Positivity for Sustained and Dynamic Regimes

Why Positivity Matters for a Longitudinal Regime

A longitudinal treatment regime specifies what treatment to give at each decision time. A sustained regime might prescribe the same treatment at every visit; a dynamic regime might prescribe treatment according to a patient’s evolving history. To learn about either intervention from observational data, the prescribed treatment must be represented among people with the histories at which the regime would prescribe it.

This requirement is called sequential positivity. It is not enough that each treatment occurs somewhere in the study, or that treatment is common overall. The relevant question is conditional: at each time, among people with a history relevant to the regime, is there a positive chance of receiving the treatment the regime specifies? If not, the data provide no direct comparison for that treatment-history combination.

Definition: Let \(A_t\) be treatment at time \(t\), and let \(H_t=(L_0,A_1,L_1,\ldots,A_{t-1},L_t)\) denote the measured history available just before that treatment. A deterministic regime \(d=(d_1,\ldots,d_K)\) prescribes treatment \(d_t(h)\) at time \(t\) for history \(h\). Sequential positivity for \(d\) requires, for discrete treatment, that \(P(A_t=d_t(H_t)\mid H_t)>0\) almost surely with respect to the distribution of \(H_t\) induced by following the regime; that is, the observational treatment mechanism must support the prescribed action on those histories. For continuous treatment, support is expressed through positive conditional density in a neighborhood of the prescribed value, rather than positive probability at one exact value.

The phrase “histories that can occur when the regime is followed” matters. A history may appear in the observed cohort but be impossible under a particular regime; positivity for that regime need not be checked there. Conversely, a rare history that the regime can produce is relevant even if it is uncommon in the full cohort. The condition concerns treatment choices along the regime’s possible paths, not every imaginable combination of covariates and past treatments.

For a static sustained regime, the prescribed value does not change with history, so the condition is that each prescribed treatment has positive conditional probability at each relevant history. A dynamic regime changes the prescribed value with history, but the logic is the same: compare the observed treatment distribution to the action specified at that history. Sequential positivity is distinct from sequential exchangeability. Exchangeability concerns whether treatment assignment is confounded after conditioning on history; positivity concerns whether the required treatment occurs at all in that history.

Assess Support One Decision at a Time

A useful assessment starts with the regime’s decisions and follows their possible histories in time order. At each decision, identify the history available just before treatment, determine the action the regime prescribes there, and ask whether the observational data include people with that history receiving that action. Do not substitute a marginal treatment percentage for this conditional check.

Write down the decision rule.
State the prescribed treatment at each time, including any history-dependent thresholds or categories.

Trace relevant histories.
List the histories that can arise along the regime’s path, including histories affected by earlier prescribed treatments.

Check conditional treatment support.
Within each relevant history, examine whether the prescribed treatment is observed or, for continuous treatment, locally supported.

Classify the concern.
Separate a genuine absence of treatment options from evidence that is merely sparse or too imprecise to establish support.

This is a conceptual assessment, not a test that can certify positivity from a finite sample. In practice, observed counts, clinical restrictions, study protocols, and the distribution of continuous treatment can all inform whether support is credible. A zero count may reflect a structural rule, or it may occur by chance in a small sample; the data alone do not always distinguish the two.

Worked Example: Support Along a Three-Decision Path

Suppose a hypothetical cohort has 1,000 people. Consider one branch of a dynamic regime. At the first decision, 400 people receive the prescribed treatment. Among those 400, the relevant second-decision history occurs and 200 receive the treatment prescribed there. Among those 200, the relevant third-decision history occurs and 40 receive the prescribed treatment. Assume these are nested groups along this branch, so each denominator is the group that followed the prior prescribed actions and reached the next decision.

The conditional treatment proportions along this branch are \(400/1000=0.40\), \(200/400=0.50\), and \(40/200=0.20\). The observed proportion following all three prescribed decisions on this branch is therefore

0.40\times\frac{200}{400}\times\frac{40}{200} =0.40\times0.50\times0.20 =0.04.

Thus, 4% of the cohort, or 40 of 1,000 people, followed this particular treatment path. Each conditional proportion is positive, so this example does not show a structural zero on the branch. However, the third decision is supported by only 40 people in the full cohort. That may leave limited information about outcomes for this path, especially if the relevant histories are more finely divided than the example shows. The calculation describes observed path frequency; it does not establish that every other history the regime could reach also has support.

Structural Violations and Practical Weakness

A structural positivity violation occurs when the prescribed treatment is unavailable or never assigned at a relevant history in the population. For example, a clinical protocol may forbid a drug above a severity threshold. If a regime prescribes that drug for patients above the threshold, no observational comparison exists for that decision. More data collected under the same restriction cannot supply the missing treatment-history combination.

A practical positivity problem occurs when the prescribed treatment is possible but uncommon in the relevant history. The population probability may be positive, while the study happens to contain few such observations. This is a matter of degree: there is no universal numerical cutoff that separates adequate from inadequate support. Small conditional treatment probabilities make estimates sensitive to a few observations and can require substantial extrapolation.

Worked Example: A Structural Zero in a Severe-History Stratum

In a hypothetical study, 120 patients have a severe-history category at the second treatment decision. A proposed dynamic regime prescribes treatment \(A_2=1\) for this category. The clinical protocol forbids \(A_2=1\) for all such patients, and the data show 0 of 120 receiving it. Therefore,

P(A_2=1\mid H_2=\text{severe})=0

in the study population governed by that protocol. This is a structural positivity violation for the proposed rule, provided the severe history can occur under the regime. Suppose, meanwhile, that 300 patients with non-severe histories received \(A_2=1\). That overall count does not repair the violation: treatment among non-severe patients does not provide support for treatment among patients with severe histories. The regime would need to be changed, or evidence from a setting where the treatment is available at that history would be needed.

Worked Example: Sparse but Nonzero Support

Now suppose 2 of 100 patients with a relevant history receive the treatment prescribed by a regime. The observed conditional proportion is

\hat{p}=\frac{2}{100}=0.02.

The sample does not show a zero: two people provide observed treatment support. But a 2% proportion is sparse support, not evidence that the treatment is well represented. For scale, the reciprocal of this estimated probability is \(1/0.02=50\); quantities that rely heavily on this treatment-history combination can consequently be highly sensitive to the few observations. This reciprocal is not itself a positivity test or a guarantee about estimator behavior. It simply illustrates how little data may represent the prescribed action in this stratum.

The data also cannot establish from these two observations alone that the true conditional treatment probability is safely bounded away from zero. Investigators would examine the protocol, the number and distribution of people in the history stratum, and uncertainty in the estimated treatment probability. Positivity is a population condition; finite-sample evidence is necessarily imperfect.

Continuous Treatments Require Local Support

When treatment is continuous, such as dose in milligrams, the probability of receiving any one exact dose is generally zero even when that dose is well represented. The relevant idea is therefore conditional density and local support. For a prescribed dose \(a\), ask whether the conditional distribution of dose given the history has positive density around \(a\), rather than asking whether the exact value \(A_t=a\) occurs with positive probability.

A broad observed range is not enough to establish local support. If many doses fall between 55 and 65 mg, they could all be clustered near 55 mg. That count alone says nothing decisive about whether the data support a rule prescribing 60 mg. Evidence needs to concern a neighborhood of the prescribed dose, and an assessment should make clear whether it relies on observed local data or on a modeling assumption about the dose distribution.

Worked Example: Checking Local Support for a Prescribed Dose

Consider a hypothetical set of 100 people with the same relevant treatment history. Fifty doses lie somewhere between 55 and 65 mg. That interval count alone does not show local support at 60 mg: all 50 doses might lie between 55 and 56 mg. Suppose instead that the recorded doses include 4 observations between 59 and 60 mg, 5 between 60 and 61 mg, and 3 between 61 and 62 mg. These data show observations in a neighborhood around 60 mg, unlike the broad interval count by itself.

The local empirical proportions are \(4/100=0.04\), \(5/100=0.05\), and \(3/100=0.03\) in the three one-milligram bands. They indicate that nearby doses were observed, but they do not prove that the underlying conditional density is positive at every point in a neighborhood; finite observations cannot prove that population property. A conclusion of positive density would need to be framed as an assumption supported by these local observations, or justified by an appropriate model for the conditional dose distribution. If the only information were “50 doses between 55 and 65 mg,” claiming support near 60 mg would be unjustified.

Why a Zero at Any Relevant Decision Matters

The sequential nature of the condition can be seen from the probability of adhering to a regime through \(K\) decisions. By the chain rule, this probability factors into conditional adherence probabilities, each given adherence at earlier decisions; the factorization applies when the preceding adherence events have positive probability (if one has probability zero, the joint probability is already zero):

P(A_1=d_1(H_1),\ldots,A_K=d_K(H_K)) = P(A_1=d_1(H_1)) \prod_{t=2}^{K} P(A_t=d_t(H_t)\mid A_1=d_1(H_1),\ldots,A_{t-1}=d_{t-1}(H_{t-1})).

This is the probability of adherence across all histories reached, not the probability of one fixed sequence of histories. Each factor includes the probability of reaching histories after earlier adherence as well as receiving the prescribed action there. If any factor is zero, the joint probability is zero; if factors are positive but very small, adherence can still be represented by very few people. This justifies checking each decision separately rather than relying only on the total frequency of a regime-like pattern.

Positivity is necessary for learning about the regime from the observed treatment process, but it is not sufficient for a causal interpretation. Sequential exchangeability is a separate requirement, and the longitudinal intervention must also be well defined. Even with those conditions, weak practical support can make conclusions imprecise or dependent on extrapolation. A regime that is clinically appealing may therefore be unsuitable for evaluation in a data source whose treatment options do not overlap with the regime’s prescriptions.

Common Mistake

Wrong reasoning: “The prescribed treatment occurs frequently in the full cohort, so positivity holds.”
Correction: The treatment must occur with positive probability at each relevant history where the regime prescribes it. Treatment given to people with different histories does not fill a conditional support gap.
Wrong reasoning: “Some doses fall in a wide interval containing the prescribed dose, so the prescribed dose is supported.”
Correction: A wide interval count does not establish local support. Examine the distribution near the prescribed value and state any density assumption explicitly.
Wrong reasoning: “No treated person was observed in a small stratum, so the population probability must be zero.”
Correction: A sample zero can reflect sparse sampling, while a protocol restriction can imply a structural zero. Distinguish the design and population mechanism from the finite-sample count.

Key takeaway: Sequential positivity is a history-specific support condition for every treatment prescribed along a regime’s possible paths. Check each time point and relevant history; treat structural absence differently from sparse evidence; and for continuous treatment, assess local support around the prescribed value.

Check Your Understanding

Use the definitions and distinctions above to answer the following questions.

A treatment is common overall but never given to patients with a particular history that a dynamic regime can reach. Does overall frequency establish positivity for that regime? Explain.
In a two-decision path, the conditional probabilities of following the prescribed actions are 0.30 and 0.10. What proportion follows both decisions along this path, assuming the stated probabilities are conditional on reaching each decision?
Give one example of a structural positivity violation and one example of a practical positivity problem.
Why does observing many doses between 55 and 65 mg not, by itself, establish support for a prescribed dose of 60 mg?
Does sequential positivity alone justify a causal interpretation of a regime comparison? Name another causal requirement relevant to longitudinal treatment assignment.