Why Distinguish Two Kinds of Positivity Problems?
A treatment can be unavailable to some patients because a clinical rule makes it impossible, or it can be available but seldom used for patients with particular characteristics. These situations can look similar in a data table: a covariate group may contain few treated patients, or none. Their underlying meanings differ. In the first situation, the data cannot contain both treatment choices for that group under the assignment process. In the second, both choices are possible, but the observed evidence for one choice may be weak.
This distinction matters when deciding what a study can support. A model cannot create observations in a group where treatment is truly impossible. When treatment is only very unlikely, some information may exist, but estimates can depend heavily on a small number of observations. Neither situation is repaired just by calculating a propensity score.
The distinction is about the treatment assignment process, not only the table of observed counts. A group with no treated patients in one sample may have a small but positive treatment probability. Conversely, a deterministic treatment policy can make treatment impossible in a group even if a fitted model returns a small positive number. For a specified target population, conditional positivity requires both treatment choices to have positive probability for covariate values represented in that population. Positivity alone does not establish a causal effect: the other causal assumptions, including appropriate control of confounding, remain necessary.
Structural Violations: Treatment Is Impossible
Structural violations often have a substantive explanation. A medication may be contraindicated for patients with a particular allergy, a procedure may be offered only to patients meeting an eligibility rule, or a policy may assign treatment deterministically based on a measured risk category. If treatment is never permitted for a group, then the true conditional probability of treatment is zero in that group. The corresponding issue arises when everyone in a group must receive treatment, so the probability of no treatment is zero.
A structural zero is not simply a small count. It is a feature of the population treatment process. To diagnose one, investigators need to understand eligibility criteria, clinical practice, data collection, and the definition of the covariates. A zero count may instead reflect a limited sample, an incomplete database, or a poorly chosen subgroup definition. Even detailed data cannot prove a probability is exactly zero solely because no one in a finite sample received treatment.
Worked Example: A Contraindicated Treatment
Suppose a hypothetical cohort has 80 patients, classified by whether they have a documented severe allergy to a drug. The treatment protocol prohibits prescribing the drug to patients with that allergy. Among 30 patients with the allergy, none receives the drug; among 50 patients without it, 20 receive the drug. Treat the protocol as applying without exceptions, and define the target population to include both groups.
For the allergy group, the protocol makes treatment impossible, so \(e(X=\text{allergy})=0\). For the group without the allergy, the observed treated proportion is \(20/50=0.40\). This sample proportion describes the observed cohort; by itself it does not establish the population propensity score exactly. The protocol does, however, justify the structural-zero conclusion for the allergy group under the stated assumptions.
Because the target population includes patients with the allergy, both treatment choices are not available throughout that target population. A causal comparison for the full population cannot be supported by observed treatment contrasts in the allergy group. Restricting the scientific question to eligible patients would define a different target population; it would not make the original group’s treatment probability positive.
Practical Violations: Treatment Is Possible but Rare
A practical violation occurs when a treatment choice is possible in principle but rare in the relevant covariate group. The propensity score is then not exactly zero or one, yet the observed sample may contain very few people receiving one of the choices. This can produce unstable estimates because a small number of observations must represent a much larger part of the target population.
For example, an inverse probability weight for a treated person in an average-treatment-effect analysis is \(1/e(X)\), while the weight for an untreated person is \(1/[1-e(X)]\). These weights grow as the relevant probability approaches zero. Large weights do not prove that the propensity score is wrong, but they signal that results can be sensitive to a few observations and to modeling decisions. The numerical threshold for calling a probability “too small” is not universal: its practical importance depends on sample size, the covariate group, the estimand, and the intended precision.
Worked Example: Rare Treatment in a High-Risk Group
Consider a hypothetical study in which the estimated treatment probability for patients in a high-risk group is \(e(X)=0.05\), and the probability of no treatment is \(0.95\). Suppose an average-treatment-effect analysis uses the unstabilized inverse probability weights described above. Calculate the weights for each treatment status.
For a treated patient, the weight is \(1/e(X)=1/0.05=20\). For an untreated patient, the weight is \(1/[1-e(X)]=1/0.95\approx1.053\). Thus, a treated patient in this group contributes about 19 times the weight of an untreated patient in the same group. The exact contribution to a final estimate also depends on the estimator and normalization, but the calculation shows why a few treated patients can have substantial influence.
The probability \(0.05\) is positive, so this is not a structural violation. It is a practical concern if the number of treated patients is too small to give stable information. The estimate also depends on whether the propensity score is adequately estimated and whether the other causal assumptions hold. A large weight is a warning to investigate, not proof of bias or proof that the treatment effect is unidentified.
Why a Sample Zero Does Not Prove a Structural Zero
Even when treatment is possible, a finite sample can contain no treated individuals in a group. A simple probability calculation shows why. Suppose \(n\) individuals have the same treatment probability \(p>0\), and, conditional on that common probability, their treatment assignments are independent. The probability that none is treated is the product of their individual probabilities of no treatment:
This probability can be substantial when \(p\) is small or \(n\) is modest. The calculation depends on the stated common-probability and independence assumptions; for unequal probabilities \(p_i\), the corresponding probability is \(\prod_{i=1}^{n}(1-p_i)\), if assignments are conditionally independent.
Worked Example: No Treated Patients in a Small Sample
In a hypothetical study, 12 patients in a covariate group each have a treatment probability of \(0.08\). Assume their assignments are independent conditional on this group, and that the probability is the same for all 12 patients. Find the expected number treated and the probability that none is treated.
The expected number treated is \(12(0.08)=0.96\) patients. The probability of no treated patients is
So there is about a 36.8% chance of observing no treated patients, even though the treatment probability is positive for each patient. Observing a zero in a group this small is therefore compatible with a practical positivity problem; it does not, by itself, demonstrate that treatment is structurally impossible. The independence and equal-probability assumptions are part of this calculation, not facts that can be inferred from the zero count.
Assessment, Alternatives, and Limitations
Assessment should combine the observed data with knowledge of how treatment was assigned. Begin by defining the target population and clinically meaningful covariates. Ask whether treatment eligibility, contraindications, or policy rules make a treatment choice impossible in any part of that population. Then examine whether some groups have so few observations under one treatment that estimates would rely on limited evidence. Counts and estimated probabilities are informative, but neither alone settles whether a zero is structural.
A useful distinction is between the data being sparse and the treatment process forbidding an option. The former can result from finite sampling; the latter requires a substantive assignment rule or compelling evidence about the population process. Conversely, observing at least one treated and one untreated person in a group does not guarantee adequate practical support: a handful of observations may still be insufficient for stable estimation.
Common responses have different costs. A more flexible outcome or propensity model may represent observed patterns better, but it cannot supply empirical treatment comparisons where treatment is impossible. A parametric model can produce predictions for such covariate values, but those predictions rely on extrapolation rather than direct treatment contrasts. Matching can leave some individuals unmatched when comparable treatment choices are scarce, and restriction to eligible or better-supported individuals changes the population to which results apply. Capping large weights may reduce their influence, but it also changes the estimator and can introduce bias; it does not turn a structural zero into positive probability.
The key diagnostic question is therefore not merely “Did the sample contain a zero?” It is “Could both treatment choices occur for this group under the treatment process, and is there enough observed evidence to estimate their contrast for the target population?” Answers depend on study design, clinical context, group size, and the causal question. Propensity scores are estimated quantities, so apparent near-boundary values can also reflect model specification or estimation error; interpretation should not treat a fitted score as a direct observation of the true probability.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A treatment is prohibited by protocol for every patient with a particular contraindication. If the target population includes those patients, is the issue structural or practical? Explain.
- For a group with \(e(X)=0.10\), calculate the unstabilized inverse probability weight for a treated person. Is the treatment probability structurally zero?
- Eight patients each have a treatment probability of \(0.20\), with independent assignments and a common probability. Calculate the probability that none is treated.
- Why does observing no treated people in a small covariate group fail to establish that treatment is impossible there?
- Would fitting a more flexible propensity-score model resolve a structural zero? Explain one limitation of that response.
Solutions. (1) It is structural: the protocol makes treatment impossible for that contraindicated group, so \(e(X)=0\). Including the group in the target population does not change that fact. (2) The treated-person weight is \(1/0.10=10\). Since \(e(X)=0.10>0\), the probability is not structurally zero, although it may be a practical concern depending on the available sample and analysis. (3) The probability is \((1-0.20)^8=0.8^8=0.1678\), approximately 16.8%. (4) A finite sample can contain no treated people even when the true probability is positive; for common probability \(p\) and independent assignments, that chance is \((1-p)^n\). (5) No. A flexible model may change estimates within observed data, but where treatment is impossible it cannot create observed treatment comparisons. Predictions there rely on extrapolating the model.