Partial R-Squared Measures of Confounding Strength

Two Conditional Associations Describe Confounding Strength

When a causal analysis omits a relevant variable, the resulting bias depends on how that variable relates to both treatment and outcome, after accounting for variables already in the model. Partial R-squared measures summarize these two conditional associations: one describes how strongly the omitted variable predicts treatment, and the other describes how strongly it predicts outcome after accounting for treatment and observed covariates.

These measures are useful because they put the two associations on a common, interpretable scale: the fraction of residual variation explained in a linear regression. They do not, by themselves, determine the size or direction of bias. In particular, they measure association, not causation.

Definition: Let \(Y\) be an outcome, \(T\) a treatment, \(X\) a vector of observed covariates, and \(U\) an omitted variable or block of omitted variables. The treatment-side partial R-squared, \(R^2_{T\sim U\mid X}\), is the proportion of the variation in \(T\) left after linearly adjusting for \(X\) that is explained by \(U\) after \(U\) is also linearly adjusted for \(X\). The outcome-side partial R-squared, \(R^2_{Y\sim U\mid T,X}\), is the proportion of the variation in \(Y\) left after linearly adjusting for \(T\) and \(X\) that is explained by \(U\) after \(U\) is also linearly adjusted for \(T\) and \(X\).

The conditioning sets differ for a reason. The treatment-side measure asks whether the omitted variable predicts treatment beyond the observed covariates. The outcome-side measure asks whether it predicts outcome beyond both treatment and the observed covariates. This pairing is intended for settings in which \(U\) is a candidate omitted confounder.

Formal Definitions Using Nested Linear Regressions

Use the same observations in each regression, and include an intercept in every model. The treatment-side measure compares a reduced regression of \(T\) on \(X\) with a full regression of \(T\) on \(X\) and \(U\). If \(\mathrm{RSS}_{\mathrm{reduced}}\) and \(\mathrm{RSS}_{\mathrm{full}}\) are their residual sums of squares, then

R^2_{T\sim U\mid X} = \frac{\mathrm{RSS}_{\mathrm{reduced}}-\mathrm{RSS}_{\mathrm{full}}} {\mathrm{RSS}_{\mathrm{reduced}}} = 1-\frac{\mathrm{RSS}_{\mathrm{full}}}{\mathrm{RSS}_{\mathrm{reduced}}}.

The outcome-side measure compares a reduced regression of \(Y\) on \(T\) and \(X\) with a full regression of \(Y\) on \(T\), \(X\), and \(U\):

R^2_{Y\sim U\mid T,X} = \frac{\mathrm{RSS}_{Y\sim T,X}-\mathrm{RSS}_{Y\sim T,X,U}} {\mathrm{RSS}_{Y\sim T,X}} = 1-\frac{\mathrm{RSS}_{Y\sim T,X,U}}{\mathrm{RSS}_{Y\sim T,X}}.

These expressions are defined when the reduced model has positive residual variation in the outcome being explained. With nested ordinary least-squares models fit to the same data, adding regressors cannot increase the residual sum of squares. Thus each partial R-squared lies between 0 and 1. A value of 0 means the added variable or block does not reduce residual variation; a value of 1 means it explains all of that residual variation.

Residualization and the Single-Variable Case

The same quantities can be understood by first removing the linear contribution of the conditioning variables. Write \(C\) for the design matrix of the relevant controls, including the intercept, and let \(M_C\) denote the operation that takes a variable or each column of a matrix to its OLS residuals after regression on \(C\). For the treatment-side measure, the controls are \(X\); for the outcome-side measure, they are \(T\) and \(X\).

If \(U\) is a single variable, residualize both the target and \(U\) against the relevant controls. The partial R-squared is the squared ordinary correlation between the two resulting residual vectors. This correlation interpretation applies to a single added variable. It does not apply as an ordinary correlation when \(U\) is a block of several variables.

R^2_{T\sim U\mid X} = \operatorname{Corr}(M_XT,M_XU)^2, \qquad R^2_{Y\sim U\mid T,X} = \operatorname{Corr}(M_{[T,X]}Y,M_{[T,X]}U)^2 \quad\text{when \(U\) is a single variable.}

For a block \(U\), residualize each column of \(U\) against the controls and regress the residualized target on those residualized columns. The partial R-squared is the proportion of target residual variation explained by that regression, equivalently its squared multiple correlation. There is no single ordinary correlation between a residual vector and a matrix of residualized variables.

Worked Example: Treatment Association for One Omitted Variable

Suppose a hypothetical analysis has already adjusted for observed covariates \(X\). In the resulting residual vectors, the treatment residuals have a sum of squares of 80, the residuals of a single candidate omitted variable \(U\) have a sum of squares of 50, and their cross-product sum is 40. Calculate the treatment-side partial R-squared.

Because \(U\) is one variable, the partial R-squared is the squared correlation of the two residual vectors. Using the residual sums, the correlation is

r = \frac{40}{\sqrt{80\cdot 50}} = \frac{40}{\sqrt{4000}} \approx 0.6325.

Therefore,

R^2_{T\sim U\mid X} = r^2 = \frac{40^2}{80\cdot 50} = \frac{1600}{4000} = 0.400.

In this hypothetical example, \(U\) explains 40.0% of the treatment variation remaining after linear adjustment for \(X\). The positive residual correlation indicates that the adjusted residuals of \(T\) and \(U\) move together in this example; the partial R-squared itself records the strength, not the sign, of that association.

Why a Block Requires a Multiple Correlation

For a block of omitted variables, the residualized predictors can point in several directions. Their joint predictive contribution is measured by projecting the target residual vector onto the space spanned by those predictors. If \(r_T=M_XT\), and \(R_U=M_XU\) is the matrix of residualized columns, then

R^2_{T\sim U\mid X} = \frac{\lVert P_{R_U}r_T\rVert^2}{\lVert r_T\rVert^2},

where \(P_{R_U}r_T\) is the OLS fitted vector from regressing \(r_T\) on all columns of \(R_U\). The analogous outcome-side expression projects \(M_{[T,X]}Y\) onto the columns of \(M_{[T,X]}U\). This follows from the ordinary least-squares identity that the total sum of squares of a response equals the sum of squares of its fitted component plus the sum of squares of its residual component. Consequently, the fraction explained is the squared length of the projection divided by the squared length of the response residual vector.

This projection definition also makes the block interpretation precise: the partial R-squared is a squared multiple correlation. It is not an average of separate squared correlations, and the individual predictors’ partial R-squared values generally cannot simply be added.

Worked Example: Treatment Association for a Block

Consider a hypothetical residualized treatment vector \(r_T\) after adjustment for \(X\), with sum of squares 100. A block \(U\) has two residualized columns, \(u_1\) and \(u_2\). Suppose these columns are orthogonal, each has sum of squares 100, and their cross-product sums with \(r_T\) are 30 and 40, respectively. Calculate the block’s treatment-side partial R-squared.

With orthogonal predictors, the OLS coefficient for each column is its cross-product with the response divided by its own sum of squares:

\hat{\beta}_1=\frac{30}{100}=0.30, \qquad \hat{\beta}_2=\frac{40}{100}=0.40.

The fitted vector is \(0.30u_1+0.40u_2\). Since the two columns are orthogonal, the fitted sum of squares is

\lVert 0.30u_1+0.40u_2\rVert^2 = (0.30)^2(100)+(0.40)^2(100) = 9+16 = 25.

The partial R-squared is the fitted sum of squares divided by the treatment residual sum of squares:

R^2_{T\sim U\mid X} = \frac{25}{100} = 0.250.

Thus, jointly, these two residualized variables explain 25.0% of the treatment variation remaining after adjustment for \(X\). The calculation uses the block’s joint fitted variation; it does not treat the block as though it were one variable correlated with a vector.

Outcome-Side Partial R-Squared

The outcome-side measure uses a different residualization step: first remove the linear contributions of both treatment and observed covariates from \(Y\) and from each column of \(U\). It then measures how much of the remaining outcome variation is explained by the residualized omitted variable or block. For a single \(U\), this reduces to a squared residual correlation; for a block, it is the squared multiple correlation from the joint regression.

Worked Example: Outcome Association for One Omitted Variable

Suppose, in a hypothetical dataset, the outcome residuals after adjustment for \(T\) and \(X\) have a sum of squares of 120. The residuals of a single omitted variable \(U\), also after adjustment for \(T\) and \(X\), have a sum of squares of 75. Their cross-product sum is \(-45\). Calculate and interpret the outcome-side partial R-squared.

The residual correlation is

r = \frac{-45}{\sqrt{120\cdot 75}} = \frac{-45}{\sqrt{9000}} \approx -0.4743.

Squaring removes the sign:

R^2_{Y\sim U\mid T,X} = r^2 = \frac{(-45)^2}{120\cdot 75} = \frac{2025}{9000} = 0.225.

In this example, \(U\) explains 22.5% of the outcome variation remaining after linear adjustment for treatment and \(X\). The negative residual correlation is relevant to the direction of the conditional association, but the partial R-squared reports only its magnitude.

What the Pair of Measures Can—and Cannot—Say

The two partial R-squared values describe different links in a linear model: \(R^2_{T\sim U\mid X}\) summarizes the association of the omitted variable with treatment conditional on \(X\), while \(R^2_{Y\sim U\mid T,X}\) summarizes its association with outcome conditional on treatment and \(X\). A candidate omitted variable can be strongly associated with one and weakly associated with the other. For a block, these are joint predictive measures for the block as a whole.

The values depend on the chosen conditioning variables, the sample, and the linear regression specification. They are not intrinsic properties of \(U\) alone. They also do not establish that \(U\) is a confounder: that is a substantive causal claim, not something a partial R-squared can prove. The measures are descriptive parameters for assessing the strength of conditional linear associations.

A partial R-squared also discards sign. The same pair of values can arise from different directions of association between \(U\), treatment, and outcome. Therefore, these measures alone do not determine whether an omitted variable would bias an estimated treatment effect upward or downward, or how large that bias would be. Their role here is to quantify association strengths; translating those strengths into adjusted estimates requires additional steps.

Common Mistakes

Treating a block as one variable. An ordinary correlation is defined for two vectors, not for a vector and a matrix of residualized predictors. For a block, use the proportion of target residual variation explained by jointly regressing on the residualized columns, or equivalently the squared multiple correlation.
Using the wrong controls. The treatment-side measure adjusts for \(X\); the outcome-side measure adjusts for both \(T\) and \(X\). These conditioning sets are part of the definitions.
Reading a partial R-squared as a signed association. Squaring removes direction. If direction matters for a particular analysis, inspect the relevant regression coefficients or residual associations as well.
Calling the measure a causal effect. A partial R-squared quantifies conditional linear predictive association. It does not establish causal relationships or, by itself, identify the direction or magnitude of omitted-variable bias.

Key takeaway: Partial R-squared measures the fraction of residual variation explained by an added variable or block. For a single variable it is a squared residual correlation; for a block it is a squared multiple correlation. The treatment-side and outcome-side measures use different conditioning sets and neither retains the sign of the association.

Check Your Understanding

Answer each question using the definitions and interpretations in this tutorial.

Which variables are conditioned on when defining \(R^2_{T\sim U\mid X}\), and which are conditioned on for \(R^2_{Y\sim U\mid T,X}\)?
For one omitted variable, residual sums of squares are 60 for the target, 40 for \(U\), and the cross-product sum is 24. What is the squared residual correlation?
Why is an ordinary correlation not the appropriate definition when \(U\) is a block of several variables?
A block has a partial R-squared of 0.10 for treatment. State its interpretation in context, conditional on \(X\).
Can the outcome-side partial R-squared alone show whether the omitted variable is positively or negatively associated with the outcome residuals? Explain.