Why Diagnose Positivity?
Inverse probability of censoring weights represent people who remain observed as standing in for comparable people who were censored. That correction can only work when histories in the target population have some chance of remaining observed. If a history almost guarantees censoring, the few people with that history who remain uncensored must represent many others, so their weights can become very large. If nobody with that history can remain observed, the data provide no such representation at all.
Positivity diagnostics assess this support requirement before relying on weighted estimates. They do not establish that censoring is conditionally independent, that the censoring model is correctly specified, or that the final estimate is unbiased. Those are distinct requirements. Here the focus is on whether the fitted retention probabilities and observed covariate support raise concerns about the availability of uncensored observations.
The almost-sure statement matters. With continuous covariates, a particular history value often has probability zero, and conditional probabilities at individual values are only defined up to sets of probability zero. A requirement phrased as “positive for every history value with positive probability” could therefore miss a whole region of continuous histories where retention is zero. Positivity instead rules out zero retention on a set of histories that has positive probability in the target population.
This is a pointwise condition over relevant times, not a guarantee that retention is comfortably large. A stronger practical condition is that \(G(t\mid H(t))\) is bounded away from zero over the histories and times that matter. Such a bound limits how large the corresponding inverse weights can be. The basic positivity condition alone does not impose a universal numerical lower threshold.
What to Examine
A useful diagnostic combines model predictions with the covariate support in the target and retained populations. For each relevant time, obtain the estimated conditional retention probabilities \(\widehat G(t\mid H_i(t))\) from the censoring model. Examine their distribution and their pattern across important covariates, time, and clinically relevant subgroups. The minimum, lower quantiles, and number of predictions near zero can flag concern, but no single cutoff is universally valid.
Also compare which histories appear in the target population with which histories appear among those still observed. For categorical covariates, cross-tabulations can reveal a target subgroup with very few or no retained individuals. For continuous covariates, compare ranges and distributions, and inspect whether a model is predicting retention in regions where there are few or no uncensored observations. A smooth fitted probability in an unsupported region may reflect model extrapolation rather than information in the data.
The model's predicted probabilities are estimates, not direct observations of the population probabilities. Consequently, a small prediction can be a sign of practical near-positivity, model misspecification, sparse data, or a combination of these. Conversely, predictions that are all moderate do not prove positivity: a misspecified model can conceal a zero-probability region. Diagnostics should therefore be reported alongside the model specification and the support information used to assess it.
Identify the target population, the histories available at each time, and the times relevant to the estimand. Positivity is assessed relative to this target, not to an unspecified population.
Summarize \(\widehat G(t\mid H(t))\) at relevant times, including low quantiles and predictions near zero. Check whether low predictions cluster in identifiable histories or emerge late in follow-up.
Compare covariate histories in the target population and among those remaining uncensored. Look for target regions with sparse or absent retained observations and for predictions that depend on extrapolation.
Distinguish observed zeros in a finite sample from a population probability of zero. Consider whether the concern is structural, practically severe, or uncertain because data are sparse.
Worked Examples
Worked Example: A Low Retention Probability in a Subgroup
Suppose a hypothetical 12-month study includes 100 people: 75 in a lower-travel-burden group and 25 in a higher-travel-burden group. At 12 months, 68 people in the first group and 3 in the second remain uncensored. Estimate each group's retention probability by its observed retained proportion, then calculate the corresponding inverse probabilities.
For the lower-burden group, the estimated probability is \(68/75=0.9067\), so the inverse probability is \[ 1/0.9067=1.103. \] For the higher-burden group, the estimated probability is \(3/25=0.1200\), so the inverse probability is \[ 1/0.1200=8.333. \]
The higher-burden group has a much smaller estimated chance of remaining observed and a much larger inverse probability. This is a practical-positivity warning: only three retained people represent the experience of 25 people in that group. However, the sample does not establish that the population retention probability is exactly zero; it is positive in the observed data, and uncertainty about its population value remains substantial. The appropriate conclusion is that retention is sparse and the resulting weights may be unstable, not that structural positivity has been proved to fail.
Worked Example: Why Near-Zero Retention Inflates Variation
Consider a single history stratum in which the true retention probability is \(g\). Let \(R\) be the indicator of remaining uncensored, so \(R\sim\operatorname{Bernoulli}(g)\). For a simple contribution equal to one when retained, the inverse-weighted contribution is \(Z=R/g\). Calculate its mean and variance when \(g=0.50\) and when \(g=0.02\).
Since \(R\) is one with probability \(g\) and zero otherwise, \[ E(Z)=g(1/g)+(1-g)0=1, \] and \[ E(Z^2)=g(1/g^2)=1/g. \] Therefore, \[ \operatorname{Var}(Z)=E(Z^2)-\{E(Z)\}^2=1/g-1. \]
When \(g=0.50\), the inverse weight for a retained person is \(1/0.50=2\), and \(\operatorname{Var}(Z)=2-1=1\). When \(g=0.02\), the retained person receives weight \(1/0.02=50\), and \(\operatorname{Var}(Z)=50-1=49\). In both cases the expected weighted contribution is one, but the second case is far more variable. This calculation isolates the mechanical instability caused by small retention probabilities; it does not rely on a particular outcome model.
Worked Example: Continuous-Covariate Support
In a hypothetical target population, a continuous covariate \(X\) ranges from 0 to 1, with 20% of people having \(0.8\leq X\leq1\). Suppose a fitted censoring model predicts retention probabilities around 0.015 for this upper range, while the observed uncensored sample contains no one with \(X\geq0.8\). What does this pattern indicate?
The fitted model's inverse probability in that region is approximately \[ 1/0.015=66.67. \] Thus, if anyone in that region were retained, their estimated weight would be about 66.67. More importantly, the target population contains a positive-probability region with no observed retained individuals. This is direct evidence of an empirical support gap, and the model's probability prediction there is largely an extrapolation from other values of \(X\).
The sample alone cannot distinguish a truly zero population retention probability from a small positive probability that happened to yield no retained observations. The diagnosis is therefore a serious concern about support and practical positivity, not proof that the population condition fails. The analyst should report the unsupported region and avoid presenting the model prediction as if it were well supported by observations.
Derivation and Interpretation of the Instability
The calculation above generalizes to a fixed outcome contribution \(Y\) in a history stratum, where \(Y\) is observed for those retained and \(R\sim\operatorname{Bernoulli}(g)\). The weighted contribution is \(RY/g\). Its expected value is \[ E(RY/g)=Y E(R)/g=Yg/g=Y, \] so inverse weighting recovers the contribution in expectation under this simplified setup. But its variance is \[ \operatorname{Var}(RY/g)=Y^2\operatorname{Var}(R/g) =Y^2\{1/g-1\}. \] As \(g\) approaches zero, this variance grows without bound for nonzero \(Y\). If \(g=0\), the weight is undefined and there are no retained observations from which to recover that history's contribution. This explains why positivity is not merely a technical condition: it determines whether a weighted representation is available and whether it can be estimated with useful precision.
In an actual analysis, outcomes vary, histories are estimated, and censoring can occur over time, so this simple variance is not a complete variance formula for an estimator. It is a justification for the diagnostic focus: small conditional retention probabilities imply potentially large weights and high variability, even when the weighting identity is otherwise appropriate.
Distinguishing Failure Modes and Choosing a Response
| Pattern | What it may mean | Diagnostic interpretation |
|---|---|---|
| Target histories have no retained observations | Structural failure or a finite-sample realization of a very small probability | Support is not demonstrated; distinguish the possibilities using design knowledge and available data. |
| Some retained observations, but fitted probabilities are very small | Practical near-positivity, sparse data, or model instability | Report how low probabilities align with histories and times; do not infer a universal cutoff. |
| Predictions appear adequate, but target histories lie outside retained-data support | Model extrapolation or misspecification may conceal the problem | Inspect support separately; model output alone is insufficient. |
| Retention differs sharply across time | Positivity may deteriorate late in follow-up | Assess the condition at the times relevant to the target estimand, not only at baseline. |
Positivity diagnostics are different from checking whether the censoring model captures dependence between censoring and observed history. They are also different from checking covariate balance or identifying the influence of individual weights; those are separate assessments. A model can have adequate support but still be misspecified, and a well-fitting model on observed data cannot create information in a region with no retained observations.
When evidence suggests a problem, possible responses depend on the scientific question. Investigators can seek longer or more complete follow-up, improve measurement of predictors that explain retention, or reconsider whether the target population or time horizon can be narrowed in a scientifically defensible way. A revised target changes the population to which the estimate applies and should be stated explicitly. Stabilized weights may change weight scale, but they do not create support where conditional retention is absent. Truncating weights can limit their magnitude, but when probabilities are near zero it changes the weighting procedure and may introduce bias; it is not proof that the original positivity problem has been solved.
Common Mistakes and Limitations
- Checking only whether a fitted probability equals zero. Probabilities can be positive yet so small that weights are practically unstable. Examine their distribution and covariate pattern.
- Declaring structural failure from one empty sample cell. An empty cell is a warning about observed support, but finite data cannot by themselves prove a population probability is exactly zero.
- Checking only individual covariates. A combination of history values can have poor support even when each covariate separately looks well represented. Assess meaningful joint histories where data permit.
- Treating smooth model predictions as evidence of support. A model can extrapolate smoothly into regions without retained observations. Compare predictions with observed support.
- Assuming a numerical threshold is universal. The practical importance of a low probability depends on sample size, outcome variability, time, and the target estimand. Justify any threshold in context.
- Equating positivity with independent censoring. Positive retention probabilities do not establish that the measured history adequately accounts for censoring. Both assumptions, and model adequacy, require attention.
Check Your Understanding
Answer each question, then compare with the solutions below.
- State the conditional censoring-positivity condition in a way that applies to continuous histories.
- If an estimated retention probability is 0.04, what is its inverse, and why is this a diagnostic concern?
- A target subgroup has no retained people in a sample. Does that establish a population retention probability of zero? Explain.
- Why can a fitted model's predicted probability fail to reveal a support problem?
- Does truncating large weights establish that positivity holds? Why or why not?
Solutions. (1) At each relevant time, \(G(t\mid H(t))>0\) almost surely under the target population's history distribution. (2) The inverse is \(1/0.04=25\); a retained person would receive a large weight, and a small number of such people may represent many censored individuals. (3) No. The empty sample subgroup signals unsupported observed data, but a small positive population probability could still yield no retained observations by chance. (4) The model can extrapolate from other histories or be misspecified; predictions should be checked against observed target and retained covariate support. (5) No. Truncation limits weight size but does not create uncensored observations in unsupported histories and may introduce bias.