Candidate for the Lee B. Lusted Student Prize Competition
It is desirable to have a short cycle length in a discrete-time Markov model, which often requires transforming transition probabilities. Our purpose was to show that the widely used formula âi =1- (1-pi)s/t for converting a probability pi over time interval t into a transition probability âi for a Markov model with shorter cycle length s is invalid for models with three or more states. We explore theoretical issues concerning the mathematically correct approach for adjusting cycle length in such models, and offer numerical approximation methods to practically solve these issues.
Method:
We present several examples of Markov models including ones that involve competing risk to highlight the inaccuracy of the traditional formulas. We formulate the problem of adjusting cycle lengths in Markov models mathematically as that of proving the existence of a unique root of a transition probability matrix and ensuring such root is stochastic (i.e. probabilities are nonnegative and sum up to 1). We use a simple Markov model for advanced liver disease to highlight the mathematically correct approach of finding the root of a matrix using eigendecomposition for determining adjusted transition probabilities. Further, using a previously published HIV Markov model of Chancellor e.t. al., we highlight scenarios where even eigendecomposition fails. Finally, we provide a framework with numerical approximation algorithms to practically change cycle lengths.
Result:
We prove that Markov models whose transition matrices are upper triangular with distinct, non-zero probabilities for the diagonal entries (which we label “progressive” Markov models) are guaranteed to have a unique matrix root. This avoids identifiability issues for transition matrices possessing multiple roots as equal candidates for the shorter cycle model. We provide conditions to determine in general if a given structure of the Markov model suffers from identifiability. We use approximation methods to convert a nonstochastic matrix to a stochastic matrix. Using the HIV and advanced liver disease example, we show that our approach leads to less bias in model outcomes when compared with the traditional (incorrect) approach.
Conclusion:
The traditional formula of converting transition probabilities to different cycle lengths leads to biased outcomes. We further highlight underlying challenges of finding unbiased outcomes and offer a unified framework that leads to more accurate outcomes than the traditional approach in medical decision models.