3 How to choose an estimand: Real-world example

3.1 1. Extent to which the risk of OUD conferred by chronic pain operates through pain management treatments

This application was explored in detail by Rudolph et al. (2025)

3.1.1 Getting specific about the question

To what extent does the total effect of chronic pain on risk of OUD operate through prescription medications for pain management and physical therapy, treated as a bundle?

What estimand do we want?

Can we set $M=m$ (i.e., same value) for everyone?
Are we interested in estimating indirect effects?

$\rightarrow$ So, not controlled direct effect.

Do we have an intermediate confounder?
- Not really, because we: 1) consider all initial treatments following chronic pain diagnosis and 2) stratify by whether or not the patient has an anxiety or depressive disorder.

$\rightarrow$ So, could estimate natural (in)direct effects

Estimands:
- Direct effect: $\E(Y_{1,M_0} - Y_{0,M_0})$
- Indirect effect: $\E(Y_{1,M_1} - Y_{1,M_0})$

3.2 2. Extent to which the risk of OUD conferred by chronic pain operates through individual pain management treatments

This application was explored in detail by Rudolph et al. (2025)

Rudolph, Kara E, Shodai Inose, Nicholas T Williams, Katherine L Hoffman, Sarah E Forrest, Rachael K Ross, Floriana Milazzo, et al. 2025. “Mediation of Chronic Pain and Disability on Opioid Use Disorder Risk by Pain Management Practices Among Adult Medicaid Patients, 2016-2019.” American Journal of Epidemiology, kwaf093.

3.2.1 Getting specific about the question

To what extent does the overall effect of chronic pain on risk of OUD operate through individual pain management treatments, controlling for other co-occurring or prior pain treatments?

What estimand do we want?

Can we set $M=m$ (i.e., same value) for everyone?
Are we interested in estimating indirect effects?

$\rightarrow$ So, not controlled direct effect.

Do we have an intermediate confounder?
- Yes, likely multiple, important ones, because we would like to control for prior or co-occurring pain treatments.
- So likely do not have a single, binary intermediate confounder.
- If we do have a binary intermediate confounder, would we assume monotonicity between the treatment and intermediate confounder?
  - No

$\rightarrow$ Randomized interventional direct and indirect effects.

Estimands: - Direct effect: $\E(Y_{1,G_0} - Y_{0,G_0})$ - Indirect effect: $\E(Y_{1,G_1} - Y_{1,G_0})$
Need to incorporate multiple and continuous intermediate confounders
BUT what about concern that this type of mediation estimand doesn’t reflect an averaging of individual-level mediation?

$\rightarrow$ Can also estimate path-specific effects (using recanting twins)

Similar in terms of interpretation to randomized interventional effects
But fully decompose the average treatment effect (ATE) - And satisfy the sharp mediational null hypothesis

What if the positivity assumption $\P(A=a\mid W)>0$ violated?

$\rightarrow$ Can’t identify or estimate any of the above effects
- But we can estimate the effect of some stochastic interventions, e.g., IPSIs
- Trade-off between feasibility and interpretation
What if the exposure variable is continuous?

$\rightarrow$ All the above effects are defined for binary exposures
- But we can estimate the effect of some stochastic interventions and general interventions on continuous exposures, like shift interventions.
What if the exposure is actually time-varying? What if the mediators and/or intermediate confounders are actually time-varying? - this is hard! but we have two estimators for this: - https://github.com/nt-williams/lcmmtp - https://github.com/nt-williams/lcm

# How to choose an estimand: Real-world example {#estimandirl} ```{r} #| label: load-renv #| echo: false #| message: false renv::autoload() library(here) ``` ## 1. Extent to which the risk of OUD conferred by chronic pain operates through pain management treatments ![](../img/topdag.pdf){width="80%"} This application was explored in detail by @rudolph2025mediation ### Getting specific about the question To what extent does the total effect of chronic pain on risk of OUD operate through prescription medications for pain management and physical therapy, treated as a bundle? 1. **What estimand do we want?** - Can we set $M=m$ (i.e., same value) for everyone? - Are we interested in estimating indirect effects? $\rightarrow$ So, *not* controlled direct effect. - Do we have an intermediate confounder? - Not really, because we: 1) consider all initial treatments following chronic pain diagnosis and 2) stratify by whether or not the patient has an anxiety or depressive disorder. $\rightarrow$ So, could estimate natural (in)direct effects - Estimands: - Direct effect: $\E(Y_{1,M_0} - Y_{0,M_0})$ - Indirect effect: $\E(Y_{1,M_1} - Y_{1,M_0})$ ![](../img/all_mediators_icd.pdf){width="40%"} ## 2. Extent to which the risk of OUD conferred by chronic pain operates through individual pain management treatments ![](../img/bottomdag.pdf){width="80%"} This application was explored in detail by @rudolph2025mediation ### Getting specific about the question To what extent does the overall effect of chronic pain on risk of OUD operate through individual pain management treatments, controlling for other co-occurring or prior pain treatments? 1. **What estimand do we want?** - Can we set $M=m$ (i.e., same value) for everyone? - Are we interested in estimating indirect effects? $\rightarrow$ So, *not* controlled direct effect. - Do we have an intermediate confounder? - Yes, likely multiple, important ones, because we would like to control for prior or co-occurring pain treatments. - So likely do not have a single, binary intermediate confounder. - If we do have a binary intermediate confounder, would we assume monotonicity between the treatment and intermediate confounder? - No $\rightarrow$ Randomized interventional direct and indirect effects. - Estimands: - Direct effect: $\E(Y_{1,G_0} - Y_{0,G_0})$ - Indirect effect: $\E(Y_{1,G_1} - Y_{1,G_0})$ - Need to incorporate multiple and continuous intermediate confounders - BUT what about concern that this type of mediation estimand doesn't reflect an averaging of individual-level mediation? $\rightarrow$ Can also estimate path-specific effects (using recanting twins) - Similar in terms of interpretation to randomized interventional effects - But fully decompose the average treatment effect (ATE) - And satisfy the sharp mediational null hypothesis 2. **What if the positivity assumption** $\P(A=a\mid W)>0$ violated? $\rightarrow$ Can't identify or estimate any of the above effects - But we can estimate the effect of some stochastic interventions, e.g., IPSIs - Trade-off between feasibility and interpretation 3. **What if the exposure variable is continuous?** $\rightarrow$ All the above effects are defined for binary exposures - But we can estimate the effect of some stochastic interventions and general interventions on continuous exposures, like shift interventions. 4. **What if the exposure is actually time-varying? What if the mediators and/or intermediate confounders are actually time-varying?** - this is hard! but we have two estimators for this: - <https://github.com/nt-williams/lcmmtp> - <https://github.com/nt-williams/lcm>