Deriving Residual Confounding After Baseline Adjustment

Why baseline adjustment can leave a calculable bias

Adjusting for baseline covariates can address confounding present before treatment begins, but it cannot automatically account for variables measured later. If a time-varying covariate predicts both a later treatment decision and the outcome, omitting it leaves a route for confounding even after baseline adjustment. A linear model lets us quantify that remaining bias and see which relationships make it larger, smaller, or reverse its sign.

The calculation here concerns the coefficient of the later treatment in a regression that includes baseline covariates and the earlier treatment but omits the time-varying confounder. It is a result for a specified additive linear model, not a general formula for every regression or treatment scale.

A two-period structural model

Let \(L_0\) denote a measured baseline covariate, \(A_0\) the earlier treatment, \(L_1\) a covariate measured after \(A_0\) but before the later treatment, \(A_1\) the later treatment, and \(Y\) the outcome. Assume the following structural equations:

\begin{aligned} L_1 &= \gamma_0+\gamma_L L_0+\gamma_0^{A}A_0+U_L,\\ A_1 &= \delta_0+\delta_L L_0+\delta_0^{A}A_0+\delta_1L_1+U_A,\\ Y &= \beta_0+\beta_L L_0+\beta_0^{A}A_0+\beta_1A_1+\theta L_1+U_Y. \end{aligned}

The superscript \(A\) on \(\gamma_0^{A}\) and \(\delta_0^{A}\) distinguishes the coefficients of \(A_0\) from the intercepts; it is not an exponent. In this model, \(\delta_1\) describes how \(L_1\) predicts the later treatment, conditional on \(L_0\) and \(A_0\). The coefficient \(\theta\) describes how \(L_1\) predicts the outcome, conditional on those variables and \(A_1\). The parameter \(\beta_1\) is the structural change in the mean outcome per one-unit increase in \(A_1\), holding \(L_0,A_0,L_1\) fixed.

Model assumptions: The equations are additive and linear, with no treatment-covariate interactions. The disturbances have mean zero and are mutually independent of one another and of \(L_0,A_0\); write \(\operatorname{Var}(U_L)=\sigma_L^2\) and \(\operatorname{Var}(U_A)=\sigma_A^2\). The outcome disturbance has mean zero and is uncorrelated with the included regressors and \(U_L,U_A\). The regression has nonzero residual variation in \(A_1\). These are strong assumptions used to isolate one source of bias; they are not guaranteed by baseline adjustment.

The confounding mechanism is explicit: \(L_1\) predicts \(A_1\) through \(\delta_1\), and predicts \(Y\) through \(\theta\). The regression of interest includes \(L_0\) and \(A_0\), but not \(L_1\). Its coefficient for \(A_1\) therefore mixes the structural \(A_1\) effect with the association between the remaining variation in \(A_1\) and the omitted \(L_1\).

Deriving the residual-confounding term

To isolate the later-treatment coefficient, remove the linear contributions of \(L_0\) and \(A_0\) from \(A_1\), \(L_1\), and \(Y\). This is the partial-regression idea: the coefficient of \(A_1\) in a regression that includes \(L_0,A_0\) is the slope relating the residualized outcome to residualized \(A_1\). From the model equations, the resulting variables are

\widetilde L_1=U_L,\qquad \widetilde A_1=\delta_1U_L+U_A,\qquad \widetilde Y=\beta_1\widetilde A_1+\theta U_L+U_Y.

Here a tilde means that the linear contribution of the included baseline covariates and earlier treatment has been removed. The omitted \(L_1\) contribution remains in the outcome residual. The fitted coefficient is the covariance of \(\widetilde A_1\) with \(\widetilde Y\), divided by the variance of \(\widetilde A_1\). Using the assumed independence of the disturbances gives

\begin{aligned} \operatorname{Cov}(\widetilde A_1,\widetilde Y) &=\operatorname{Cov}(\widetilde A_1,\beta_1\widetilde A_1+\theta U_L+U_Y)\\ &=\beta_1\operatorname{Var}(\widetilde A_1) +\theta\operatorname{Cov}(\widetilde A_1,U_L),\\ \operatorname{Var}(\widetilde A_1) &=\operatorname{Var}(\delta_1U_L+U_A) =\delta_1^2\sigma_L^2+\sigma_A^2,\\ \operatorname{Cov}(\widetilde A_1,U_L) &=\operatorname{Cov}(\delta_1U_L+U_A,U_L) =\delta_1\sigma_L^2. \end{aligned}

The \(U_Y\) covariance term is zero under the model assumptions. Dividing the covariance by the variance yields the population coefficient from the baseline-adjusted regression that omits \(L_1\).

Result: Under the stated model, the coefficient of \(A_1\) in a linear regression of \(Y\) on \(L_0,A_0,A_1\), omitting \(L_1\), is

\beta_1^{\mathrm{omit}} =\beta_1+ \underbrace{\frac{\theta\delta_1\sigma_L^2} {\delta_1^2\sigma_L^2+\sigma_A^2}}_{\text{residual-confounding bias}}.

The bias is the difference between this coefficient and \(\beta_1\). Its denominator is the residual variance in \(A_1\); its numerator reflects the association of \(A_1\) with the omitted covariate and the covariate’s outcome effect.

In words, the bias is the outcome association of \(L_1\), multiplied by the slope relating residual \(A_1\) to residual \(L_1\). This is the usual omitted-variable logic expressed in terms of the variation left after adjusting for \(L_0\) and \(A_0\). The formula concerns a population linear projection; a finite-sample estimate will also vary because of sampling.

Worked calculations

Worked Example: Positive Confounding Moves the Coefficient Upward

Suppose, hypothetically, that the later-treatment equation has \(\delta_1=0.8\), the residual variance of \(L_1\) after accounting for \(L_0,A_0\) is \(\sigma_L^2=4\), and the residual treatment-assignment variance is \(\sigma_A^2=1\). Suppose a one-unit increase in \(L_1\) raises the outcome by \(\theta=3\) outcome units, while the structural later-treatment effect is \(\beta_1=-2\) outcome units per treatment unit. All variances and coefficients refer to the model above.

The residual-confounding bias is

\frac{\theta\delta_1\sigma_L^2} {\delta_1^2\sigma_L^2+\sigma_A^2} = \frac{(3)(0.8)(4)}{(0.8)^2(4)+1} = \frac{9.6}{3.56} \approx 2.697.

Thus the baseline-adjusted coefficient that omits \(L_1\) is \(\beta_1^{\mathrm{omit}}=-2+2.697=0.697\) outcome units per treatment unit. The positive bias is large enough to change the sign of the coefficient: it is positive even though the structural effect in the model is negative. This is not evidence that adjustment itself reversed the treatment effect; it is a consequence of the specified omitted-confounder mechanism.

Worked Example: Negative Confounding Pulls the Coefficient Down

Now suppose, hypothetically, that higher \(L_1\) makes the later treatment more likely, with \(\delta_1=0.5\), but predicts a lower outcome, with \(\theta=-1.5\). Let \(\sigma_L^2=4\), \(\sigma_A^2=3\), and \(\beta_1=1.2\) outcome units per treatment unit. These values satisfy the same linear-model setup.

The bias is

\frac{(-1.5)(0.5)(4)}{(0.5)^2(4)+3} = \frac{-3}{4} =-0.75.

The regression coefficient omitting \(L_1\) is therefore \(1.2-0.75=0.45\) outcome units per treatment unit. Here \(L_1\) is positively associated with treatment but negatively associated with outcome, so the omitted-variable bias is negative. The adjusted coefficient remains positive, but it understates the structural effect by 0.75 units per treatment unit under this model.

Worked Example: Computing Bias From Residual Covariances

For another hypothetical model, suppose that after removing \(L_0\) and \(A_0\), the residualized later treatment has variance 5 and covariance 1.5 with the residualized \(L_1\). Suppose the structural outcome effect is \(\beta_1=2\), \(L_1\) has outcome coefficient \(\theta=4\), and the residualized outcome disturbance is uncorrelated with residualized treatment. Then

\operatorname{Cov}(\widetilde A_1,\widetilde Y) =\beta_1\operatorname{Var}(\widetilde A_1) +\theta\operatorname{Cov}(\widetilde A_1,\widetilde L_1) =(2)(5)+(4)(1.5)=16.

The coefficient from the regression omitting \(L_1\) is \(16/5=3.2\) outcome units per treatment unit. Its bias relative to the structural coefficient is \(3.2-2=1.2\). Equivalently, the omitted contribution is \(\theta\operatorname{Cov}(\widetilde A_1,\widetilde L_1)/\operatorname{Var}(\widetilde A_1)=(4)(1.5)/5=1.2\). This version makes clear that the key quantity is the residual association after baseline adjustment, not the raw association before adjustment.

When is this bias term zero?

The formula identifies several ways this particular residual-confounding term can vanish. If \(\delta_1=0\), \(L_1\) does not predict \(A_1\) after conditioning on \(L_0,A_0\). If \(\theta=0\), it does not predict \(Y\) after conditioning on the other terms in the outcome equation. If \(\sigma_L^2=0\), no residual variation in \(L_1\) remains after the included covariates. In each case, the product in the numerator is zero. These are statements about this model and this omitted-variable path; they do not establish that every possible source of confounding has been removed.

The denominator also matters. Holding the numerator’s components fixed, more independent treatment-assignment variation, represented by a larger \(\sigma_A^2\), reduces the residual association between \(A_1\) and \(L_1\), and therefore reduces this bias term. That does not mean randomized treatment assignment is the only route to unbiased estimation, or that a small calculated bias guarantees a good design. It describes only the specified linear model.

Common mistake and interpretation

Wrong reasoning: “I adjusted for baseline covariates and earlier treatment, so the later-treatment coefficient must be free of confounding.” Correction: those variables do not remove the residual association created when \(L_1\) predicts both \(A_1\) and \(Y\). In the model, the remaining bias is \(\theta\delta_1\sigma_L^2/(\delta_1^2\sigma_L^2+\sigma_A^2)\).
Wrong reasoning: “The bias must be positive because \(L_1\) predicts treatment.” Correction: its sign depends on both \(\delta_1\) and \(\theta\). If they have opposite signs, the bias is negative; if either is zero, this term is zero.
Wrong reasoning: “This formula applies to any outcome regression.” Correction: it is derived for the stated additive linear model, with its error and variance assumptions. Nonlinear outcome models, interactions, misspecification, and other unmeasured confounding require different analysis.

The structural coefficient \(\beta_1\) describes a model-based treatment contrast holding the other variables in the outcome equation fixed. The coefficient from the regression that omits \(L_1\) generally does not equal it, even when baseline covariates and earlier treatment are included. In applications, the model clarifies why baseline-only adjustment can leave residual confounding; it does not by itself establish that the assumptions hold or that a causal effect is identified.

Check Your Understanding

Use the model and the residual-confounding formula to answer the following.

If \(\delta_1\) is positive and \(\theta\) is negative, what is the sign of the residual-confounding bias, assuming positive residual variance in \(L_1\)?
Suppose \(\theta=2\), \(\delta_1=0.5\), \(\sigma_L^2=4\), and \(\sigma_A^2=2\). Calculate the bias and explain its units relative to the treatment coefficient.
Why does including \(L_0\) and \(A_0\) not, by itself, make the residual covariance between \(A_1\) and \(L_1\) equal to zero?
Under what model condition can the bias term be zero even when \(L_1\) predicts the outcome?
Which assumptions make the derived expression a linear-model result rather than a universal formula for omitted-confounder bias?