Workshop: Audit A Longitudinal Regression Analysis

Why Audit the Timeline Before the Regression

A longitudinal regression can contain several treatment measurements and covariates collected between them. Its formula may look familiar, but the causal meaning of adjustment depends on when each variable was measured and on how earlier treatment may have changed later covariates. A covariate that predicts a later treatment can be important for confounding control and, at the same time, lie on a pathway from an earlier treatment to the outcome.

The purpose of an audit is not to decide from the regression formula alone that a causal effect has been identified. It is to line up the study timeline, the covariate list, and the fitted model; identify likely consequences of the included adjustments; and report the treatment coefficients at the level they support. The central question is: which contrast does the model estimate after its included variables are held fixed?

Definition: In a two-wave setting, let \(A_0\) be treatment at the first wave, \(L_1\) a covariate measured after that treatment but before the second treatment, \(A_1\), and \(Y\) the outcome. Treatment-confounder feedback occurs when an earlier treatment affects a later covariate, and that covariate in turn predicts subsequent treatment and the outcome. Thus \(L_1\) can be both treatment-affected and a confounder of the later treatment-outcome relationship.

For example, initial treatment may change disease severity at a follow-up visit. That severity may influence the clinician’s next treatment decision and also predict the final outcome. Adjusting for severity may help compare patients with similar indications for the second treatment, yet it also conditions on a variable affected by the first treatment. Earlier material on affected confounders and collider bias explains why this can distort a total-effect analysis.

A Practical Audit Sequence

Before interpreting a coefficient, write down the actual measurement order. Use the protocol, visit schedule, and data definitions rather than relying on variable names alone. A covariate called “baseline severity” is not baseline for a treatment decision if it was measured after treatment began.

Order the measurements.
List baseline covariates, each treatment wave, covariates between waves, and the outcome. Record when each value is measured relative to each treatment decision.

Trace plausible arrows.
Ask whether an earlier treatment could affect a later covariate, and whether that covariate predicts a subsequent treatment and the outcome. If so, flag feedback.

Match the model terms to the timeline.
Identify which post-treatment variables are adjusted for, omitted, or combined across visits. Check whether treatment interactions or nonlinear terms change the meaning of a coefficient.

State the coefficient’s contrast and limits.
Describe what is compared while the model’s other terms are held fixed. Then explain why that conditional contrast is not, without further justification, the total causal effect of a treatment history.

This audit relies on a basic model interpretation: in a correctly specified linear regression, a treatment coefficient describes a fitted conditional mean difference for a one-unit change in that treatment, holding the other included predictors fixed (and, in an additive model, assuming the same difference at every covariate value). It is first a statement about the fitted model. Causal interpretation additionally requires suitable design and assumptions, including adequate control of confounding for the particular causal contrast. A regression formula does not establish those assumptions.

Worked Audits

Worked Example: A Post-Treatment Severity Measure in the Model

Hypothetical study. Patients begin treatment \(A_0\) at baseline. Disease severity \(L_1\) is measured at the next visit, a second treatment \(A_1\) is then chosen, and the final symptom score \(Y\) is recorded. Baseline severity \(C\) is measured before \(A_0\). Investigators fit the additive linear model

E(Y\mid A_0,L_1,A_1,C) =\beta_0+\beta_{A_0}A_0+\beta_{L_1}L_1+\beta_{A_1}A_1+\beta_C C.

Suppose the hypothetical fitted coefficients are \(\beta_0=5.0\), \(\beta_{A_0}=1.2\), \(\beta_{L_1}=0.6\), \(\beta_{A_1}=0.8\), and \(\beta_C=0.3\). The outcome is a symptom score, with higher values indicating worse symptoms. The model predicts, for \(C=1\), \(L_1=4\), and \(A_1=1\), the following conditional means:

\begin{aligned} \widehat{E}(Y\mid A_0=0,L_1=4,A_1=1,C=1) &=5.0+(1.2)(0)+(0.6)(4)+(0.8)(1)+(0.3)(1)\\ &=8.5,\\ \widehat{E}(Y\mid A_0=1,L_1=4,A_1=1,C=1) &=5.0+(1.2)(1)+(0.6)(4)+(0.8)(1)+(0.3)(1)\\ &=9.7. \end{aligned}

The fitted conditional difference is \(9.7-8.5=1.2\) symptom-score points. It compares \(A_0=1\) with \(A_0=0\) at the same modeled values of later severity, later treatment, and baseline severity. Because \(L_1\) is measured after \(A_0\), it may carry part of the effect of \(A_0\) to \(Y\). Holding it fixed therefore does not estimate the overall contrast that allows severity to evolve under each initial treatment. In addition, \(L_1\) may influence \(A_1\), making its adjustment relevant to the later treatment comparison. The coefficient alone does not establish either a total effect or a direct causal effect.

Audit conclusion: the model adjusts for a post-\(A_0\) covariate that may be both a mediator of \(A_0\)'s effect and a confounder of the \(A_1\)-\(Y\) relationship. Report \(1.2\) as the model’s conditional association unless a separate causal argument supports a more specific interpretation.

Worked Example: Conditioning on a Treatment-Affected Collider

Hypothetical illustration. Suppose \(A_0\) is randomized and binary, and an unmeasured binary factor \(U\) is independent of \(A_0\). A later measured variable \(L_1\) is determined by \(L_1=A_0+U\), while the outcome is \(Y=10+U\). A later treatment \(A_1\) is randomized independently and has no effect on \(Y\). Consider the regression of \(Y\) on \(A_0,L_1,A_1\).

By construction, changing \(A_0\) while leaving \(U\) unchanged does not change \(Y\), since \(Y=10+U\). The marginal mean outcome for either value of \(A_0\) is

E(Y\mid A_0=0)=E(Y\mid A_0=1)=10+E(U)=10.5,

because \(U\) is equally likely to be 0 or 1 and is independent of randomized \(A_0\). But the regression adjusts for \(L_1\), which is caused by both \(A_0\) and \(U\). Since \(U=L_1-A_0\), the outcome can be written exactly as

$$ Y=10+L_1-A_0. $$

Thus the fitted regression has coefficient \(-1\) for \(A_0\), coefficient \(1\) for \(L_1\), and coefficient \(0\) for the independent \(A_1\). For instance, among observations with \(L_1=1\), \(A_0=1\) implies \(U=0\), so \(Y=10\); \(A_0=0\) implies \(U=1\), so \(Y=11\). The adjusted contrast is \(10-11=-1\) point, despite the zero marginal causal effect of \(A_0\).

Here \(L_1\) is a collider on the path \(A_0\rightarrow L_1\leftarrow U\rightarrow Y\). Conditioning on it creates an association between \(A_0\) and \(U\), which distorts the treatment comparison. This deliberately simple construction isolates the collider mechanism; actual data require subject-matter reasoning to determine whether a measured covariate has this role.

Worked Example: An Interaction Changes the Coefficient’s Meaning

Hypothetical study. Researchers regress a final outcome score \(Y\) on two treatment waves, an intermediate severity measure \(L_1\), and a treatment interaction:

\widehat{E}(Y\mid A_0,A_1,L_1) =12+0.4A_0+1.1A_1+0.9(A_0A_1)+0.6L_1.

Assume both treatments are coded 0 or 1 and that higher \(Y\) is worse. In this model, the coefficient \(0.4\) is the fitted \(A_0\) contrast when \(A_1=0\). When \(A_1=1\), the contrast is \(0.4+0.9=1.3\) points. For example, at \(L_1=2\) and \(A_1=1\), the predicted outcomes are

\begin{aligned} \widehat{E}(Y\mid A_0=0,A_1=1,L_1=2) &=12+0+1.1+0+1.2=14.3,\\ \widehat{E}(Y\mid A_0=1,A_1=1,L_1=2) &=12+0.4+1.1+0.9+1.2=15.6,\\ 15.6-14.3&=1.3\text{ points}. \end{aligned}

So it would be incorrect to report \(0.4\) as a uniform fitted contrast for \(A_0\) across both levels of \(A_1\). Nor do these conditional predictions, by themselves, describe the effect of changing an entire treatment history: \(L_1\) is held fixed, and the model has not established that its conditional comparisons are causal.

What the Audit Can and Cannot Establish

The audit can reveal that a regression’s adjustment set does not correspond to a simple “control for all confounders” strategy. In a feedback structure, a later covariate may be necessary to address confounding of a later treatment while also being affected by an earlier treatment. Conventional adjustment can then block part of an earlier treatment’s pathway or, depending on the causal structure, induce collider bias. Which problem is present depends on the actual arrows, not merely on the variable’s label.

The audit cannot determine from the regression equation alone whether a causal effect is identified. It does not verify that all relevant common causes were measured, that measurements precede the decisions they are meant to explain, or that the functional form is adequate. It also does not turn a conditional coefficient into a total effect. A careful report names the outcome and units, the treatment contrast, the variables held fixed, any interaction-dependent interpretation, and the causal limitation.

Common Mistakes

Calling every adjusted covariate a confounder. A variable measured after treatment may be affected by that treatment. Trace its timing and plausible causes before deciding what adjustment means.
Reading a coefficient as the total effect. In a model that holds later severity fixed, the coefficient does not represent a contrast that allows that severity to change naturally under different earlier treatments.
Assuming more adjustment must reduce bias. Conditioning on a treatment-affected collider can create an association that was absent before adjustment.
Ignoring interactions. With a treatment interaction, a main-effect coefficient is a contrast at the reference value of the interacting treatment, not necessarily a common contrast for everyone.
Inferring causality from model fit. A well-fitting regression can still have an inappropriate adjustment set or unmeasured confounding.

Key takeaway: Audit the time order first, then interpret each treatment coefficient as a model-based conditional contrast. When an intermediate covariate is affected by earlier treatment and predicts later treatment and outcome, ordinary regression adjustment can have competing consequences; the coefficient is not automatically a total causal effect.

Check Your Understanding

For each question, use the timeline and model terms to explain the interpretation rather than relying on the variable names.

A covariate is measured after \(A_0\) and before \(A_1\). What two roles can it have if it predicts both \(A_1\) and \(Y\), and \(A_0\) can affect it?
In an additive regression containing \(A_0,L_1,A_1\), what does the coefficient of \(A_0\) describe in the fitted model?
Why can adjusting for a treatment-affected collider create a treatment-outcome association even when the treatment was randomized?
In a model containing \(A_0A_1\), how would you calculate the fitted \(A_0\) contrast when \(A_1=1\)?
Name two conclusions a timeline-and-model audit can support and one causal conclusion it cannot establish by itself.