# Overview

Before performing a decision analysis, (discounted) costs and quality-adjusted life-years (QALYs) must be simulated. In cohort models, they are simulated as a function of previously simulated state occupancy probabilities. In individual-level models, they are simulated as a function of simulated trajectories characterizing disease progression. In hesim, the discrete time state transition and partitioned survival models are cohort models and the continuous time state transition model is an individual-level model.

# Cohort Models

Costs and QALYs in cohort models are computed by integrating the “weighted” probability of being in each state. Weights are a function of the discount factor and state values (e.g., annualized costs and utility) predicted using either the cost or utility model. Mathematically, for a time horizon $$T$$, discounted costs and QALYs in health state $$h$$ are computed as,

$\int_0^{T} z_h(t) e^{-rt} P_h(t) dt,$

where $$z_h(t)$$ is the predicted cost or utility value at time $$t$$, $$r$$ is the discount rate, and $$P_h(t)$$ is the probability of being in a given health state. Note that the state values, $$z_h(t)$$, can depend on time since the start of the model but not on time since entering a new health state.

## Integration methods

Three types of approaches are currently available for numerical integration given values of state probabilities at distinct discrete times.

• Left Riemann sum: The function is approximated by its value at the left most point (i.e., start) of each time interval.
• Right Riemann sum: The function is approximated by its value at the right most point (i.e., end) of each time interval.
• Trapezoid rule: The function is approximated by the average of its values at the left and right endpoints. Specifically, for an interval [$$t_1$$, $$t_2$$] with value $$y_1$$ at the left endpoint and $$y_2$$ at the right endpoint, the function is approximated as $$\frac{t_2 - t_1}{2}(y_1 + y_2)$$.

The Riemann sum rules approximate the area under the curve using rectangles in each time interval whiles the trapezoid rule approximates the area under the curve using a trapezoid. In general, the left Riemann sum will underestimate costs and QALYs whereas the right Riemann sum will overestimate them.

# Individual-level models

In individual-level models, costs and QALYs are computed using the continuous time present value given a flow of state values, which change as patients transition between health states or as costs vary as a function of time. The state values can be partitioned into $$M$$ time intervals indexed by $$m = 1,\ldots, M$$ where interval $$m$$ contains times $$t$$ such that $$t_m\leq t \leq t_{m+1}$$ and values for state $$h$$ are equal to $$z_{hm}$$ during interval $$m$$. $$z_{hm}$$ will equal zero during time intervals in which a patient is not in state $$h$$. Discounted costs and QALYs for health state $$h$$ are then given by,

$\sum_{m = 1}^M \int_{t_m}^{t_m+1} z_{hm}e^{-rt}dt = \sum_{m = 1}^M z_{hm} \left(\frac{e^{-r{t_{m}}} - e^{-r{t_{m+1}}}}{r}\right),$

where $$r > 0$$ is the discount rate. If $$r = 0$$, then the present value simplifies to $$\sum_{m = 1}^M z_{hm}(t_{m+1} - t_{m})$$.

Note that while state values in cohort models can depend on time since the start of the model, state values in individual-level models can depend on either time since the start of the model or time since entering the most recent health state. Individual-level models consequently not only afford more flexibility than cohort models when simulating disease progression, but when simulating costs and/or QALYs as well.