Abstract
Following its introduction over three decades ago, the cohort model has been used extensively to model population trajectories over time in decision-analytic modeling studies. However, the stochastic process underlying cohort models has not been properly described. In this study, we explicate the stochastic process underlying a cohort model, by carefully formulating the dynamics of populations across health states and assigning probability rules on these dynamics. From this formulation, we explicate a mathematical representation of the system, which is given by the master equation. We solve the master equation by using the probability generation function method to obtain the explicit form of the probability of observing a particular realization of the system at an arbitrary time. The resulting generating function is used to derive the analytical expressions for calculating the mean and the variance of the process. Secondly, we represent the cohort model by a difference equation for the number of individuals across all states. From the difference equation, a continuous-time cohort model is recovered and takes the form of an ordinary differential equation. To show the equivalence between the derived stochastic process and the cohort model, we conduct a numerical exercise. We demonstrate that the population trajectories generated from the formulas match those from the cohort model simulation. In summary, the commonly-used cohort model represent the average of a continuous-time stochastic process on a multidimensional integer lattice governed by a master equation. Knowledge of the stochastic process underlying a cohort model provides a theoretical foundation for the modeling method.
Highlights
Decision models have been used in various applications from clinical decision making to screening guideline development
NjSj may refer to a partitioning of a cohort of individuals into the numbers of healthy, sick and, dead individuals. The convolution of these two different processes in decision modeling literature lead to the second issue: practitioners are taught that cohort models capture the average behavior of the individuals. [7, 8] this claim is often stated without any clear reference to which stochastic process
The cohort model provides an easy-to-implement method for modeling recurrent events over time and has been used extensively in many applications ranging from clinical decision making to policy evaluations
Summary
Decision models have been used in various applications from clinical decision making to screening guideline development. A theoretical foundation for state-transition cohort models in health decision analysis making They applied standard methods from Markov chain theory [2, 3] to simulate a life history of an individual which is structured as transitions across various heath states over time, i.e. a stochastic process on a finite state space (S). According to [1] and the subsequent published tutorials and textbook [4,5,6,7], given a matrix of transition probabilities and an initial distribution of counts of individuals across health states, a cohort model generates the life trajectory of a cohort of identical individuals by repeated multiplications of the vector of population counts by the transition probability matrix This matrix operation alludes to a stochastic process on a much larger state space, i.e. NjSj (compared to S), where N and jSj denote the set of natural numbers and the number of states, respectively. Wide acceptance of methodologies does not automatically imply veracity. [10] In the context of cohort models, the wide acceptance is ingrained by the ISPOR-SMDM Modeling Good Research Practices Task Force-3. [11] The published best practices cites the work of Beck et al [1, 4] as the main and only references for cohort models, thereby standardizing cohort models as the recommended method for simulating population trajectories despite of the lack of a proper description of the theoretical support in the decision modeling area
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