Joint Modeling of Multidimensional Outcomes

(Take Your Markov Model to 11!)

John Graves

Vanderbilt University

2024-05-07

Collaborators and Thanks

Support for this work is graciously acknowledged from the Data to Policy initiative administered by Vital Strategies and funded by Bloomberg Philanthropies and the CDC Foundation.

  • Jinyi Zhu, Ph.D. (Vanderbilt)
  • Ashley Leech, Ph.D. (Vanderbilt)
  • Shawn Garbett, MS (Vanderbilt)
  • Hanxuan (Astrid) Yu (Vanderbilt … but soon Harvard!)
  • Marie Martin, Ph.D. (Vanderbilt)

Contribution

  1. New tools for structuring Markov models for multidimensional outcomes (QALYs, DALYs, one-time costs, etc.).
  2. Advanced matrix-based demographic methods for discrete time Markov cohort modeling.

Draft manuscript (with R code) available online at https://graveja0.github.io/dalys/

Background: Payoffs and Rewards

  1. Occupancy-based payoffs:
  • Utility/DALY weight applied for a time period / step.
  • Treatment/disease cost per time period / step.
  1. Transition-based payoffs:
  • One-time event-based cost (e.g., disease-related death, initial Dx, etc.).
  • One-time health outcome (e.g., years of life lost to premature mortality)

Disability-Adjusted Life Years (DALYs)

  • Reflect both occupancy- and transition-based payoffs.
  • There’s also very little guidance on how to structure a decision model for DALY outcomes.

Years of Life Lost to Disease

For a given condition c,

YLD(c) = D_c \cdot L_c

  • D_c is the condition’s disability weight
  • L_c is the time lived with the disease.

Years of Life Lost to Premature Mortality

  • YLL are defined by by a loss function.
  • Drawn from a reference life table, indicating remaining life expectancy at age a.
  • YLL(a)= Ex(a)

DALYs

DALY(c,a) = YLD(c) + YLL(a)

Evolution of DALY Calculations

  • Historical Practice: Initial GBD studies applied age-weighting and 3% annual time discounting.
  • Changes Post-2010: Discontinuation of these practices for a more descriptive DALY measure.

Current Discounting Practices

  • WHO-CHOICE Recommendations: Time discounting of health outcomes.
  • Methodology: Continuous-time discounting from original GBD equations is retained, though you can also input a \approx 0 discount rate.

Additional slides below (hit down button).

Bottom Line

  • You’re going to see a lot of math expressions that take care of continuous time discounting in discrete time.
  • This math adds some complexity but not a ton of insight, so I’ll gloss over it a bit.

Sources of Discounting

Model Time Horizon

1. Present Value of YLL (Continuous Time)

2. Present Value of YLD (Continuous Time)

2. Present Value of YLL (Discrete Time)

2. Present Value of YLL & YLD (t=0)

Years of Life Lost to Premature Mortality (YLL)



At age of death a, and based on discount rate r, YLL(a)= \frac{1}{r}\left(1-e^{-r Ex(a)}\right)

Years of Life Lived with Disability (YLD)



At cycle t, and for cycle duration \Delta_t YLD(c,\Delta_t) = D_c \bigg ( \frac{1}{r_{\Delta_t}}(1-e^{-r_{\Delta_t}}) \bigg ) \Delta_t

This approach applies a discount factor over time within a discrete time cycle to maintain the continuous time discounting approach used by the GBD.

Overview of Decision Problem

Overview of Decision Problem

  • Basic CVD model for the UK.
  • Strategies:
    • Natural History
    • Prevention (£50/yr; HR for CVD incidence = 0.90)
    • Treatment (£1,000/yr; HR for CVD death = 0.85)
    • Prevention and Treatment