Deriving the Transport Formula by Target Standardization

Why the Target Covariate Distribution Matters

A treatment effect can vary across covariate groups. If those groups occur in different proportions in a study and in the population of interest, the study’s overall average effect need not equal the target population’s average effect—even when the effect within each group is the same in both populations. The target average must give each group the weight it has in the target.

This tutorial derives that standardization formula. The key step is to average the source population’s conditional treatment effects over the target covariate distribution. The result concerns the average treatment effect on an additive mean-difference scale; it does not require the source and target populations to have the same outcome levels.

Definition: Let $A$ be a binary treatment, with values $1$ and $0$, and let $Y(1)$ and $Y(0)$ be the potential outcomes under those treatment levels. Let $S=1$ indicate membership in the source population and $S=0$ membership in the target population. The target average treatment effect is $$ \tau_T = E\{Y(1)-Y(0)\mid S=0\}. $$ For covariates $X$, the conditional treatment effect in population $s$ is $$ \tau_s(x)=E\{Y(1)-Y(0)\mid X=x,S=s\}. $$

For a binary outcome, this definition gives an average risk difference. For a continuous outcome, it gives a difference in mean outcomes. The units are the units of $Y$, and the sign depends on the order $Y(1)-Y(0)$. The target effect is an average over people in the target population, not an average over people in the source study.

Assumptions Behind the Formula

The derivation uses two distinct links. First, source data must identify the source conditional treatment effect. Second, a substantive assumption must connect that conditional effect to the target population. Neither link follows merely from having data on covariates in both populations.

Assumptions:

Consistency and a well-defined treatment. For a person receiving $A=a$, the observed outcome equals $Y(a)$. The treatment versions being compared must be sufficiently well defined for this equality to be meaningful.
Conditional exchangeability in the source. For $a\in\{0,1\}$, $Y(a)\perp A\mid X,S=1$. Conditional on the measured covariates, source treatment assignment must not be associated with the potential outcomes.
Source treatment positivity. For covariate values represented in the target, both treatment levels occur with positive probability in the source: $P(A=a\mid X=x,S=1)>0$ for each $a$.
Conditional effect invariance. For target-relevant values of $x$, $\tau_0(x)=\tau_1(x)$. The average treatment contrast at a given $x$ is the same in the two populations.
Covariate alignment and target distribution. $X$ has the same meaning and measurement in both populations, and the target distribution of $X$ is available for averaging.

Conditional exchangeability and positivity make the source conditional effect identifiable from observed source outcomes. Conditional effect invariance then says that this identified contrast is also the target conditional contrast. The invariance assumption is generally substantive: observed source data alone cannot verify what the treatment effect would have been in the target population.

Deriving Target Standardization

Condition on the target population’s covariates. The law of iterated expectations expresses the target average effect as an average of target conditional effects. Conditional effect invariance replaces each target conditional effect with the corresponding source conditional effect. Finally, source exchangeability and consistency identify that source contrast using observed outcomes.

\begin{aligned} \tau_T &=E\{Y(1)-Y(0)\mid S=0\}\\ &=\int E\{Y(1)-Y(0)\mid X=x,S=0\}\,dF_T(x)\\ &=\int E\{Y(1)-Y(0)\mid X=x,S=1\}\,dF_T(x)\\ &=\int \left[ E(Y\mid A=1,X=x,S=1) - E(Y\mid A=0,X=x,S=1) \right]\,dF_T(x), \end{aligned}

Here $F_T$ is the distribution of $X$ in the target population. The second line is the iterated-expectations step; the third uses conditional effect invariance; the last uses source exchangeability and consistency. In a discrete covariate setting, the integral is a weighted sum:

\tau_T = \sum_x \left[ E(Y\mid A=1,X=x,S=1) - E(Y\mid A=0,X=x,S=1) \right]P(X=x\mid S=0).

This is target standardization: calculate a treatment contrast within each covariate value using source data, then average those contrasts using target-population weights. It is important that the final weights come from the target distribution, not the source distribution. Also, the formula transports a contrast. Conditional effect invariance alone does not assert that either potential-outcome mean separately is equal across populations.

Worked Example: Standardizing a Risk Difference

Suppose a hypothetical source study compares a treatment with control and records a binary outcome. Let $X$ indicate a lower-risk or higher-risk covariate group. Assume treatment is randomized within each group, both treatment levels occur in both groups, and the source includes both groups that appear in the target. These design facts support source exchangeability and positivity for this example. Also assume, substantively, that the risk difference within each group is invariant across source and target populations.

The hypothetical source risks and target group proportions are:

Covariate group	Source risk if $A=0$	Source risk if $A=1$	Target proportion
Lower risk	0.10	0.16	0.75
Higher risk	0.30	0.50	0.25

The source conditional risk differences are $0.16-0.10=0.06$ in the lower-risk group and $0.50-0.30=0.20$ in the higher-risk group. Standardizing these effects to the target proportions gives

\tau_T=(0.75)(0.06)+(0.25)(0.20) =0.045+0.050 =0.095.

The estimated target average effect represented by these hypothetical values is a risk increase of 0.095, or 9.5 percentage points, under treatment versus control. The interpretation depends on the stated conditional effect invariance assumption; the source data and target proportions do not establish that assumption by themselves.

For comparison, if the source population were half lower-risk and half higher-risk, its marginal risk difference would be $(0.50)(0.06)+(0.50)(0.20)=0.13$, or 13 percentage points. That differs from the target effect because the two populations use different weights for the conditional effects.

What Must—and Need Not—Be Invariant

Conditional effect invariance is weaker than equality of the populations’ conditional outcome means. For example, target patients could have higher baseline outcome levels than source patients within every covariate group, yet still have the same treatment-minus-control contrast within each group. The standardization formula needs invariance of the contrast, not invariance of those levels.

This distinction also clarifies why one must define the effect scale. The formula above averages differences in conditional means. If the scientific target is a risk difference, the within-group contrast is a difference in risks. The expression should not be silently changed to a different effect scale: invariance of one kind of contrast does not by itself establish invariance of another.

Worked Example: Weighting Effects Across Age Groups

Consider a hypothetical study of time to symptom resolution, measured in days. Suppose the source conditional mean outcomes are as follows, and assume conditional exchangeability within age group and positive representation of both treatment levels in every listed group. Each target age group also appears in the source. For transport, assume the treatment effect in each age group is invariant across populations.

Age group	Mean days if $A=0$	Mean days if $A=1$	Target proportion
20–39 years	18	16	0.20
40–59 years	20	14	0.50
60 years or older	24	12	0.30

The source conditional effects, in treated-minus-control order, are $16-18=-2$, $14-20=-6$, and $12-24=-12$ days. The target-standardized effect is

\tau_T=(0.20)(-2)+(0.50)(-6)+(0.30)(-12) =-0.40-3.00-3.60 =-7.00\text{ days}.

Under the assumptions, treatment reduces mean symptom-resolution time by 7 days on average in the target population. The negative sign indicates fewer days under treatment. If the source had different age proportions, averaging with those source proportions would answer a source-population question instead.

Continuous Covariates and the Target Distribution

When $X$ is continuous, standardization averages the conditional effect function over the full target distribution. In general, knowing only the target mean of $X$ is not enough: nonlinear effect functions can depend on other features of that distribution. A mean alone is sufficient in the next example only because the conditional effect is linear.

Worked Example: A Linear Effect Function and Target Mean Age

Suppose a hypothetical source population has age $X$, measured in years, and outcome $Y$, measured as days until symptom resolution. Over ages 20 through 60, assume the source conditional mean outcomes are $E(Y\mid A=0,X=x,S=1)=10+0.05x$ days and $E(Y\mid A=1,X=x,S=1)=8-0.05x$ days. Assume consistency, source conditional exchangeability and treatment positivity over the target’s age range, and conditional effect invariance. Suppose the target age distribution is supported within this range and has mean 40 years.

The source conditional effect is the treated mean minus the control mean:

\tau_1(x)=(8-0.05x)-(10+0.05x)=-2-0.10x\text{ days}.

By conditional effect invariance this is also the target conditional effect. Averaging over the target age distribution gives

\begin{aligned} \tau_T &=E(-2-0.10X\mid S=0)\\ &=-2-0.10E(X\mid S=0)\\ &=-2-0.10(40)\\ &=-6.00\text{ days}. \end{aligned}

Thus the target average treatment effect is a reduction of 6 days in mean resolution time. The calculation uses the target mean age because the conditional effect is linear in age. If the conditional effect were nonlinear, the full target age distribution—or enough information to evaluate the expectation of that nonlinear function—would be needed.

Derivation as a Practical Sequence

The algebra separates the conceptual target from the information used to identify it. Keeping those pieces distinct helps prevent an unsupported transport claim.

Define the target contrast.
Specify the treatment levels, outcome, population, and effect scale. Here the target is $E\{Y(1)-Y(0)\mid S=0\}$.

Condition on covariates.
Express the target average as the target-distribution-weighted average of target conditional effects.

Link target and source effects.
Apply conditional effect invariance for the covariate values that occur in the target.

Identify source conditional effects.
Use source exchangeability, consistency, and treatment positivity to represent each effect as a difference in observed source conditional means.

Average with target weights.
Integrate or sum the source conditional contrasts using the target covariate distribution.

Common Mistakes and Why the Formula Matters

Using source weights for a target estimand. The wrong reasoning is that an overall source effect is automatically the effect in the target if conditional effects are invariant. The correction is to average the conditional effects with $P(X=x\mid S=0)$, the target distribution.
Assuming invariance of outcome levels. The formula does not require $E\{Y(a)\mid X,S=1\}=E\{Y(a)\mid X,S=0\}$ separately for each treatment level. It requires equality of the conditional treatment contrasts.
Ignoring support. If a covariate value occurs in the target but not in the source, the source conditional effect there cannot be identified from source observations without additional assumptions. Likewise, source treatment positivity is needed to compare both treatment levels within relevant covariate values.
Treating invariance as a consequence of standardization. Standardization is the averaging step; it does not prove that conditional effects are transportable. That link must be justified by subject-matter knowledge and the choice of covariates.

The formula matters because it states precisely how a conditional causal result can answer a population-level question under a change in covariate composition. It also makes the limits of that answer visible: the source must identify the conditional effects, the target distribution must be known, and the conditional effect invariance assumption must be credible for the target-relevant covariate values.

Key takeaway: Under source conditional exchangeability and positivity, consistency, and conditional effect invariance, the target average treatment effect is the source conditional treatment effect averaged over the target covariate distribution.

Check Your Understanding

Use the definitions and derivation above to answer the following.

In the discrete transport formula, why are the weights $P(X=x\mid S=0)$ rather than $P(X=x\mid S=1)$?
Suppose source conditional risk differences are 0.04 and 0.16 in two groups, and the target proportions are 0.60 and 0.40. What is the target-standardized risk difference?
Does conditional effect invariance require the source and target control-group means to be equal within each covariate value? Explain.
Why is source treatment positivity needed for covariate values represented in the target?
If the conditional effect is a nonlinear function of a continuous covariate, why might the target mean of that covariate be insufficient for standardization?