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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.