Regime-Specific Counterfactual Means

What Does a Regime-Specific Mean Describe?

A longitudinal treatment question concerns more than whether treatment was ever received. It may ask what the population outcome would be if everyone followed one treatment sequence rather than another. The quantity answering that question is a regime-specific counterfactual mean: the expected outcome in a specified population under a specified sequence of treatments.

To define one precisely, specify the treatment opportunities, the sequence imposed at those times, the outcome and its measurement horizon, and the population over which the mean is taken. These choices are part of the estimand, not details to be supplied after analysis.

Definition: Let treatment be assigned at times $t=0,\ldots,K$, and let $\bar a=(a_0,\ldots,a_K)$ denote a specified treatment sequence. For each individual, $Y^{\bar a}$ is the potential outcome that would be observed at the chosen outcome horizon if that individual followed sequence $\bar a$. For a target population $P^*$, the regime-specific mean is $\mu_{P^*}(\bar a)=E_{P^*}[Y^{\bar a}]$, provided the outcome is integrable under that regime. If $Y$ is binary and coded 1 for the event, this mean is the regime-specific risk: $P_{P^*}(Y^{\bar a}=1)$.

The sequence notation makes the intervention explicit. For example, $(0,1,1)$ means no treatment at time 0 followed by treatment at times 1 and 2; it is not the same intervention as $(1,1,0)$, even though both include two treated time points. The outcome $Y^{\bar a}$ is defined under the whole sequence, so it can reflect effects of earlier treatment on later covariates and on response to later treatment.

A potential outcome is a conceptual quantity, not necessarily an observed value for that individual. An individual follows only one realized treatment history, while the other sequence-specific potential outcomes remain counterfactual. The population mean averages the relevant potential outcome over the target population; it is not the average among only those who happened to follow that sequence in the observed data.

Means, Risks, and Contrasts

The word “mean” applies to quantitative outcomes and also to binary outcomes. With a binary outcome coded 0 or 1, the mean equals the probability of observing the event, so it is also called a risk. For a continuous outcome, such as a symptom score at 12 months, the mean is in the outcome’s units. Naming the outcome scale prevents a mean difference from being mistaken for a risk difference.

Definition: For two regimes $\bar a$ and $\bar a'$ in the same target population, the marginal mean difference is $\mu_{P^*}(\bar a)-\mu_{P^*}(\bar a')$. If the outcome is binary, this is the marginal risk difference. The risk ratio is $\mu_{P^*}(\bar a)/\mu_{P^*}(\bar a')$, when the denominator risk is positive. Each contrast compares population quantities under two specified regimes; it is not the distribution of individual-level effects.

“Marginal” means averaged over the target population, including its distribution of baseline characteristics. A conditional mean, such as $E[Y^{\bar a}\mid X=x]$, instead refers to a particular baseline stratum. Conditional comparisons can be informative, but they are not automatically the same estimand as a population-average contrast. In particular, when conditional effects vary across strata, a marginal contrast depends on how those strata are represented in the target population.

Worked Example: Comparing Risks Under Two Sequences

Suppose $Y$ indicates a hospitalization by one year, and the target population is all eligible patients at baseline. The regime-specific risks under two sequences are $P(Y^{(1,1)}=1)=0.18$ and $P(Y^{(0,0)}=1)=0.25$.

The first quantity is an 18% risk under treatment at both opportunities; the second is a 25% risk under no treatment at either opportunity. The risk difference, using the first regime minus the second, is

$$ 0.18-0.25=-0.07. $$

Thus the marginal risk under $(1,1)$ is 7 percentage points lower. The risk ratio is $0.18/0.25=0.72$, or 28% lower relative to the risk under $(0,0)$. These are two different summaries of the same pair of risks. Neither number means that every patient’s hospitalization probability falls by that amount.

Target-Population Averaging

The target population must be held fixed across regimes for a marginal contrast to compare like with like. For example, one might target the population represented by eligible patients at baseline, rather than separately averaging each regime over the different people who actually received it. When a baseline variable $X$ divides the target population into strata, the marginal mean is a weighted average of the stratum-specific means, using the target population’s stratum proportions.

Result: Suppose $X$ takes a finite set of values $x$, with $P_{P^*}(X=x)=p_x$. If $E_{P^*}[|Y^{\bar a}|]<\infty$, then $$ \mu_{P^*}(\bar a)=\sum_x p_x E_{P^*}[Y^{\bar a}\mid X=x]. $$ For two regimes, it follows that $$ \mu_{P^*}(\bar a)-\mu_{P^*}(\bar a') =\sum_x p_x\left\{E_{P^*}[Y^{\bar a}\mid X=x]-E_{P^*}[Y^{\bar a'}\mid X=x]\right\}. $$

Proof. Because the possible values of $X$ form a finite partition, the expectation of the integrable counterfactual outcome can be decomposed across that partition:

E_{P^*}[Y^{\bar a}] =\sum_x E_{P^*}\!\left[Y^{\bar a} I(X=x)\right] =\sum_x P_{P^*}(X=x)E_{P^*}[Y^{\bar a}\mid X=x].

Terms with $P_{P^*}(X=x)=0$ contribute zero and can be omitted, so conditional expectations are needed only on positive-probability strata. Substituting $p_x=P_{P^*}(X=x)$ proves the first identity. Apply that identity separately to $\bar a$ and $\bar a'$, then subtract the finite sums term by term to obtain the second identity. $\square$

This is a decomposition of the estimand, not an identification result: it says how a population mean relates to its stratum-specific means, not how those counterfactual means can be recovered from observed data.

Worked Example: Averaging Stratum-Specific Mean Differences

Suppose the outcome is a symptom score, and the target population is 40% high-risk and 60% low-risk at baseline. Under regime $\bar a$, the stratum-specific means are 8 for high-risk patients and 4 for low-risk patients. Under regime $\bar a'$, they are 6 and 3, respectively.

First calculate each regime’s marginal mean using the same target-population proportions:

\mu_{P^*}(\bar a)=0.40(8)+0.60(4)=3.2+2.4=5.6, \qquad \mu_{P^*}(\bar a')=0.40(6)+0.60(3)=2.4+1.8=4.2.

The marginal mean difference is $5.6-4.2=1.4$ score units. Equivalently, the high-risk stratum difference is $8-6=2$, and the low-risk difference is $4-3=1$. Weighting these differences by the target proportions gives

$$ 0.40(2)+0.60(1)=0.80+0.60=1.40. $$

Both calculations agree. The high-risk contribution is $0.80$, not $0.40$: the stratum weight must be multiplied by its difference. Changing the target population’s proportions could change the marginal contrast even though the two stratum-specific differences stayed the same.

Binary Outcomes and Risks

For a binary potential outcome, the equivalence between mean and risk follows directly from its possible values. No modeling assumption is needed for this identity.

Result: If $Y^{\bar a}\in\{0,1\}$, then $E_{P^*}[Y^{\bar a}]=P_{P^*}(Y^{\bar a}=1)$.

Proof. Since $Y^{\bar a}$ equals 1 on the event $\{Y^{\bar a}=1\}$ and 0 otherwise, it can be written as the indicator $I(Y^{\bar a}=1)$. Therefore

E_{P^*}[Y^{\bar a}] =E_{P^*}[I(Y^{\bar a}=1)] =1\cdot P_{P^*}(Y^{\bar a}=1)+0\cdot P_{P^*}(Y^{\bar a}=0) =P_{P^*}(Y^{\bar a}=1).

This identity is why the mean-difference formula for a binary outcome is a risk difference. If the event is coded differently, or the outcome is not binary, that interpretation may not hold. $\square$

Worked Example: A Sequence-Specific Event Risk

Let $Y$ indicate whether a patient experiences a complication by day 90. Consider $\bar a=(1,0,1)$: treatment at the first and third opportunities, but not the second. Suppose the target-population potential outcome under this sequence is binary and the fraction of its distribution equal to 1 is 0.12.

By the binary-outcome identity, the regime-specific mean is

E_{P^*}[Y^{(1,0,1)}] =1(0.12)+0(0.88)=0.12.

It is therefore also the 90-day risk, 12%, under this exact sequence. A risk under $(1,1,1)$ would be a different quantity; the 12% figure cannot be assigned to it without additional information.

Defining and Interpreting the Estimand Carefully

A regime-specific mean is defined whether or not the available observed data identify it. To give it a causal interpretation, the treatment sequences must be well-defined interventions with sufficiently clear versions, and the potential outcomes must be meaningful for the individuals in the target population. Connecting these counterfactual quantities to observed data additionally requires causal identification assumptions, including consistency, sequential exchangeability, and positivity. Those assumptions are distinct from the definition of the estimand; the earlier material on positivity emphasizes that support must be considered for the particular regime contrast and target population.

A useful specification names four things: the target population, the full treatment sequence, the outcome, and the time horizon. For instance, “the 90-day risk of complication in all eligible patients if everyone followed $(1,0,1)$” is more precise than “the treatment effect.” If the outcome is only defined for people who survive to a later time, the target quantity needs further clarification: an outcome defined for everyone, a composite outcome, and an outcome conditional on survival are not interchangeable estimands.

Name the target population.
State whose counterfactual outcomes are averaged, such as all eligible individuals at baseline.

Write each regime in full.
List the treatment at every opportunity; do not replace a sequence with an ambiguous label such as “treated.”

Specify outcome and horizon.
State what is measured, how it is coded, and when it is measured.

Name the summary and contrast.
Distinguish a mean or risk under one regime from a difference or ratio between regimes.

Common Pitfalls, Diagnostics, and Alternatives

A common mistake is to compare outcomes among people observed to follow different sequences and call that comparison the regime-specific contrast. Those groups may differ in baseline and evolving characteristics, and observed adherence is not itself the definition of a counterfactual regime. Estimation requires a method and assumptions that connect observed data to the target quantities.

A second mistake is to average conditional effects using the wrong population proportions. The decomposition uses the same $p_x$ for both regimes: the proportions in the stated target population. Using one set of proportions for $\bar a$ and another for $\bar a'$ compares quantities averaged over different populations. This can produce a numerical difference that is not the marginal contrast defined here.

A third pitfall is to report a conditional model coefficient as if it were automatically a marginal risk difference or mean difference. A model coefficient may describe a conditional association on a particular scale; the marginal estimand is formed by averaging the regime-specific potential outcomes over the target population and then taking the chosen contrast. The two summaries need not agree, particularly when effects vary across strata or the scale is nonlinear.

Check that each regime is clinically interpretable and that the relevant sequence has support in the population of interest. A risk ratio also requires a positive denominator risk; if that risk is zero, the ratio is undefined even though both risks and their difference can still be defined. When follow-up, outcome coding, or the target population is unclear, the estimand is not fully specified. No estimator can repair an ambiguous target.

Regime-specific means are useful for comparing full longitudinal strategies, while conditional means answer stratum-specific questions and observed-group summaries describe what happened under actual treatment patterns. These alternatives can be relevant for different purposes, but they answer different questions. The mean under a regime is also not an individual treatment effect: it averages a regime-specific counterfactual outcome, whereas an individual effect would compare that individual’s potential outcomes under two regimes.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For binary $Y$, a regime-specific mean is 0.08. What is the corresponding risk?
In a target population, a stratum has proportion 0.25 and a regime-specific mean difference of 4 units. What is its contribution to the marginal mean difference?
Why is a comparison between people observed to follow $(1,1)$ and people observed to follow $(0,0)$ not, by itself, the regime-specific contrast?
Under regime $\bar a$, a binary outcome risk is 0.10; under $\bar a'$, it is 0.20. Give the risk difference in the order $\bar a-\bar a'$, and the risk ratio in the order $\bar a/\bar a'$.

Solutions. (1) The risk is $0.08$, or 8%, because the mean of a binary 0/1 outcome equals its event probability. (2) The contribution is $0.25(4)=1$ outcome unit. (3) The observed groups may differ in characteristics related to treatment history and outcome; the observed comparison is not automatically an average of the two counterfactual outcomes in one common target population. (4) The risk difference is $0.10-0.20=-0.10$, or 10 percentage points lower under $\bar a$; the risk ratio is $0.10/0.20=0.50$.