Proxy Measurements and Latent Confounders

Why Proxy Measurements Need a Different Argument

When a confounder is measured, adjustment can compare treated and untreated units with the same confounder value, subject to exchangeability and positivity. A proxy for an unobserved confounder is not that confounder: two people with the same proxy value can still differ in their latent confounding. Simply including the proxy in a regression or standardization does not generally remove confounding.

Proxy-based methods use a more specific structure. One proxy, denoted $Z$, carries information about the latent confounder that helps distinguish treatment groups. Another, denoted $W$, carries information about the latent confounder relevant to outcomes. A bridge function links these proxies. The assumptions connecting that function to the latent confounder are essential: fitting an equation among observed variables does not, by itself, establish that the bridge recovers a causal effect.

Definition: Let $A$ be a binary treatment, $Y$ an outcome, $X$ measured covariates, and $U$ an unobserved confounder. Let $Y(a)$ denote the potential outcome under treatment level $a$. A treatment-inducing proxy $Z$ is an observed variable informative about $U$ in the treatment process; an outcome-inducing proxy $W$ is an observed variable informative about $U$ in the outcome process. These names describe their roles in the proxy strategy, not a claim that either proxy is itself a sufficient confounder.

Latent Bridges and Observed Bridge Equations

A useful sufficient set of conditions can be stated for each treatment level $a$. Assume consistency, so $Y=Y(A)$; latent conditional exchangeability, so treatment assignment is independent of potential outcomes given $U,X$; and positivity, so each treatment level has positive probability at every relevant $U,X$. For the proxy argument below, use the stronger condition $Y(a)\perp (A,Z)\mid U,X$. This includes the needed exchangeability and also requires that, conditional on $U,X$, $Z$ contain no additional information about $Y(a)$.

Assume also that $W\perp(A,Z)\mid U,X$. This says that, after conditioning on the latent confounder and measured covariates, the distribution of $W$ does not depend on treatment or on $Z$. It does not say that $W$ is independent of outcomes without conditioning on $U,X$. Finally, require a bridge function $h(W,a,X)$ with the latent relationship shown below.

Definition: A latent outcome bridge for treatment level $a$ is a function $h(W,a,X)$ satisfying $$ E[h(W,a,X)\mid U,X]=E[Y(a)\mid U,X]. $$ An observed-data bridge equation is $$ E[Y\mid A=a,Z,X]=E[h(W,a,X)\mid A=a,Z,X]. $$ The latent relationship directly links the bridge to potential outcomes. The observed equation is checkable or estimable from observed variables, but alone it need not imply the latent relationship.

Under the stated conditional independence assumptions, a latent bridge implies the observed equation. Given $A=a$, consistency gives $Y=Y(a)$. Conditional on $U,X$, the distribution of $Y(a)$ is unchanged by conditioning on $A,Z$, and the distribution of $W$ is unchanged by conditioning on $A,Z$. Thus both sides of the observed equation are conditional averages of the same latent quantity.

\begin{aligned} E[Y\mid A=a,Z,X] &=E[E[Y(a)\mid U,X]\mid A=a,Z,X]\\ &=E[E[h(W,a,X)\mid U,X]\mid A=a,Z,X]\\ &=E[h(W,a,X)\mid A=a,Z,X]. \end{aligned}

The first equality uses consistency and $Y(a)\perp(A,Z)\mid U,X$; the second uses the latent bridge; the third uses $W\perp(A,Z)\mid U,X$. The direction matters: these assumptions show that a latent bridge yields an observed bridge equation. The observed equation does not automatically prove that a proposed $h$ is a latent bridge.

Identification Result and the Role of Completeness

If a valid latent bridge exists, averaging it over the population identifies the potential-outcome mean. The distribution of $W$ conditional on $U,X$ is the same whether or not we condition on treatment and $Z$, and the latent bridge equates its conditional average to the mean potential outcome given $U,X$.

Theorem: Under consistency, positivity, $Y(a)\perp(A,Z)\mid U,X$, $W\perp(A,Z)\mid U,X$, and existence of a latent outcome bridge $h$ for level $a$, $$ E[Y(a)]=E[h(W,a,X)]. $$ If the target is the average treatment effect, then $E[Y(1)-Y(0)]=E[h(W,1,X)-h(W,0,X)]$, provided valid latent bridges exist for both levels.

The result follows by iterated expectation:

\begin{aligned} E[h(W,a,X)] &=E[E[h(W,a,X)\mid U,X]]\\ &=E[E[Y(a)\mid U,X]]\\ &=E[Y(a)]. \end{aligned}

In practice, analysts often solve the observed-data bridge equation rather than impose the latent relationship directly. To justify this route, an additional condition is needed. One sufficient condition is completeness: for the relevant $a$ and target support, if a function $q(U,X)$ satisfies $E[q(U,X)\mid A=a,Z,X]=0$ almost surely, then $q(U,X)=0$ almost surely. Suppose the conditional-independence assumptions hold and a function $h$ satisfies the observed bridge equation. The difference $q(U,X)=E[h(W,a,X)\mid U,X]-E[Y(a)\mid U,X]$ then has conditional mean zero given $A=a,Z,X$. Completeness forces this difference to be zero, giving the latent bridge. Positivity and support conditions are needed so this argument covers the target population.

Completeness is a substantive identification assumption, not a consequence of having two proxies or of obtaining a good numerical fit. It can fail, for example, when $Z$ cannot distinguish some latent-confounder patterns. Without a latent bridge assumption or conditions such as completeness that establish it, the observed bridge equation alone does not justify the causal formula.

Worked Example: Ordinary Adjustment for a Measured Confounder

Suppose a hypothetical study measures a binary confounder $C$ without error. Assume consistency, conditional exchangeability $Y(a)\perp A\mid C$, and treatment positivity within both $C$ strata. The target population has $P(C=1)=0.40$. The observed outcome risks are:

Confounder stratum	Risk if treated	Risk if untreated
$C=1$	0.50	0.30
$C=0$	0.20	0.10

Here $C$ is the confounder itself, so standardization uses the conditional risks and the target distribution of $C$. The treated potential-outcome mean is

$$ E[Y(1)]=(0.40)(0.50)+(0.60)(0.20)=0.20+0.12=0.32. $$

Similarly,

$$ E[Y(0)]=(0.40)(0.30)+(0.60)(0.10)=0.12+0.06=0.18. $$

The standardized risk difference is $0.32-0.18=0.14$, or 14 percentage points. This calculation depends on exchangeability within the measured $C$ strata and positivity; it does not show that the same calculation with a noisy proxy in place of $C$ would remove confounding.

Worked Example: Solving a Discrete Proxy Bridge

Consider a hypothetical population with binary $U$, $P(U=1)=P(U=0)=0.50$, and binary proxies $W,Z$. Conditional on $U$, each proxy correctly classifies $U$ with probability $0.80$, and their errors are independent of treatment and potential outcomes. Let treatment assignment satisfy $P(A=1\mid U=1)=0.80$ and $P(A=1\mid U=0)=0.20$; both treatment levels therefore have positive probability in each latent stratum. Suppose potential-outcome means are $E[Y(1)\mid U=1]=8$, $E[Y(1)\mid U=0]=4$, $E[Y(0)\mid U=1]=6$, and $E[Y(0)\mid U=0]=3$. Assume the conditional-independence conditions stated above and that the latent bridge exists.

For $a=1$, write $h(1,1)=h_1$ and $h(0,1)=h_0$. The latent bridge requires

0.80h_1+0.20h_0=8,\qquad 0.20h_1+0.80h_0=4.

Subtracting the second equation from the first gives $0.60h_1-0.60h_0=4$, so $h_1-h_0=6.6667$. Substituting into the first equation gives $h_0=2.6667$, and hence $h_1=9.3333$. For $a=0$, the corresponding equations are $0.80h_1+0.20h_0=6$ and $0.20h_1+0.80h_0=3$. Their solution is $h_1=7$, $h_0=2$.

Because $U$ is equally prevalent and $W$ has symmetric 80% accuracy, $P(W=1)=0.50$. Therefore,

$$ E[h(W,1)]=(0.50)(9.3333)+(0.50)(2.6667)=6.0000, $$

and

$$ E[h(W,0)]=(0.50)(7)+(0.50)(2)=4.500. $$

The identified average treatment effect is $6.0000-4.500=1.500$ outcome units. The calculation is valid because this example specifies the latent bridge and proxy conditions, not merely because the bridge values solve an observed-data equation.

Worked Example: Why an Observed Bridge Alone Can Mislead

Consider a hypothetical population in which $U$ is equally likely to be 1, 2, or 3; $W=U$; $Z=1\{U=1\}$; and $P(A=1\mid U)=(0.80,0.80,0.20)$. Let $Y(0)=Y(1)=U$ for every unit. Treatment positivity and latent exchangeability hold. Conditional on $U$, $W$ is fixed and $Z$ carries no additional outcome information.

For treatment level 0, set $h(W,0)=W$. For treatment level 1, define $h(W,1)=W+g(W)$, where $g(1)=0$, $g(2)=1$, and $g(3)=-4$. For $A=1,Z=0$, only $U=2$ and $U=3$ are possible. Their conditional probabilities are $0.80$ and $0.20$, because the $Z=0$ group contains $U=2,3$ in equal population proportions and their treatment probabilities are $0.80,0.20$. Thus

E[Y\mid A=1,Z=0]=(0.80)(2)+(0.20)(3)=2.20,

while

E[h(W,1)\mid A=1,Z=0]=(0.80)(3)+(0.20)(-1)=2.20.

For $A=1,Z=1$, $U=1$, so both conditional means equal 1. The level-0 bridge $h(W,0)=W$ also satisfies the observed equation. Yet $E[Y(1)]=(1+2+3)/3=2$, whereas

E[h(W,1)]=\frac{1+3+(-1)}{3}=1.

The observed bridge equation holds, but the proposed function is not a latent bridge at level 1: at $U=2$, $E[h(W,1)\mid U=2]=3$, while $E[Y(1)\mid U=2]=2$. Completeness fails because $Z$ only separates $U=1$ from the combined $U=2,3$ group; it cannot detect every difference across latent strata. This example shows why the observed equation alone is insufficient.

Interpreting and Using Proxy Adjustment

Proxy adjustment is useful when direct measurement of a confounder is unavailable, but its credibility depends on substantive and statistical conditions. The proxies must contain adequate information about the latent structure, the conditional independence claims must be defensible, treatment must have support across relevant latent strata, and a suitable bridge must exist. Identification does not guarantee that the bridge can be estimated precisely from a finite sample; weak proxy information or near failure of completeness can make estimates unstable.

A common mistake is to treat $Z$ or $W$ as though it were simply a measured confounder and adjust for it in an ordinary outcome regression. The correction is that proxy methods use the joint structure of the proxies and a bridge relationship. A proxy's association with treatment and outcome does not make it an adequate confounder adjustment set. Conversely, a bridge equation that fits the observed data does not certify its latent interpretation.

Key takeaway: A causal mean can be recovered from $E[h(W,a,X)]$ when $h$ is a valid latent bridge. To obtain that bridge from observed data, add sufficient identifying conditions—such as appropriate completeness—to the observed bridge equation. Do not infer the latent relationship from the observed equation alone.

Check Your Understanding

Use the assumptions and distinctions developed above to answer the following questions.

What role does $Z$ play, and why is it not automatically a sufficient confounder adjustment variable?
State the latent bridge relationship and the observed-data bridge equation. Which one directly links the proxy function to potential outcomes?
In the discrete example, why could the bridge function be averaged over $W$ to obtain each causal mean?
What does completeness rule out when using an observed bridge equation to infer a latent bridge?
Why does solving an observed bridge equation not, by itself, prove that $E[h(W,a,X)]$ equals $E[Y(a)]$?

Proxy Measurements and Latent Confounders

What You'll Learn

Why Proxy Measurements Need a Different Argument

Latent Bridges and Observed Bridge Equations

Identification Result and the Role of Completeness

Worked Example: Ordinary Adjustment for a Measured Confounder

Worked Example: Ordinary Adjustment for a Measured Confounder

Worked Example: Solving a Discrete Proxy Bridge

Worked Example: Solving a Discrete Proxy Bridge

Worked Example: Why an Observed Bridge Alone Can Mislead

Worked Example: Why an Observed Bridge Alone Can Mislead

Interpreting and Using Proxy Adjustment

Check Your Understanding

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