Deriving the Two-Time Identification Formula

Why Two Decisions Require Two Stages of Standardization

With treatment at two decision times, the outcome under a fixed treatment sequence depends on more than the people who happened to receive that sequence. The intermediate covariate may be affected by the first treatment and may influence both the second treatment and the outcome. To identify a regime-specific mean, we must average over the intermediate covariate distribution that would arise under the first treatment in the regime.

Let $L_0$ denote baseline covariates, $A_0$ the first treatment, $L_1$ the intermediate covariates measured before the second treatment, $A_1$ the second treatment, and $Y$ the final outcome. Write $L_1^{a_0}$ for the intermediate covariates under first treatment $a_0$, and $Y^{a_0,a_1}$ for the outcome under the treatment sequence $(a_0,a_1)$. The target is $E[Y^{a_0,a_1}]$, for a fixed regime $(a_0,a_1)$.

Definition: The two-time identification formula, also called the two-time g-formula for a fixed treatment regime, is $$ E[Y^{a_0,a_1}] = \int E[Y\mid L_0=l_0,A_0=a_0,L_1=l_1,A_1=a_1]\, dF(l_1\mid l_0,A_0=a_0)\,dF(l_0). $$ For discrete covariates, the integrals are sums. The inner distribution is the distribution of the intermediate covariates among people receiving $a_0$, conditional on baseline covariates; the outer distribution is the baseline covariate distribution in the target population.

This formula is an identification statement, not an assumption-free equality. For the chosen regime, it requires well-defined treatment versions and consistency; sequential exchangeability at both decisions; treatment support for the prescribed choices on the relevant histories; and finite expectations. In particular, $P(A_0=a_0\mid L_0)>0$ for almost every baseline value, and $P(A_1=a_1\mid L_0,A_0=a_0,L_1)>0$ for almost every history that can arise under $a_0$. These support conditions ensure that the observed data contain information about both prescribed decisions along the regime’s histories.

Assumptions Used in the Derivation

Consistency means that, for a person who receives $A_0=a_0$, the observed intermediate covariates equal $L_1^{a_0}$; if that person also receives $A_1=a_1$, the observed outcome equals $Y^{a_0,a_1}$. Sequential exchangeability for this fixed regime can be written as

(L_1^{a_0},Y^{a_0,a_1})\perp A_0\mid L_0, \qquad Y^{a_0,a_1}\perp A_1\mid L_0,A_0=a_0,L_1.

The first condition says that, conditional on baseline covariates, first treatment assignment provides no further information about the potential intermediate covariates and outcome in question. The second says that, among people with the specified first treatment and observed history, second treatment assignment provides no further information about the potential outcome. These are assumptions about assignment conditional on measured history; they do not follow merely from having measured $L_0$ and $L_1$.

Deriving the Two-Time Formula

The key is to condition first on $L_0$ and the potential intermediate covariates under $a_0$. This preserves the distribution of $L_1$ induced by the first treatment, rather than substituting the distribution observed in the whole sample.

Theorem: Suppose consistency holds for the treatment regime $(a_0,a_1)$, the two sequential exchangeability conditions above hold, the required treatment probabilities are positive on the relevant histories, and $E[|Y^{a_0,a_1}|]<\infty$. Then the two-time identification formula holds.

Condition on the regime’s intermediate history.
By iterated expectation, average the conditional mean of $Y^{a_0,a_1}$ given $L_0$ and $L_1^{a_0}$ over the distributions of those variables.

Use exchangeability and consistency at the first decision.
Exchangeability identifies the distribution of $L_1^{a_0}$ given $L_0$ with the observed distribution of $L_1$ among those with $A_0=a_0$. It also lets the conditional potential-outcome mean be evaluated in that first-treatment group.

Use exchangeability and consistency at the second decision.
Within the specified first-treatment history, exchangeability identifies the potential-outcome mean with the observed outcome mean among those with $A_1=a_1$.

In detail, iterated expectation gives

E[Y^{a_0,a_1}] = \int E[Y^{a_0,a_1}\mid L_0=l_0,L_1^{a_0}=l_1]\, dF(l_1\mid L_0=l_0)\,dF(l_0).

By first-decision exchangeability, the conditional distribution of $(L_1^{a_0},Y^{a_0,a_1})$ given $L_0=l_0$ is the same as its distribution given $L_0=l_0,A_0=a_0$. Consistency then makes $L_1^{a_0}=L_1$ in that group. Consequently,

dF(l_1\mid L_0=l_0) \ \text{for }L_1^{a_0} = dF(l_1\mid L_0=l_0,A_0=a_0),

and the conditional potential-outcome mean can be written as the mean of $Y^{a_0,a_1}$ given $L_0=l_0,A_0=a_0,L_1=l_1$. By second-decision exchangeability, this conditional mean equals the same mean with $A_1=a_1$ additionally conditioned on. Positivity makes that conditional comparison available in the observed data. Consistency at the second decision then replaces $Y^{a_0,a_1}$ by $Y$. Substitution yields the displayed formula. The integrability condition permits these conditional expectations and their averaging. $\square$

Worked Calculations

Worked Example: Standardizing Over a Binary Intermediate Covariate

Suppose $L_0$ is binary, with $P(L_0=0)=P(L_0=1)=0.5$. Under $a_0$, suppose $P(L_1=1\mid L_0=0,A_0=a_0)=0.2$ and $P(L_1=1\mid L_0=1,A_0=a_0)=0.8$. For the second treatment $a_1$, suppose the observed conditional outcome means are 2 and 6 when $L_0=0$, for $L_1=0$ and $L_1=1$, respectively; when $L_0=1$, they are 4 and 10.

For $L_0=0$, the standardized conditional mean is $0.8(2)+0.2(6)=1.6+1.2=2.8$. For $L_0=1$, it is $0.2(4)+0.8(10)=0.8+8=8.8$. Averaging over baseline covariates gives

E[Y^{a_0,a_1}] =0.5(2.8)+0.5(8.8) =1.4+4.4 =5.8.

The weights for $L_1$ differ across baseline groups because the intermediate covariate distribution is conditioned on $L_0$ and $A_0=a_0$. Using a single pooled distribution of $L_1$ would generally change the answer.

Worked Example: A One-Stage Simplification

Suppose $L_0$ has only one value, so it can be omitted. Under $a_0$, let $P(L_1=1\mid A_0=a_0)=0.3$. Under $a_1$, let the conditional outcome means be 10 for $L_1=0$ and 20 for $L_1=1$. Then the regime mean is

E[Y^{a_0,a_1}] = P(L_1=0\mid A_0=a_0)(10) +P(L_1=1\mid A_0=a_0)(20) =0.7(10)+0.3(20)=7+6=13.

The number 13 is not the outcome mean for a hypothetical person with an “average” value of $L_1$. It is an average of the two conditional means, weighted by the intermediate covariate distribution under the first treatment.

Worked Example: Why the Observed Sequence Mean Can Mislead

Let $L_1$ be binary with $P(L_1=1\mid A_0=a_0)=0.5$. Suppose $P(A_1=a_1\mid L_1=1,A_0=a_0)=0.9$ and $P(A_1=a_1\mid L_1=0,A_0=a_0)=0.1$. Under $a_1$, let the conditional outcome means be 20 for $L_1=1$ and 0 for $L_1=0$. Assume the sequential exchangeability and consistency conditions hold.

The g-formula gives the regime mean $0.5(20)+0.5(0)=10$. But among those who actually receive $a_1$, the fraction with $L_1=1$ is

P(L_1=1\mid A_0=a_0,A_1=a_1) = \frac{0.9(0.5)}{0.9(0.5)+0.1(0.5)} =\frac{0.45}{0.50}=0.9.

Thus the observed mean among people receiving both $a_0$ and $a_1$ is $0.9(20)+0.1(0)=18$, not 10. The difference arises because the second treatment was more common for $L_1=1$, and $L_1$ predicts the outcome. The regime mean averages over the $L_1$ distribution under $a_0$, not over the selected distribution among people who also happened to receive $a_1$.

An Equivalent Weighted Expectation

The g-formula is a direct standardization: it combines observed conditional outcome means with the covariate distributions along the specified regime. Under the same assumptions, it also has an inverse-probability weighted representation. This representation provides an algebraic check on the formula, not a different identification assumption.

Derived result: Define $p_0(L_0)=P(A_0=a_0\mid L_0)$ and $p_1(L_0,L_1)=P(A_1=a_1\mid L_0,A_0=a_0,L_1)$. If the assumptions for the two-time formula hold and the weighted expectation is finite, then $$ E\left[ \frac{I(A_0=a_0)I(A_1=a_1)Y} {p_0(L_0)p_1(L_0,L_1)} \right] =E[Y^{a_0,a_1}]. $$

To verify this identity, condition on $L_0$, then sum over the treatment and intermediate-covariate histories. Histories with $A_0\ne a_0$ contribute zero. For $A_0=a_0$, the factor $p_0(L_0)$ cancels the probability of that first treatment; for $A_1=a_1$, $p_1(L_0,L_1)$ cancels the probability of the second treatment. The remaining expectation is

\int E[Y\mid l_0,a_0,l_1,a_1]\, dF(l_1\mid l_0,a_0)\,dF(l_0),

which equals $E[Y^{a_0,a_1}]$ by the proved identification formula. This derivation also shows why both treatment probabilities appear in the denominator. In practice, small probabilities can make the weighted representation unstable even when the identifying formula is mathematically valid.

Diagnostics, Comparisons, and Limits

The g-formula relies on estimating the conditional outcome mean and the intermediate-covariate distribution under $A_0=a_0$. A practical analysis should check whether the prescribed treatment levels occur across the relevant histories, whether the fitted conditional distributions are plausible, and whether results change substantially under reasonable alternative model specifications. Sparse histories, extreme estimated treatment probabilities, or predictions outside plausible outcome ranges are warnings that the data or models may not support reliable estimation.

The most tempting alternative is the raw mean among people observed to receive $A_0=a_0,A_1=a_1$. As the third example shows, that mean averages over a treatment-selected distribution of $L_1$, so it need not target the regime mean. A second approach is to fit a conditional outcome model and average its predictions over baseline and intermediate histories under the regime. When that model and the required covariate distribution are correctly specified, this is a way to implement the g-formula. Neither approach removes the need for consistency, sequential exchangeability, and support.

The formula does not correct unmeasured confounding, ill-defined treatment versions, or a lack of treatment support. Nor does it guarantee that estimated means will be accurate: misspecified conditional models and limited data can cause substantial error. The formula identifies a population quantity under assumptions; it does not by itself establish that those assumptions are credible.

Common pitfall: A common error is to condition on both observed treatments and report $E[Y\mid A_0=a_0,A_1=a_1]$ as the regime mean. This implicitly uses the intermediate-history distribution among people selected into $A_1=a_1$. The correction is to standardize the conditional outcome mean over $F(L_1\mid L_0,A_0=a_0)$, then average over $F(L_0)$.

Check Your Understanding

Try each question, then check the solution immediately below it.

In the two-time formula, which distribution supplies the weights for the intermediate covariates?
What does consistency identify when $A_0=a_0$ and $A_1=a_1$?
In the third worked example, why is the observed sequence mean 18 while the regime mean is 10?
What two treatment-support conditions are needed for the fixed regime?

Solutions. (1) The weights are $F(L_1\mid L_0,A_0=a_0)$, averaged over the baseline distribution $F(L_0)$. (2) It identifies the observed intermediate covariates with $L_1^{a_0}$ after the first decision, and the observed outcome with $Y^{a_0,a_1}$ after both prescribed decisions. (3) The people observed to receive $a_1$ are disproportionately those with $L_1=1$, whose conditional outcome mean is higher; the regime instead weights $L_1$ by its distribution under $a_0$. (4) The probability of $A_0=a_0$ must be positive across baseline histories, and the probability of $A_1=a_1$ must be positive across histories arising under $a_0$.