Quantify Bias from an Omitted Confounder

Why quantify an omitted confounder?

An unadjusted regression estimate can combine an exposure association with an association carried by a confounder. Quantifying that contribution makes the possible distortion concrete: it shows how large the shift could be under stated assumptions, and which direction it would move the estimate. It does not, by itself, establish that the adjusted estimate is causal or that all important confounding has been removed.

Let \(A\) denote an exposure, \(Y\) an outcome, and \(C\) a measured confounder that is left out of an unadjusted regression. For example, \(A\) might be the number of weekly therapy sessions, \(Y\) a recovery score, and \(C\) baseline illness severity. The calculation below concerns one confounder and an additive linear model. A confounder is a variable associated with exposure and outcome that, if not appropriately handled, can distort the exposure-outcome association.

Definition: In the additive linear model \(Y=\beta_0+\beta_A A+\beta_C C+\varepsilon\), the omitted-variable bias in the unadjusted exposure coefficient is the difference between that coefficient and the exposure coefficient from the model including \(C\). Under the model and conditions below, this difference is \(\beta_C\delta\), where \(\beta_C\) is the outcome change per unit of \(C\) at fixed exposure and \(\delta\) is the slope of \(C\) on \(A\).

The model and the bias formula

Assume the conditional mean of the outcome is linear and additive in \(A\) and \(C\):

E(Y\mid A,C)=\beta_0+\beta_A A+\beta_C C.

Here, \(\beta_A\) is the adjusted exposure coefficient: the expected outcome change for a one-unit increase in \(A\), comparing observations with the same value of \(C\). The coefficient \(\beta_C\) is the expected outcome change for a one-unit increase in \(C\), at fixed \(A\). The model has no \(A\)-by-\(C\) interaction, so these slopes are constant over the values of the other variable. For the statistical model, assume the remaining error has conditional mean zero given \(A\) and \(C\), and that the linear forms are appropriate for the variables being analyzed.

Now describe the exposure-confounder relationship with the linear regression of \(C\) on \(A\):

E(C\mid A)=\gamma_0+\delta A.

Thus, \(\delta\) is the expected difference in \(C\) associated with a one-unit increase in \(A\). Substituting this relationship into the outcome model gives the expected outcome at a given exposure level after averaging over the values of \(C\) associated with that exposure:

E(Y\mid A) =\beta_0+\beta_A A+\beta_C(\gamma_0+\delta A) =(\beta_0+\beta_C\gamma_0)+(\beta_A+\beta_C\delta)A.

The slope in this unadjusted relationship is therefore \(\beta_A+\beta_C\delta\). The portion beyond the adjusted slope, \(\beta_A\), is the omitted-variable bias. This calculation describes the linear association induced by the model; it is not a general guarantee that a regression omitting \(C\) has a causal interpretation.

Result: Under the additive linear model, the omitted-variable bias for the exposure coefficient is

\text{Bias}=\beta_A^{\text{unadjusted}}-\beta_A^{\text{adjusted}} =\beta_C\delta.

The bias is positive when \(\beta_C\) and \(\delta\) have the same sign, negative when they have opposite signs, and zero if either relationship is zero.

In a sample, there is a corresponding exact regression identity when the unadjusted model \(Y\) on \(A\) and the adjusted model \(Y\) on \(A\) and \(C\) use the same observations and include intercepts. Let \(b_C\) be the adjusted sample coefficient of \(C\), and let \(d\) be the sample slope from regressing \(C\) on \(A\). Then:

b_A^{\text{unadjusted}}=b_A^{\text{adjusted}}+b_Cd.

This is an algebraic identity for those regressions. In a particular sample, the product \(b_Cd\) is the difference between the two fitted exposure coefficients. Interpreting the adjusted coefficient as a causal effect requires additional causal identification assumptions; regression algebra alone does not supply them.

A short derivation using covariances

The same result can be seen from the usual slope formula for a regression with an intercept. Write the adjusted outcome model in sample form as \(Y=b_0+b_A^{\text{adjusted}}A+b_CC+e\), where the adjusted regression residual \(e\) has sample covariance zero with \(A\). Taking the sample covariance with \(A\) on both sides gives:

\operatorname{Cov}(A,Y) =b_A^{\text{adjusted}}\operatorname{Var}(A) +b_C\operatorname{Cov}(A,C).

The unadjusted slope is \(\operatorname{Cov}(A,Y)/\operatorname{Var}(A)\). Dividing the equation by \(\operatorname{Var}(A)\), and noting that the slope from regressing \(C\) on \(A\) is \(d=\operatorname{Cov}(A,C)/\operatorname{Var}(A)\), yields:

b_A^{\text{unadjusted}} =b_A^{\text{adjusted}}+ b_C\frac{\operatorname{Cov}(A,C)}{\operatorname{Var}(A)} =b_A^{\text{adjusted}}+b_Cd.

This derivation explains why the confounder’s outcome association alone is not enough to determine bias. Its association with exposure matters too.

Worked Example: A Continuous Confounder and a Recovery Score

Suppose, hypothetically, an analysis uses weekly therapy sessions as \(A\), recovery points as \(Y\), and baseline severity points as \(C\). Assume the additive model is adequate, the adjusted coefficient of baseline severity is \(b_C=-1.80\) recovery points per severity point, and the slope from regressing severity on sessions is \(d=-0.60\) severity points per additional weekly session. The adjusted exposure estimate is \(b_A^{\text{adjusted}}=2.40\) recovery points per session. These values are assumed for illustration, not reported study findings.

The estimated omitted-variable contribution is:

b_Cd=(-1.80\ \text{recovery points/severity point}) (-0.60\ \text{severity points/session}) =1.08\ \text{recovery points/session}.

The units of severity cancel, leaving recovery points per session. The unadjusted estimate implied by the identity is:

b_A^{\text{unadjusted}} =2.40+1.08 =3.48\ \text{recovery points/session}.

The direction also makes sense under these assumptions. More sessions are associated with lower baseline severity (\(d<0\)), and greater severity predicts a lower recovery score (\(b_C<0\)). The product is positive, so omitting severity shifts the session coefficient upward. The comparison quantifies this assumed shift; it does not establish that 2.40 is a causal effect.

Binary confounders: use a prevalence difference

When \(A\) is binary, the slope \(\delta\) in the regression of \(C\) on \(A\) is the difference in mean confounder values between the exposed and unexposed groups. If \(C\) is also binary, its mean is its prevalence. Therefore, \(\delta\) is the confounder prevalence among exposed observations minus its prevalence among unexposed observations. The product formula applies when the outcome model is additive and has no exposure-confounder interaction.

Worked Example: A Binary Baseline Risk Factor

Imagine a hypothetical comparison of a training program (\(A=1\)) with no program (\(A=0\)), where \(Y\) is improvement in a score and \(C=1\) indicates high baseline risk. Assume the adjusted model estimates that high baseline risk is associated with 5.0 fewer improvement points (\(b_C=-5.0\) points), at either program status. Suppose high risk is present in 25% of the program group and 55% of the no-program group. The adjusted program estimate is 4.0 improvement points.

The difference in high-risk prevalence is:

d=P(C=1\mid A=1)-P(C=1\mid A=0) =0.25-0.55=-0.30.

The omitted-variable contribution is:

b_Cd=(-5.0\ \text{points})(-0.30)=1.50\ \text{points}.

Thus the unadjusted difference implied by these assumptions is:

b_A^{\text{unadjusted}} =b_A^{\text{adjusted}}+b_Cd =4.0+1.50=5.50\ \text{points}.

Because the program group has a lower prevalence of a risk factor associated with less improvement, its unadjusted advantage includes the contribution of that group difference. The calculation uses the stated prevalence difference and outcome association; it does not prove that the adjusted estimate is free of other sources of bias.

Use covariance information when relationships are summarized that way

For a continuous exposure and confounder, the exposure-confounder slope can also be calculated from their covariance and the exposure variance: \(\delta=\operatorname{Cov}(A,C)/\operatorname{Var}(A)\). This is useful when a hypothetical scenario supplies a correlation and standard deviations rather than a regression slope. The covariance must retain the variables’ units.

Worked Example: Calculate Bias From a Correlation

Suppose a hypothetical analysis relates air-pollution exposure \(A\), measured in units of concentration, to a health score \(Y\), measured in points. Let \(C\) be an access-to-care index. Assume the adjusted outcome coefficient is \(b_C=2.0\) health-score points per index unit, the correlation between \(A\) and \(C\) is \(-0.50\), the sample standard deviation of \(A\) is 2.0 concentration units, and the sample standard deviation of \(C\) is 3.0 index units. Suppose the adjusted exposure estimate is \(-0.80\) health-score points per concentration unit.

First calculate the covariance from the correlation and standard deviations:

\operatorname{Cov}(A,C) =r_{A,C}s_As_C =(-0.50)(2.0)(3.0) =-3.0\ \text{concentration units}\cdot\text{index units}.

The exposure-confounder slope is:

d=\frac{\operatorname{Cov}(A,C)}{\operatorname{Var}(A)} =\frac{-3.0}{(2.0)^2} =-0.75\ \text{index units/concentration unit}.

The estimated bias and unadjusted coefficient are therefore:

b_Cd=(2.0)(-0.75)=-1.50\ \text{health-score points/concentration unit},

b_A^{\text{unadjusted}} =-0.80+(-1.50) =-2.30\ \text{health-score points/concentration unit}.

Here the confounder is positively associated with the outcome but negatively associated with exposure, producing negative bias. Under the assumed model, omitting access to care makes the exposure coefficient more negative by 1.50 score points per concentration unit.

What the calculation does—and does not—tell you

An omitted-variable calculation is informative because it makes assumptions visible. To apply it, state the assumed outcome association of \(C\), its association with \(A\), the resulting product, and the units. If the confounder’s association with either exposure or outcome is close to zero, the product is close to zero under this model. If both associations are substantial, the product may also be substantial. The signs determine direction, but the sizes determine magnitude.

The calculation does not establish that the inputs are correct. They might come from an adjusted regression, external evidence, or a deliberately chosen scenario. If they are uncertain, the resulting bias estimate is uncertain too; it is often more informative to calculate several plausible combinations than to present one product as known. Also distinguish the bias calculation from sampling uncertainty: a large or small standard error does not, by itself, tell you whether omitted-confounder bias is present.

The simple product depends on the additive linear setup. If the outcome relationship is nonlinear, or the effect of \(A\) varies with \(C\), one constant product may not describe the omitted-variable shift. Model misspecification, measurement problems, selection, and other unmeasured variables can also affect an observational estimate. In particular, adjusting for one confounder does not guarantee that all confounding has been controlled.

Common mistake

Wrong reasoning: “The confounder predicts the outcome, so omitting it must bias the exposure coefficient upward.”
Correction: The direction depends on both relationships. Calculate \(\beta_C\delta\). If the signs match, the bias is positive; if they differ, it is negative. A confounder-outcome association alone does not determine the direction.
Wrong reasoning: “The adjusted coefficient is automatically the true causal effect.”
Correction: The adjusted coefficient is a regression estimate conditional on the included variables. A causal interpretation needs the relevant identification assumptions, not just the act of adding a covariate.

Key takeaway: In an additive linear model with one omitted confounder, the exposure-coefficient shift is the confounder’s outcome coefficient multiplied by its slope on exposure. Report the assumptions, both component relationships, the product with units, and the limits of the adjusted-versus-unadjusted comparison.

Check Your Understanding

For each question, identify the assumptions or calculation needed before interpreting the result.

If \(b_C=3.0\) outcome units per confounder unit and \(d=0.4\) confounder units per exposure unit, what is the omitted-variable contribution, including its units and direction?
A binary confounder is present in 60% of exposed observations and 35% of unexposed observations. What is its slope on the binary exposure when coded 1 for exposed?
Why is a confounder’s association with the outcome alone insufficient to determine the direction of omitted-variable bias?
In the binary-risk-factor example, what would happen to the calculated bias if the risk factor had the same prevalence in both exposure groups?
Does agreement between an adjusted estimate and an assumed bias calculation prove that the adjusted estimate is causal? Explain briefly.