Choosing Among Multiple Plausible Adjustment Sets

Why Compare Adjustment Sets?

In multiple regression, an adjusted association describes the relationship between an exposure and an outcome after accounting for selected covariates. Choosing those covariates is not just a matter of adding every available variable. A variable may be a common cause of exposure and outcome, a mediator on the pathway between them, or a common effect of other variables. Those roles matter: adjusting for a common cause may help address confounding, while adjusting for a mediator can change the association of interest. Conditioning on a common effect can also introduce bias.

Sometimes subject-matter knowledge and the timing of measurements do not settle a variable’s role. In that situation, it is useful to identify a small number of plausible adjustment sets, state the assumptions behind each, and compare the exposure coefficient from each fitted model. The comparison documents how the adjusted association depends on the assumptions; it does not, by itself, tell us which assumptions are true.

What the Coefficient Means

For a continuous outcome \(Y\), an exposure \(X\), and covariates \(Z_1,\ldots,Z_k\), an additive multiple regression model can be written as:

Here, \(\beta_X\) is the model’s exposure coefficient. Under this model, it describes the difference in the fitted mean outcome associated with a one-unit difference in \(X\), comparing observations with the same included covariate values. It is an adjusted association. To interpret it as a causal effect requires further assumptions, including an appropriate adjustment set and adequate control of confounding; regression alone does not establish those assumptions.

For the coefficient interpretation to be appropriate, the model’s conditional mean must be represented adequately by its chosen form. In the equation above, that means a linear, additive relationship in the included predictors. The observations must also be analyzed on a consistent basis: comparisons of candidate sets should use the same outcome, exposure scale, and analysis sample. Otherwise, a coefficient difference could reflect a change in data or model specification unrelated to the covariate choice. Exact linear dependence among predictors must not prevent the coefficients from being separately estimated.

The coefficient is not required to stay the same across adjustment sets. It can change because the included variables account for different sources of association, because a variable lies on a pathway of interest, or because the model’s assumptions are imperfect. A changed estimate does not identify which explanation is correct.

Build Candidate Sets From Competing Assumptions

A useful comparison begins with the causal question and a diagram or verbal account of how the variables might relate. Earlier material on using subject-matter knowledge to select covariates and on distinguishing confounders, mediators, and prognostic variables provides the relevant concepts. Here the aim is to use them to organize a comparison when a variable’s role remains uncertain.

For each candidate set, write down what must be true for it to support the intended interpretation. For example, one set might adjust for a variable because it is assumed to precede and affect both exposure and outcome. A second might omit that variable because it is assumed to occur after exposure and lie on a pathway to the outcome. If the variable’s timing or causal role is unknown, neither choice should be presented as settled fact.

Keep the comparison focused. Each model should use the same outcome, exposure coding, and observations, and should differ only in the covariates whose roles are genuinely in question. This makes the comparison easier to interpret. It also avoids treating a broad collection of arbitrary models as if it were a principled analysis.

Interpreting Differences Without Overclaiming

When the exposure coefficient changes across candidate sets, describe the change in the coefficient’s units and explain which covariate was added or removed. Then connect each model to its assumptions. Avoid saying that adjustment “revealed the true effect” or that the model with the most covariates is automatically least biased. The comparison is informative because it makes dependence on assumptions visible, not because regression can determine the causal structure from the coefficient estimates alone.

It can be tempting to select whichever model has the largest \(R^2\), the smallest p-value, or an exposure coefficient that matches expectations. These criteria do not establish that a set is causally appropriate. Adding predictors can improve prediction of the outcome while including a mediator or common effect that is unsuitable for the intended question. Likewise, an estimate’s direction or magnitude cannot prove that the model’s adjustment choices are correct.

A Practical Comparison and a Common Pitfall

A concise comparison can be organized in four steps:

Common mistake: “The model with the most covariates must control the most confounding, so I will report that model.” This reasoning treats all covariates as harmless adjustments. A post-exposure mediator or common effect can make an expanded set inappropriate for the target question. The correction is to justify each covariate’s inclusion using its timing and plausible causal role, then present alternative estimates when reasonable assumptions remain unsettled.

A clear report need not imply that all candidate sets are equally plausible. It can identify one set as primary because it best matches the available subject-matter knowledge and the target association, while showing how alternatives change the estimate. The limitation should be stated plainly: if the variable roles cannot be determined from the available information, the adjusted association remains dependent on those assumptions.