1 Parametric Sensitivity Analysis For Cancer Survival Models Using Large- Sample Normal Approximations To The Bayesian Posterior Distribution Gordon B. Hazen, PhD Northwestern University

2 Cancer survival models as components of many analyses Col et al. (2002), “Survival impact of tamoxifen use for breast cancer risk reduction”, Medical Decision Making 22,

3 A simple cancer survival model – the Conditional Cure model p = probability of cure  = mortality rate if not cured Survival function:

4 Fitting CCure Model to DATA

5 Fitting CCure Model to DATA (cont.)

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9 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

10 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.

11 Bayesian model with censoring Posterior distribution Posterior distribution is analytically awkward

12 Bayesian model with censoring Ovarian Cancer Stage II: Posterior on p, 

13 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

14 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.

15 Hessian of the log posterior…

16 Hessian of the log posterior The Hessian is the matrix of second partial derivatives with respect to p and  :

17 Large-Sample Bayesian Posteriors Using 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

18 Large-sample Bayesian posterior Ovarian Cancer Stage II True posterior density Approximate normal density

19 Two-way sensitivity analysis on p,  Vary p and  simultaneously two standard deviations along the principal component of the approximate normal posterior density.

20 Two-way sensitivity analysis on p,  cont.) The resulting variation in stage II ovarian cancer survival:

21 Two-way sensitivity analysis on p,  (cont.) The resulting variation in survival for a 50-year- old white female with stage II ovarian cancer:

22 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.

23 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.