Quantile Treatment Effects and Individual Effect Distributions

Why Quantiles Describe More Than Average Effects

An average treatment effect can conceal how treatment shifts different parts of an outcome distribution. A program might leave the lower tail nearly unchanged while substantially raising outcomes in the upper tail; a single mean difference would not reveal that pattern. Quantile treatment effects (QTEs) compare selected locations in the two marginal potential-outcome distributions.

That comparison is useful, but it has a precise interpretation: it is a difference between two population quantiles, not automatically the effect for a particular person or the quantile of effects across people. The distinction matters because the marginal distributions do not, by themselves, reveal which treated potential outcome belongs together with which untreated potential outcome for each individual.

Definition: Let $Y(1)$ and $Y(0)$ be the potential outcomes under treatment and control, respectively. For $d\in\{0,1\}$, let $F_d(y)=P(Y(d)\leq y)$, and define the generalized quantile function $Q_d(\tau)=\inf\{y:F_d(y)\geq\tau\}$, for $0<\tau<1$. The unconditional quantile treatment effect at quantile index $\tau$ is $$ \operatorname{QTE}(\tau)=Q_1(\tau)-Q_0(\tau). $$ It compares the $\tau$-quantile of the population’s treated potential-outcome distribution with the $\tau$-quantile of its control potential-outcome distribution.

“Unconditional” means that these are the marginal distributions for the target population, not distributions restricted to a particular covariate value. The subtraction is in the units of the outcome. For example, a QTE of 4 at the median means the treated and control marginal medians differ by 4 outcome units; it does not mean every person, or the median person, gains 4 units.

Identification Before Calculation

Only one potential outcome is observed for each unit. If $D$ denotes treatment assignment, consistency says $Y=Y(D)$: the observed outcome is the potential outcome corresponding to the treatment actually received. In a randomized experiment, assignment is independent of the potential outcomes, so the observed outcome distribution in each assignment arm identifies the corresponding marginal potential-outcome distribution. The QTE can then be calculated by taking the two marginal quantiles and subtracting.

For observational data, a standard identification route assumes conditional exchangeability, $Y(d)\perp D\mid X$, and positivity, $0<P(D=d\mid X=x)$ for relevant covariate values and both treatment states. Together with consistency, these conditions identify each marginal distribution by standardizing over the target population’s covariate distribution:

F_d(y)=\int P(Y\leq y\mid D=d,X=x)\,dF_X(x).

This formula first averages conditional outcome probabilities across the target population and then the marginal quantile is obtained from the resulting distribution. In general, averaging conditional quantiles is not a substitute. The causal interpretation depends on the identification assumptions; the arithmetic definition of a QTE alone does not establish them.

Worked Example: Calculating Unconditional Quantile Effects

Consider a hypothetical randomized trial with an outcome measured in points. Suppose the population potential-outcome distributions are known for this illustration. Under control, 25% of outcomes equal 0, 25% equal 10, 25% equal 20, and 25% equal 30. Under treatment, the corresponding probabilities are 25% at 5, 25% at 10, 25% at 25, and 25% at 40. The randomization and consistency conditions justify reading the two assignment-arm distributions as the marginal potential-outcome distributions.

Using $Q_d(\tau)=\inf\{y:F_d(y)\geq\tau\}$, the lower quantile at $\tau=0.25$ is the first outcome value at which the cumulative probability reaches 0.25. Thus $Q_0(0.25)=0$ and $Q_1(0.25)=5$, giving

\operatorname{QTE}(0.25)=5-0=5\text{ points}.

At $\tau=0.50$, the cumulative probability first reaches 0.50 at 10 in both distributions, so $\operatorname{QTE}(0.50)=10-10=0$ points. At $\tau=0.75$, the quantiles are 20 under control and 25 under treatment, giving $\operatorname{QTE}(0.75)=25-20=5$ points. The distributional comparison therefore shows positive differences at the first and third quartiles, but no difference at the median. These are comparisons of marginal quantiles, not claims about the gains of individuals at those ranks.

Why a QTE Is Not Generally an Individual-Effect Quantile

The individual treatment effect is $\Delta=Y(1)-Y(0)$. Its distribution depends on the joint distribution of $Y(1)$ and $Y(0)$: to calculate each person’s difference, we need to know how their two potential outcomes pair up. The two marginal distributions determine the QTE, but they do not generally determine this pairing. Consequently, the QTE need not equal a quantile of $\Delta$.

The mean is a special case. Provided the expectations exist, linearity gives $E[\Delta]=E[Y(1)]-E[Y(0)]$, so the average treatment effect is the difference in means, regardless of the pairing. Quantiles do not share this property: taking a difference of two marginal quantiles is not generally the same as taking a quantile of paired differences.

Worked Example: Same Marginals, Different Individual-Effect Distributions

Suppose a hypothetical population contains two equally common types. Every person’s control potential outcome is either 0 or 10, each with probability 0.5, and every person’s treated potential outcome is either 10 or 20, each with probability 0.5. The marginal distributions are fixed, so at $\tau=0.5$, $Q_0(0.5)=0$ and $Q_1(0.5)=10$. Under the stated inverse-CDF convention, the median QTE is $10-0=10$.

Now consider two possible pairings of those same marginal outcomes. In the first, pair $(Y(0),Y(1))=(0,10)$ and $(10,20)$, one for each type. The individual effects are 10 and 10, so their median is 10. In the second pairing, use $(0,20)$ and $(10,10)$. The individual effects are 20 and 0. Their distribution puts probability 0.5 at each value, so its generalized median is $Q_\Delta(0.5)=0$. The marginal QTE remains 10, but the individual-effect median is now 0.

Both pairings have the same potential-outcome marginals and hence the same QTEs. They imply different distributions of individual effects. This shows why marginal distributional comparisons alone cannot identify the individual-effect distribution. In both pairings the average effect is 10: the first has mean $(10+10)/2=10$, and the second has mean $(20+0)/2=10$.

Conditional Quantile Contrasts Are Different Targets

A conditional quantile contrast at covariate value $x$ compares $Q_{Y\mid D=1,X=x}(\tau)$ with $Q_{Y\mid D=0,X=x}(\tau)$. Under suitable conditional identification assumptions, it can describe a treatment-related contrast in the outcome distributions within that covariate stratum. But it is not the unconditional QTE: the latter compares quantiles after combining outcomes across the target population’s covariate distribution.

Quantiles are nonlinear summaries. In general, the quantile of a mixture of distributions is not the mixture, or average, of the component quantiles. Thus conditional contrasts at each $x$ do not usually aggregate into the unconditional QTE by taking their average. Nor is a conditional quantile contrast automatically the quantile of individual effects among people with that $x$; that interpretation also concerns a joint distribution of potential outcomes.

Worked Example: A Conditional Contrast Does Not Aggregate to the Marginal QTE

Suppose a hypothetical target population has two equally common strata, A and B. Within each stratum, outcomes are constant under each treatment state: in A, $Y(0)=0$ and $Y(1)=100$; in B, $Y(0)=100$ and $Y(1)=101$. These values describe the stratum-specific potential outcomes, so the example does not require inferring them from an observed nonrandomized comparison.

Within A, the conditional quantile contrast is $100-0=100$ outcome units at every $0<\tau<1$. Within B, it is $101-100=1$ outcome unit at every such $\tau$. The equally weighted average of those two conditional contrasts is $(100+1)/2=50.5$ units.

For the unconditional distributions, the control outcomes are 0 and 100 with probability 0.5 each, while the treated outcomes are 100 and 101 with probability 0.5 each. Using the generalized quantile definition, at $\tau=0.5$, $Q_0(0.5)=0$ and $Q_1(0.5)=100$, so the unconditional median QTE is $100-0=100$ units, not 50.5. The conditional contrasts and their average do not recover this marginal quantile contrast.

Rank Invariance and Individual-Effect Interpretations

A stronger assumption can supply the missing pairing. Under rank invariance, each individual has the same rank in the treated and control potential-outcome distributions. In a continuous-outcome formulation, this means there is a common $U\sim\mathrm{Uniform}(0,1)$ such that $Y(0)=Q_0(U)$ and $Y(1)=Q_1(U)$. This assumption specifies a particular joint distribution from the two marginals: individuals at rank $U$ have both potential outcomes at that rank.

Under rank invariance, the individual effect is $g(U)$, where $g(u)=Q_1(u)-Q_0(u)$. The distribution of individual effects is therefore the distribution of $g(U)$. For the QTE at $\tau$ to equal the $\tau$-quantile of individual effects at every rank, a useful sufficient condition is that $g$ is nondecreasing. Then applying the nondecreasing function $g$ to a uniform rank preserves the rank ordering of effects, and the effect quantile at $\tau$ is $g(\tau)$.

g\text{ nondecreasing} \quad\Longrightarrow\quad Q_{\Delta}(\tau)=g(\tau)=Q_1(\tau)-Q_0(\tau) \quad\text{for every }0<\tau<1.

Rank invariance alone does not guarantee this equality: $g$ may be nonmonotone, so the ordering of individual effects can differ from the ordering of ranks. Also, monotonicity of $g$ is sufficient for equality at every rank, not necessary for equality at one particular rank. A coincidence at a single $\tau$ does not establish monotonicity or identify the full individual-effect distribution.

Worked Example: Equality at One Rank Without Monotone Effects

Let $U\sim\mathrm{Uniform}(0,1)$, and suppose a hypothetical rank-invariant population has $Q_0(u)=10u$ and $Q_1(u)=10u+\sin(2\pi u)$. These are valid increasing quantile functions: the derivative of $Q_0$ is 10, and the derivative of $Q_1$ is $10+2\pi\cos(2\pi u)$, which is at least $10-2\pi\approx3.717>0$. Under rank invariance, the individual effect is

\Delta=g(U)=\sin(2\pi U).

At $\tau=0.5$, the QTE is $Q_1(0.5)-Q_0(0.5)=5-5=0$. The effect distribution is symmetric about zero: the transformation $U\mapsto 1-U$ preserves the uniform distribution and changes $\sin(2\pi U)$ to $-\sin(2\pi U)$. Moreover, $\sin(2\pi U)\leq0$ on half of the unit interval and $\sin(2\pi U)\geq0$ on the other half, so zero is a median of the individual effects; under the generalized quantile convention, $Q_\Delta(0.5)=0$.

Thus the QTE and the individual-effect median coincide at this rank. Yet $g(u)=\sin(2\pi u)$ is not nondecreasing: for instance, $g(0.25)=1$ and $g(0.75)=-1$. This example illustrates the precise scope of the monotonicity condition: it is sufficient for equality across all ranks, but equality at one rank can occur without it.

Common Mistakes and Practical Meaning

Calling a QTE the effect on a particular person or rank. The QTE compares marginal quantiles. Without extra assumptions on the joint potential outcomes, it is not an individual-effect quantile.
Treating the difference in quantiles as the quantile of differences. Quantiles do not generally commute with subtraction. The pairing example shows that the individual-effect median can vary while the marginal QTE remains fixed.
Averaging conditional quantile effects to get an unconditional QTE. The quantile of a mixture is generally not the average of its component quantiles. Construct the target-population marginal distributions before taking their quantiles.
Assuming rank invariance makes every QTE an individual-effect quantile. Rank invariance supplies a common rank pairing, but a nonmonotone effect function can reorder individuals by effect. Nondecreasing $g$ is sufficient for equality at every rank; it is not required for an isolated equality.

QTEs are valuable when the question concerns distributional shifts rather than only changes in the mean. Their interpretation should stay tied to that target: a QTE reports how two marginal outcome distributions differ at a chosen quantile. An individual-effect distribution is a distinct object, requiring information or assumptions about how each person’s two potential outcomes are jointly related.

Check Your Understanding

Answer each question using the definitions and distinctions developed above.

In one sentence, define the unconditional QTE at quantile index $\tau$, and state its units.
Why can two populations with identical marginal potential-outcome distributions have different individual-effect distributions?
What assumptions allow observed treatment and control outcome distributions to identify the marginal potential-outcome distributions in a randomized experiment?
Why is an average of conditional quantile contrasts generally not the unconditional QTE?
Under rank invariance, what additional condition on $g(u)=Q_1(u)-Q_0(u)$ is sufficient for the QTE to equal the individual-effect quantile at every rank? Is it necessary for equality at one rank?