censored 3 14
Dead 4 40 Dead 5 26 Censored 6 13
Censored
1 48 0 2 22 0 3 14 1 4
40 1 5 26 0 6 13 0
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ANALYSIS
ID TIME EVENT
Prevalence 2/6
1 48 0 2 22 0 3 14 1 4
40 1 5 26 0 6 13 0
Incidence density 2/163 person-months
Proportion of surviving at month t
Median survival time
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RESULTS
ID TIME EVENT
Incidence density 1.2 per100 person-months
(95CI 0.1 to 4.4)
1 48 0 2 22 0 3 14 1 4
40 1 5 26 0 6 13 0
Proportion of surviving at 24 month 80
(95CI 20 to 97)
Median survival time 40 Months
(95CI 14 to 48)
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Type of Censoring

1. Left censoring When the patient experiences the
   event in question before the beginning of the
   study observation period.
2. Interval censoring When the patient is followed
   for awhile and then goes on a trip for awhile and
   then returns to continue being studied.
3. Right censoring
4. single censoring does not experience event
   during the study observation period
5. A patient is lost to follow-up within the study
   period.
6. Experiences the event after the observation
   period
7. multiple censoring May experience event multiple
   times after study observation ends, when the
   event in question is not death.

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Summary description of survival data setstdes

• This command describes summary information about
  the data set. It provides summary statistics
  about the number of subjects, records, time at
  risk, failure events, etc.

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Computation of S(t)

1. Suppose the study time is divided into periods,
   the number of which is designated by the letter,
   t.
2. The survivorship probability is computed by
   multiplying a proportion of people surviving for
   each period of the study.
3. If we subtract the conditional probability of the
   failure event for each period from one, we obtain
   that quantity.
4. The product of these quantities constitutes the
   survivorship function.

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Kaplan-Meier Methods
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Kaplan-Meier survival curve
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Median survival time
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Survival Function

• The number in the risk set is used as the
  denominator.
• For the numerator, the number dying in period t
  is subtracted from the number in the risk set.
  The product of these ratios over the study time

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Survival experience
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Survival curve more than one group
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Comparing survival between groups
ID TIME DEAD DRUG
1 48 0 1
2 22 0 1
3 14 1 1
4 40 1 1
5 26 0 1
6 13 0 1
7 13 0 0
8 6 1 0
9 12 1 0
10 14 1 0
11 22 1 0
12 13 1 0
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Kaplan-Meier surve
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Log-rank test

• t Time
• n Number at risk for both groups at time t
• n1 Number at risk for group 1 at time t
• n2 Number at risk for group 2 at time t
• d Dead for both groups at time t
• c Censored for both groups at time t
• O1 Number of dead for group 1 at time t
• O2 Number of dead for group 2 at time t
• E1 Number of expected dead for group 1 at time
  t
• E2 Number of expected dead for group 2 at time
  t

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Log-rank test example

• DRUG1 48, 22, 26, 13,14,40
• DRUG0 13, 6, 12, 14, 22, 13

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Hazard Function
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Survival Function vs Hazard Function
H(t) -ln(S(t)) (S(t)) EXP(-H(t))
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Hazard rate

• The conditional probability of the event under
  study, provided the patient has survived up to an
  including that time period
• Sometimes called the intensity function, the
  failure rate, the instantaneous failure rate

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Formulation of the hazard rate
The HR can vary from 0 to infinity. It can
increase or decrease or remain constant over
time. It can become the focal point of much
survival analysis.
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Cox Regression

• The Cox model presumes that the ratio of the
  hazard rate to a baseline hazard rate is an
  exponential function of the parameter vector.
• h(t) h0(t) ? EXP(b1X1 b2X2 b3X3 . . .
  bpXp )

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Hazard ratio
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Testing the Adequacy of the model

1. We save the Schoenfeld residuals of the model and
   the scaled Schoenfeld residuals.
2. For persons censored, the value of the residual
   is set to missing.

borrowed from Professor Robert A. Yaffee
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A graphical test of the proportion hazards
assumption

• A graph of the log hazard would reveal 2 lines
  over time, one for the baseline hazard (when x0)
  and the other for when x 1
• The difference between these two curves over time
  should be constant B

If we plot the Schoenfeld residuals over the line
y0, the best fitting line should be parallel to
y0.
borrowed from Professor Robert A. Yaffee
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Graphical tests

• Criteria of adequacy
• The residuals, particularly the rescaled
  residuals, plotted against time should show no
  trend(slope) and should be more or less constant
  over time.

borrowed from Professor Robert A. Yaffee
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Other issues

• Time-Varying Covariates
• Interactions may be plotted
• Conditional Proportional Hazards models
• Stratification of the model may be performed.
  Then the stphtest should be performed for each
  stratum.

borrowed from Professor Robert A. Yaffee
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Suggested Readings for beginners
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Suggested Readings for advanced learners
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Survival analysis in practice

• What is the type of research question that
  survival analysis should be used?

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Stata for one-group survival analysis

• stset time, failure(event)
• stdescribe
• tab event
• stsum
• strate
• stci
• sts list, at(12 24)

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Stata for one-group survival analysis (cont.)

• sts g
• sts g, atrisk
• sts g, lost
• sts g, enter
• sts g, risktable
• sts g, cumhaz
• sts g, cumhaz ci
• sts g, hazard

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Stata for multiple-group survival analysis

• stset time, failure(event)
• stdescribe
• stsum, by(group)
• sts test group
• sts test group, wilcoxon
• strate group
• stci , by(group)
• sts g, by(group) atrisk
• sts g, by(group) risktable
• sts g, by(group) cumhaz lost
• sts g, by(group) hazard ci

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Stata for multiple-group survival analysis

• sts list, , by(group) at(12 24)
• sts list, , by(group) at(12 24) compare
• ltable group, interval()
• ltable group, graph
• ltable group, hazard
• stmh group
• stmh group, by(strata)
• stmc group
• stcox group
• stir group

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Stata for Model Fitting

• Continuous covariate
• xtile newvar varlist , nq(4)
• tabstat varlist, stat(n min max) by(newvar)
• xistcox i.newvar
• stsum, by(newvar)
• Categorical covariate
• tab exposure outcome, col
• xistcox i.exposure

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Sample size for Cox Model

• stpower cox, failprob(.2) hratio(0.1 0.3) sd(.3)
  r2(.1) power(0.8 0.9) hr