Why Work Backward?
With two treatment decisions, a fixed regime specifies a first treatment \(a_1\) and a second treatment \(a_2\). A baseline covariate \(W\) is measured before the first decision, an intermediate covariate \(L\) is measured between decisions, and \(Y\) is the final outcome. The goal is the mean outcome that would be observed if everyone followed the regime \((a_1,a_2)\).
Backward iterated regression expresses that mean through two conditional expectations. First, estimate the outcome mean at the terminal treatment decision. Then average that regression over the distribution of \(L\) among people with first treatment \(a_1\), separately within baseline covariate levels. Finally, average over the baseline distribution in the target population. The order matters: \(L\) is affected by the first treatment, so its distribution in the second step must correspond to that treatment.
The definition of \(Q_1\) uses the observed conditional distribution of \(L\) given \(A_1=a_1,W=w\). Thus it averages the terminal regression over the intermediate-covariate values that occur, in the observed data, among people who received the regime’s first treatment. In a discrete setting, this is a weighted sum; in a continuous setting, it is a conditional integral.
Identification Conditions and the Backward-Recursion Result
Write \(Y^{a_1,a_2}\) for the outcome under the two specified treatments. The following conditions give the causal interpretation of \(\psi(a_1,a_2)\). Consistency means that a person who receives \(A_1=a_1,A_2=a_2\) has observed outcome \(Y=Y^{a_1,a_2}\); the intermediate covariate under first treatment \(a_1\) is observed as well. Assume no interference between individuals.
- Sequential exchangeability at the first decision: \(Y^{a_1,a_2}\) is independent of \(A_1\) conditional on \(W\).
- Sequential exchangeability at the second decision: among histories with \(A_1=a_1\), \(Y^{a_1,a_2}\) is independent of \(A_2\) conditional on \(W,L\).
- Positivity: \(P(A_1=a_1\mid W=w)>0\) for baseline values in the target population, and \(P(A_2=a_2\mid A_1=a_1,L=l,W=w)>0\) for histories occurring with positive probability under \(A_1=a_1\).
- Integrability: the conditional expectations exist and \(E[|Y^{a_1,a_2}|]<\infty\).
The exchangeability conditions are statements about the counterfactual outcome, not statements that the intermediate covariate’s counterfactual distribution is identified. The derivation below therefore uses the second-decision condition to identify the terminal regression, and uses the first-decision condition only after averaging over \(L\).
Proof. For a regime-relevant history \((w,l)\), consistency identifies the observed outcome with the counterfactual among individuals receiving both specified treatments. The second-decision exchangeability condition then removes conditioning on \(A_2=a_2\):
Average this equality over the observed conditional distribution of \(L\) given \(A_1=a_1,W=w\). The law of iterated expectations gives
The last equality follows from first-decision exchangeability. Averaging over the marginal distribution of \(W\) proves the claimed population mean identity. This argument does not assert that the distribution of \(L^{a_1}\) conditional on \(W\) equals the observed distribution of \(L\) among those with \(A_1=a_1\); it does not need that assertion. \(\square\)
Calculating the Recursion
For discrete \(L\), the conditional expectation in \(Q_1\) can be written explicitly:
For continuous \(L\), replace the sum by an integral with respect to the conditional distribution of \(L\mid A_1=a_1,W=w\). In either case, the outer step averages \(Q_1(a_1,W)\) over the target population’s baseline distribution. The following examples show each operation in concrete terms.
Worked Example: Binary Intermediate Covariate
Suppose there is no baseline covariate. The target regime is \(A_1=1,A_2=0\), and \(L\) is binary. In the observed data, among people with \(A_1=1\), \(P(L=1\mid A_1=1)=0.40\). The fitted terminal regression gives \(Q_2(1,0,0)=0.20\) and \(Q_2(1,1,0)=0.60\). Calculate the backward-regression mean.
The probability of \(L=0\) in this group is \(1-0.40=0.60\). Thus the earlier regression is
There is no \(W\) over which to average, so \(\psi(1,0)=0.36\). The answer is not the unweighted average \((0.20+0.60)/2=0.40\): the two values of \(L\) do not occur equally often under the relevant first treatment.
Worked Example: Standardizing Over Baseline Covariates
Now let \(W\) be binary, with \(P(W=1)=0.40\), and consider regime \((1,0)\). Suppose the fitted terminal regressions and the conditional distribution of \(L\) among those with \(A_1=1\) are:
| \(W\) | \(P(L=1\mid A_1=1,W)\) | \(Q_2(1,0,0,W)\) | \(Q_2(1,1,0,W)\) |
|---|---|---|---|
| 0 | 0.25 | 0.10 | 0.50 |
| 1 | 0.60 | 0.30 | 0.70 |
For \(W=0\), the probability of \(L=0\) is \(0.75\), so
For \(W=1\), the probability of \(L=0\) is \(0.40\), giving
Finally, average these two earlier-regression values using the baseline distribution, not the distribution of \(W\) among people who received \(A_1=1\):
The result is \(0.336\). If \(Y\) is binary, this is the regime-specific risk; if \(Y\) is continuous, it is the regime-specific mean on the outcome scale.
A Conditioning Pitfall
A common error is to claim that first-decision exchangeability identifies the distribution of \(L^{a_1}\mid W\) with the observed distribution of \(L\mid A_1=a_1,W\). That exchangeability condition concerns \(Y^{a_1,a_2}\), not \(L^{a_1}\). The recursion instead averages \(Q_2\) over the observed \(L\mid A_1=a_1,W\) distribution and then applies first-decision exchangeability to the resulting conditional mean.
Worked Example: Outcome Exchangeability Does Not Identify \(L^{a_1}\)
Let \(U\) be Bernoulli with probability \(0.50\), set \(A_1=U\), and define \(L^1=U\). Let every potential outcome equal zero, \(Y^{a_1,a_2}=0\), and let \(A_2\) be randomized with probability \(0.50\) within each observed history. Both first-treatment levels occur with positive probability. Because every potential outcome is constant, first-decision exchangeability holds, and second-decision exchangeability also holds.
In the population, \(P(L^1=1)=P(U=1)=0.50\). But among people with observed \(A_1=1\), \(U=1\), so consistency gives \(L=1\) for everyone in that group:
The outcome regression for the regime is zero on the observed regime-relevant history, so averaging it over the observed \(L\mid A_1=1\) distribution gives \(Q_1(1)=0\), the correct regime-specific mean. This example shows both why the claimed distributional equality is unjustified and why the backward-recursion proof does not require it.
Implementation, Diagnostics, and Alternatives
In an analysis, backward iterated regression is a sequence of conditional-mean calculations: estimate \(Q_2\) for the specified terminal treatment and relevant histories; average its predictions over the observed \(L\mid A_1=a_1,W\) distribution to form \(Q_1\); then average \(Q_1\) over the empirical baseline distribution. This describes the regression-standardization calculation, not a guarantee that a chosen model is correctly specified.
Model the conditional mean of \(Y\) given \(W,L,A_1,A_2\), then evaluate it at \(A_1=a_1,A_2=a_2\).
For each baseline value, average the terminal predictions using the observed distribution of \(L\) among those with \(A_1=a_1\) and that baseline value.
Average the resulting \(Q_1\) values over the target population’s distribution of \(W\).
Diagnostics should focus on whether the data support these operations. Check the numbers of observations receiving each treatment at relevant histories, and inspect the distribution of fitted terminal means over \(W,L\). Sparse or absent treatment combinations indicate positivity problems; very small treatment probabilities imply that estimates may be sensitive to a few observations. For binary outcomes, predicted means should lie between zero and one. Compare fitted and observed outcome summaries in well-supported history strata, while recognizing that such checks cannot establish exchangeability or prove that a regression model is correct.
Regression standardization and inverse-probability weighting are alternative ways to represent the same identified regime mean under the assumptions above. The regression route models conditional outcome means and averages predictions; weighting instead reweights observations according to treatment probabilities. Regression can be sensitive to outcome-model misspecification, while weighting can be unstable when treatment probabilities are small. A marginal structural model can summarize means across regimes, whereas this recursion calculates a particular regime-specific mean. None of these approaches removes the need for causal identification assumptions or adequate treatment support.
A further limitation is that the displayed calculation assumes the needed conditional means and distributions can be estimated with adequate accuracy. A highly flexible regression may be unstable in sparse histories; an overly restrictive regression may smooth over important interactions between \(W\) and \(L\). The recursion also targets the population represented by the outer distribution of \(W\). Changing that target population changes the final average even when the conditional regressions remain the same.
Check Your Understanding
Answer each question, then compare with the solutions below.
- If \(P(L=1\mid A_1=a_1,W=w)=0.30\), \(Q_2(a_1,0,a_2,w)=0.10\), and \(Q_2(a_1,1,a_2,w)=0.50\), what is \(Q_1(a_1,w)\)?
- If \(Q_1(a_1,0)=0.20\), \(Q_1(a_1,1)=0.50\), and \(P(W=1)=0.25\), what is the marginal regime mean?
- Which exchangeability condition identifies the terminal regression as a conditional mean of \(Y^{a_1,a_2}\) given the first-treatment history and \(L\)?
- Why is it incorrect to use first-decision exchangeability to assert \(L^{a_1}\mid W\) has the same distribution as observed \(L\mid A_1=a_1,W\)?
Solutions. (1) \(Q_1=(0.10)(0.70)+(0.50)(0.30)=0.07+0.15=0.22\). (2) The baseline probabilities are \(0.75\) and \(0.25\), so the mean is \((0.20)(0.75)+(0.50)(0.25)=0.15+0.125=0.275\). (3) Second-decision exchangeability, together with consistency, identifies it on regime-relevant histories. (4) That exchangeability condition concerns the counterfactual outcome, not the counterfactual intermediate covariate; the recursion needs the observed \(L\mid A_1=a_1,W\) distribution instead.