Using Proxy Variables for Unmeasured Common Causes

Why Use a Proxy?

A common cause can influence both an exposure and an outcome, creating a backdoor association that makes their observed relationship differ from the causal effect of interest. Sometimes the common cause is not recorded: disease severity, for example, may influence both treatment choice and later health, while the available data contain only a symptom score related to severity. The symptom score is a proxy. It may provide useful information about the unobserved cause, but it is not automatically a substitute for measuring that cause.

The key question is not merely whether the proxy predicts the common cause. It is whether adjustment for the proxy blocks the relevant noncausal path, and whether the adjustment creates any new problems. A proxy can reduce confounding when it captures some of the information that made exposure and outcome differ. In general, however, adjusting for a proxy alone does not guarantee an unbiased causal estimate.

Definition: A proxy variable is an observed variable used as an indirect indicator of another variable that is not observed. In this tutorial, \(U\) is an unmeasured common cause of exposure \(A\) and outcome \(Y\), and \(Z\) is a measured proxy for \(U\). A proxy is informative when it contains information about \(U\); being called a proxy does not by itself establish how informative it is or whether adjusting for it is appropriate.

What Adjustment Can Do

Consider a causal diagram with \(U \rightarrow A\), \(U \rightarrow Y\), and \(U \rightarrow Z\). The path \(A \leftarrow U \rightarrow Y\) is a backdoor path: it connects exposure and outcome through their common cause. Measuring and adjusting for \(U\) blocks that path. Adjusting for its descendant \(Z\), however, does not generally block it under the backdoor criterion. Within a given value of \(Z\), people can still differ in \(U\), and those differences can still relate to both \(A\) and \(Y\).

Even when the path remains open, conditioning on an informative proxy can make the remaining association between \(A\) and \(U\) smaller. If \(U\) affects \(Y\), that can reduce the part of the exposure–outcome association attributable to \(U\). Whether it does so, and by how much, depends on the relationships among \(U\), \(Z\), \(A\), and \(Y\). The regression calculation below makes this distinction precise for an additive linear model.

Model for the calculation: Suppose the outcome follows \(Y=\beta_0+\tau A+\gamma U+\varepsilon\), where \(\tau\) is the causal effect of \(A\) on \(Y\) for a one-unit increase in \(A\), and \(\gamma\) describes the effect of \(U\) on \(Y\). Assume this additive model is appropriate, \(U\) is the only unadjusted common cause under consideration, and the outcome error has conditional mean zero given \(A\), \(U\), and \(Z\). These assumptions allow the examples to isolate the role of the proxy; they do not establish that a real analysis meets them.

In a regression of \(Y\) on \(A\) and \(Z\), the population coefficient of \(A\) can be written as

\beta_{A\mid Z} = \tau + \gamma \frac{\operatorname{Cov}(A-\widehat{A}_Z,\ U-\widehat{U}_Z)} {\operatorname{Var}(A-\widehat{A}_Z)}.

Here \(\widehat{A}_Z\) and \(\widehat{U}_Z\) are the linear predictions of \(A\) and \(U\), respectively, from an intercept and \(Z\). The fraction is the coefficient of \(A\) in the regression of \(U\) on \(A\) and \(Z\). Thus, the adjusted coefficient equals the causal effect plus a remaining bias term: the effect of \(U\) on \(Y\), multiplied by how strongly \(A\) still carries information about \(U\) after accounting for \(Z\).

Why the Formula Holds

The short derivation uses residualization: the coefficient of \(A\) in a regression that also includes \(Z\) is the slope relating the parts of \(A\) and \(Y\) left after their linear relationships with \(Z\) are removed. Let \(A^*=A-\widehat{A}_Z\), \(U^*=U-\widehat{U}_Z\), and \(Y^*=Y-\widehat{Y}_Z\). Residualizing the outcome model on \(Z\) gives \(Y^*=\tau A^*+\gamma U^*+\varepsilon^*\). The slope is therefore \(\operatorname{Cov}(A^*,Y^*)/\operatorname{Var}(A^*)\). Under the stated conditional-mean assumption, the residual outcome error contributes zero covariance with \(A^*\), so

\frac{\operatorname{Cov}(A^*,Y^*)}{\operatorname{Var}(A^*)} = \tau+\gamma\frac{\operatorname{Cov}(A^*,U^*)}{\operatorname{Var}(A^*)}.

This is a population relationship for the specified linear model. It is not a guarantee that an estimated regression coefficient in a real study has a causal interpretation. That interpretation also depends on whether the model and causal assumptions are reasonable.

Worked Example: A Perfect Proxy Removes This Confounding

Suppose a hypothetical population follows \(A=U+e_A\) and \(Y=3+2A+4U+\varepsilon\). Let \(U\) have variance \(1\), let \(e_A\) have variance \(3\), and assume \(U\), \(e_A\), and \(\varepsilon\) are mutually independent and centered. Suppose the measured proxy is exact: \(Z=U\). Find the exposure slope with and without adjustment.

Without adjustment, the coefficient of \(A\) in the regression of \(Y\) on \(A\) is the causal effect plus omitted-variable bias. Since \(\operatorname{Var}(A)=1+3=4\) and \(\operatorname{Cov}(A,U)=1\), the bias is

4\frac{\operatorname{Cov}(A,U)}{\operatorname{Var}(A)} = 4\frac{1}{4}=1.

The unadjusted slope is \(2+1=3\), rather than the causal effect of \(2\). With \(Z=U\), adjusting for \(Z\) is equivalent to adjusting for the common cause itself. The residual variation in \(A\) is \(e_A\), which is independent of \(U\); consequently \(\operatorname{Cov}(A^*,U^*)=0\). The bias term is \(4(0)/3=0\), so the adjusted slope is \(2\). This example illustrates the special case in which the proxy contains the common cause exactly.

An Informative Proxy Can Reduce, but Leave, Bias

A more typical proxy is related to the common cause but does not identify it exactly. The following calculation uses a hypothetical population model and known population variances and covariances, so it illustrates the regression relationship rather than reporting an estimate from a sample.

Worked Example: A Noisy Proxy Partly Reduces Confounding

Let \(U\), \(v_A\), and \(v_Z\) be independent, centered variables with variance \(1\) each. Suppose \(A=0.6U+v_A\), \(Z=U+v_Z\), and \(Y=2+1.5A+2U+\varepsilon\), where \(\varepsilon\) has conditional mean zero. The causal effect is \(\tau=1.5\); \(U\) is a common cause because it affects both \(A\) and \(Y\), and \(Z\) is an informative but noisy proxy. Compare the population exposure slopes with and without adjustment for \(Z\).

First, \(\operatorname{Var}(A)=0.6^2(1)+1=1.36\) and \(\operatorname{Cov}(A,U)=0.6\). The unadjusted omitted-variable bias is therefore \(2(0.6/1.36)=0.8824\), giving slope \(1.5+0.8824=2.382\).

For adjustment, \(\operatorname{Var}(Z)=1+1=2\), \(\operatorname{Cov}(A,Z)=0.6\), and \(\operatorname{Cov}(U,Z)=1\). Removing each variable's linear relationship with \(Z\) gives

\operatorname{Cov}(A^*,U^*) =0.6-\frac{(0.6)(1)}{2} =0.3000, \qquad \operatorname{Var}(A^*) =1.36-\frac{0.6^2}{2} =1.180.

The remaining bias is \(2(0.3000/1.180)=0.5085\), so the adjusted slope is \(1.5+0.5085=2.008\). Adjustment reduces the bias from \(0.8824\) to \(0.5085\), but does not remove it. The calculation demonstrates what “reduces confounding” means here: the proxy lowers the residual association between exposure and the unobserved common cause, while some remains.

Adjustment Can Also Create Bias

A proxy may have causes other than the unmeasured common cause. If the proxy is a common effect of \(U\) and another variable, adjusting for it can open a path that was previously blocked. For example, if \(U \rightarrow Z \leftarrow V\), conditioning on \(Z\) can associate \(U\) and \(V\). If \(V\) also affects the outcome, this induced association can contribute bias. This is why the proxy's position and other causes in the causal diagram matter, not just its correlation with \(U\).

Worked Example: A Proxy That Is Also a Common Effect

Let \(U\) and \(V\) be independent, centered variables with variance \(1\). Let \(e_A\) be independent of them with variance \(1\), and suppose \(A=U+e_A\), \(Z=U+V\), and \(Y=2+2A+2U+4V+\varepsilon\), with a mean-zero outcome error independent of these variables. Here \(Z\) is related to \(U\), but it is also a common effect of \(U\) and \(V\); \(V\) affects the outcome. Compare the unadjusted and proxy-adjusted slopes.

Without adjustment, \(\operatorname{Var}(A)=2\), \(\operatorname{Cov}(A,U)=1\), and \(\operatorname{Cov}(A,V)=0\). The bias from \(U\) is \(2(1/2)=1\), while \(V\) contributes no unadjusted bias. The unadjusted slope is therefore \(2+1=3\).

For adjustment, \(\operatorname{Var}(Z)=2\), \(\operatorname{Cov}(A,Z)=1\), \(\operatorname{Cov}(U,Z)=1\), and \(\operatorname{Cov}(V,Z)=1\). After removing the linear relationship with \(Z\),

\operatorname{Var}(A^*)=2-\frac{1^2}{2}=1.5,\quad \operatorname{Cov}(A^*,U^*)=1-\frac{1(1)}{2}=0.5,\quad \operatorname{Cov}(A^*,V^*)=0-\frac{1(1)}{2}=-0.5.

The remaining bias is \(\{2(0.5)+4(-0.5)\}/1.5=-0.6667\), giving an adjusted slope of \(2-0.6667=1.333\). In this hypothetical example, adjustment moves the slope past the causal effect of \(2\), rather than closer to it. Conditioning on \(Z\) has associated \(A\) with \(V\), producing an additional bias contribution. The example shows why a proxy should not be added automatically just because it is related to a suspected confounder.

Choosing and Interpreting Proxy Adjustment

A proxy is most useful when subject-matter knowledge supports a clear relationship between it and the unmeasured common cause, and the causal diagram gives no reason to expect adjustment to open a problematic path. Its measured values may help make exposed and unexposed people more comparable with respect to the hidden cause. But the observed data alone do not prove that the proxy captures enough information, nor that no other unmeasured common causes remain.

A careful analysis therefore treats proxy adjustment as a reasoned strategy, not as a correction guaranteed by regression. State which unmeasured cause the proxy is intended to represent, explain why it should carry information about that cause, and consider other causes of the proxy. Interpret the adjusted coefficient as an adjusted association unless the causal assumptions needed for a causal interpretation are justified.

Key takeaway: A proxy can reduce confounding when it reduces the exposure's remaining association with an unmeasured common cause. It generally does not block the common-cause path merely by being adjusted for, and adjustment can introduce bias if the proxy's causal structure makes it a collider.

Check Your Understanding

Use the model, calculations, and causal diagrams in this tutorial to answer the following questions.

In the diagram \(U \rightarrow A\), \(U \rightarrow Y\), and \(U \rightarrow Z\), why does adjusting for \(Z\) not generally block the path from \(A\) to \(Y\) through \(U\)?
In the linear model, what two quantities determine the remaining bias after adjustment for \(Z\)?
In the noisy-proxy example, what was the unadjusted slope, and what was the slope after adjustment for \(Z\)? What explains the difference?
What special feature of \(Z\) allowed it to remove the confounding in the first worked example?
In the common-effect example, which path did conditioning on \(Z\) open, and how did that affect the adjusted slope?