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!

Summary

IPW and other g-methods, but not traditional methods, are suited to handle time-varying confounding affected by past treatment (feedback)
As with IPW for time-fixed confounding, default standard error estimators of many software packages are inappropriate for IPW regressions, because the weights are falsely assumed to reflect actual (indpendent!) observation frequenties!
IPW can (and sometimes needs) to be combined with marginal structural modelling.

Mediation

What part of the effect “goes via” or is “mediated by” a third variable?

What part of the effect of cognitive behavioural therapy (CBT) on depression symptoms is mediated by antidepressant use?
To what extent is the effect of genetic variants on incident lung cancer mediated by smoking behaviour?
To what extent is the effect of maternal smoking on infant mortality mediated by birth weight?

Notation

Treatment (e.g., CBT): A
Mediator (e.g., antidepressant use): M
Mediator had A been a: M^a
Outcome (e.g., depression symptoms): Y
Outcome had A = a and M = m: Y^a,m

In this notation, the counterfactual outcome under a point intervention on A
is written Y^{a,M^a}

Mediation analyis presumes some sort of decomposition of the totaleffect into direct effects and indirect effect:

Total effect = E[Y^1,M¹ − Y^0,M⁰]
Total effect = E[Y^1,M¹ − Y^1,M⁰ + Y^1,M⁰ − Y^0,M⁰]

Natural direct and indirect effects

The natural indirect effect: E[Y^1,M¹ − Y^1,M⁰]
The natural direct effect: E[Y^1,M⁰ − Y^0,M⁰]

Difficult interpretation! Y^1,M⁰ is a cross-world counterfactual!

Instead of natural direct effects, we can also look at controlled direct effects:

Controlled direct effect

Controlled direct effect (m): E[Y^1,m − Y^1,m]

Natural versus controlled direct effects

Conceptually different
- NDE: “How much of the effect flows through the mediator path?”
- CDE: “How much of the effect would be removed/preserved by fixing the mediator to a constant?”
CDE more interesting from policy-making perspective!
Possibly numerically different

We’ve discussed diferent estimands, but
we’ve yet to look into identification/estimation …

Traditional approach to mediation analysis

Consists of fitting two linear regression models:
- First, linearly regress Y on A only — under trial-like conditions (DAG), the coefficient for A can be interpreted as total effect
- Second, linearly regress Y on A and M — idea: block path through M; the coefficient for A is interpreted as direct effect, which is valid under following model …

Model underlying traditional approach

Y^a,m = α₀ + α₁a + α₂m + U, for all a and m, with mean-zero U ⫫ (A,M)
(N.B.: since U is used for all a,m, the individual effects Y^1,m − Y^0,m are constant — strong “rank-preservation”/cross-world assumption)

Traditionally, people made no distinction between the NDE and the CDE, because under the (highly restrictive) underlying model, they didn’t need too — the NDE and CDE are the same!

Problems with the traditional approach

No interaction term!
It relies on strong rank-preservation/cross-world assumption.
It assumes no (uncontrolled) mediator-outcome confounding
(formally, Y^a,m ⫫ M^a given A = a) — absence not guaranteed by design in typical trial!

As with traditional methods for time-varying confounding control, neither conditioning on L nor not conditioning on L works!

The study of causal mediation can be seen as a special case of causal inference with time-varying treatments … Rather than having a single treatment that takes different values over time, in mediation analysis we have two different variables — the treatment of interest and the mediator — at different times.

Hernán and Robins, 2020, Causal Inference: What If

IPW for inference about controlled directed effects

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

Y ^a,m ⫫ A | L₀ and Y ^a,m ⫫ M | L₀, L₁, A=a

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

E[Y ^a,m] = E_{pseudopopulation}[Y | A=a, M=m],