# Simulating costs and quality-adjusted life-years

#### 2020-05-31

# 1 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.

# 2 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.

## 2.1 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.

# 3 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.