What Does a Regime-Specific Mean Measure?
A longitudinal treatment regime specifies what treatment would be given at each decision time. The population mean under that regime answers a direct question: what would the average outcome be if everyone in the target population followed this treatment rule? The answer is a causal parameter, defined through counterfactual outcomes. It is not, by itself, a formula for estimating that parameter from observed data.
Let \(Y\) be the outcome measured at the end of follow-up, and let \(Y^d\) denote the outcome that would be observed if a person followed regime \(d\). A static regime assigns a fixed treatment sequence, such as \(d=(0,1)\). A dynamic regime assigns treatment using the history available at each decision: \(d_t(H_t)\), where \(H_t\) is the information observed before treatment at time \(t\). For example, a rule might assign treatment at the second decision only if a specified clinical measurement is above a threshold.
The target population matters: the expectation averages over the people represented by the target population, including their baseline characteristics. The same two regimes can therefore have different regime-specific means in different populations. The parameter is also distinct from an estimator, such as a statistic calculated from a sample.
Conditions for a Meaningful Causal Parameter
The definition \(E[Y^d]\) is a counterfactual definition, not an assertion that the outcome under each regime is directly observed. A clear causal interpretation requires a sufficiently well-defined intervention: the regime must specify treatment in a way that makes \(Y^d\) meaningful. When treatment has versions that can affect the outcome differently, the intervention must distinguish those versions or justify treating them as equivalent.
Finite expectation is needed for the mean difference to be finite. A mean ratio additionally requires a nonzero denominator; in many applications, a positive denominator is needed for a useful relative interpretation. If \(Y\) is binary, \(\psi(d)\) is the risk under regime \(d\). If \(Y\) is a cost or a continuous measurement, the mean is on that outcome’s scale.
Defining the parameter does not require assumptions about how treatment was assigned. Estimating it from observational data does. Common identification conditions include consistency (the observed outcome agrees with the potential outcome for the treatment history actually received), sequential exchangeability (treatment at each time is independent of the relevant future potential outcomes conditional on the observed history before that treatment), and positivity (each action specified by the regime has positive probability at histories that occur under the regime). These conditions, together with appropriate handling of loss to follow-up, support identification from observed data. They do not change what \(\psi(d)\) means.
Contrasts on Difference and Ratio Scales
The mean difference preserves the units of \(Y\). If \(Y\) is measured in days, \(\psi(d)-\psi(d')=3\) means that the population average outcome under \(d\) is three days greater than under \(d'\). A mean ratio is unitless when both means use the same units. A ratio of \(1.25\) means that the mean under \(d\) is 1.25 times the mean under \(d'\), provided the denominator gives the ratio a meaningful interpretation.
To verify the identity, finite absolute expectations allow linearity of expectation:
This identity concerns an average of differences. It does not say that every individual experiences the population-average effect, and it does not imply that the individual counterfactual difference is observed for any one person. \(\square\)
Worked Example: Two Static Regimes and an Absolute Contrast
Suppose a target population consists of four equally likely types of people. The table gives their potential outcomes under two static regimes, \(d=(0,0)\) and \(d'=(1,1)\).
| Type | Probability | \(Y^d\) | \(Y^{d'}\) | \(Y^d-Y^{d'}\) |
|---|---|---|---|---|
| 1 | 0.25 | 2 | 3 | -1 |
| 2 | 0.25 | 4 | 5 | -1 |
| 3 | 0.25 | 6 | 8 | -2 |
| 4 | 0.25 | 8 | 12 | -4 |
The regime-specific means are $$ \psi(d)=0.25(2+4+6+8)=5,\qquad \psi(d')=0.25(3+5+8+12)=7. $$ The mean difference for \(d\) versus \(d'\) is \(5-7=-2\) outcome units. Checking by averaging the individual-level differences gives $$ 0.25(-1-1-2-4)=-2. $$ The two calculations agree, as the result requires. The negative sign indicates that the mean under \(d\) is lower than the mean under \(d'\); it does not imply that every person has the same difference.
Baseline Standardization
Regime means can be expressed as averages of conditional regime means over baseline characteristics. This is an identity about the counterfactual distribution, not an identification formula for observed data. Let \(W\) be a baseline variable with finitely many values \(w\), and suppose \(E[|Y^d|]<\infty\). Then:
For the first equality, partition the expectation according to the mutually exclusive values of \(W\). For each value with positive probability, the definition of conditional expectation gives \(E[Y^d\mathbf{1}(W=w)]=E[Y^d\mid W=w]P(W=w)\). Summing over \(w\) yields $$ E[Y^d]=\sum_w E[Y^d\mathbf{1}(W=w)] =\sum_w E[Y^d\mid W=w]P(W=w). $$ Apply the same identity to \(Y^{d'}\), then subtract the two finite sums to obtain the contrast formula. \(\square\)
The weights are the baseline proportions in the target population. They are not generally the proportions in the groups observed to receive each treatment. This distinction matters whenever the regimes’ conditional means vary across baseline groups.
Worked Example: Different Conditional Effects Across Baseline Groups
Suppose 60% of the target population is in group \(W=0\), and 40% is in group \(W=1\). The conditional potential-outcome means are:
| Baseline group | \(P(W=w)\) | \(E[Y^d\mid W=w]\) | \(E[Y^{d'}\mid W=w]\) |
|---|---|---|---|
| \(W=0\) | 0.60 | 10 | 8 |
| \(W=1\) | 0.40 | 16 | 12 |
Standardizing regime \(d\) gives $$ \psi(d)=0.60(10)+0.40(16)=6+6.4=12.4. $$ For \(d'\), $$ \psi(d')=0.60(8)+0.40(12)=4.8+4.8=9.6. $$ The marginal mean difference is \(12.4-9.6=2.8\). The conditional differences are \(10-8=2\) and \(16-12=4\); their population-weighted average is \(0.60(2)+0.40(4)=1.2+1.6=2.8\), confirming the standardization identity.
Ratios, Interpretation, and Alternatives
If both regime means are positive, the mean ratio \(\psi(d)/\psi(d')\) describes their relative size. For a binary outcome, it is a risk ratio. In the preceding example the ratio is \(12.4/9.6\approx1.292\): the mean under \(d\) is about 1.292 times the mean under \(d'\). The absolute difference of 2.8 remains important because it conveys the change in outcome units.
A ratio of marginal means is generally not the average of conditional mean ratios. In the preceding example, the conditional ratios are \(10/8=1.25\) and \(16/12\approx1.333\). Their weighted average is \(0.60(1.25)+0.40(1.333)\approx1.283\), whereas the marginal ratio is about 1.292. These differ because taking a ratio and averaging do not generally commute.
The difference scale is usually more directly interpretable in outcome units and remains defined when a regime mean is zero. The ratio scale can communicate relative change, but becomes undefined if its denominator is zero and can be unstable when that mean is close to zero. Neither scale is universally preferable; the scientific question and outcome scale should guide the choice. An odds ratio is a contrast of odds, not a ratio of means, and should not be substituted for a mean ratio.
Worked Example: Regime-Specific Risks and a Risk Ratio
Let \(Y\) be a binary outcome. Suppose 80% of the target population has \(W=0\) and 20% has \(W=1\). Under regime \(d\), the conditional risks are 0.10 and 0.40; under \(d'\), they are 0.05 and 0.20.
The marginal risks are $$ \psi(d)=0.80(0.10)+0.20(0.40)=0.08+0.08=0.16, $$ and $$ \psi(d')=0.80(0.05)+0.20(0.20)=0.04+0.04=0.08. $$ Therefore, the risk difference is \(0.16-0.08=0.08\), or 8 percentage points, and the risk ratio is \(0.16/0.08=2\). The marginal risk under \(d\) is twice the risk under \(d'\). The conditional risk ratios happen to equal 2 in both groups here, but that need not hold in other examples.
Common Mistakes and Practical Checks
- Confusing an observed group mean with a regime mean. The mean among people who happened to receive a treatment is not automatically \(E[Y^d]\). It may reflect differences in who received each treatment.
- Changing the target population without noticing. Standardization weights must correspond to the population named by the parameter. Using different weights can produce a different marginal mean and contrast.
- Interpreting a mean ratio as an average individual ratio. A ratio of population means is not generally an average of person-specific ratios; the latter may not even be defined.
- Reporting a ratio without checking its denominator. A zero denominator makes the ratio undefined, and a very small denominator can make it unstable.
- Assuming a well-defined parameter is automatically identified. Counterfactual means can be defined even when observational data lack treatment support or confounding control needed to estimate them.
Before interpreting a regime contrast, check that both regimes are explicit, the target population is stated, the outcome has a finite mean, and the contrast scale matches the question. For an observational analysis, also examine whether the data support the regimes’ treatment choices across relevant histories; violations of positivity can undermine estimation even though the target parameter remains well-defined.
Check Your Understanding
Answer each question using the definitions and identities above. Complete solutions follow.
- If \(\psi(d)=18\) and \(\psi(d')=24\), what are the mean difference for \(d\) versus \(d'\) and the mean ratio?
- A target population has two baseline groups with proportions 0.25 and 0.75. The conditional means under \(d\) are 4 and 10. Find \(\psi(d)\).
- Why can an observed mean among people who received regime \(d\) differ from \(\psi(d)\)?
- When is the mean ratio \(\psi(d)/\psi(d')\) undefined, and what additional issue arises when the denominator is close to zero?
Solutions. (1) The difference is \(18-24=-6\) outcome units, and the ratio is \(18/24=0.75\). (2) Standardize using the target-population proportions: \(0.25(4)+0.75(10)=1+7.5=8.5\). (3) Treatment receipt may be associated with baseline or time-varying characteristics that also predict outcomes, so the observed treatment group need not represent the counterfactual outcomes under that regime for the whole target population. (4) The ratio is undefined when \(\psi(d')=0\). If \(\psi(d')\) is close to zero, small changes in either mean can produce large changes in the ratio, making it unstable and difficult to interpret.