Download Predicting the Future: ATAC Trial

Using Weibull Model to Predict
the Future: ATAC Trial
Anna Osmukhina, PhD
Principal Statistician, AstraZeneca
15 April 2010
Survival Analysis
Name
Formula
Example:
exponential
distribution
T
Time to event random variable
f (t )
Probability density function
 exp  t 
1  exp  t 
Cumulative distribution function F (t )
S (t )  1  F (t )
exp  t 
Survival function
h(t )  f (t )

Hazard function
S (t )
Rate
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 0
2
Example: Exponential Time to Event
S (t )  exp  t 
Constant hazard
h(t )  f (t )
S (t )

f (t )   exp  t 
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3
Events in Early Breast Cancer
Overall Survival
Initial
treatment:
surgery,
chemotherapy,
radiotherapy
Disease-Free-Survival: time from randomization
to first recurrence or death
No disease No disease No disease
Randomization
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New
lesions
Recurrence
Death
4
A Little Bit of History: Tamoxifen
• “Tamoxifen for early breast cancer: an
overview of the randomised trials “
– Early Breast Cancer Trialists' Collaborative Group
• The Lancet, V 351, 1998, pp 1451-67
• Meta-analysis of 55 trials, ~37000 women
• In women with hormone receptor +-ve
disease, tamoxifen  5 years 
– Recurrence  43%
– Death (any cause)  23%
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ATAC Trial
• Anastrozole, Tamoxifen, Alone or in
Combination
• >9000 early breast cancer patients;
• 5 years of treatment + 5 years follow up
• Analyses:
– 2001: Major analysis (DFS event-driven)
– 2004: Treatment completion
– 2007: 5+2
– (2009)
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Presenting the Results: KM Plot for
DFS, 2004
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ATAC Results by 2004
(Hormone Receptor Positive Subgroup)
Analysis data
cut off date
Endpoint
29 June 2001
Analysis results*
Comment
Hazard ratio , A/T (95% CI )
P-value
DFS
0.78 (0.65, 0.93)
0.005
Superior
OS
Not reported
NR
NR
0.83 (0.73, 0.95)
0.005
Superior
0.97 (0.83, 1.14)
Not sig
Non-inferior**
31 March 2004 DFS
OS
* Cox proportional hazards model: semi-parametric
**Rothman approach
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Questions About the Future
2001
• DFS:
superiority
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2004
• DFS:
superiority
• OS: Noninferiority
2007
• DFS: Keep?
• OS: Gain
superiority?
Lose NI?
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Weibull Distribution for Survival
Analysis
Name
Formula
Exponential
distribution
Weibull
distribution
 exp  t 
exp  t 
t  1 exp  t  
exp  t 

t  1
TTE random variable T
PDF
Survival function
Hazard function
f (t )
S (t )  1  F (t )
h(t )  f (t )
S (t )


Constant hazard
Rate
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  0,   0
“Accelerated
failure time”
Scale (Shape)
10
Exponential Time to Event
S (t )  exp  t 
Constant hazard
h(t )  f (t )
S (t )

f (t )   exp  t 
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Weibull Time to Event

S (t )  exp  t 

 1
Accelerated hazard
h(t )  t  1

f (t )  t  1 exp  t 
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
12
Weibull Time to Event

S (t )  exp  t 
0   1

Decelerated hazard
h(t )  t  1

f (t )  t  1 exp  t 
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
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Weibull Distribution in SAS PROC
LIFEREG
Name
Formula
TTE random variable
T
PDF
Survival function
f (t )
Hazard function
t  1 exp  t  

S (t )  1  F (t ) exp  t 
 1
h(t )  f (t )

t
S (t )

Rates in ith individual:
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Weibull
distribution

covariates
1



     xi 
i  exp  




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Questions About the Future
2001
• DFS:
superiority
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2004
• DFS:
superiority
• OS: Noninferiority
2007
• DFS: Keep?
• OS: Gain
superiority?
Lose NI?
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Predictions Using Weibull Model
B
U
I
L
D
Individual
patient data
so far
SIMULATE
EXPLORE
Future data for
each patient
x1000
Weibull model
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Fit Weibull Model to the Data So Far
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Fitting Weibull Model
• SAS PROC LIFEREG
• Model events using baseline characteristics
– Demography
– Disease characteristics
• Version 1: separately for each treatment
• Version 2: treatment arms combined
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Weibull Models for the Data So Far
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Predictions Using Weibull Distribution
B
U
I
L
D
Individual
patient data
so far
SIMULATE
EXPLORE
Future data for
each patient
x1000
Weibull model
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Future Assumptions: 3 Scenarios
• Optimistic: Trend continues
• Middle: no difference from now on
• Conditional HR=1.0
• Pessimistic: “A” worse from now on
– Conditional HR=1.1
• Very optimistic (for OS only)
– Conditional HR = 0.9
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Predictions Using Weibull Distribution
B
U
I
L
D
Individual
patient data
so far
SIMULATE
Weibull model
Future
assumptions
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EXPLORE
Future data for
each patient
x1000
ANALYZE
1000 versions
of the study
future/
scenario 22
Predicting the Future, 31 March 2004
Endpoint
Scenario
DFS
OS
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Total events,
simulated mean
HR, A/T (95% CI)
Now
921
0.83 (0.73, 0.95)
3 years later: Optimistic
1385
0.83 (0.75, 0.92)
3 years later: Middle
1385
0.88 (0.80, 0.98)
3 years later: Pessimistic
1407
0.92 (0.82, 1.02)
Now
597
0.97 (0.83, 1.14)
3 years later: Very Optimistic
971
0.94 (0.83, 1.07)
3 years later: Middle
989
0.98 (0.87, 1.11)
3 years later: Pessimistic
1007
1.02 (0.90, 1.15)
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Another Way to Look at It
Endpoint Scenario
DFS
OS
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Probability of…
Superiority
Non-inferiority
(Rothman)
Inferiority
Now (2004)
100%
Not useful
0%
3 years later: Optimistic
99.4%
Not useful
<0.1%
3 years later: Middle
71.5%
Not useful
<0.1%
3 years later: Pessimistic
29.9%
Not useful
<0.1%
Now (2004)
0%
100%
0%
3 years later: Very Optimistic
5.5%
99.2%
<0.1%
3 years later: Middle
0.6%
89.7%
<0.1%
3 years later: Pessimistic
<0.1%
66.0%
0.2%
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Predictions About the Future
2001
• DFS:
superiority
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2004
• DFS:
superiority
• OS: Noninferiority
2007
• DFS: Likely
to keep
superiority
• OS:
Superiority
very
unlikely;
Likely to
keep NI
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So, How Did That Work Out?
Endpoint
Scenario
DFS
OS
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Total events,
simulated mean
HR, A/T (95% CI)
Now
921
0.83 (0.73, 0.95)
3 years later: Optimistic
1385
0.83 (0.75, 0.92)
3 years later: Middle
1385
0.88 (0.80, 0.98)
3 years later: Pessimistic
1407
0.92 (0.82, 1.02)
3 years later: Actual
1320
0.85 (0.76-0.94)
Now
597
0.97 (0.83, 1.14)
3 years later: Very Optimistic
971
0.94 (0.83, 1.07)
3 years later: Middle
989
0.98 (0.87, 1.11)
3 years later: Pessimistic
1007
1.02 (0.90, 1.15)
3 years later: Actual
949
0.97 (0.86-1.11)
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Revisiting: Fitting Weibull Model
• Model events using baseline characteristics
– Demography
– Disease characteristics
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Side Note: Loss to Follow Up
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Predictions Using Weibull Distribution
B
U
I
L
D
Individual
patient data
so far
SIMULATE
Weibull model
Future
assumptions
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EXPLORE
Future data for
each patient
x1000
ANALYZE
1000 versions
of the study
future/
scenario 29
Revisiting: Fitting Weibull Model
• Model events using baseline characteristics
– Demography
– Disease characteristics
• Model discontinuation with time-dependent
covariate: (time</>5 years)
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Future Event Prediction
Good
• Good HR (CI) estimates
– Thanks to mature data?
Bad
• Overestimated number of
new events
• Individual risk factors
• Scenarios, complex questions
• Describe/manage expectations
• Complex models
– Loss to follow up,
administrative censoring
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• Is as good as assumptions
– More parameters = More
assumptions (correct or not)?
• Adjusting for emergent risk
factors?
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References
• Early Breast Cancer Trialists' Collaborative Group
– Lancet 1998; 351: 1451-67
• ATAC trialists’ group
– Lancet 2002; 359: 2131–39
– Lancet 2005; 365: 60–62
– Lancet Oncol 2008; 9: 45–53
• Carroll K, “On the use and utility of the Weibull model
in the analysis of survival data”
– Controlled Clinical Trials 24 (2003) 682–701
• Rothman M, “Design and analysis of non-inferiority
mortality trials in oncology”
– Statist. Med. 2003; 22:239–264
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