lecture_friday

Introduction to Causal Inference and Causal Data Science

Day 5:
Mediation and Complex Longitudinal Settings

Bas B.L. Penning de Vries
Dept. Data Science & Biostatistics
Julius Center, UMC Utrecht

Topics

A taxonomy of estimands on time-varying treatments (incl. intention-to-treat and per-protocol effects) and mediation
Why/when traditional methods fail for inference about time-varying treatments or mediation
G-methods for time-varying treatments and mediation

Inference about time-varying treatments

If treatment/exposure is time-varying, there are many possible causal contrasts …

single- versus multiple-point interventions
2^{# time points} static rules
Static versus dynamic

Single-point (baseline) intervention

Decision at baseline only — individuals are allowed to deviate

Assign/initiate versus withhold drug treatment at baseline (intention-to-treat)

Multiple-point (joint) intervention

Decisions/randomisation at multiple points

Sustained/daily/weekly/monthly drug use versus continuous non-use (per-protocol)

Static treatment rule/regime/protocol/…

… assigns the same treatment option to everyone

Assign versus withhold treatment programme at baseline (intention-to-treat)
Always treat versus never treat (per-protocol)

Dynamic (individualised) treatment rule

… assigns treatment based on the then-available information

Choose dose depending on baseline covariates
Start when blood marker first drops below threshold
Stop when toxicity occurs

Static

Single-point

Multiple-point

Dynamic

Inference about always-/never-treat strategies

Trials with baseline randomisation naturally answer questions on point interventions.
Questions about always-/never-treat strategies are most naturally addressed by trials with (full adherence or) sequential randomisation.

For simplicity, let’s assume there are only two intervention times, t₀ and t₁:

A₀ and A₁ are indiciator variales for treatment at t₀ and t₁, respectively
Y ^a₀,a₁ is the outcome if, possibly contrary to fact, A₀ = a₀ and A₁ = a₁

Identification of always- versus never-treat effect (ATE) under sequential randomisation

ATE

= E[Y ^1,1] − E[Y ^0,0]

(causal estimand, assuming two intervention times)

= E[Y ^1,1 | A₀ = 1] − E[Y ^0,0 | A₀ = 0]

(randomisation at t₀)

= E[Y ^1,1 | A₀ = 1, A₁ = 1] − E[Y ^0,0 | A₀ = 0, A₁ = 0]

(randomisation at t₁)

= E[Y | A₀ = 1, A₁ = 1] − E[Y | A₀ = 0, A₁ = 0]

(consistency)

But what if treatment is not assigned by (sequential) randomisation? How to address such departures from target trial?

DAG after Naimi et al. (2017). Here, A₀ may represent anti-retroviral treatment at time 0, L HIV viral load just prior to the second round of treatment, A₁ anti-retroviral treatment status at time 1, Y the CD4 count measured at the end of follow-up, U and an unmeasured common cause of HIV viral load and CD4.

🧩 Condition on (“adjust for”) L? Hint: we want to block non-causal paths, while leaving all direct (causal) paths from A₀ or A₁ to Y open!

⚠️ Traditional methods for confounding adjustment

Traditional methods (multivariable regression modelling) are not suited for time-varying confounding adjustment when there is covariate-treatment feedback!

Methods that can handle treatment-covariate feedback and adjust for time-varying confounding:

IPW
G-computation
G-estimation

IPW for inference about sustained treatment strategies

If, in addition to consistency and positivity, there is sequential conditional exchangeability of the form

Y ^a₀,a₁ ⫫ A₀ | L₀ and Y ^a₀,a₁ ⫫ A₁ | L₀, L₁, A₀=a₀

(i.e., L₀ blocks all backdoor paths from A₀ to Y, and L₀, L₁, A₀ together block all backdoor paths from from A₁ to Y), then

E[Y ^a₀,a₁] = E_{pseudopopulation}[Y | A₀=a₀, A₁=a₁],

where the pseudopopulation is obtained by weighting everyone by W₀W₁ with

W₀ = A₀ / Pr(A₀ = 1 | L₀) + (1 − A₀) / Pr(A₀ = 0 | L₀)
and W₁ = A₁ / Pr(A₁ = 1 | L₀, L₁, A₀) + (1 − A₁) / Pr(A₁ = 0 | L₀, L₁, A₀)

💡 IPW doesn’t adjust by conditioning — think of it as surgically removing arrows!

Introduction to Causal Inference and Causal Data Science

Day 5:Mediation and Complex Longitudinal Settings

Topics

Inference about time-varying treatments

Inference about always-/never-treat strategies

Day 5:
Mediation and Complex Longitudinal Settings