Average Direction Treatment Effect

• ADTE is defined as the difference between counterfactual circular mean directions, using trigonometric moments to capture treatment-induced location shifts.
• It employs inverse-probability and Hájek-type weighting to estimate sine and cosine moments, ensuring proper identification in circular causal inference.
• ADTE distinguishes directional shifts from changes in concentration, emphasizing the importance of defining both the contrast and the target population.

Average Direction Treatment Effect (ADTE) is a causal effect summary designed for settings in which the outcome is circular, such as times of day, compass bearings, orientations, or phases. In that setting, ordinary Euclidean contrasts such as $E\{Y^{(1)}-Y^{(0)}\}$ or $E\{Y^{(1)}\}-E\{Y^{(0)}\}$ are not appropriate, because circular outcomes obey wrap-around geometry: $0$ and $2\pi$ represent the same direction, and naive subtraction or arithmetic averaging can be misleading. The formulation developed in “Causal Inference for Circular Data” replaces linear means by mean resultant directions and defines ADTE as the difference between the mean directions of two counterfactual circular outcomes (Wu, 26 Jul 2025). The broader arXiv literature also uses nearby language for distinct estimands, including average derivative effects for continuous exposures and direct average treatment effects under interference. This suggests that “ADTE” is best treated as a context-dependent acronym whose precise meaning must be fixed by the outcome space and causal model (Hines et al., 2021, Cattaneo et al., 18 Feb 2025).

1. Circular definition and basic geometry

For a circular random variable $\Theta$, the basic summaries are its trigonometric moments. At first order, the cosine and sine components are

$$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$

and they determine the mean resultant length and mean resultant direction,

$$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$

Here $\mu$ is the circular analogue of location, while $\rho$ is the analogue of concentration. In the circular-data formulation, potential outcomes are $\Theta^{(1)}$ and $E\{Y^{(1)}\}-E\{Y^{(0)}\}$0, with corresponding counterfactual moments

$E\{Y^{(1)}\}-E\{Y^{(0)}\}$1

$E\{Y^{(1)}\}-E\{Y^{(0)}\}$2

The ADTE is then defined as

$E\{Y^{(1)}\}-E\{Y^{(0)}\}$3

Equivalently, if

$E\{Y^{(1)}\}-E\{Y^{(0)}\}$4

then ADTE is the difference between the arguments of the treated and control mean resultant vectors. The same paper defines the Average Length Treatment Effect (ALTE) as

$E\{Y^{(1)}\}-E\{Y^{(0)}\}$5

so ADTE and ALTE decompose circular treatment effects into location and concentration components (Wu, 26 Jul 2025).

This construction differs fundamentally from averaging raw angular differences. ADTE is not the average of individual wrapped treatment effects; it is a contrast between two population-level circular centers. That distinction is essential because circular means and circular differences are nonlinear functions of the sine and cosine moments.

2. Potential outcomes, observed data, and identification

The observed data are

$E\{Y^{(1)}\}-E\{Y^{(0)}\}$6

where $E\{Y^{(1)}\}-E\{Y^{(0)}\}$7 is a binary treatment, $E\{Y^{(1)}\}-E\{Y^{(0)}\}$8 are covariates, and $E\{Y^{(1)}\}-E\{Y^{(0)}\}$9 is a circular outcome. The circular ADTE framework assumes consistency,

$0$0

strong ignorability conditional on $0$1, and positivity

$0$2

It also assumes i.i.d. sampling and a condition ensuring that the mean direction is well defined, stated in the paper as $0$3 (Wu, 26 Jul 2025).

Identification proceeds through inverse-probability-weighted moment identities. For measurable functions $0$4 and $0$5, the paper states

$0$6

and

$0$7

Setting $0$8 or $0$9, with $2\pi$0, yields

$2\pi$1

$2\pi$2

ADTE is therefore identified as a deterministic function of four identified counterfactual trigonometric moments. In this respect, the estimand is closer to a functional of weighted first moments than to a Euclidean mean difference.

3. Inverse-probability-weighted estimation

The paper estimates the propensity score by logistic regression,

$2\pi$3

with $2\pi$4 solving

$2\pi$5

The estimated propensity score is

$2\pi$6

Using Horvitz–Thompson-type weights,

$2\pi$7

the paper defines weighted estimators of the counterfactual cosine and sine moments: $2\pi$8 The plug-in ADTE estimator is

$2\pi$9

The paper also introduces Hájek-type normalized weights,

$\Theta$0

with corresponding normalized moment estimators $\Theta$1 and $\Theta$2, and the estimator

$\Theta$3

A distinctive result is that the Horvitz–Thompson and Hájek point estimators of ADTE are identical, because

$\Theta$4

and $\Theta$5 depends on direction, not scale. The paper emphasizes that this identity is specific to ADTE and does not extend to ALTE, since ALTE depends on vector length rather than only vector angle (Wu, 26 Jul 2025).

4. Large-sample theory and empirical behavior

The circular ADTE paper proves asymptotic normality in two stages. First, the estimated propensity-score coefficients satisfy

$\Theta$6

where

$\Theta$7

Second, the weighted trigonometric-moment estimators are asymptotically normal, and the multivariate delta method then yields

$\Theta$8

and

$\Theta$9

for $$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$0 (Wu, 26 Jul 2025).

The Jacobian of the transformation from trigonometric moments to $$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$1 makes the geometry of ADTE explicit. The derivative of the ADTE component with respect to $$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$2 contains terms proportional to

$$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$3

This implies that asymptotic uncertainty for ADTE becomes large when either $$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$4 or $$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$5 is small. In practical terms, when a counterfactual circular distribution is nearly uniform, its mean direction is unstable, and the same is true of the direction-treatment effect.

The simulation study in the same paper examines three wrapped-Cauchy data-generating scenarios with true ADTE equal to $$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$6 and true ALTE equal to $$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$7. For $$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$8, ADTE biases range roughly from $$\alpha \equiv E(\cos \Theta), \qquad \beta \equiv E(\sin \Theta),$$9 to $$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$0, with standard errors around $$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$1 to $$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$2. For $$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$3, biases are essentially zero and standard errors are about $$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$4 to $$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$5. The reported coverage rates are close to $$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$6, and the Horvitz–Thompson and Hájek ADTE estimators coincide numerically, exactly as predicted by the theory.

The empirical illustration studies dispatchers’ sleep timing in Federal Railroad Administration data, with treatment $$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$7 for assistant chief dispatchers and $$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$8 for trick dispatchers. The estimated ADTE is

$$\rho \equiv (\alpha^2+\beta^2)^{1/2}, \qquad \mu \equiv \operatorname{atan2}(\beta,\alpha).$$9

which the paper converts to approximately $\mu$0 minutes. Because the application uses a counterclockwise polar system whereas clock time is conventionally read clockwise, the negative sign corresponds there to later sleep onset for assistant chief dispatchers.

5. Relation to ALTE, derivative effects, and direct effects under interference

Within circular-data causal inference, ADTE is paired with ALTE. The two objects answer different questions. ADTE,

$\mu$1

captures a treatment-induced shift in circular location, whereas ALTE,

$\mu$2

captures a treatment-induced change in concentration. A treatment can therefore alter average direction without changing concentration, or vice versa (Wu, 26 Jul 2025).

Outside circular outcomes, the closest analogue to an “average directional” effect appears in the continuous-exposure literature on average derivative effects. “Parameterising the effect of a continuous exposure using average derivative effects” defines the average derivative effect as

$\mu$3

where $\mu$4, and more generally considers weighted derivative effects

$\mu$5

That paper interprets these as the average causal effect of infinitesimal shifts in exposure, and in a one-dimensional continuous-treatment setting this is the closest formal object to a directional treatment effect. It is, however, a derivative-based estimand for continuous exposure rather than a circular mean-direction contrast (Hines et al., 2021).

A different nearby usage appears in interference settings. “Robust Inference for the Direct Average Treatment Effect with Treatment Assignment Interference” studies

$\mu$6

which is a conditional direct average treatment effect under network interference. This is an own-treatment contrast averaged over the random assignment of others’ treatments. It is “direct” in the interference sense, not “directional” in the circular or derivative sense (Cattaneo et al., 18 Feb 2025).

A third adjacent development is the generalized ATT literature for multi-valued and continuous treatments. “Longitudinal Generalizations of the Average Treatment Effect on the Treated for Multi-valued and Continuous Treatments” defines

$\mu$7

a family of finite-shift modified-treatment-policy effects over treated-like subsets. That paper explicitly does not define an infinitesimal ADTE, but it shows how treated-subgroup shift effects can be formulated for continuous and longitudinal treatments. In that literature, an ADTE-like object would arise only after differentiating a finite-shift policy effect with respect to the shift magnitude, a step that the paper does not develop (Susmann et al., 2024).

6. Terminological ambiguity, target populations, and common misconceptions

The surveyed literature supports a narrow but important caution: “Average Direction Treatment Effect” is not yet a universally standardized acronym. In circular-outcome causal inference it denotes a difference in counterfactual mean directions. In continuous-exposure work, the closest objects are average derivative effects. In network-interference work, related phrasing points instead to a direct average treatment effect. This suggests that any use of “ADTE” should specify at least four ingredients: the treatment type, the outcome geometry, the interference structure, and the averaging measure.

A common misconception is to treat all “average effects” as if they differed only by notation. The ATT-overlap literature shows that average causal estimands are defined jointly by a causal contrast and a target population. In “Average treatment effect on the treated, under lack of positivity,” trimming, truncation, and overlap weighting for ATT are shown to alter the estimand by changing the effective target population rather than merely stabilizing estimation. The paper’s lesson is that one must ask not only “what contrast?” but also “averaged over whom?” (Liu et al., 2023). This point generalizes to ADTE-oriented work: even when the contrast is a mean-direction difference, weighting and overlap remedies can modify the population over which that direction contrast is averaged.

A second misconception is to interpret circular ADTE as the average of unit-level angular differences. The circular-data paper does not define it that way. Its estimand is

$\mu$8

a contrast between two counterfactual circular centers. Because $\mu$9 is a nonlinear functional of the sine and cosine moments, ADTE is fundamentally a population-level geometric contrast.

A third misconception is to treat directional and direct effects as interchangeable. They are not. A derivative-based directional effect for continuous exposure, a mean-direction contrast for circular outcomes, and a direct effect under interference answer different causal questions even when all are “average” treatment effects. The substantive value of ADTE therefore depends on disciplined specification of its mathematical object, not on the acronym alone.