Worked Algebra: A Two-Period Linear Feedback Model

Why the Algebra Matters

In a longitudinal study, a first-period treatment can change a covariate measured before the next treatment decision. That covariate can then influence both the second-period treatment and the outcome. It is therefore a confounder of the later treatment–outcome relationship, but it is also affected by earlier treatment. A regression that adjusts for it answers a different question from the total effect of changing the first-period treatment.

A linear model makes this distinction visible. We can calculate the changes in the covariate and later treatment under an intervention, carry those changes through the outcome equation, and then compare the resulting total effect with the coefficient of first-period treatment in an adjusted regression.

A Two-Period Structural Model

Let \(A_1\) denote treatment in period 1, \(L\) a covariate measured after period 1 but before the next treatment decision, \(A_2\) treatment in period 2, and \(Y\) the final outcome. Consider the following additive structural equations:

The equations specify the order of events: \(A_1\) changes \(L\), then \(A_1\) and \(L\) help determine \(A_2\), and all three variables may affect \(Y\). In particular, \(L\) is treatment-affected and predicts both \(A_2\) and \(Y\). The model has no interactions, so each coefficient represents a constant additive change holding the other variables in that equation fixed.

For a total effect of \(A_1\), intervene on \(A_1\) but let \(L\) and \(A_2\) follow their structural equations. For an increase of \(\delta\) units in \(A_1\), the equations imply a change in \(L\), followed by a change in \(A_2\), followed by a change in \(Y\). The total effect includes all of these pathways.

Deriving the Total Effect

Write \(L(a)\), \(A_2(a)\), and \(Y(a)\) for the values that the variables would take if \(A_1\) were set to \(a\), while the later variables were generated by their equations. First, the covariate changes by \(\alpha_1\delta\) when \(A_1\) increases by \(\delta\). Substituting that change into the second-period treatment equation gives the change in \(A_2\):

The outcome equation then gives the total change. The direct contribution of \(A_1\) is \(\beta_1\delta\); the changes through \(A_2\) and \(L\) contribute \(\beta_2\Delta A_2\) and \(\beta_3\Delta L\), respectively. Combining them and dividing by \(\delta\) gives the total effect per unit increase in \(A_1\):

This is a finite contrast for any \(\delta\) under the stated linear, additive model. The term \(\beta_2\gamma_1\) is the pathway from \(A_1\) to \(A_2\) that does not pass through \(L\). The term \(\beta_2\gamma_2\alpha_1\) passes through both \(L\) and \(A_2\), and \(\beta_3\alpha_1\) passes from \(A_1\) through \(L\) to \(Y\). The remaining term, \(\beta_1\), is the direct contribution holding \(L\) and \(A_2\) fixed.

What the Fully Adjusted Regression Coefficient Represents

Now consider the regression of \(Y\) on \(A_1\), \(A_2\), and \(L\). Under the model assumptions, the conditional mean of \(Y\) given those variables is

The error term has conditional mean zero under the stated independence assumption. Consequently, the population coefficient of \(A_1\) in this correctly specified adjusted regression is \(\beta_1\). It compares outcome predictions at the same values of \(L\) and \(A_2\), changing \(A_1\) while holding both later variables fixed. It is not, in general, the total effect of intervening on \(A_1\) and allowing those later variables to respond.

Adjusting for the Covariate but Not the Later Treatment

A different regression adjusts for \(L\) but omits \(A_2\). Its coefficient can also be derived from the structural equations. Conditional on \(A_1\) and \(L\), the expected second-period treatment is \(\gamma_0+\gamma_1A_1+\gamma_2L\). Substituting that expectation into the outcome equation gives

Under these assumptions, the coefficient of \(A_1\) in this regression is \(\beta_1+\beta_2\gamma_1\). It includes the pathway from \(A_1\) to \(A_2\) that remains when \(L\) is held fixed. It does not include the pathways through changes in \(L\), because the regression conditions on \(L\).

Why This Distinction Is Useful

The algebra prevents a common error: treating “adjusted for more variables” as if it automatically meant “closer to the total effect.” In this model, full adjustment for \(L\) and \(A_2\) produces a coefficient that holds both treatment-affected variables fixed. It removes the contributions of the pathways through them rather than adding those contributions into the total effect.

The calculations also depend on the model and assumptions. If the outcome equation has interactions, such as an \(A_1A_2\) term, the effect of changing \(A_1\) while holding \(A_2\) fixed can depend on the value of \(A_2\); a single constant coefficient may no longer summarize the contrast. If the disturbances are associated, the regression conditional mean may not equal the structural outcome equation. In either case, the simple coefficient identities above need not hold.

Common Mistakes

• Calling the fully adjusted coefficient the total effect. That coefficient holds \(L\) and \(A_2\) fixed. The total-effect contrast allows both to change according to their equations.
• Counting only one pathway. The change in \(A_2\) includes both \(\gamma_1\delta\) and \(\gamma_2\alpha_1\delta\). Leaving out either term omits a pathway.
• Adding coefficients without following the equations. Coefficients from different equations must be multiplied along a pathway and added across pathways. For example, the \(A_1\to L\to Y\) contribution is \(\alpha_1\beta_3\), not \(\alpha_1+\beta_3\).
• Assuming the identity is automatic in observational data. The derivation relies on the specified model and disturbance assumptions. A fitted regression alone does not establish those assumptions or identify a causal effect.