Censoring Positivity Along Follow-Up

Why Positivity Is a Follow-Up Condition

In longitudinal studies, participants may stop being observed at different times. The chance of remaining observed can depend on their recorded history: for example, participants with worsening symptoms may be more likely to drop out. To learn about outcomes for the target population, it is not enough that some participants remain under observation overall. At each follow-up decision, the study must retain a positive chance of observation for every history that matters to the target.

This condition is called censoring positivity. It is distinct from sequential independent censoring. Independence concerns whether, conditional on the observed history, censoring is related to future outcomes. Positivity concerns whether observation continues at all in each relevant history. Independence cannot supply information for a history in which nobody can remain observed.

Definition at Each Follow-Up Time

Consider discrete follow-up times \(t=1,\ldots,K\). Let \(R_t=1\) indicate that a participant is observed through time \(t\), and let \(R_t=0\) indicate that censoring has occurred by then. Everyone begins observed, so \(R_0=1\), and \(R_t=1\) implies \(R_{t-1}=1\). Let \(H_t\) denote the history available immediately before the censoring decision at time \(t\), including the relevant past treatment, covariate, and outcome information. The conditional probability of continuing observation is

The conditioning describes a decision among people still observed just before time \(t\). It does not condition on future histories. In particular, a history \(H_t\) is considered at the time it is available, not as part of a later complete history.

For discrete histories, “almost every” means that each target-relevant history with positive target probability must have \(g_t(h)>0\). For continuous histories, a requirement at every mathematically possible value would be too strong: values outside the target support do not matter. The relevant support is determined by the target estimand, not merely by the histories that happen to appear among observed participants.

The condition is sequential because it applies to each decision among those still at risk of censoring. A history relevant at a later time may only be reached if earlier observation continues. Thus positivity must hold along the follow-up process, rather than only for the overall probability of being observed at the final time.

What a Zero Probability Means

A structural zero occurs when \(g_t(h)=0\) for a target-relevant history. No participant with that history who is at risk remains observed through the decision at time \(t\). This is a failure of support, not merely a small-sample problem. Increasing the sample size under the same design does not create observations in that history.

A simple identity makes the point precise. For any history event \(B\) determined by \(H_t\),

Here \(I(A)\) is 1 when event \(A\) occurs and 0 otherwise. To verify the identity, note that \(R_t=1\) implies \(R_{t-1}=1\), and take the conditional expectation of \(I(R_t=1)\) given \(R_{t-1}\) and \(H_t\). On the event \(R_{t-1}=1\), that conditional expectation is \(g_t(H_t)\). Multiplication by \(I(H_t\in B)\) and taking expectations gives the displayed equality. Therefore, if \(g_t(h)=0\) throughout a target-relevant set \(B\), the probability of being observed at time \(t\) with a history in \(B\) is zero, even if that set matters to the target.

The identity also clarifies the risk-set qualification. A conditional probability \(g_t(h)\) is defined by the censoring mechanism among people with \(R_{t-1}=1\). If a history is never reached among those people because observation failed earlier, that earlier decision must be examined. Positivity is checked decision by decision, including the prior opportunities to remain observed.

Worked Examples

Formal Consequences and a Useful Stronger Condition

The following result gives a direct way to see how small continuation probabilities affect the size of later risk sets. It is a probability statement, not a formula for correcting outcome estimates.

Proof. Conditioning on \(R_{t-1}\) and \(H_t\), and using \(R_t=1\) only when \(R_{t-1}=1\), gives

On the event \(R_{t-1}=1\), the assumed lower bound gives \(g_t(H_t)\geq\epsilon\). Hence

This establishes the result. \(\square\)

Iterating the inequality shows that if the same lower bound \(\epsilon\) holds at each of \(K\) decisions, then \(P(R_K=1)\geq\epsilon^K P(R_0=1)=\epsilon^K\). This is a deliberately strong, uniform condition. Ordinary positivity requires probabilities to be positive on relevant histories, but does not require a common lower bound. In finite samples, however, very small probabilities can leave later risk sets sparse even when formal positivity holds.

For a finite set of relevant histories at a fixed time, pointwise positivity does imply a positive minimum: if the probabilities are \(g_t(h_1),\ldots,g_t(h_m)\), each is positive, so their minimum is positive. This fact may not extend to continuous histories or to an infinite set of discrete histories. A sequence of positive probabilities can approach zero without ever equaling zero.

Diagnostics, Alternatives, and Limitations

Positivity is a property of the study design and the distribution of histories, not a goodness-of-fit statistic. Diagnostics should therefore examine censoring by time and by substantively important history features. Useful checks include tabulating the numbers at risk and the numbers continuing within history strata; plotting or summarizing fitted continuation probabilities; and checking whether important target histories are absent from the risk sets. Sparse cells and fitted probabilities near zero are warnings, not proof of a structural zero. Conversely, an adequate overall retention rate can conceal a zero in a subgroup.

A common mistake is to say, “Some people remain observed at every time, so positivity holds.” This checks only the overall follow-up rate. If every person with a particular target-relevant history is censored at one decision, positivity fails for that history even while many others continue. The correction is to assess conditional continuation probabilities at each decision across the relevant history support.

Sequential independent censoring and positivity answer different questions. Independent censoring says, in effect, that after conditioning on the specified history, censoring does not carry additional information about future outcomes. Positivity says that there are observed participants at the next time for the histories to which that claim is applied. Both conditions are needed for the usual nonparametric identification arguments, but neither implies the other.

When positivity fails, possible responses depend on the scientific question. A study can improve follow-up procedures for histories with high dropout, collect additional data, or define a narrower target population that excludes histories without follow-up support. Restricting the target changes the estimand and should be justified explicitly. Parametric extrapolation may yield a numerical answer, but it does not restore information absent from the data; its conclusions then depend on the extrapolation model. Truncating extreme quantities in later estimation procedures may reduce numerical instability, but it cannot turn a structural zero into positive support.

The central limitation is that positivity is always relative to a target. A history irrelevant to the target need not satisfy the condition for that analysis. A history relevant to a broader population cannot be ignored merely because it is rare in the observed data. Define the target first, then examine continuation probabilities at every follow-up decision for the histories that target requires.