Partial-R-Squared Sensitivity for Regression Effects

Why Measure Confounder Strength?

A regression-adjusted treatment effect can depend on which covariates were measured and included. When a relevant confounder is unobserved, the treatment coefficient in the fitted regression may differ from the coefficient that would have been obtained had that confounder been included. Sensitivity analysis asks how strong the omitted variable's relationships with treatment and outcome would have to be to materially change the result.

Partial R-squared values put those relationships on a common scale: each measures the fraction of residual variation explained by the omitted variable after conditioning on specified observed variables. The resulting calculations do not reveal whether a particular unmeasured confounder exists. They quantify how much bias would be possible under stated strength assumptions.

Regression Setup and Partial R-Squared

Let \(Y\) be an outcome, \(D\) a treatment or exposure, and \(X\) a vector of observed covariates that includes an intercept in the regression. The observed-data regression is \(Y\) on \(D\) and \(X\). Write its treatment coefficient as \(\hat{\tau}\), its conventional homoskedastic standard error as \(SE(\hat{\tau})\), and its residual degrees of freedom as \(df\). Suppose a scalar omitted variable \(U\), or a block of omitted variables, could be added to this regression. The sensitivity calculation compares the fitted treatment coefficient with the coefficient after controlling for \(U\).

These are conditional associations, not ordinary correlations between \(U\) and \(D\) or \(Y\). A variable may be strongly related to the outcome overall but have a small \(R^2_{Y\sim U\mid D,X}\) if the observed covariates already explain the relevant outcome variation. The two partial R-squared values play distinct roles: the treatment partial R-squared captures how much \(U\) can shift the treatment coefficient, while the outcome partial R-squared captures how strongly \(U\) predicts the outcome after accounting for treatment and observed covariates.

The algebra below uses the same observations and controls in the observed and augmented regressions, with ordinary least squares and conventional homoskedastic standard errors. It describes coefficient bias from omitting \(U\); it does not establish that the regression coefficient is a causal effect. A causal interpretation additionally requires a suitable design and assumptions about confounding, treatment, and the target population.

Bias Bound in Standard-Error Units

Given specified partial R-squared values, the maximum absolute omitted-variable bias, measured in units of the observed regression's standard error, is

For a single omitted variable under the stated linear-regression setup, this expression gives the magnitude implied by the two partial R-squared values (with the sign determined by the directions of the relationships). For an omitted block, it is a bound: it allows the block's relationships with treatment and outcome to align in the direction that produces the greatest bias. A larger outcome partial R-squared or treatment partial R-squared increases the possible bias. The denominator shows why treatment predictability matters especially as \(R^2_{D\sim U\mid X}\) approaches 1.

A short derivation explains the scaling. First residualize \(D\) and \(U\) with respect to \(X\). The omitted-variable coefficient shift is the coefficient on \(D\) attributable to the part of \(U\) associated with residualized treatment. By the definition of \(R^2_{D\sim U\mid X}\), the squared correlation of those residualized variables is \(R^2_{D\sim U\mid X}\). By the definition of \(R^2_{Y\sim U\mid D,X}\), the outcome variation attributable to the part of \(U\) orthogonal to \(D\) and \(X\) is that fraction of the outcome's residual sum of squares. Combining these two projections gives the squared bias divided by the squared conventional standard error:

The factor \(df\) enters because the conventional residual variance estimate is the residual sum of squares divided by \(df\). This relationship is an algebraic sensitivity result, not a claim that the omitted variable's strength is known.

Estimating Partial R-Squared Values for Benchmarks

In practice, the omitted variable's partial R-squared values are unknown. One useful approach is to benchmark them against observed covariates or blocks of covariates. The purpose is not to assert that an observed variable is a perfect stand-in for an unobserved confounder, but to make the assumed strength scale interpretable.

For an auxiliary regression test of a block of \(k\) variables, let \(F\) be the usual partial F statistic for the block, with residual degrees of freedom \(df_{\mathrm{res}}\) in the full auxiliary regression. The block's partial R-squared is

This identity follows from the partial F statistic: \(F=(R^2_{\mathrm{partial}}/k)/((1-R^2_{\mathrm{partial}})/df_{\mathrm{res}})\). Rearranging gives the displayed expression. For one variable, \(k=1\), and if the coefficient's test statistic is \(t\), then \(F=t^2\). This calculation must use the degrees of freedom from the relevant auxiliary regression, not automatically the \(df\) from another model.

Robustness Value for Moving an Estimate to Zero

A robustness value summarizes the equal-strength case: how large would both partial R-squared values need to be for the maximum possible bias to equal the absolute observed coefficient? Here “equal strength” means \(R^2_{D\sim U\mid X}=R^2_{Y\sim U\mid D,X}=r\). Let \(t=\hat{\tau}/SE(\hat{\tau})\) be the observed coefficient's t statistic. Setting the bias bound equal to \(|\hat{\tau}|/SE(\hat{\tau})=|t|\) gives

Define \(f^2=t^2/df\). The equation becomes \(r^2+f^2r-f^2=0\). Its nonnegative solution is the robustness value:

This value is a partial R-squared, not a percentage change in the coefficient. If both partial R-squared values equal \(RV_0\), the maximum bias bound is just large enough to move the point estimate to zero. Smaller values in both dimensions do not suffice under this equal-strength bound; unequal strengths can also produce enough bias, but their combination must be assessed with the general formula.

Assumptions, Scope, and a Common Pitfall

The formulas quantify sensitivity within a linear regression framework. They require a common analysis sample and consistent definitions of treatment and controls across the observed and augmented models. The conventional standard-error scaling shown here relies on ordinary least squares with homoskedastic standard errors. With heteroskedasticity-robust or clustered standard errors, the coefficient-bias calculation may still be informative, but this exact conversion to conventional standard-error units should not be treated as an inferential result without an appropriate extension.

Partial R-squared values do not supply the direction of confounding. The sign of bias depends on how the omitted variable relates to treatment and outcome after conditioning on the specified variables. A magnitude bound allows the direction most adverse to the estimated effect; a directional claim needs additional information. Nor does a small robustness value prove that the effect is absent: it indicates sensitivity to confounders at or above a stated strength, not evidence about whether those confounders are present.

• Common mistake: “The omitted confounder has a 10% partial R-squared with the outcome, so it can explain 10% of the treatment estimate.” These are different quantities: an outcome partial R-squared alone does not determine coefficient bias.
• Correction: Use both \(R^2_{Y\sim U\mid D,X}\) and \(R^2_{D\sim U\mid X}\), together with \(df\), in the bias formula. Then convert the result from standard-error units to outcome units using \(SE(\hat{\tau})\).
• Common mistake: Interpreting a robustness value as the probability that the estimate is causal or that no confounder exists.
• Correction: A robustness value is a threshold in partial-R-squared units under a specified sensitivity calculation. It is not a probability and does not replace a causal identification argument.