Why One Conventional Regression Cannot Generally Resolve Feedback

Why Feedback Creates a Regression Trade-Off

In a longitudinal study, treatment decisions occur at different times, and information collected between decisions can affect what happens next. A covariate measured after an earlier treatment may therefore have two roles: it can carry part of that treatment’s effect forward, and it can confound the relationship between a later treatment and the outcome. This is treatment-confounder feedback.

The difficulty is not that regression is always unsuitable. It is that a conventional regression adjustment typically conditions on a chosen set of covariates, and that same conditioning choice can have different consequences for treatment effects at different times. Controlling a covariate may help address later confounding while blocking a pathway through which earlier treatment affects the outcome.

The treatment sequence \((a_0,a_1)\) specifies the treatment assigned at both decision times. Write \(Y(a_0,a_1)\) for the outcome that would be observed under that sequence. A total effect of changing the first treatment compares outcomes under different first-treatment assignments while allowing consequences of that assignment—including changes in \(L_1\) and later treatment—to contribute. It is different from an effect that holds \(L_1\) fixed.

For observational data, interpreting an adjusted regression causally also requires suitable assumptions. In particular, treatment assignment at each decision must be unconfounded given the relevant observed history, treatment options must have positive probability across histories being compared, and the observed outcome under a treatment sequence must agree with the corresponding potential outcome (consistency). These assumptions do not make every regression coefficient a total effect; the adjustment and the target still have to match.

The Two Roles of the Intermediate Covariate

The relevant paths can be read directly from the DAG. Because \(L_1\) predicts both \(A_1\) and \(Y\), it can create confounding in the later-treatment–outcome relationship. For example, if greater disease severity increases the chance of a dose increase and also predicts a worse outcome, a comparison of patients receiving different later doses can partly reflect their severity rather than the dose.

At the same time, if \(A_0\) changes \(L_1\), then \(L_1\) lies on a pathway from the earlier treatment toward the outcome. That pathway can run directly through \(L_1\) to \(Y\), or continue from \(L_1\) to \(A_1\) and then to \(Y\). Conditioning on \(L_1\) blocks those routes in the usual path-based interpretation of regression adjustment.

Thus a regression such as \(Y \sim A_0 + A_1 + L_1\) uses \(L_1\) as an adjustment variable for both treatment terms. Its coefficient for \(A_0\) describes a comparison holding \(L_1\) and \(A_1\) fixed, under the model’s assumptions. That is not generally the total effect of changing \(A_0\), because the comparison excludes effects carried through \(L_1\). If \(L_1\) is omitted instead, the regression may fail to control confounding of the \(A_1\)-outcome relationship.

Why a Single Adjustment Set Does Not Generally Solve Both Problems

A conventional adjustment set is a collection of variables conditioned on together. When the same regression includes treatment terms for multiple decision times, it does not ordinarily assign a different causal role to a covariate for each treatment term. Including \(L_1\) conditions on it in the comparison of \(A_0\) as well as in the comparison of \(A_1\); leaving it out means it is not controlled for either treatment term.

This is the key limitation: a set that is useful for removing later confounding can be inappropriate for estimating an earlier total effect. Conversely, a set that avoids blocking the earlier treatment’s mediated effect can leave later confounding open. The problem is about the target effect and the causal structure, not merely about choosing more or fewer predictors or improving a model’s fit.

A Numerical Illustration of Later Confounding

The following hypothetical calculation isolates the later-treatment problem. Let \(L_1=1\) indicate high severity and \(L_1=0\) low severity. Suppose, within a fixed \(A_0\) group, half the individuals have high severity. Let the probability of receiving \(A_1=1\) be 0.8 for high-severity individuals and 0.2 for low-severity individuals. For illustration, suppose the mean outcome follows the stated relationship \(E[Y\mid A_0,L_1,A_1]=10+A_0+4L_1+2A_1\). Thus, holding \(A_0\) and \(L_1\) fixed, the outcome difference associated with changing \(A_1\) is 2 units.

Deriving the Trade-Off From the Graph

The logic can be summarized as a short path argument. Suppose the graph contains the four arrows \(A_0 \rightarrow L_1\), \(L_1 \rightarrow A_1\), \(L_1 \rightarrow Y\), and \(A_1 \rightarrow Y\). For the later treatment, the path \(A_1 \leftarrow L_1 \rightarrow Y\) is a backdoor path. Conditioning on the non-collider \(L_1\) blocks it. For the earlier treatment, the directed paths \(A_0 \rightarrow L_1 \rightarrow Y\) and \(A_0 \rightarrow L_1 \rightarrow A_1 \rightarrow Y\) pass through that same variable. Conditioning on \(L_1\) blocks those directed paths. A single adjustment set that includes \(L_1\) therefore has opposing consequences for these two targets.

This is a graph-based justification, not a claim about every possible data-generating process. If \(A_0\) does not affect \(L_1\), for example, the specific earlier-treatment pathways just described are absent. If \(L_1\) does not cause \(Y\) or later treatment, its role may also differ. Whether conventional adjustment is adequate depends on the actual graph and target; the general lesson is that one must not treat a post-treatment covariate as an ordinary baseline confounder.

Common Mistake

A frequent mistake is to reason, “\(L_1\) predicts both later treatment and the outcome, so it is a confounder; therefore I should adjust for it in the regression.” The first sentence may correctly describe its role for \(A_1\), but it omits the fact that \(L_1\) occurs after and may be caused by \(A_0\). Conditioning on it can remove part of the earlier treatment’s total effect. The opposite shortcut—“never adjust for post-treatment variables”—also fails, because leaving \(L_1\) uncontrolled can leave the later-treatment backdoor path open.

The correction is to state the effect being targeted, draw the time-ordered causal graph, and assess the consequence of conditioning on each covariate for each treatment time. No variable’s label alone determines whether it belongs in an adjustment set.