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Using Weibull Model to Predict the Future: ATAC Trial
Anna Osmukhina, PhDPrincipal Statistician, AstraZeneca
15 April 2010
Survival Analysis
Name Formula Example: exponential distribution
Time to event random variableProbability density functionCumulative distribution functionSurvival functionHazard function
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)()()( tStfth
T
)(tf
)(tF
)(1)( tFtS
t exp1
t exp
texp
0Rate
Example: Exponential Time to Event
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ttS exp)(
ttf exp)(
)()()( tStfth
Constant hazard
Events in Early Breast Cancer
Randomization Death
Overall Survival
No diseaseNo disease
Disease-Free-Survival: time from randomization to first recurrence or death
No diseaseNo disease New New lesionslesions
Recurrence
No diseaseNo disease
Initial treatment: surgery, chemotherapy, radiotherapy
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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 Analysis results* Comment
Hazard ratio , A/T (95% CI ) P-value
29 June 2001 DFS 0.78 (0.65, 0.93) 0.005 Superior
OS Not reported NR NR
31 March 2004 DFS 0.83 (0.73, 0.95) 0.005 Superior
OS 0.97 (0.83, 1.14) Not sig Non-inferior*** Cox proportional hazards model: semi-parametric**Rothman approach
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Questions About the Future
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Weibull Distribution for Survival Analysis
Name Formula Exponential distribution
Weibull distribution
TTE random variablePDFSurvival functionHazard function
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)()()( tStfth
T
)(tf
)(1)( tFtS t exp
texp
Constant hazard
texp
tt exp1
0,0
1t
“Accelerated failure time”
Rate Scale (Shape)
Exponential Time to Event
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ttS exp)(
ttf exp)(
)()()( tStfth
Constant hazard
Weibull Time to Event
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ttS exp)(
tttf exp)( 1
1)( tth
1
Accelerated hazard
Weibull Time to Event
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ttS exp)(
tttf exp)( 1
1)( tth
10
Decelerated hazard
Weibull Distribution in SAS PROC LIFEREG
Name Formula Weibull distribution
TTE random variablePDFSurvival functionHazard function
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)()()( tStfth
T
)(tf
)(1)( tFtS texp
tt exp1
1t
ii
xexp
1
Rates in ith individual:
covariates
Questions About the Future
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Predictions Using Weibull Model
Future data for each patient
x1000
Individual patient data
so far
Weibull model
SIMULATE
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EXPLORE
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
Future data for each patient
x1000
Individual patient data
so far
Weibull model
SIMULATE
EXPLORE
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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
Future data for each patient
x1000
Individual patient data
so far
Weibull model
SIMULATE
Future assumptions
ANALYZE
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1000 versions of the study
future/ scenario
EXPLORE
Predicting the Future, 31 March 2004Endpoint Scenario Total events,
simulated meanHR, A/T (95% CI)
DFS 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)
OS 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 ItEndpoint Scenario Probability of…
Superiority Non-inferiority (Rothman)
Inferiority
DFS 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%
OS 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
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So, How Did That Work Out?Endpoint Scenario Total events,
simulated meanHR, A/T (95% CI)
DFS 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)
OS 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
Future data for each patient
x1000
Individual patient data
so far
Weibull model
SIMULATE
Future assumptions
ANALYZE
4/15/2010 29
1000 versions of the study
future/ scenario
EXPLORE
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?
• Individual risk factors
• Scenarios, complex questions
• Describe/manage expectations
• Complex models– Loss to follow up,
administrative censoring
Bad• Overestimated number of
new events
• 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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