Identifying Contrasts Between Treatment Regimes

From Regime-Specific Means to Causal Contrasts

An identified mean under one treatment strategy is often not the final scientific target. A clinical question usually compares strategies: for example, what is the difference in average outcome if everyone follows one schedule rather than another? The contrast is marginal because it compares outcomes averaged over the target population, not outcomes conditional on a selected covariate value.

Let \(Y\) be the outcome measured after \(K+1\) treatment decisions, \(A_0,\ldots,A_K\), with covariate history \(L_0,\ldots,L_K\). Write \(\bar A_t=(A_0,\ldots,A_t)\) and \(\bar L_t=(L_0,\ldots,L_t)\). A regime \(g\) specifies the treatment at each decision; it may prescribe fixed treatments or let treatment depend on the observed history available at that decision. Write \(Y^g\) for the potential outcome if the regime were followed by everyone.

These are population-level causal estimands. The mean difference is suitable for a quantitative outcome; the risk difference is a difference in probabilities for a binary outcome. Neither definition requires a linear, logistic, or other parametric outcome model.

Identification and the Contrast Formula

To identify each mean from observed data, the treatment strategies must be well-defined, consistency must connect observed outcomes to the potential outcomes under the treatment actually received, and sequential exchangeability must hold at each decision. In words, conditional on the measured history before a decision, treatment assignment must be independent of the relevant future potential outcomes. The specified treatment choices must also have support at the histories where the regime requires them. Assume additionally that \(E[|Y^g|]\) and \(E[|Y^{g'}|]\) are finite; for a binary outcome this is automatic.

Under these conditions, the longitudinal g-formula identifies each regime mean. For a deterministic regime, set \(a_t=g_t(\bar l_t,\bar a_{t-1})\) at each history. With \(m(\bar l_K,\bar a_K)=E[Y\mid \bar L_K=\bar l_K,\bar A_K=\bar a_K]\), the formula is

Here the treatment values in each factor are those prescribed by \(g\), and the integral includes summation over discrete covariates. Thus the covariate histories are averaged according to their distribution under the regime, not according to an arbitrary observed subgroup. For fixed regimes, this is the same identification principle expressed using a particular treatment sequence. The inductive identification result for multiple decision times supplies these regime-specific means; constructing a contrast requires no new outcome model.

Proof. The identification assumptions and g-formula give observed-data functionals \(\mu(g)\) and \(\mu(g')\) equal to the respective counterfactual means. Subtracting these equalities gives \(E[Y^g]-E[Y^{g'}]=\mu(g)-\mu(g')\). If \(Y^g\) is binary, its expectation is the probability that it equals one, so the same identity gives the risk difference. The finite-mean condition ensures both expectations and their difference are defined. \(\square\)

Worked Marginal Contrasts

Derivation. By the law of total expectation, \(E[Y^a]=\sum_x P(X=x)E[Y^a\mid X=x]\). Consistency and conditional exchangeability identify \(E[Y^a\mid X=x]\) as \(E[Y\mid A=a,X=x]\) on supported strata. Subtract the two expressions for \(a=1\) and \(a=0\), and collect terms using the common weights \(P(X=x)\). For binary \(Y\), each conditional expectation is the corresponding risk, proving the displayed identity. \(\square\)

Interpretation, Diagnostics, and Limits

A contrast should always be reported with its orientation, scale, and target population. “The risk difference is 7 percentage points for treatment versus control” specifies more than “the effect is 0.07.” For a binary outcome, a risk ratio is another possible contrast, \(P(Y^g=1)/P(Y^{g'}=1)\), when the denominator is positive. It answers a different question and should not be confused with a risk difference.

The key diagnostic question is whether each regime mean was computed with the history distribution that regime implies. In longitudinal settings, prior treatment can change later covariates, and later treatment depends on those histories. Check that the g-formula uses the appropriate treatment-specific covariate distributions at each time and that the outcome conditional means correspond to the prescribed treatment histories. Also check that the two reported means refer to the same outcome definition, follow-up time, and target population.

• Wrong weights: Averaging both regimes over the covariate distribution observed in one treatment group generally changes the target contrast.
• Conditional is not marginal: A difference between conditional model coefficients is not automatically the population risk or mean difference.
• Identification is not estimation: The g-formula identifies a functional of the observed-data law. Estimating its conditional distributions or means still requires data and appropriate statistical methods.
• Parametric convenience is not required: A correctly specified parametric model can be used in estimation, but the causal contrast itself does not assume a parametric outcome response. Flexible estimation does not remove the need for identification assumptions.

The principal limitations are the assumptions themselves and the information in the data. Unmeasured confounding, ill-defined treatment versions, or absent support for a required treatment choice can invalidate identification. Even with identification, finite samples, model misspecification in an estimator, and uncertainty in estimated covariate distributions can affect results. A contrast is therefore not a guarantee that the strategies are feasible, beneficial for every individual, or identified without assumptions.