Bounds Under Unmeasured Informative Censoring

Why Bound the Risk?

A fixed-horizon event risk is often the quantity of interest: for example, the probability that a person experiences a particular event by five years. When some people are censored before five years, their event status at that horizon is unknown. If censoring may depend on unmeasured outcomes, the observed people with known status need not represent those whose status is missing. A single risk estimate may then rely on assumptions that the data cannot verify.

Bounds make this uncertainty explicit. They give a range of risks compatible with the observed outcomes and stated assumptions, rather than choosing one value for the unobserved outcomes. The range may be wide; that is informative, not a calculation failure. Its width shows how much the conclusion depends on censoring assumptions.

For this definition, an observed competing event before \(\tau\) gives \(Y=0\): it rules out cause \(k\) as the first event by the horizon. A person known to remain event-free through \(\tau\) also has \(Y=0\). A person censored before their status through \(\tau\) is known has an unknown \(Y\). The target is the cumulative incidence of cause \(k\) by \(\tau\), not a cause-specific hazard or a hypothetical risk in a world where competing events are removed.

This setup assumes that the target population and horizon are defined, event type and timing are recorded correctly when observed, and censoring is the reason status is unknown. It does not assume that censoring is independent of the unobserved outcome. The bounds below describe what is identified at the population level under these conditions. Their sample analogues are estimates of the bound endpoints and have sampling uncertainty.

Worst-Case Bounds

The observed data identify the outcome risk among people with known status, but not the risk among those censored early. Since a binary risk must lie between zero and one, \(0\leq p_{\mathrm{mis}}\leq1\). The law of total probability gives the target risk as the weighted average of these two risks:

Setting the unknown risk to its smallest possible value gives the lower bound; setting it to its largest gives the upper bound. These are sharp bounds: without further restrictions, both endpoint assignments are compatible with the observed data.

The lower endpoint counts the observed cause-\(k\) events and assumes no censored person has cause \(k\) by \(\tau\). The upper endpoint adds every person with unknown status to those events and assumes all censored people have cause \(k\) by \(\tau\). These are deliberately extreme assignments, not claims that either is plausible.

How Assumptions Narrow the Bounds

A narrower interval requires information or assumptions restricting \(p_{\mathrm{mis}}\). Any such restriction should be stated in a form that can be examined and varied. For instance, suppose subject-matter knowledge supports that the risk among censored people is within \(\delta\) of the observed-group risk, where \(\delta\) is a chosen nonnegative risk difference. Then

Substitution into the total-risk identity yields bounds. The restriction is a sensitivity assumption, not a consequence of the observed data. At \(\delta=1\), it imposes no effective restriction and recovers the worst-case interval. At \(\delta=0\), it assumes equal risks in the two groups and identifies the risk as \(p_{\mathrm{obs}}\).

A different restriction is an ordering assumption. For example, \(p_{\mathrm{mis}}\leq p_{\mathrm{obs}}\) says that, on average, censored people are no more likely to have the event by \(\tau\) than people with known status. Combining that assumption with \(p_{\mathrm{mis}}\geq0\) gives

The reverse ordering, \(p_{\mathrm{mis}}\geq p_{\mathrm{obs}}\), instead gives the interval from \(p_{\mathrm{obs}}\) to \(q\,p_{\mathrm{obs}}+(1-q)\). Either ordering may be plausible in a particular setting, but neither follows merely from observing censoring. An assumed direction of selection should not be presented as a finding established by the observed outcomes.

Derivation and Interpretation

The bounds follow directly from partitioning the population into those with known and unknown horizon outcomes. Let \(R\) indicate whether the outcome is known. Then \(Y=1\) occurs either among people with \(R=1\) or among people with \(R=0\), so

Because \(Y\) is binary, the only restriction available without further assumptions is \(0\leq p_{\mathrm{mis}}\leq1\). Multiplication by the nonnegative fraction \(1-q\) preserves these inequalities, giving

This justification also explains sharpness. One can complete the unobserved outcomes by assigning \(Y=0\) to every censored person to attain the lower endpoint, or \(Y=1\) to every censored person to attain the upper endpoint, without changing any observed outcome. Additional restrictions exclude some such completions and can narrow the interval.

Diagnostics, Alternatives, and Limitations

Before interpreting a bound, check what counts as an unknown outcome. A competing event observed before censoring is a known zero for a cause-specific cumulative incidence; it must not be relabelled as censoring. Likewise, people known to be event-free through \(\tau\) have known zero outcomes. Counting either group as missing needlessly widens the bound. Conversely, treating a person censored before \(\tau\) as known event-free understates uncertainty.

The worst-case interval is especially informative when few people are censored before the horizon: its width is \(1-q\). When censoring is extensive, the interval may be too wide for a useful conclusion. Check the number and fraction with unknown status, the event coding, and whether the chosen horizon leaves enough follow-up to observe the outcome. If bounds are calculated within covariate groups, inspect whether each group has observed outcomes; a group with no known outcomes has no observed-group risk to support an ordering restriction.

These bounds answer a different question from approaches that produce a point estimate. Inverse probability of censoring weighting and related methods use assumptions such as conditional independent censoring, together with positivity, to estimate a risk. Selection models and pattern-mixture models make different assumptions about unobserved outcomes. Such assumptions can yield narrower estimates, but the observed data alone do not verify them. Bounds can complement those approaches by making the role of assumptions visible; they do not automatically validate a point estimate or provide a sampling confidence interval.

The basic bounds also have limitations. They use only the binary outcome at a chosen horizon, so they do not identify the full event-time distribution or the timing of events after censoring. They can be wide, and assumptions used to narrow them may be difficult to justify. In a multi-state setting, the same logic applies to a binary indicator such as being in a specified state at \(\tau\): known state membership contributes observed outcomes, while censored individuals have unknown membership. The target and what constitutes a known status must be defined for that estimand.

Common Mistake

• Wrong reasoning: “Censoring was unrelated to the measured covariates, so the observed event proportion is the true risk.” This does not rule out dependence between censoring and unmeasured outcomes.
• Correction: Independence from measured covariates is not, by itself, evidence that \(p_{\mathrm{mis}}=p_{\mathrm{obs}}\). Without a justified censoring assumption or a restriction on \(p_{\mathrm{mis}}\), report the worst-case bounds. If an assumption is used to narrow them, state it and show how results change under alternatives.