1 parametric sensitivity analysis for cancer survival models using large- sample normal...
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Parametric Sensitivity Analysis For Cancer Survival Models Using Large-
Sample Normal Approximations To The Bayesian Posterior Distribution
Gordon B. Hazen, PhDNorthwestern University
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Cancer survival models as components of many analyses
Tamoxifen vs. No tamoxifen
Breast cancer incidence
Breast cancer?
Breast cancer survival
Endometrial cancer incidence
Endometrial cancer?
Endometrial cancer survival
Venous Thrombosis?
Pulmonary embolism?
Background mortality
Overall survival
Col et al. (2002), “Survival impact of tamoxifen use for breast cancer risk reduction”, Medical Decision Making 22, 386-393.
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A simple cancer survival model – the Conditional Cure model
( ) (1 ) tS t p p e 0 10 20
0
0.5
1
Survival Curvep
p = probability of cure
= mortality rate if not cured
Survival function:
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Fitting CCure Model to DATA
Breast Cancer Survival, Age 50-60
50%
60%
70%
80%
90%
100%
0 5 10 15 20 25 30
Years
Su
rviv
al
Pc
t
SEER Data
CCure Model
ˆ ˆ0.56, 0.010 /p yr
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Fitting CCure Model to DATA (cont.)
Endometrial Cancer Survival, Age 50-60
80%
90%
100%
0 5 10 15 20 25 30
Years
Su
rviv
al P
ct
SEER
CCure Model
ˆ ˆ0.84, 0.26 /p yr
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Fitting CCure Model to DATA (cont.)
Ovarian Cancer Stage II
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16
Years
Su
rviv
al P
ct
SEER 1985-2001
CCure Model
ˆ ˆ0.42, 0.15 /p yr
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Fitting CCure Model to DATA (cont.)
Ovarian Cancer Stage IIl
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16
Years
Su
rviv
al P
ct
SEER 1985-2001
p,mu Model
ˆ ˆ0.21, 0.28 /p yr
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Fitting CCure Model to DATA (cont.)
Ovarian Cancer Stage lV
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 2 4 6 8 10 12 14
Years
Su
rviv
al P
ct
SEER 1985-2001
CCure Model
ˆ ˆ0.090, 0.53/p yr
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Question
• It is easy to choose p, to fit a Conditional Cure survival curve to SEER survival data, but …
• How should we conduct a sensitivity analysis on the resulting estimates ˆ ˆ, ?p
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The Bayesian approach
• Treat the unknowns p, as random variables with specified prior distribution.
• Use Bayes’ rule to calculate the posterior distribution of p, given SEER or other data.
• Use this posterior distribution to guide a sensitivity analysis, or to conduct a probabilistic sensitivity analysis.
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Bayesian model with censoring
• Posterior distribution
unrelateddeath or censored
( , ) (1 ) (1 )
number of observed cancer deaths
average time of cancer death
itn n n t
i
f p p e p p e
n
t
• Posterior distribution is analytically awkward
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Bayesian model with censoring
True Posterior Distribution
Ovarian Cancer Stage II: Posterior on p,
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Bayesian model with censoring
• Awkward analytical form makes the posterior distribution on p, difficult to use for sensitivity analysis:– Where is a 95% credible region?
– How to generate random p, for probabilistic sensitivity analysis?
• Solution: Large-sample Bayesian posterior distributions are approximately normal
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Large-sample Bayesian posteriors
• Fundamental result: For large samples, the Bayesian posterior distribution is approximately multivariate normal with – mean equal to the posterior mode (under a
uniform prior, this is the maximum likelihood estimate)
– covariance matrix equal to the matrix inverse of the Hessian of the log-posterior evaluated at the posterior mode.
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Hessian of the log posterior…
unrelateddeath or censored
number of observed cancer deaths
average time of cancer death
log ( , ) const log(1 ) log
log (1 ) it
i
n
t
f p n p n
n t p p e
Log posterior :
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Hessian of the log posterior
-- 2
2 2 2
- -2
2 2 2
-
(1- )- -(1- ) ( ) ( )
- (1- )( ) ( )
( ) (1- )
ii
i i
i
ttD i
i ii i
t ti D i
i ii i
ti
H
n t ee
p S t S t
t e n t ep p
S t S t
S t p p e
The Hessian is the matrix of second partial derivatives with respect to p and :
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Large-Sample Bayesian PosteriorsUsing Excel’s Solver to calculate mle and covariance matrix for p,
Value of log posterior at
p,
Mle’s
Hessian H
p, covariance
matrix
SEER data
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Large-sample Bayesian posterior
True Posterior Distribution
Bivariate normal approximation
Ovarian Cancer Stage II
True posterior density
Approximate normal density
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Two-way sensitivity analysis on p,
Bivariate normal approximation
Vary p and simultaneously two standard deviations along the principal component of the approximate normal posterior density.
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Two-way sensitivity analysis on p,cont.)
0 10 20 30 400
0.5
1
MeanMinus 2 SDPlus 2 SDSEER Data
Years
Cau
se-S
peci
fic
Sur
viva
l Pro
babi
lity
The resulting variation in stage II ovarian cancer survival:
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Two-way sensitivity analysis on p, (cont.)
0 10 20 30 40 500
0.5
1
MeanMinus 2 SDPlus 2 SD
Years
Sur
viva
l Pro
babi
lity
The resulting variation in survival for a 50-year-old white female with stage II ovarian cancer:
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Summary
• The Conditional Cure model for cancer survival.
• A method for using a large-sample normal approximation to the Bayesian posterior distribution to guide a sensitivity analysis of parameter estimates for this model.
• Appears to be a useful and practical method.
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Potential Pitfalls
• Large-sample normal approximation requires mle to be an interior maximum – estimates p = 0, p = 1, or = 0 do not yield approximate normal posteriors
• If sample size is very large, then posterior distribution will be so tight that sensitivity analysis is unnecessary.