Why Positivity Matters
A causal contrast may ask what would happen if everyone in a target population received treatment \(A=1\), rather than \(A=0\). But if some covariate profiles occur only among people who received one treatment, the observed data cannot directly show how people with those profiles fare under the other treatment. This is a support problem: a causal question may describe an intervention that the data do not represent for part of the target population.
The issue also arises when the intervention changes how a competing event is handled, or when a treatment is decomposed into components that can be assigned separately in the causal question. Positivity is therefore not a claim that an intervention is clinically sensible or ethically acceptable. It is a condition about whether the observed data include relevant people, treatment assignments, and event histories needed to learn about that intervention.
The phrase “for almost every \(l\)” matters. A condition need not be checked at covariate values that have probability zero in the target population. Conversely, a contrast for a restricted target population should be assessed against that population’s covariate distribution, not automatically against every covariate value in a broader source population.
Positivity is separate from exchangeability. Exchangeability concerns whether groups are comparable with respect to their potential outcomes, conditional on measured covariates. Positivity concerns whether each treatment being compared occurs at those covariate values. Even if exchangeability is plausible, an absent treatment group within a covariate stratum leaves no observed outcome information for that treatment in that stratum.
From a Condition to an Identification Problem
For a fixed time \(t\), let \(Y^a(t)\) indicate whether the event of interest would have occurred by \(t\) under treatment \(a\), with competing events handled as specified by the causal question. Under consistency, conditional exchangeability given \(L\), and treatment positivity, the risk under treatment \(a\) can be related to observed data by
The outer expectation averages over the covariate distribution in the target population. If \(L\) is discrete, that expectation can be written as a sum over its values. If \(L\) is continuous, it is an integral against the target distribution of \(L\); the expectation notation covers both cases. When \(P(A=a\mid L=l)=0\), the observed conditional risk \(P\{Y(t)=1\mid A=a,L=l\}\) is not identified from data at that \(l\). Positivity makes the required conditional comparison available in principle.
This relation is a justification for why positivity matters, not a promise that the risk can be estimated precisely. It also does not say that positivity alone is sufficient: exchangeability, consistency, an appropriate definition of the outcome and intervention, and any additional assumptions needed for the particular competing-risk contrast remain important.
Worked Example: Treatment Support Across Covariate Strata
Consider hypothetical data from 200 people in a target population. A binary baseline covariate \(L\) has two strata. Among the 100 people with \(L=0\), 50 received \(A=1\) and 50 received \(A=0\). Among the 100 with \(L=1\), 0 received \(A=1\) and 100 received \(A=0\).
Assess the sample support. The observed treatment proportions are \(50/100=0.50\) for \(A=1\) and \(50/100=0.50\) for \(A=0\) in the \(L=0\) stratum. For \(L=1\), the corresponding sample proportions are \(0/100=0\) and \(100/100=1\). Thus the sample contains no treated people with \(L=1\).
What follows for the population condition? The counts establish that empirical support for \(A=1\) in \(L=1\) is absent in this sample. They do not establish that the population probability \(P(A=1\mid L=1)\) is zero. A finite sample can contain no treated people even when the true probability is positive. Population positivity fails in this stratum if the true conditional probability is zero; these counts alone cannot decide whether that structural zero exists.
Implication for analysis. A direct estimate of the treated conditional risk for \(L=1\) cannot be calculated from these data because there are no observed treated people in that cell. If the underlying probability is positive but small, more data might reveal treated people; if it is structurally zero, simply increasing the sample under the same assignment process will not provide them. Either way, an analyst should report the observed support problem and avoid treating the missing cell as if its outcome risk had been measured.
Positivity for Competing-Event Interventions
A competing-event intervention changes the intervention being considered, not merely the label attached to an outcome. For example, a causal question might consider a hypothetical intervention that prevents the competing event, or might compare outcomes under components of treatment that affect different event processes. In such settings, ordinary treatment positivity may not be the only support requirement.
This definition is deliberately tied to the intervention and identification strategy. “Positivity for preventing the competing event” does not simply mean that the data must contain people who naturally experience no competing event. The intervention may set the competing-event process to zero by definition. The support question is whether the observed data provide information about the event-of-interest process in the covariate and at-risk histories needed to identify the risk under that intervention. A particular approach that treats competing events as censoring may, for example, require adequate probability of remaining under observation and free of the competing event through relevant times. That is a support requirement for that approach, not a universal formula for every intervention that affects competing events.
With a time-varying history, support must be considered at each decision or event time where the intervention requires a conditional quantity. If \(H_k\) denotes the observed history just before time \(k\), a regime assigning action \(a_k\) requires positive probability of that action among histories with positive probability under the regime. A history that can arise under the intervention but is absent from the observed data can make a required conditional event risk unavailable. Positivity at baseline alone does not automatically solve this later-history problem.
Worked Example: A Hypothetical Competing-Event Intervention
Suppose a hypothetical study follows 120 people who are all event-free at the start of a one-year interval. Investigators consider an intervention that prevents a competing event during that interval and ask about the event-of-interest risk. For illustration, divide baseline covariates into \(L=0\) and \(L=1\). Among 60 people with \(L=0\), 12 have the competing event before the end of the interval, 6 have the event of interest first, and 42 remain free of either event through the interval. Among 60 with \(L=1\), the respective counts are 30, 6, and 24.
Check the arithmetic. In each stratum the counts sum to 60. The observed competing-event proportions are \(12/60=0.20\) for \(L=0\) and \(30/60=0.50\) for \(L=1\). The observed event-of-interest proportions are \(6/60=0.10\) in both strata.
Interpret the support information. There are observed event-of-interest outcomes in both baseline strata, so these counts do not show a baseline stratum with no observed event-of-interest cases. However, the counts alone do not establish that the event-of-interest risk under an intervention eliminating the competing event is identified. The intervention creates a different event process, and an identification strategy must specify which observed conditional quantities it uses. If that strategy needs event-of-interest information among people who would otherwise experience the competing event, their counterfactual event-of-interest outcomes under prevention are not directly observed merely because some other people had the event of interest.
The example therefore illustrates a diagnostic, not a causal estimate: inspect the observed histories and ask whether the intervention’s required conditional quantities are represented. Do not infer the risk under competing-event prevention by simply dividing the six observed event-of-interest cases by the 120 people or by excluding those with competing events. Those calculations do not, by themselves, answer the intervention-specific causal question.
Component Interventions and Joint Support
For separable-effect questions, treatment may be represented conceptually by two components, such as \(A_Y\) and \(A_D\), intended to act through different event processes. An intervention might set these components to a combination such as \((A_Y=1,A_D=0)\). If the observed treatment only ever assigns the components together, that combination may not occur in the data.
In that case, positivity for the original treatment levels \(A=0\) and \(A=1\) does not imply positivity for every component combination. The relevant check is the probability of each required joint assignment conditional on covariates and, where needed, histories. Even if all combinations have positive support, that fact alone does not establish that the component-specific causal contrast is identified; assumptions about how the components relate to the event processes are also required.
Worked Example: Missing Component Combinations
Suppose 300 hypothetical participants have the same baseline stratum. A treatment has two conceptual binary components, \(A_Y\) and \(A_D\). The observed assignments are 150 participants with \((0,0)\) and 150 with \((1,1)\); no one receives \((1,0)\) or \((0,1)\).
Check ordinary treatment support. If the actual treatment is recorded only as \(A=0\) for \((0,0)\) and \(A=1\) for \((1,1)\), both observed treatment levels have sample proportions \(150/300=0.50\). The sample has support for that binary treatment contrast in this stratum.
Check support for a component intervention. The empirical proportions are \(0/300=0\) for \((1,0)\) and \(0/300=0\) for \((0,1)\). Thus this sample provides no direct observations under either discordant component combination. If the causal contrast requires one of those combinations, it has an empirical support problem. As in the previous example, an empty sample cell does not by itself prove a population probability of exactly zero. But if the treatment system structurally enforces \(A_Y=A_D\), then the discordant combinations have zero probability under that system, so positivity for those interventions fails.
This distinction helps locate the problem correctly: support for the observed whole-treatment contrast is not the same as support for interventions that assign its components separately.
Diagnostics, Near-Violations, and Alternatives
A practical assessment combines knowledge of how treatment and follow-up were assigned with summaries of the observed data. Useful checks include:
- Tabulate treatment assignments across important covariate strata, and check whether the intended target population includes strata with no observed treatment level.
- For continuous covariates, inspect estimated treatment probabilities across their observed range and identify regions with little overlap. A finite sample cannot establish a population structural zero simply because an estimated probability is zero.
- For intervention-specific questions, list the treatment combinations and event histories required by the intervention and compare them with the combinations and histories actually represented.
- In longitudinal data, repeat the support assessment at relevant decision times, conditional on histories that can occur under the intervention.
A probability near zero is not the same as a structural zero, but it can still be consequential. If only a few people receive a treatment in a covariate region, estimates relying on those people can be highly variable and sensitive to individual outcomes. In weighting analyses, for instance, very small assignment probabilities can produce very large weights. Trimming or restricting a target population may reduce instability, but it changes the analysis or target and does not create information about unsupported interventions.
The main alternatives are not interchangeable repairs. Restricting the target population to a region with adequate support can make a narrower contrast estimable, if that restricted population is scientifically relevant. Modeling outcomes beyond observed support requires extrapolation and depends on modeling assumptions; it does not remove the underlying lack of data support. Collecting data under an assignment process that includes the missing treatment or component combinations can address the empirical gap, subject to ethical and practical constraints. Each option should be described transparently rather than presented as proof that positivity was satisfied.
A common mistake is to see a zero cell in a sample table and announce that population positivity fails. The correction is to state what the data show—no observed units in that cell—and distinguish it from the population claim that the true conditional probability is zero. The converse mistake is also serious: observing at least one person in every cell does not guarantee good practical support or precise estimation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a hypothetical sample, no participant with \(L=1\) received treatment. What can and cannot be concluded about population positivity?
- Write the general averaging expression for the risk under \(A=a\) when \(L\) could be continuous. What does the outer averaging distribution represent?
- A study observes only component combinations \((0,0)\) and \((1,1)\). What support issue arises for an intervention assigning \((1,0)\)?
- Why does observing some event-of-interest cases in every baseline stratum not automatically establish positivity for an intervention that prevents a competing event?
Solutions. (1) The sample has no empirical support for treatment in the \(L=1\) stratum. The counts alone do not establish that the population probability \(P(A=1\mid L=1)\) is zero; a structural zero would imply a population positivity failure. (2) Under the relevant identification assumptions, \(P\{Y^a(t)=1\}=E[P\{Y(t)=1\mid A=a,L\}]\). The expectation averages over the target population’s distribution of \(L\); for continuous \(L\), this is integration rather than a sum. (3) The sample contains no observed participants assigned \((1,0)\), so an intervention requiring that combination lacks empirical support. If the assignment mechanism structurally forbids it, population positivity fails for that intervention; an empty finite-sample cell alone does not prove this. (4) The intervention changes the competing-event process. Identification may require event-of-interest information for histories that are not observed under the natural process, so baseline-stratum case counts alone do not establish the needed intervention-specific support.