Questions: Time-Varying Confounders and Longitudinal Exposure

Question 1

A study of AZT and HIV survival includes CD4 count as a time-varying covariate in a Cox regression model. CD4 count is affected by prior AZT and also independently predicts mortality. What bias does this introduce?

• A. No bias — including more covariates always reduces confounding
• B. Upward bias only, because sicker patients received more AZT
• C. Collider bias by conditioning on a mediator — blocking part of AZT's causal effect through CD4 improvement while still failing to fully adjust for confounding
• D. Measurement error bias, because CD4 count is imprecisely measured

Answer: (C)

CD4 count at each follow-up visit is simultaneously a confounder for future treatment (sicker patients get more AZT) and an intermediate outcome of past treatment (AZT improves CD4). Conditioning on it in regression blocks the causal pathway from prior AZT through CD4 improvement to survival — this is conditioning on a mediator and biases the treatment effect estimate downward. But omitting it leaves the confounding by CD4 unaddressed, biasing upward. No choice in standard regression resolves both problems simultaneously.

Question 2

Why does time-varying confounding create a structural problem that standard regression adjustment cannot solve, even in principle?

• A. Because regression models cannot include more than one covariate measured at multiple time points
• B. Because time-varying confounders are always unmeasured in practice
• C. Because the same variable is both a confounder (for future exposure) and a mediator (of past exposure), so conditioning on it is simultaneously required and prohibited
• D. Because survival analysis methods like Cox regression do not allow time-varying covariates

Answer: (C)

Standard regression divides variables into confounders (condition on them) and mediators (do not condition on them). A time-varying confounder that is also affected by prior exposure belongs to both categories sequentially: it must be conditioned on to remove confounding for future treatment assignments, but doing so blocks the causal effect of prior exposure. This is a structural, not technical, limitation. Cox regression can include time-varying covariates; the issue is not a modeling constraint but a causal identification problem.

Question 3

Marginal structural models handle time-varying confounding by including most time-varying covariates directly as predictors in the outcome model.

• T. True
• F. False

Answer: False

Marginal structural models solve the problem by *reweighting* observations rather than conditioning on time-varying covariates. Each subject is assigned an inverse probability of treatment weight (IPTW) based on their covariate history, creating a pseudo-population in which treatment assignment is independent of confounders. The outcome model is then fit in this reweighted pseudo-population without including the time-varying confounder as a covariate. Including the time-varying confounder as a predictor is precisely the mistake that standard regression makes.

Question 4

A time-varying confounder is structurally different from a baseline confounder because it can simultaneously be a confounder for future exposure and an intermediate outcome of past exposure.

• T. True
• F. False

Answer: True

This is the defining characteristic of time-varying confounding. A baseline confounder measured before any exposure begins can only be a confounder — it cannot be caused by the exposure. A time-varying covariate that changes during follow-up can be affected by prior exposure while also affecting future exposure and the outcome. It is this temporal dual role — caused by past exposure, causing future treatment decisions and the outcome — that standard regression cannot accommodate.

Question 5

Why does marginal structural model estimation require correctly specifying a model for the *probability of treatment* rather than a model for the *outcome*, and what happens if this model is misspecified?

Model answer: MSMs work by assigning inverse probability of treatment weights (IPTW) to each observation based on their probability of receiving their actual treatment given their covariate history. The weights are derived from a treatment probability model, not the outcome model. If this treatment model is misspecified, the weights are wrong: they fail to fully break the association between confounders and treatment in the pseudo-population, leaving residual confounding in the outcome estimate. The causal estimate then depends directly on how well the treatment model is specified.

This is a crucial practical point: MSMs shift the modeling burden from the outcome equation to the treatment equation. The analyst must correctly model how treatment was assigned given covariates at each time point. Misspecification of this propensity model propagates into the weights and thus into all downstream causal estimates. This is analogous to how standard regression requires correctly specifying the outcome model — you have traded one modeling assumption for another, though the MSM assumption is often more tractable when the causal structure is well-understood.