Learning Treatment Rules From CATE Estimates

From Conditional Effects to Decisions

A conditional average treatment effect (CATE) describes how the mean treatment effect varies with observed characteristics. A treatment rule turns that information into an action: for a person or unit with covariates $X=x$, assign treatment or withhold it. The central decision is not simply whether the predicted effect is positive. Treatment may have a cost, and that cost can vary across people. The relevant comparison is between the expected gain and the cost.

This tutorial considers a binary treatment and a single outcome measured so that both gains and costs can be expressed on a common scale. A rule based on an estimated CATE is a practical decision rule, not a guarantee that the estimate is correct. Its assignments can change when estimation error moves a prediction across the treatment threshold.

Definition: Let $Y(1)$ and $Y(0)$ denote the potential outcomes under treatment and no treatment, respectively, and let $X$ be observed covariates. The conditional average treatment effect is $$ \tau(x)=E[Y(1)-Y(0)\mid X=x]. $$ A deterministic treatment rule is a function $d(x)\in\{0,1\}$, where $d(x)=1$ assigns treatment and $d(x)=0$ withholds it. If treatment has covariate-specific cost $c(x)$, expressed in the same units as the outcome, the conditional expected net gain from treatment rather than no treatment is $\tau(x)-c(x)$.

The causal interpretation of $\tau(x)$ depends on valid identification and estimation. For example, it may rely on random assignment or on appropriate conditional exchangeability, positivity, and consistency conditions. Learning a treatment rule from CATE estimates does not itself establish those conditions or repair a biased CATE estimate. The decision step takes the effect estimates as inputs and asks how they compare with costs.

The Threshold Rule

Suppose the decision-maker is risk-neutral with respect to the outcome scale being used and aims to maximize conditional expected net outcome. Withholding treatment is the reference action, with incremental net gain zero. Treatment has conditional expected incremental net gain $\tau(x)-c(x)$. It follows that treatment is preferable when this quantity is positive.

Definition: A threshold-optimal rule is $$ d^*(x)=\mathbb{1}\{\tau(x)>c(x)\}, $$ where $\mathbb{1}\{\cdot\}$ equals 1 when the condition is true and 0 otherwise. This convention withholds treatment at an exact tie, $\tau(x)=c(x)$; either action has the same conditional expected net outcome at a tie.

If treatment cost is constant, the rule uses a common threshold. If the cost depends on covariates, the threshold varies with $x$. Costs may represent resource use, monetary expense translated into the outcome's utility scale, or another explicitly defined loss. Comparing a treatment effect measured in one unit with a cost measured in an unrelated unit is not meaningful until a conversion or utility scale is specified.

Worked Example: Applying a Cost Threshold

Suppose a hypothetical program produces an outcome measured in utility points, and its cost for each profile has already been translated into the same units. Estimated effects for three profiles are shown below. For this example, treat when the estimated gain strictly exceeds cost.

Profile	Estimated CATE	Cost	Estimated net gain	Assignment
A	8.0 points	3.0 points	5.0 points	Treat
B	2.5 points	3.0 points	-0.5 points	Withhold
C	4.0 points	4.0 points	0.0 points	Withhold at tie

For profile A, the estimated net gain is $8.0-3.0=5.0$ points, so the rule assigns treatment. For profile B, it is $2.5-3.0=-0.5$ points, so the rule withholds treatment. For profile C, the gain and cost are both 4.0 points; the rule withholds at equality by its stated convention, although treating would have the same estimated net outcome. These are hypothetical predictions, not observed effects for real people.

Using Estimated CATEs

In practice, the CATE is unknown. Let $\widehat{\tau}(x)$ be an estimated conditional treatment effect, obtained, for example, from a CATE learner. A direct plug-in rule replaces the unknown effect with its estimate and compares that estimate with the cost:

\widehat{d}(x)=\mathbb{1}\{\widehat{\tau}(x)>c(x)\}.

This rule is easy to compute and makes its decision criterion explicit. It does not imply that $\widehat{\tau}(x)$ is the true effect, nor that every assigned person will benefit. The CATE is a conditional average, so it concerns a mean difference for units with covariates $x$, rather than a known individual treatment effect. As with any estimated decision rule, performance depends on the accuracy of the estimates in the regions where treatment decisions are made.

Worked Example: Estimation Error Changes an Assignment

Consider a hypothetical profile with cost $c(x)=4.0$ utility points. Suppose, only to illustrate the decision consequence, that its true CATE is $6.0$ points but the fitted CATE estimate is $3.4$ points. The true threshold rule assigns treatment because $6.0>4.0$. The plug-in rule withholds treatment because $3.4<4.0$.

The true conditional net gain from treatment is $6.0-4.0=2.0$ points. Thus the mistaken assignment sacrifices 2.0 points of conditional expected net outcome. The absolute estimation error is $$ |\widehat{\tau}(x)-\tau(x)|=|3.4-6.0|=2.6\text{ points}. $$ Here a 2.6-point error crosses a threshold only 2.0 points below the true CATE. By contrast, if the true CATE were $2.0$ points and the estimate were $5.0$ points at the same 4.0-point cost, the rule would wrongly treat: the foregone net gain is $4.0-2.0=2.0$ points, while the absolute estimation error is $|5.0-2.0|=3.0$ points. In both cases, the assignment error is consequential because it changes which side of the cost threshold the prediction occupies.

Disagreement and Decision Regret

The size of a CATE estimation error alone does not determine whether an assignment changes. An error can be large while leaving an estimate on the same side of the threshold, or small while crossing a threshold when the true CATE is close to cost. The distance between the true CATE and the threshold helps quantify the consequences of a disagreement.

Result: For a threshold-optimal rule $d^*(x)$ and a plug-in rule $\widehat{d}(x)$ using the same cost and tie convention, the conditional regret is $$ \bigl(d^*(x)-\widehat{d}(x)\bigr)\bigl(\tau(x)-c(x)\bigr) = |\tau(x)-c(x)|\,\mathbb{1}\{\widehat{d}(x)\ne d^*(x)\}. $$ In particular, disagreement implies $$ |\tau(x)-c(x)|\leq|\widehat{\tau}(x)-\tau(x)|. $$ The inequalities allow either the estimate or the true CATE to equal the threshold.

Here regret means the conditional expected net outcome under the optimal rule minus the conditional expected net outcome under the estimated rule. If the two rules agree, it is zero. If they disagree, the rule chooses the action with lower conditional net outcome, so the loss is the distance $|\tau(x)-c(x)|$ from the threshold. At an exact tie that distance is zero.

The distance inequality follows directly from the threshold comparison. If $\tau(x)>c(x)$ but the plug-in rule withholds treatment, then $\widehat{\tau}(x)\leq c(x)<\tau(x)$, so $\tau(x)-c(x)\leq\tau(x)-\widehat{\tau}(x)$. If $\tau(x)\leq c(x)$ but the plug-in rule treats, then $\tau(x)\leq c(x)<\widehat{\tau}(x)$, so $c(x)-\tau(x)\leq\widehat{\tau}(x)-\tau(x)$. These cases prove the inequality. The conclusion also accommodates a different common tie-breaking convention: disagreement places the true CATE and estimate on different sides of the threshold, allowing one of them to equal it. For instance, if treatment is assigned at a tie, a true CATE below cost and an estimate equal to cost can disagree; the distance inequality still holds.

E\!\left[\text{regret}(X)\right] \leq E\!\left[|\widehat{\tau}(X)-\tau(X)|\right].

This bound follows by applying the pointwise inequality and noting that regret is zero when the rules agree. The expectation averages over the covariate distribution; if the estimator is itself random, it can also average over its training-sample variation. The bound explains why CATE accuracy near the decision threshold is especially relevant: errors far from the threshold may not change assignments, whereas errors near it can.

Worked Example: A Disagreement at the Threshold

Suppose treatment is assigned at a tie, so the rule is $d(x)=1$ when the effect is at least the cost. For a hypothetical profile, let $c(x)=1.0$ utility point, $\tau(x)=0.9$ points, and $\widehat{\tau}(x)=1.0$ point. The true rule withholds treatment because $0.9<1.0$, while the estimated rule treats because its estimate equals the threshold and the stated convention treats at a tie.

The rules disagree, although the estimate is not strictly above the threshold. The true net gain from treatment is $0.9-1.0=-0.1$ point, so the regret from treating is $0.1$ point. The estimation error is $$ |\widehat{\tau}(x)-\tau(x)|=|1.0-0.9|=0.1\text{ point}, $$ and the threshold distance is $$ |\tau(x)-c(x)|=|0.9-1.0|=0.1\text{ point}. $$ Thus $|\tau(x)-c(x)|\leq|\widehat{\tau}(x)-\tau(x)|$ holds with equality. This example shows why a disagreement should be described as the estimate and true CATE lying on different sides of the threshold, allowing one to equal it, rather than requiring them to be strictly on opposite sides.

Uncertainty Near the Threshold

A point estimate just above cost and one far above cost lead to the same plug-in assignment, but they need not support equally stable decisions. If an estimate has substantial uncertainty, plausible CATE values may fall on both sides of the threshold. A confidence interval for a pointwise CATE can help describe that uncertainty, provided it is interpreted as an interval procedure for the conditional effect and not as a probability statement about a fixed parameter.

Worked Example: An Interval That Crosses Cost

Suppose a hypothetical estimated CATE is $5.2$ utility points with standard error $1.4$ points, and treatment cost is $4.0$ points. Using a normal-approximation 95% pointwise interval, the calculation is $$ 5.2\pm1.96(1.4)=5.2\pm2.744, $$ which gives the interval $[2.456,\,7.944]$ utility points. The plug-in rule treats because $5.2>4.0$, but the interval includes values below and above cost. It therefore indicates uncertainty about which side of the threshold the true CATE occupies; it does not make the plug-in assignment certain.

Under repeated-sampling conditions supporting this approximation, the interval procedure has 95% coverage for the pointwise CATE. It is not correct to say that there is a 95% probability that this fixed CATE lies in this realized interval. Nor does the interval alone specify a new treatment rule: a decision-maker would need an explicit criterion for how uncertainty, consequences, and possibly risk aversion enter the decision.

Common Mistakes and Practical Interpretation

Comparing only with zero. A positive estimated effect does not automatically justify treatment when cost is positive. Compare $\widehat{\tau}(x)$ with $c(x)$.
Mixing units. A CATE of 5 outcome points cannot be directly compared with a cost of 100 dollars unless both have been expressed on a common decision scale.
Treating an estimate as an individual effect. A CATE is a conditional mean effect, not a known effect for a particular person.
Ignoring the tie convention. State whether equality leads to treatment or withholding. A disagreement can occur when the estimate equals the threshold if the true CATE is on the other side.
Equating assignment with certainty. A plug-in rule supplies an action based on a point estimate; it does not remove estimation uncertainty or guarantee positive realized outcomes.

The threshold rule provides a transparent baseline for converting estimated heterogeneity into action. Its most important practical implication is that effect-estimation accuracy should be considered in relation to costs and decision boundaries, not only through an overall prediction error summary. Whether a more complex rule is appropriate depends on additional constraints and on how the resulting policy is assessed.

Check Your Understanding

Answer each question using the treatment threshold and tie convention stated in it.

A program has cost 2.5 outcome units and an estimated CATE of 2.1 units. What does the strict plug-in rule assign, and what is the estimated net gain?
Why must treatment cost and the CATE be expressed on a common scale before they can be compared?
With a strict rule that treats only when $\widehat{\tau}(x)>c(x)$, explain how a true CATE above cost can still lead to withholding treatment.
If the estimated CATE equals cost, can the estimate and true CATE disagree in their treatment assignment under a rule that treats at a tie? Give a numerical illustration.
Why does the pointwise CATE describe a conditional average effect rather than a known treatment effect for each person?