Causal Assumptions in Individual-Level Exposure-Response

From an Exposure-Response Association to a Causal Question

An individual-level exposure-response analysis relates a person's exposure to an outcome. A pattern in observed data may be useful for prediction, but it is not automatically the effect that would occur if exposure were deliberately changed. People with different exposures can also differ in baseline characteristics that affect the outcome. A causal interpretation therefore depends on assumptions about the intervention, the measured covariates, and which exposures are represented in the data.

Let \(A\) denote an exposure metric selected for the clinical question, \(L\) a set of baseline covariates measured before exposure, and \(Y\) the outcome over a specified follow-up period. For each exposure level \(a\), let \(Y(a)\) denote the outcome that would be observed under an intervention setting exposure to \(a\). Only one of these potential outcomes is observed for each individual. The target may be a population mean \(E[Y(a)]\), or a contrast such as \(E[Y(a_1)]-E[Y(a_0)]\).

The intervention must be meaningful for the exposure metric and outcome being studied. For example, “set exposure to \(a\)” must describe a sufficiently well-defined intervention, rather than combining materially different ways of attaining the same measured exposure that could have different effects. The estimand also depends on the target population and outcome definition; those choices should be made before interpreting the exposure-response curve.

The Assumptions and Their Consequences

Consistency states that if an individual actually receives exposure \(A=a\), then their observed outcome equals the potential outcome under that exposure: \(Y=Y(a)\) when \(A=a\). It presumes that the observed exposure corresponds to the intervention represented by \(a\), including relevant versions of that intervention. Consistency does not say that exposure was randomly assigned.

Conditional exchangeability states that, within levels of the measured baseline covariates, exposure is independent of the potential outcomes:

This assumption means that after conditioning on \(L\), there are no remaining common causes of exposure and outcome that would make people at different exposures incomparable for the causal question. It cannot be verified from the observed data alone. It depends on substantive knowledge, measurement quality, and a defensible covariate set.

Positivity requires that, for each covariate profile in the target population, the exposure level or levels needed for the comparison are possible and represented. For a discrete exposure, this means \(P(A=a\mid L=l)>0\) for each relevant \(l\) and each exposure level \(a\) being compared. For a continuous exposure, the probability of exactly one value is generally zero. Positivity is therefore expressed using the conditional exposure density: for an exact point intervention at \(a\), the density must be positive in a neighborhood of \(a\) for the relevant covariate profiles. Local support is necessary, but, by itself, it does not identify the response at the exact point; regularity conditions are also needed, as discussed below.

Adjustment should use covariates measured before the exposure intervention and selected to make exchangeability plausible. Relevant baseline common causes of exposure and outcome are important candidates. A variable that occurs after exposure may be part of the causal pathway, so adjusting for it can remove part of the effect of interest; other post-exposure variables can introduce bias. A variable is not automatically appropriate simply because it predicts the outcome. The covariate-confounding material earlier in the course provides useful background for assessing these choices.

Standardization for Discrete Exposure

When exposure levels are discrete, the assumptions give a direct identification result. Averaging over the baseline covariate distribution of the target population gives the mean potential outcome at level \(a\). Exchangeability permits replacement of the potential-outcome mean within each covariate profile by the corresponding mean among individuals observed at exposure \(a\). Consistency then links their observed outcome to \(Y(a)\).

For continuous covariates, the sums are replaced by integrals over the target population's covariate distribution. This is the standardization, or g-formula, expression. It averages conditional outcome means at exposure \(a\) using the target population's covariate distribution, rather than the covariate distribution among only those observed at \(a\). The result is an identification statement under the assumptions; estimating the conditional means and averaging them introduces additional modeling and sampling uncertainty.

Continuous Exposure: Exact Values Need More Than Local Support

For continuous \(A\), \(P(A=a\mid L=l)=0\) in the usual case. Thus, the discrete-exposure argument cannot simply be reused by conditioning on the event \(A=a\): conditional means at exactly \(a\) are defined only up to versions on probability-zero sets. Observations near \(a\) do not, without further assumptions, determine the conditional mean at precisely \(a\). A function can be changed at that single point without changing the observed-data distribution.

A common sufficient route to identifying an exact-point response adds regularity. Suppose the exposure is continuously distributed, its conditional density is positive in a neighborhood of \(a\) for the covariate profiles in the target population, and the conditional causal response \(\mu(a,l)=E[Y(a)\mid L=l]\) has a continuous version in exposure near \(a\). Suppose also that the observed conditional mean \(m(a,l)=E[Y\mid A=a,L=l]\) has a continuous version there and, by exchangeability and consistency, \(m(x,l)=\mu(x,l)\) for almost every \(x\) in that neighborhood. Continuity then extends the equality to \(a\). These are additional identifying assumptions, not consequences of positivity alone.

The expression is therefore justified for a continuous exposure only under appropriate regularity and a well-defined point intervention, in addition to consistency, exchangeability, and local support. Another option is to define a causal question that does not rely on a single exact exposure value, such as the effect of an exposure range or a stochastic intervention that shifts the exposure distribution. The intervention and estimand must be stated precisely in either case.

Adjustment Strategies, Diagnostics, and Alternatives

Standardization can be implemented by estimating conditional outcome means for each exposure and covariate profile, predicting outcomes under the exposure level of interest for every member of the target population, and averaging those predictions. Regression adjustment is one way to estimate the conditional means, but a well-fitting regression alone does not establish exchangeability or positivity. Stratification is useful when the covariate set is small and discrete; weighting or matching can also seek comparability, but neither creates information in unsupported parts of the exposure range.

If exposure groups barely overlap, estimates can become unstable or depend heavily on modeling choices. A lack of support is not repaired by adding more covariates or fitting a more flexible model; the target population or causal contrast may need to be narrowed. For continuous exposure, inspect density or exposure-range plots within important covariate profiles, not only the overall exposure histogram. Sensitivity analyses can assess how conclusions change under alternative plausible models, but they cannot prove that unmeasured confounding is absent.

The principal alternative is to report an adjusted association rather than a causal effect when the assumptions are not credible. That may still be useful for prediction or description, provided the interpretation is explicit. A stochastic exposure intervention or an effect over a supported range may also be preferable to an exact-point effect when point support or the meaning of setting exposure exactly is problematic.

Common Pitfalls and Limits

• Calling an adjusted association causal by default. Adjustment supports causal interpretation only if the intervention is meaningful and the assumptions, especially exchangeability, are credible.
• Using the discrete positivity condition for a continuous exposure. \(P(A=a\mid L=l)>0\) is generally false for a continuous variable. Use local density support, and recognize that exact-point identification also needs regularity.
• Confusing nearby observations with proof of an exact-point effect. Nearby exposure values do not determine the conditional mean at \(a\) without a continuity or other identifying assumption.
• Adjusting for every available variable. Post-exposure variables can change the causal quantity being estimated or introduce bias; covariates should be chosen in light of their timing and causal role.
• Assuming good fit verifies causal assumptions. Diagnostics can reveal poor fit or support problems, but observed data alone cannot confirm the absence of unmeasured common causes.

Even when the assumptions are stated carefully, estimated causal exposure-response functions remain uncertain because of sampling variation, covariate measurement, model specification, and possible departures from the identifying assumptions. Conclusions should therefore be limited to the defined intervention, outcome, follow-up, and population rather than generalized automatically to other settings.