using weibull model to predict the future: atac trial anna osmukhina, phd principal statistician,...

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

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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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