Confounding Functions for Average Treatment Effects

Why Use Treatment-Specific Confounding Functions?

When treatment is not randomized, people receiving treatment may differ from those not receiving it in ways that affect their outcomes. Adjustment for measured covariates can address differences represented by those covariates, but it does not by itself establish that the remaining groups are comparable. Treatment-specific confounding functions provide a direct way to describe the remaining differences in potential-outcome means and show how they affect an average treatment effect.

The key is to distinguish outcomes that are observed from outcomes that are counterfactual for a particular person. For each person, only the outcome under the treatment actually received is observed. A confounding function compares mean potential outcomes across treatment groups within a covariate stratum, including the potential outcome that is counterfactual for one of those groups.

Definition: Let $A\in\{0,1\}$ denote treatment, $X$ the measured covariates, and $Y(1)$ and $Y(0)$ the potential outcomes under treatment and control. For a covariate value $x$, define the treatment-specific confounding functions $$ \delta_1(x)=E[Y(1)\mid A=1,X=x]-E[Y(1)\mid A=0,X=x], $$ $$ \delta_0(x)=E[Y(0)\mid A=1,X=x]-E[Y(0)\mid A=0,X=x]. $$ Each function is a conditional difference in the mean of one potential outcome between the treated and untreated groups. The subscript identifies which potential outcome is being compared.

The first term in $\delta_1(x)$ is the mean of the observed outcome among treated people in stratum $x$, by consistency; the second is the mean outcome those untreated people would have had under treatment. Conversely, $\delta_0(x)$ compares the mean outcome those treated people would have had under control with the observed control-group mean. Positive values mean the treated group has the higher mean for that potential outcome, conditional on $X=x$; negative values mean it has the lower mean.

From Confounding Functions to the Average Treatment Effect

Write $e(x)=P(A=1\mid X=x)$ for the treatment probability, $m_1(x)=E[Y\mid A=1,X=x]$, and $m_0(x)=E[Y\mid A=0,X=x]$. The covariate-standardized observed contrast averages $m_1(x)-m_0(x)$ over the covariate distribution in the target population. It is an associational contrast unless additional assumptions justify a causal interpretation.

The target average treatment effect (ATE) is $E[Y(1)-Y(0)]$. To express it using observed conditional means and the confounding functions, first consider a fixed stratum $X=x$. The mean of $Y(1)$ in that stratum combines the treated group’s observed mean with the untreated group’s counterfactual mean. The mean of $Y(0)$ combines the untreated group’s observed mean with the treated group’s counterfactual mean.

E[Y(1)\mid X=x] =e(x)m_1(x)+\{1-e(x)\}\{m_1(x)-\delta_1(x)\},

E[Y(0)\mid X=x] =\{1-e(x)\}m_0(x)+e(x)\{m_0(x)+\delta_0(x)\}.

Subtracting the second expression from the first gives the conditional ATE. Averaging that result over the target population’s distribution of $X$ gives the overall ATE.

Result: Under consistency and conditional positivity, the ATE can be written as $$ E[Y(1)-Y(0)] =E_X\left[m_1(X)-m_0(X)-\{1-e(X)\}\delta_1(X)-e(X)\delta_0(X)\right]. $$ Here $E_X$ denotes averaging over the target population’s covariate distribution. Thus the covariate-standardized observed contrast $E_X[m_1(X)-m_0(X)]$ differs from the ATE by $$ E_X\left[\{1-e(X)\}\delta_1(X)+e(X)\delta_0(X)\right]. $$ This difference is the bias of the standardized contrast relative to the ATE, with bias defined as contrast minus ATE.

The factors $1-e(x)$ and $e(x)$ arise because the unobserved counterfactual mean for $Y(1)$ belongs to the untreated group, while the unobserved counterfactual mean for $Y(0)$ belongs to the treated group. The two functions need not be equal, and their contributions can reinforce or offset one another. A single generic “confounding amount” can therefore miss important differences between the two potential outcomes.

Assumptions and What the Functions Can Tell Us

Consistency means that $Y=Y(a)$ for a person who actually receives treatment level $A=a$. Conditional positivity means that for covariate values represented in the target population, $0<e(x)<1$; both treatment groups must be possible in those strata to define and estimate their observed conditional means. The identity above does not assume conditional exchangeability, $Y(a)\perp A\mid X$. Instead, the confounding functions describe how conditional mean exchangeability may fail.

The functions involve potential outcomes that are not observed for everyone, so they are generally not identified from the observed data alone. The observed data can estimate $e(x)$, $m_1(x)$, and $m_0(x)$ subject to the usual data and modeling limitations, but it cannot directly reveal the missing potential-outcome means in $\delta_1(x)$ and $\delta_0(x)$. Using the formula for sensitivity analysis therefore requires specifying or otherwise justifying values for these functions. If both functions equal zero in every stratum, the two groups have equal conditional means for each potential outcome, and the standardized contrast equals the ATE. That is a mean-exchangeability condition, not a conclusion established merely by observing the data.

Worked Example: Adjusting Within One Covariate Stratum

Suppose a hypothetical study compares a treatment with control, and every person in the target population belongs to the same covariate stratum. The treatment probability is $e=0.30$. The observed mean outcome is 12 units among treated people and 8 units among untreated people. Suppose subject-matter assumptions assign $\delta_1=3$ units and $\delta_0=1$ unit.

The observed standardized contrast is $12-8=4$ units. Its estimated bias under the specified functions is $$ (1-0.30)(3)+(0.30)(1)=0.70(3)+0.30(1)=2.40\text{ units}. $$

Subtracting this bias gives the ATE: $$ 4-2.40=1.60\text{ units}. $$ The calculation can also be checked by reconstructing the two potential-outcome means. For treatment, $$ E[Y(1)]=0.30(12)+0.70(12-3)=3.60+6.30=9.90\text{ units}. $$ For control, $$ E[Y(0)]=0.70(8)+0.30(8+1)=5.60+2.70=8.30\text{ units}. $$ Their difference is $9.90-8.30=1.60$ units, as before. Under these hypothetical assumptions, the standardized observed contrast exceeds the ATE by 2.40 units.

Standardizing Across Covariate Strata

When covariates define multiple strata, calculate the conditional ATE and bias within each stratum, then average across strata using the target population’s stratum proportions. Do not average with the treatment-group proportions unless those happen to equal the target distribution. The treatment probability within a stratum affects the bias adjustment, while the target stratum proportion determines how much that stratum contributes to the overall ATE.

Worked Example: Combining Two Covariate Strata

Consider a hypothetical target population with 60% in a lower-risk stratum and 40% in a higher-risk stratum. The following table gives the treatment probability, observed conditional means, and assumed treatment-specific confounding functions. All outcomes are measured in points.

Stratum	Target proportion	$e(x)$	$m_1(x)$	$m_0(x)$	$\delta_1(x)$	$\delta_0(x)$
Lower-risk	0.60	0.25	15	10	2	1
Higher-risk	0.40	0.75	13	10	1	-1

In the lower-risk stratum, the observed contrast is $15-10=5$ points. The bias is $$ (1-0.25)(2)+(0.25)(1)=1.50+0.25=1.75\text{ points}, $$ so the conditional ATE is $5-1.75=3.25$ points.

In the higher-risk stratum, the observed contrast is $13-10=3$ points. The bias is $$ (1-0.75)(1)+(0.75)(-1)=0.25-0.75=-0.50\text{ points}, $$ so the conditional ATE is $3-(-0.50)=3.50$ points. The negative bias means the observed contrast is 0.50 points below the conditional ATE in this stratum.

Average the conditional effects using the target proportions: $$ 0.60(3.25)+0.40(3.50)=1.95+1.40=3.35\text{ points}. $$ As a check, the standardized observed contrast is $$ 0.60(5)+0.40(3)=4.20\text{ points}, $$ and the standardized bias is $$ 0.60(1.75)+0.40(-0.50)=1.05-0.20=0.85\text{ points}. $$ Therefore, $4.20-0.85=3.35$ points, matching the weighted conditional ATE. The two strata contribute biases in opposite directions, so simply assuming that confounding must increase the observed contrast would be incorrect.

Reading the Signs and Magnitudes

The functions are measured in the outcome’s units. If the outcome is a score, $\delta_a(x)$ is a difference in score points; if it is a biomarker, it is a difference in the biomarker’s units. Their signs describe differences between treatment groups for a particular potential outcome, conditional on $X$; they do not, on their own, describe the treatment effect.

For example, $\delta_1(x)>0$ says that treated people have a higher mean $Y(1)$ than untreated people in the same stratum. It does not say that treatment increases the outcome by $\delta_1(x)$. That treatment effect compares $Y(1)$ with $Y(0)$ for the target population. Similarly, $\delta_0(x)$ is not a treatment effect. Both functions describe selection differences across treatment groups, each for a fixed potential outcome.

Worked Example: Opposing Treatment-Specific Differences

In a hypothetical single-stratum study, suppose $e=0.60$, $m_1=20$ units, $m_0=15$ units, $\delta_1=-2$ units, and $\delta_0=4$ units. The standardized observed contrast is $20-15=5$ units. The bias is $$ (1-0.60)(-2)+(0.60)(4)=-0.80+2.40=1.60\text{ units}. $$ Thus the ATE is $5-1.60=3.40$ units.

To verify, the mean $Y(1)$ among untreated people is $m_1-\delta_1=20-(-2)=22$ units. Hence $$ E[Y(1)]=0.60(20)+0.40(22)=12+8.80=20.80\text{ units}. $$ The mean $Y(0)$ among treated people is $m_0+\delta_0=15+4=19$ units. Hence $$ E[Y(0)]=0.40(15)+0.60(19)=6+11.40=17.40\text{ units}. $$ The resulting ATE is $20.80-17.40=3.40$ units. Here the negative $\delta_1$ contribution partially offsets the positive $\delta_0$ contribution; treating either function alone as the total bias would give the wrong adjustment.

Common Mistake

A frequent error is to call $m_1(x)-m_0(x)$ the conditional treatment effect, then treat a confounding function as a correction to just one observed mean. The observed contrast compares outcomes in different groups; it is not automatically a contrast of potential outcomes for the same target population. The correction must account for both missing potential-outcome means, using $\{1-e(x)\}\delta_1(x)$ and $e(x)\delta_0(x)$, and then average over the target covariate distribution. Another error is to reverse the sign convention for $\delta_0$: as defined here, it is the treated-minus-untreated difference in $Y(0)$, so the counterfactual control mean among treated people is $m_0(x)+\delta_0(x)$.

Key takeaway: Treatment-specific confounding functions quantify, within covariate strata, how the two treatment groups differ in the mean of each potential outcome. The ATE equals the covariate-standardized observed contrast minus the weighted contributions of those differences. The functions are sensitivity inputs, not quantities identified from observed outcomes alone.

Check Your Understanding

Use the definitions and sign convention in this tutorial.

In words, what does a positive $\delta_1(x)$ say about the two treatment groups within covariate stratum $x$?
If $e(x)=0.40$, $\delta_1(x)=2$ units, and $\delta_0(x)=-1$ unit, what is the stratum-specific bias of the standardized contrast?
Why is the covariate-standardized observed contrast not automatically the ATE?
In a multi-stratum target population, which weights should be used to average stratum-specific ATEs?
Which assumptions support using observed conditional means for the corresponding treatment-group potential-outcome means, and what role does positivity play?