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CALIBRATION OF PIECEWISE MARKOV MODELS USING A BAYESIAN CHANGE-POINT ANALYSIS THROUGH AN ITERATIVE CONVEX OPTIMIZATION ALGORITHM

Candidate for the Lee B. Lusted Student Prize Competition

** Purpose: **Relative survival, as reported by the Surveillance,
Epidemiology, and End

**Results**(SEER) Program, represents cancer survival in the absence of other causes of death. Often, cancer Markov models have a distant metastasis state, a state not directly observed in SEER, from which cancer deaths are presumed to occur. The aim of this research is to use a novel approach to calibrate the transition probabilities to and from an unobserved state of a Markov model to fit a relative survival curve.

** Methods: ** We
modeled relative survival through a three-piecewise Markov model (i.e., with a
specific Markov chain within each specified pieces) for stage 3 colorectal cancer
patients. For each piece we used a constant transition matrix with three
states: 1) recurrence free, 2) metastatic recurrence and 3) dead from cancer. We
estimated the optimal cutoff time points using a Bayesian Markov chain Monte
Carlo (MCMC) change-point model. This technique allowed us to estimate the time
points at which the slope of the relative survival changes. We calibrated the
transition probabilities using a two-step iterative convex optimization algorithm
previously published. The dynamics of the disease can be defined as

*x*, where

_{t+1}= x_{t}M*x*is the state vector that results from the transformation given by the monthly transition matrix

_{t+1}*M*. The matrix

*M*is a piecewise block-diagonal matrix that includes in each piece a block-diagonal matrix for each Markov chain.

** Results****:** We applied our method
to calibrate a Markov model to fit a relative survival curve for stage 3
colorectal cancer patients younger than 75 years old. We compared our piecewise
calibration method to a single-piece approach (i.e., a Markov chain). While the
single-piece converged faster, the piecewise method improved the goodness of
fit by 60%. The mean of the change points estimated from the Bayesian
change-point model was at months 3 and 24 (see figure). The observed, and the
piecewise and single-piece calibrated relative survival curves are shown in the
figure.

** Conclusions: **

**By estimating the change points in the relative survival curve we were able to find the optimal transition probabilities for a piecewise Markov model that allowed us to impose a particular structure defined by the progression of the disease. We propose a piecewise calibration method that produces more accurate solutions compared to a single-piece approach.**