Why Positivity Needs Its Own Diagnosis
A hypothetical effect under competing-event elimination asks what the risk of a target event would be if the competing event were prevented during follow-up. Estimation needs data that can inform the histories created by that intervention. In particular, the data must include relevant patients receiving each treatment and, at each relevant history, some support for continuing follow-up without the competing event. Those are separate requirements: treatment groups can overlap while event histories do not, or the reverse.
Tables of treatment assignments and event histories are useful because they make these failures visible before fitting a model. A table cannot, by itself, establish that a population probability is exactly zero: an empty cell can arise by chance in a small sample. The diagnosis therefore combines the table with the study design, the way events occur, and knowledge of the plausible data-generating process.
Two Distinct Positivity Conditions
Let \(A\) be a binary treatment, \(L\) baseline covariates, and \(t=1,\ldots,K\) discrete follow-up intervals. At each interval, a participant who has had neither event remains at risk for a target event or a competing event. Let \(Y_t\) and \(C_t\) indicate the target and competing event, respectively, at interval \(t\), among those at risk at its start. Let \(H_t\) denote the measured history available just before that interval, including baseline covariates and relevant prior information. Define \(q_t(a,h)=P(C_t=0\mid A=a,H_t=h,\text{at risk at }t)\).
The elimination intervention sets \(C_t=0\) whenever a participant is at risk, while leaving the target-event process to evolve. The event-history condition matters because an individual whose observed history ends in a competing event does not subsequently reveal what their target-event outcome would have been had that competing event been prevented. Identification from observed data therefore needs, at each relevant history, some people who continue without that event.
These conditions are part of a larger identification argument, not substitutes for it. Consistency connects observed outcomes to the relevant potential outcomes when the observed treatment and event history match the intervention. Treatment exchangeability requires no unmeasured confounding of treatment given the adjustment variables. Sequential exchangeability for competing-event elimination requires that, given the measured history, those who avoid the competing event provide information about the target-event outcomes under elimination. Positivity supplies the required support. A table can reveal an apparent support failure, but cannot verify exchangeability or consistency.
What to Look for in the Tables
For treatment positivity, cross-tabulate treatment by baseline covariates or clinically meaningful covariate strata. A stratum containing only one treatment level has no within-stratum comparison for the missing level. For elimination positivity, tabulate competing-event and no-competing-event transitions among participants at risk, separately by treatment and relevant history. The denominator must be the risk set for that interval, not the original sample size: people who already had either event are no longer at risk for a first target or competing event.
A structural failure means that the probability of a required treatment or transition is zero in the target population. Examples include a treatment assigned deterministically by a covariate, or a clinical rule that makes a competing event inevitable in a particular history. A practical or empirical warning is a very small estimated probability, or a cell with few observations. These cases are not interchangeable. Sparse data can produce unstable estimates even when the true probability is positive; increasing the sample could help. A genuine structural zero cannot be repaired by adding more observations from the same design.
| Table pattern | What it suggests | What it does not establish alone |
|---|---|---|
| No participants receive one treatment in a baseline stratum | Possible treatment positivity failure | That the population treatment probability is zero, unless the design or assignment rule confirms it |
| All at-risk participants experience the competing event in a history stratum | Possible failure of support for elimination in that history | That the no-competing-event probability is truly zero rather than merely small |
| Very few participants follow a required treatment or event-free transition | Practical positivity concern and potentially unstable weights | Structural failure |
Worked Diagnostics
Worked Example: Treatment Overlap Does Not Guarantee Elimination Support
Consider a hypothetical study with binary baseline covariate \(L\). In each \(L\) stratum, 30 participants receive each treatment. At the first interval, the event-history table is:
| Baseline stratum | Treatment | At risk | No competing event | Competing event |
|---|---|---|---|---|
| \(L=0\) | 0 | 30 | 27 | 3 |
| \(L=0\) | 1 | 30 | 24 | 6 |
| \(L=1\) | 0 | 30 | 30 | 0 |
| \(L=1\) | 1 | 30 | 0 | 30 |
Treatment positivity has empirical support in both strata: both treatment groups are represented at \(L=0\) and \(L=1\). For the \(L=1,A=1\) history, however, the observed no-competing-event fraction is \(0/30=0\), while for \(L=1,A=0\) it is \(30/30=1\). If the underlying event rule makes a competing event certain for \(L=1,A=1\), then \(q_1(1,h)=0\) and elimination positivity fails in that stratum. If the table is a finite sample from a population with a small positive probability of avoiding the event, it is instead an alarming empirical warning. The study design or substantive knowledge is needed to distinguish those explanations.
The distinction affects estimation directly. The inverse probability factor for an event-free transition is \(1/q_1(1,h)\). It is undefined if \(q_1(1,h)=0\), and potentially very large if \(q_1(1,h)\) is merely close to zero. Treatment overlap does not remove this problem: there are treated and untreated participants at \(L=1\), but no observed treated participants in that stratum who continue without the competing event.
Worked Example: A Missing Treatment Level in a Covariate Stratum
Suppose a hypothetical target population contains two baseline strata, \(L=0\) and \(L=1\). A treatment table shows 40 participants with \(L=0,A=0\), 40 with \(L=0,A=1\), 0 with \(L=1,A=0\), and 25 with \(L=1,A=1\). Assume \(L=1\) occurs in the target population and the causal contrast is intended to compare both treatments across that population.
At \(L=0\), both treatment levels are represented. At \(L=1\), however, the empirical probability of treatment 0 is \(0/25=0\). If treatment is deterministically assigned to level 1 for everyone with \(L=1\), then the population condition \(P(A=0\mid L=1)>0\) fails. The outcome risk under treatment 0 for that stratum is not identified by directly comparing observed groups, because there is no observed comparison group.
Restricting the analysis to \(L=0\) changes the target population; it does not identify the original population contrast. A model might produce a predicted risk for \(A=0,L=1\), but that prediction depends on extrapolation beyond the observed treatment support. It is not evidence that positivity holds. If the empty cell is only a consequence of a small sample and the true assignment probability is positive, more data may help, but this table alone cannot make that determination.
Worked Example: Reading Event-Free Weights Across Two Intervals
In a hypothetical treatment-history stratum, suppose the estimated probabilities of avoiding the competing event are \(q_1=18/20=0.90\) at the first interval and \(q_2=16/20=0.80\) at the second interval, conditional on remaining at risk and on the history at each interval. For a person who avoids the competing event at both intervals, the cumulative inverse probability factor is:
The product form follows because the probability of avoiding the competing event through both intervals, conditional on the successive histories, is \(q_1q_2\). The reciprocal upweights observed histories that remain event-free, compensating for the fact that some comparable histories end in the competing event, under the required exchangeability and model assumptions. If instead \(q_2=0\) in a required history, the product is zero and the reciprocal is undefined. If \(q_2\) is positive but tiny, the factor can be extreme and make estimates highly variable.
The fractions in this example are estimates from a small table, not proof that the corresponding population probabilities equal 0.90 or 0.80. They illustrate how to calculate and interpret the diagnostic and the weight.
Why the Weight Requires Support
The one-interval logic shows why a zero probability is not a numerical nuisance that can be fixed by a different weighting formula. Let \(F\) be a quantity depending on the outcome under the event-free transition. If \(q_t(a,h)>0\), then among people with the specified treatment and history,
The cancellation is valid only when the denominator is positive. Under sequential exchangeability, the conditional event-free outcomes can inform the corresponding outcomes under elimination at that history; applying the argument across intervals yields the product of transition weights. If \(q_t(a,h)=0\), the numerator indicator is always zero in that stratum, and the observed data contain no event-free outcome there to reweight. Assigning a finite value to \(1/0\) cannot create that information.
Diagnostics, Alternatives, and Limits
A practical diagnosis should inspect both the design and the empirical support. Use tables with denominators that reflect the relevant treatment and event-history risk sets. Include important baseline and time-varying history variables in the stratification or model; aggregating across histories can hide a zero transition in a subgroup. Review the treatment-assignment protocol and clinical rules for deterministic decisions. Also inspect fitted treatment and event-free probabilities and the resulting weight distribution: probabilities near zero and a few dominating weights are signs of practical positivity problems, even when no cell is exactly empty.
A fitted probability model can smooth over empty cells, but smoothing is not a cure for structural nonpositivity. If the model predicts a positive probability where the design makes the transition impossible, the resulting weight rests on model extrapolation. Conversely, a zero cell in a small table does not warrant automatically declaring structural failure. The key question is whether the required transition has positive probability in the target population, not whether a particular finite sample happened to contain an example.
The main alternatives address different issues. Standardization or the competing-risk g-formula models transition and outcome distributions, but also requires support for the histories under the intervention; a model cannot identify a transition that never occurs in the relevant population. Inverse probability weighting exposes the support requirement through its denominators and weights. If support is genuinely absent, one may redefine the target population or intervention so it uses supported histories, or use partial-identification bounds and sensitivity analysis rather than present an unsupported point estimate. These choices change what can be learned; none justifies silently extrapolating as though positivity held.
A stochastic intervention on the competing event can sometimes avoid demanding support for a transition it never assigns. Its support condition is that every transition assigned positive probability by the intervention also has positive probability in the observed data at that history. For deterministic elimination, the assigned transition is always \(C_t=0\), so its observed probability must be positive wherever the intervention reaches. This is a support check, not a guarantee that the other causal assumptions hold.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A study includes both treatment groups in every baseline stratum, but in one treatment-history stratum every at-risk participant has the competing event. Which positivity condition is implicated?
- In a stratum, 0 of 8 participants avoid the competing event. Does this prove the population probability of avoiding it is zero? What additional evidence would help?
- At a history, the estimated probability of avoiding the competing event is 0.25. What is the one-interval inverse probability factor, and what does its size indicate?
- Why does restricting an analysis to covariate strata with both treatments represented not automatically estimate the original population contrast?
Solutions. (1) Competing-event-elimination positivity is implicated: the observed support for continuing without the competing event is absent in that history, even though treatment positivity holds. (2) No. A finite sample of eight could have no event-free transitions even if the population probability is positive. The assignment and clinical rules, the history-specific event mechanism, and additional data can help distinguish structural absence from sparse sampling. (3) The factor is \(1/0.25=4\). Such a factor indicates that the transition is relatively uncommon and that estimates may be sensitive to a small number of event-free observations; the probability remains positive in this example. (4) Restriction changes the population being averaged over. It yields a contrast for the supported subset, not necessarily the contrast for the original population, which includes strata lacking one treatment level.