Introduction to Survival Analysis

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Introduction to Survival Analysis
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Consider the following clinical out-come
evaluation situation

• Goal To determine the effectiveness of a new
  therapeutic intervention with sexually abused
  children.

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Consider the following clinical outcome
evaluation

• Procedure
• Identify N9 children who are referred to your
  agency after being sexually abused.

• 2. Obtain a baseline measure on each (e.g.,
  number of behavioral or emotional symptoms).
• 3. Provide therapeutic intervention.
• 4. Obtain post-treatment measurement.

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Results
ID Name PRE- of symptoms POST- of symptoms Improvement ? in of symptoms
1 Bob 54 42 12
2 Susan 62 46 16
3 Richard 58 44 14
4 Debbie 64 43 21
5 Todd 55 57 -2 (symptom increase)
6 Cindy 60 52 8
7 Sam 67 52 15
8 Carrie 62 48 14
9 Jim 61 51 10
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How would we summarize the data?
Summary Pre and Post Data Summary Pre and Post Data Summary Pre and Post Data Summary Pre and Post Data Summary Pre and Post Data
Group N Mean St. Dev. St. Error Mean
PRE 9 60 4.15 1.38
POST 9 48 5.42 1.81
Analysis Change in of symptoms Analysis Change in of symptoms Analysis Change in of symptoms Analysis Change in of symptoms Analysis Change in of symptoms
N Mean St.Dev. Min Max
9 12.00 6.42 -2.00 21.00
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Outcome Analysis

• Are the children showing significant improvement
  in number of symptoms?
• What type of statistical test should I do?
• Answer A paired t-test (pre/post data)

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Pre/Post Findings
Analysis Paired Sample T-test Analysis Paired Sample T-test Analysis Paired Sample T-test Analysis Paired Sample T-test
t value df P-value (2 tail)
Pair One 5.444 8 0.001
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Pre/Post Findings
Analysis Paired Sample T-test Analysis Paired Sample T-test Analysis Paired Sample T-test Analysis Paired Sample T-test
t value df P-value (2 tail)
Pair One 5.444 8 0.001

• Interpretation There was a significant
  improvement in number of symptoms between the
  beginning of treatment and the end of treatment.

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Pre/Post Findings

• Concern But would the children have improved the
  same amount without our treatment?

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Pre/Post Findings

• Concern But would the children have improved the
  same amount without our treatment?
• We need a control group!

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How Do We Compare Pre/Post Data Between Groups?
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How Do We Compare Pre/Post Data Between Groups?

• Answer Independent t-test on change in number of
  symptoms.

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Analysis
Rx Versus Control Rx Versus Control N Mean St.Dev
Symptom Reduction Cntrl 9 12.00 6.42
Symptom Reduction RX 9 8.78 5.59
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Analysis
Rx Versus Control Rx Versus Control N Mean St.Dev
Symptom Reduction Cntrl 9 12.00 6.42
Symptom Reduction RX 9 8.78 5.59
Independent t-test (Ctrl versus Rx Groups) Independent t-test (Ctrl versus Rx Groups) Independent t-test (Ctrl versus Rx Groups) Independent t-test (Ctrl versus Rx Groups)
Symptom Reduction t-value df P-value (2 tailed)
Symptom Reduction 1.136 16 .273
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Findings

• Interpretation No significant difference in
  symptom reduction between control and treatment
  groups

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New Type of Problem

• How would you analyze the following problem?

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TIME TO FAILUREDid our client fail over time?
Yes/No

• Recidivism
• Rehospitalization
• Acting Out
• Relapse
• Dropping Out
• Death

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Time is a Variable

• The question is not simply, Did our client
  experience a failure?

• Rather, Did our
• client last longer before failure?
• e.g. Did our clients go longer before being
  rehospitalized?

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Chi-Square??

• Problem What time period are you talking about?

Failure Failure Failure
New Treatment Yes No
New Treatment Yes 60 40
New Treatment No 80 20
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Particularly Difficult In Agencies

• Unlike controlled experiments, it is difficult to
  study cohort groups.
• Clients come and go in unpredictable manner.
• How do you randomize over time?

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Path Of Failure Over Time Is Important
Rx as Usual
100
of clients NOT rehospitalized
6 months
1 year
Time to Rehospitalization
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Additionally, Path Of Failure Is Important
Rx as Usual
100
of clients NOT rehospitalized
New Rx
6 months
1 year
Time to Rehospitalization
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Survival Analysis

• Compares number of failures over TIME
• Recorded as binary yes/no
• Path of that Failure

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Instead Of Looking at Symptom Reduction, Lets
Look at the Symptom Avoidance?

• EXAMPLE
• Sexually Reactive Children
• Sexually Reactive" children are pre-pubescent
  boys and girls who have been exposed to, or had
  contact with, inappropriate sexual activities.
  The sexually reactive child may engage in a
  variety of age-inappropriate sexual behaviors as
  a result of his or her own exposure to sexual
  experiences, and may begin to act out, or engage
  in, sexual behaviors or relationships that
  include excessive sexual play, inappropriate
  sexual comments or gestures, mutual sexual
  activity with other children, or sexual
  molestation and abuse of other children.

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Suppose we are interested in decreasing or
avoiding sexual reactivity.

• What kinds of things are going to change with
  respect to our methods for evaluating the data?

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What would the data look like?
ID Name Months Status
1 Bob 5 Reactive
2 Susan 10 Reactive
3 Richard 3 Reactive
4 Debbie 11 Non-reactive
5 Todd 4 Non-reactive
6 Cindy 8 Non-reactive
7 Sam 7 Non-reactive
8 Carrie 5 Reactive
9 Jim 2 Non-reactive
Summary Median survival 10 months or 6-month
survival 58 /- 19
S start 0 last measurement of observed
non-reactive behavior x Reactive behavior
occurred
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Summary What is Survival Analysis?

• Outcome variable Time until an event occurs
• Time Survival time
• Event (Assume single event) failure
• Death
• Relapse
• Sexual reactivity
• Assault
• Rehospitalization

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Censored Data

• Why censor?
• Study ends (no event)
• Lost to follow-up
• Withdraws

• Censoring Dont know survival time exactly

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Terminology and Notation

• t observed survival time
• (0,1) random variable
• 1 if failure
• 0 if censored
• S(t) survivor function
• h(t) hazard function
• t(j) time period

Probability of survival beyond a certain point in
time
Failure rate
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How do we use this data in a survival analysis?
ID Name Months Status
1 Bob 5 Reactive
2 Susan 10 Reactive
3 Richard 3 Reactive
4 Debbie 11 Non-reactive
5 Todd 4 Non-reactive
6 Cindy 8 Non-reactive
7 Sam 7 Non-reactive
8 Carrie 5 Reactive
9 Jim 2 Non-reactive
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• t observed survival time
• (0,1) random variable
• 1 if failure
• 0 if censored)

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t(j) failure time period t month when
failure occurred m of failures q
censored R risk set
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Based upon this data set, what is the probability
that a child entering our program will remain
non-sexually reactive for each failure time
period?
S(3) 7/8 .875 Interpretation Having started
the program, there is a 87.5 chance that a given
child will survive to 3 months without becoming
reactive
S(5) 4/6 .66.7 Interpretation Having survived
3 months, there is a 66.7 chance that a given
child will survive to 5 months without becoming
reactive
S(10) 1/2 .500 Interpretation Having survived
5 months, there is a 50.0 chance that a given
child will survive to 10 months without becoming
reactive
S(0) 9/9 1 Interpretation Having been
admitted to the program, there is 100 chance all
children will start the Rx
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Probability Paths
What are your chances of flipping heads the 1st
time?
Answer 50
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Probability Paths
Having already flipped heads once, what is the
chance your next flip is heads?
Answer 50
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Probability Paths
Having already flipped heads twice, what is the
chance your next flip is heads?
Answer 50
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Probability Paths
What are your chances of flipping heads three
times in a row?
Answer .50 x .50 x .50 .125 or there is a
12.5 chance of flipping heads three times in a
row.
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Probability Paths
What are your chances of flipping two heads and
one tail in that order?
Answer .50 x .50 x .50 .125 or there is a
12.5 chance of flipping heads three times in a
row.
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Kaplin-Mier Method Using Probability Paths
S(3) 7/8 .875 Interpretation Having started
the program, there is a 85.5 chance that a given
child will survive to 3 months without becoming
reactive
S(5) 4/6 .66.7 Interpretation Having survived
3 months, there is a 66.7 chance that a given
child will survive to 5 months without becoming
reactive
S(10) 1/2 .500 Interpretation Having survived
5 months, there is a 50.0 chance that a given
child will survive to 10 months without becoming
reactive
S(0) 9/9 1 Interpretation Having been
admitted to the program, there is 100 chance all
children will start the Rx
What is the chance of a child surviving (being
non-reactive) until 3 months after being assigned
to the program?
Answer S(t(3)) 1x .875 .875 or there is a
87.5 chance of a child being non-reactive for 3
months after being assigned to the program.
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Kaplin-Mier Method Using Probability Paths
S(3) 7/8 .875 Interpretation Having started
the program, there is a 85.5 chance that a given
child will survive to 3 months without becoming
reactive
S(5) 4/6 .66.7 Interpretation Having survived
3 months, there is a 66.7 chance that a given
child will survive to 5 months without becoming
reactive
S(10) 1/2 .500 Interpretation Having survived
5 months, there is a 50.0 chance that a given
child will survive to 10 months without becoming
reactive
S(0) 9/9 1 Interpretation Having been
admitted to the program, there is 100 chance all
children will start the Rx
What is the chance of a child surviving (being
non-reactive) until 5 months after being assigned
to the program?
Answer S(t(5)) 1 x .875 x .667 .584 or
there is a 58.4 chance of a child being
non-reactive for 5 months after being assigned to
the program.
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Kaplin-Mier Method Using Probability Paths
S(3) 7/8 .875 Interpretation Having started
the program, there is a 85.5 chance that a given
child will survive to 3 months without becoming
reactive
S(5) 4/6 .66.7 Interpretation Having survived
3 months, there is a 66.7 chance that a given
child will survive to 5 months without becoming
reactive
S(10) 1/2 .500 Interpretation Having survived
5 months, there is a 50.0 chance that a given
child will survive to 10 months without becoming
reactive
S(0) 9/9 1 Interpretation Having been
admitted to the program, there is 100 chance all
children will start the Rx
What is the chance of a child surviving (being
non-reactive) until 3 months after being assigned
to the program?
Answer S(t(10)) 1 x .875 x .667 x .500 .292
or there is a 29.2 chance of a child being
non-reactive for 10 months after being assigned
to the program.
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Use survival data summary table to make survival
curve
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Theoretical S(t)
S(t)
0
8
t
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t m q R S(t(j))
t(0) 0 0 1 9 1
t(1) 3 1 1 8 .875
t(2) 5 2 2 6 .584
t(3) 10 1 1 2 .292
Additionally, we know that 1 child survived gt (at least 11 months) Additionally, we know that 1 child survived gt (at least 11 months) Additionally, we know that 1 child survived gt (at least 11 months) Additionally, we know that 1 child survived gt (at least 11 months) Additionally, we know that 1 child survived gt (at least 11 months) Additionally, we know that 1 child survived gt (at least 11 months)
1
.80
.60
Survival Probability
S(t) in practice The Kaplin-Mier Survival Curve
.40
.20
0
2
4
6
8
10
12
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We can compare the Kaplin-Mier survival curves of
two Rx groups
1
Are these two curves significantly
different? Simpliest method is known as the
Log-rank Test
.80
.60
Survival Probability
.40
.20
0
2
4
6
8
10
12
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We can compare the Kaplin-Mier survival curves of
two Rx groups
1
Note For one group to have a significantly
different outcome from the other, the curves
cannot cross
.80
.60
Survival Probability
.40
.20
0
2
4
6
8
10
12
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Logrank Test (aka Mantel-Cox Test)
The Logrank Test statistic compares estimates of
the hazard functions of the two groups at each
observed event time. The Logrank statistic
compare each O1j to its expectation E1j under the
null hypothesis and is defined as
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Logrank Test (aka Mantel-Cox Test)
The Logrank Test statistic compares estimates of
the hazard functions of the two groups at each
observed event time. The Logrank statistic
compare each O1j to its expectation E1j under the
null hypothesis and is defined as