A Recursive Linear Model Of Confounder Feedback

Why Write the Model Recursively?

With repeated treatment, the variables at one visit help determine what happens at the next. A prior treatment can change an intermediate covariate, and that covariate can both predict later treatment and affect the outcome. The intermediate covariate is therefore a confounder of the later treatment–outcome relationship, but it is also a consequence of prior treatment. This is the feedback pattern that makes ordinary adjustment difficult.

The earlier material on treatment-confounder feedback and direct and total effects motivates the path distinctions. Here the aim is to make those paths explicit in one recursive linear model and solve its equations. The model is a tool for understanding the consequences of adjustment, not a claim that all longitudinal data follow linear relationships.

Definition: A recursive linear structural model specifies each variable as a linear function of variables that precede it in time, plus an error term. In the two-visit model below, $A_0$ is prior treatment, $L_1$ is an intermediate covariate, $A_1$ is later treatment, $Y$ is the outcome, and $X$ is a measured baseline covariate. The ordering is $X,A_0,L_1,A_1,Y$.

For compactness, write the equations for variables centered at their means. Intercepts can be restored without changing the path coefficients or the contrasts derived below:

\begin{aligned} A_0 &= \alpha_X X+\varepsilon_0,\\ L_1 &= \beta_A A_0+\beta_X X+\varepsilon_L,\\ A_1 &= \gamma_A A_0+\gamma_L L_1+\gamma_X X+\varepsilon_1,\\ Y &= \tau_A A_0+\tau_L L_1+\tau_1 A_1+\tau_X X+\varepsilon_Y. \end{aligned}

The errors represent variation not explained by the displayed predictors. For the cleanest causal interpretation, suppose the errors have mean zero, finite variances, and are mutually independent and independent of $X$. Also suppose the equations remain valid under the interventions being considered: setting treatment does not itself change the other equations except through their displayed inputs. These are strong assumptions. In observational work, the equations are often approximations to conditional means; causal conclusions additionally require adequate control of confounding, well-defined interventions, and sufficient support for the treatment choices being compared.

The key feedback feature is $\beta_A\ne 0$: prior treatment changes $L_1$. The covariate $L_1$ then predicts later treatment through $\gamma_L$, and predicts the outcome through $\tau_L$. Thus $L_1$ is both downstream of $A_0$ and a common cause, in the model, of $A_1$ and $Y$.

Solving the Equations Reveals the Paths

Consider setting $A_0$ to a value $a_0$, while allowing later treatment to follow its structural equation. First substitute $a_0$ into the equation for $L_1$, then substitute the resulting expression into the equation for $A_1$, and finally substitute both into the outcome equation. The result is

\begin{aligned} L_1(a_0) &= \beta_A a_0+\beta_X X+\varepsilon_L,\\ A_1(a_0) &= (\gamma_A+\gamma_L\beta_A)a_0 +(\gamma_L\beta_X+\gamma_X)X+\gamma_L\varepsilon_L+\varepsilon_1,\\ Y(a_0) &= \{\tau_A+\tau_L\beta_A +\tau_1(\gamma_A+\gamma_L\beta_A)\}a_0\\ &\quad+\{\tau_L\beta_X+\tau_1(\gamma_L\beta_X+\gamma_X)+\tau_X\}X\\ &\quad+(\tau_L+\tau_1\gamma_L)\varepsilon_L+\tau_1\varepsilon_1+\varepsilon_Y. \end{aligned}

Result: Under the stated model and intervention-invariance assumptions, the mean outcome contrast from changing prior treatment by one unit, while allowing later treatment to evolve naturally, is $$ \frac{\partial E[Y(a_0)]}{\partial a_0} =\tau_A+\tau_L\beta_A+\tau_1\gamma_A+\tau_1\gamma_L\beta_A. $$ The four terms correspond to the direct path $A_0\to Y$, the path $A_0\to L_1\to Y$, the path $A_0\to A_1\to Y$, and the path $A_0\to L_1\to A_1\to Y$, respectively.

Derivation. Under an intervention setting $A_0=a_0$, the first equation for $L_1$ gives a change of $\beta_A$ units in $L_1$ per unit change in $a_0$. The equation for $A_1$ then gives a change of $\gamma_A+\gamma_L\beta_A$ units in later treatment: $\gamma_A$ is the direct dependence on $A_0$, and $\gamma_L\beta_A$ is the change transmitted through $L_1$. In the outcome equation, a unit change in $A_0$ contributes $\tau_A$ directly, $\tau_L\beta_A$ through $L_1$, and $\tau_1(\gamma_A+\gamma_L\beta_A)$ through $A_1$. Adding these contributions gives the stated expression. Because the errors have mean zero and do not change under the intervention, their expectations contribute no term to the contrast. $\square$

Holding later treatment fixed gives a different contrast. If both treatments are set externally to $a_0$ and $a_1$, the outcome equation and the equation for $L_1$ imply

E[Y(a_0,a_1)\mid X] =(\tau_A+\tau_L\beta_A)a_0+\tau_1a_1 +(\tau_L\beta_X+\tau_X)X.

The controlled contrast for $A_0$ is therefore $\tau_A+\tau_L\beta_A$, not the natural-course contrast above. It includes the route through $L_1$, but not the routes through $A_1$, because $A_1$ is held fixed. Being precise about which treatment is allowed to respond is essential when interpreting a coefficient or a contrast.

Worked Paths and Adjustment Consequences

Worked Example: Decomposing the Prior-Treatment Effect

Suppose $\beta_A=0.5$, $\gamma_A=0.2$, $\gamma_L=0.8$, $\tau_A=1$, $\tau_L=3$, and $\tau_1=2$. Treat $X$ as fixed. Find the natural-course effect of increasing $A_0$ by one unit and identify each path contribution.

The direct path contributes $\tau_A=1$. The route through $L_1$ directly to the outcome contributes $\tau_L\beta_A=3(0.5)=1.5$. The route through $A_1$ without $L_1$ contributes $\tau_1\gamma_A=2(0.2)=0.4$. The route through both $L_1$ and $A_1$ contributes $\tau_1\gamma_L\beta_A=2(0.8)(0.5)=0.8$.

Adding the four contributions gives $1+1.5+0.4+0.8=3.7$. Equivalently, the later-treatment response is $\gamma_A+\gamma_L\beta_A=0.2+0.8(0.5)=0.6$, so the total is $1+3(0.5)+2(0.6)=3.7$. The two calculations agree.

The same equations show why adjustment for $L_1$ can answer a different question from the total effect of $A_0$. In a regression containing $A_0,L_1,A_1,X$, the coefficient of $A_0$ equals $\tau_A$ when the outcome equation is correctly specified and its error is uncorrelated with the included predictors. That coefficient holds $L_1$ and $A_1$ fixed. It omits both the $A_0\to L_1\to Y$ route and the routes through $A_1$. This can be appropriate for a controlled contrast, but it is not the natural-course effect derived above.

Worked Example: A Regression Coefficient That Blocks Paths

Use the coefficients in the previous example. Suppose the outcome is generated exactly by $Y=A_0+3L_1+2A_1+\varepsilon_Y$, with the specified equations for $L_1$ and $A_1$. Compare the coefficient of $A_0$ in the outcome regression that includes $L_1,A_1$ with the natural-course effect.

In the regression including $A_0,L_1,A_1$, the equation for $Y$ shows that the coefficient of $A_0$ is $1$, provided $X$ is also included if it affects the outcome and the errors satisfy the model assumptions. This is the direct coefficient $\tau_A$.

But allowing the later variables to respond to prior treatment gives a contrast of $3.7$, calculated as $1+3(0.5)+2(0.2+0.8(0.5))$. The difference is $3.7-1=2.7$. It consists of $1.5$ through $L_1$ directly to $Y$, $0.4$ through $A_1$ directly, and $0.8$ through $L_1$ and then $A_1$. Conditioning on those downstream variables removes these pathways from the comparison.

Adjustment for $L_1$ has a different role when the target is the effect of later treatment. Within levels of $A_0,X,L_1$, the model compares $A_1$ values among individuals with the same modeled confounder history. Omitting $L_1$, however, can leave the association between $A_1$ and the outcome partly attributable to their shared dependence on $L_1$.

Result: In the linear model $A_1=\gamma_L L_1+\varepsilon_1$ and $Y=\tau_1 A_1+\tau_L L_1+\varepsilon_Y$, suppose the errors have mean zero, are uncorrelated with $L_1$, and are uncorrelated with each other. If $\operatorname{Var}(A_1)>0$, the population slope from regressing $Y$ on $A_1$ alone is $$ \tau_1+\tau_L\frac{\operatorname{Cov}(A_1,L_1)}{\operatorname{Var}(A_1)}. $$ Thus, when $\tau_L\ne 0$ and $\operatorname{Cov}(A_1,L_1)\ne 0$, that slope generally differs from $\tau_1$.

Proof. The population least-squares slope from regressing $Y$ on $A_1$ is $\operatorname{Cov}(A_1,Y)/\operatorname{Var}(A_1)$. Substitute the outcome equation into its numerator:

\begin{aligned} \operatorname{Cov}(A_1,Y) &=\operatorname{Cov}(A_1,\tau_1A_1+\tau_L L_1+\varepsilon_Y)\\ &=\tau_1\operatorname{Var}(A_1) +\tau_L\operatorname{Cov}(A_1,L_1) +\operatorname{Cov}(A_1,\varepsilon_Y). \end{aligned}

The last covariance is zero under the stated assumptions: $A_1=\gamma_L L_1+\varepsilon_1$, and both $L_1$ and $\varepsilon_1$ are uncorrelated with $\varepsilon_Y$. Dividing the remaining terms by $\operatorname{Var}(A_1)>0$ proves the formula. $\square$

Worked Example: Omitting the Intermediate Confounder

After accounting for $A_0$ and $X$, suppose the residual variation satisfies $A_1=0.8L_1+\varepsilon_1$, $Y=2A_1+3L_1+\varepsilon_Y$, with $\operatorname{Var}(L_1)=1$, $\operatorname{Var}(\varepsilon_1)=1$, and independent mean-zero errors. Find the slope from the regression of $Y$ on $A_1$ alone, and compare it with the coefficient when $L_1$ is included.

The covariance is $\operatorname{Cov}(A_1,L_1)=0.8\operatorname{Var}(L_1)=0.8$. Independence gives $\operatorname{Var}(A_1)=0.8^2(1)+1=1.64$. The omitted-confounder formula gives the slope $2+3(0.8/1.64)=2+60/41=142/41$, approximately $3.46$. Including $L_1$ in the correctly specified outcome regression gives the $A_1$ coefficient $2$. The difference arises because the unadjusted association also reflects the contribution of $L_1$ to $Y$.

When the Algebra Is Useful—and When It Is Not Enough

The recursive equations provide a map of which variables transmit an effect and which variables confound a later treatment comparison. They do not, by themselves, establish that a regression coefficient is a causal effect. A misspecified linear mean, correlated errors from unmeasured common causes, treatment-dependent error distributions, or inadequate overlap can all undermine the interpretation. In particular, the independence assumptions above are not guaranteed by fitting the equations.

A practical comparison is between three approaches. A baseline-only regression avoids conditioning directly on the intermediate covariate, but it can fail to control confounding of later treatment. A conventional regression that includes time-varying covariates can control modeled confounding for a later-treatment comparison, yet can also block pathways from earlier treatment when the target is a total effect. Marginal structural models, commonly estimated using inverse-probability weights, are designed to address this time-varying-confounder problem under their own assumptions; they do not make confounding, positivity, or model specification irrelevant.

Useful diagnostics begin with the data and the sequential model. Check whether the fitted mean relationships are plausibly linear, whether important interactions or nonlinearities are evident, and whether residual patterns suggest omitted structure. Examine how strongly $L_1$ predicts $A_1$, and whether treatment choices overlap across relevant histories. Very sparse treatment histories or extreme fitted treatment probabilities signal that the model may rely on extrapolation. Large changes between baseline-adjusted and time-varying-adjusted coefficients are worth investigating, but are not by themselves proof of bias: the models may be targeting different contrasts.

Common pitfall: “$L_1$ predicts both $A_1$ and $Y$, so I should adjust for it in every treatment-effect regression.” The correction is to specify the target contrast first. Adjusting for $L_1$ can be important for comparing later treatment at a given history, but it also conditions on a consequence of $A_0$ and can remove earlier-treatment pathways. The right adjustment set depends on the treatment and effect being studied.

The linear model is especially useful for seeing the mechanism because each path contributes a product of coefficients. Its limitations are equally important: it assumes additive linear relations, constant path coefficients, and no interactions in the equations shown. If effects vary with $X$ or $L_1$, a single path product need not describe the average contrast. Treat the calculations as a transparent model-based derivation, then assess whether its assumptions are defensible in the application.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $\beta_A=0$, which paths from $A_0$ through $L_1$ vanish in the natural-course effect formula?
Suppose $\tau_A=0.5,\ \tau_L=2,\ \beta_A=0.4,\ \tau_1=1.5,\ \gamma_A=0.1,\ \gamma_L=0.5$. Calculate the natural-course effect of a one-unit increase in $A_0$.
In the stated structural outcome equation, what does the coefficient $\tau_A$ represent when $L_1$ and $A_1$ are held fixed?
In the omitted-confounder result, what happens to the unadjusted slope if $\operatorname{Cov}(A_1,L_1)=0$?
Name one diagnostic that can reveal a limitation of the fitted recursive model, and explain what it would indicate.

Solutions. (1) The paths $A_0\to L_1\to Y$ and $A_0\to L_1\to A_1\to Y$ vanish because each contains the factor $\beta_A$; the direct route through $A_1$ may remain through $\gamma_A$. (2) The effect is $0.5+2(0.4)+1.5\{0.1+0.5(0.4)\}=0.5+0.8+1.5(0.3)=1.75$. (3) It is the direct outcome-equation effect of changing $A_0$ while holding the later variables fixed, not the natural-course total effect. (4) It equals $\tau_1$, assuming the other conditions of the result hold. (5) For example, residual patterns against fitted values may reveal curvature or changing variance, suggesting that a linear conditional-mean model is inadequate; sparse treatment histories may instead indicate poor overlap and reliance on extrapolation.

A Recursive Linear Model Of Confounder Feedback

What You'll Learn