Pattern-Mixture Models for Informative Censoring

Why Model the Missing Outcomes by Censoring Pattern?

When a participant is censored, the event time and cause after censoring are unobserved. If censoring is informative, people censored at a particular time or for a particular reason may have different subsequent event distributions from otherwise similar people who remain under observation. A pattern-mixture model makes that unobserved distribution explicit: it describes post-censoring outcomes separately for groups defined by their observed censoring patterns.

The distribution after censoring is not identified from the observed data alone. A model therefore needs a reference distribution and sensitivity parameters that describe how the censored group might differ from it. The aim is not to claim that the missing outcomes have been recovered exactly. It is to show how conclusions change under transparent, scientifically interpretable departures from a benchmark.

Patterns and a Sensitivity Parameterization

Definition: A censoring pattern \(G\) is a recorded feature of incomplete follow-up used to group censored observations, such as censoring time, withdrawal reason, or a combination of these. A pattern-mixture model specifies the distribution of unobserved post-censoring outcomes conditional on the observed pattern and relevant covariates or history. A sensitivity parameter indexes departures of that distribution from a chosen reference distribution.

For competing risks, let cause \(k\) range over \(1,\ldots,K\). In discrete time, let \(q_{k,t}\) be the reference probability of an event from cause \(k\) in interval \(t\), conditional on being event-free at its start and on the covariates and history used to construct the reference. Let \(q_{0,t}=1-\sum_{k=1}^K q_{k,t}\) be the reference probability of remaining event-free through that interval. These probabilities are nonnegative and sum to one.

For a person censored under pattern \(g\), a cause-specific sensitivity parameter \(\delta_{k,g}\) tilts the reference probabilities after censoring. The tilt multiplies the odds of cause \(k\) versus no event in that interval by \(\exp(\delta_{k,g})\), then renormalizes all probabilities so they still sum to one:

q_{k,t}^{(g,\delta)} = \frac{\exp(\delta_{k,g})q_{k,t}} {q_{0,t}+\sum_{\ell=1}^{K}\exp(\delta_{\ell,g})q_{\ell,t}}, \qquad q_{0,t}^{(g,\delta)} = \frac{q_{0,t}} {q_{0,t}+\sum_{\ell=1}^{K}\exp(\delta_{\ell,g})q_{\ell,t}}.

When every \(\delta_{k,g}=0\), the post-censoring distribution equals the reference distribution. This is the conditionally independent-censoring benchmark if the reference model correctly represents the event distribution for censored individuals given their covariates and history. A positive \(\delta_{k,g}\) increases the interval odds of cause \(k\) relative to remaining event-free; a negative value decreases them. These are odds tilts, not direct multipliers of event probabilities. When sensitivity parameters differ across causes, changing one cause’s probability also changes the normalized probabilities of the other causes.

Deriving a Post-Censoring Cumulative Incidence

Under this model, the probability of cause \(k\) occurring by the end of a sequence of post-censoring intervals is obtained by adding the probability of cause \(k\) in each interval, multiplied by the probability of surviving event-free through all preceding intervals. This is the discrete-time competing-risks calculation applied to the pattern-specific probabilities.

F_k^{(g,\delta)}(m) = \sum_{t=1}^{m} \left\{ \prod_{u=1}^{t-1}q_{0,u}^{(g,\delta)} \right\} q_{k,t}^{(g,\delta)}.

For \(t=1\), the empty product is defined as 1, so the first term is simply the probability of cause \(k\) in the first interval. Each later term includes the chance of having had no event in all earlier intervals. This derivation also explains why one cannot simply add interval event probabilities: a person who has already had a competing event is no longer at risk for a later event in the modeled process.

Worked Example: Two Causes After Early Censoring

Consider a hypothetical reference model for people event-free at an early censoring time. In the first post-censoring interval, the reference probabilities are \(q_1=0.10\), \(q_2=0.05\), and \(q_0=0.85\). In the next interval, conditional on reaching it event-free, they are \(q_1=0.20\), \(q_2=0.10\), and \(q_0=0.70\). Suppose both causes have \(\delta=\log(2)\) in each interval, so each cause’s odds relative to no event are doubled.

In the first interval, the denominator for renormalization is \(0.85+2(0.10)+2(0.05)=1.15\). Thus the tilted probabilities are \(q_1^{(\delta)}=0.20/1.15=0.1739\), \(q_2^{(\delta)}=0.10/1.15=0.0870\), and \(q_0^{(\delta)}=0.85/1.15=0.7391\).

In the second interval, the denominator is \(0.70+2(0.20)+2(0.10)=1.30\). The probabilities are \(q_1^{(\delta)}=0.40/1.30=0.3077\), \(q_2^{(\delta)}=0.20/1.30=0.1538\), and \(q_0^{(\delta)}=0.70/1.30=0.5385\).

The two-interval cumulative incidence for cause 1 is \(0.1739+(0.7391)(0.3077)=0.4013\). For cause 2 it is \(0.0870+(0.7391)(0.1538)=0.2007\). The probability of remaining event-free through both intervals is \((0.7391)(0.5385)=0.3980\). The three probabilities sum to approximately 1, as required. Under the reference distribution (\(\delta=0\)), the corresponding cumulative incidences are \(0.10+(0.85)(0.20)=0.2700\) for cause 1 and \(0.05+(0.85)(0.10)=0.1350\) for cause 2. These figures illustrate sensitivity to the chosen departures; they do not establish which distribution is true.

Combining Observed Outcomes and Pattern-Specific Risks

For an overall cumulative incidence by a fixed horizon \(\tau\), observed events before censoring contribute their known event indicators. A person censored before \(\tau\), while event-free, contributes the model-based probability of experiencing cause \(k\) between censoring and \(\tau\), conditional on their observed history and censoring pattern. Averaging these contributions across the original sample gives a standardized estimate:

\widehat F_k(\tau) = \frac{1}{n}\sum_{i=1}^n \left[ I(T_i\leq C_i,\ T_i\leq \tau,\ J_i=k) + I(C_i<\min(T_i,\tau))\, \widehat P_{\delta}\{T_i\leq\tau,\ J_i=k\mid T_i>C_i,\ G_i,\text{ observed history}\} \right].

Here \(T_i\) and \(J_i\) denote event time and cause when observed, \(C_i\) is censoring time, and \(I(\cdot)\) is an indicator. A participant known to be followed through \(\tau\) without the event contributes zero for that cause. The probability in the second term is calculated by accumulating the pattern-specific post-censoring probabilities over the remaining intervals. In practice, the reference probabilities may be estimated with a suitable model using observed event histories; uncertainty in the estimated reference model should also be reflected in inference.

Worked Example: Aggregating Two Censoring Patterns

Suppose a hypothetical sample has 100 participants and the target horizon is \(\tau\). Sixty have observed outcomes through \(\tau\): 12 experience cause 1, 8 experience cause 2, and 40 remain event-free. The other 40 are censored event-free before \(\tau\): 10 under an early-censoring pattern and 30 under a later-censoring pattern. Under a stated sensitivity scenario, the modeled probabilities of cause 1 after censoring by \(\tau\) are 0.30 for the early pattern and 0.12 for the later pattern. These conditional probabilities are assumed to come from the relevant pattern-specific interval calculations.

The estimated overall cause 1 cumulative incidence is the observed cause 1 contribution plus the expected cause 1 contributions among censored participants, divided by all 100 participants: \((12+10(0.30)+30(0.12))/100=(12+3+3.6)/100=0.186\). Thus the estimated cause 1 risk by \(\tau\) is 18.6% under this sensitivity scenario.

If the corresponding post-censoring cause 2 probabilities are 0.10 for the early pattern and 0.08 for the later pattern, its estimated cumulative incidence is \((8+10(0.10)+30(0.08))/100=(8+1+2.4)/100=0.114\), or 11.4%. The contribution from each pattern is weighted by how many sampled participants have that pattern; the two patterns should not be pooled as if their post-censoring distributions were necessarily identical.

Assumptions, Diagnostics, and Comparisons

A pattern-mixture analysis requires assumptions even though it is designed to examine informative censoring. The observed event and censoring records must be correctly classified; the event model must appropriately condition on measured covariates and history; and the reference probabilities must be supported by observed data for comparable participants. The sensitivity parameters then describe the remaining, unobserved difference. Because that difference is not identified by the observed data, its plausible range must be justified substantively rather than selected by a significance test.

Useful diagnostics include checking the number of participants and observed events in each pattern, examining whether reference-model predictions are plausible across covariate and time strata, and verifying that every interval’s cause probabilities and event-free probability are nonnegative and sum to one. Sparse patterns, extreme predicted probabilities, or large changes in results under small changes to \(\delta\) signal that conclusions depend heavily on extrapolation. If no comparable individuals remain observed after a censoring time, the post-censoring reference distribution is particularly weakly supported. Sensitivity analysis does not repair that lack of information; it makes the extrapolation explicit.

Censoring-weighted estimation models the censoring process and reweights observed follow-up, typically relying on conditional independent censoring and adequate positivity. A pattern-mixture analysis instead specifies what the unobserved event distribution might be within censoring patterns. These approaches answer related questions but encode assumptions differently. A pattern-mixture analysis is useful when investigators can articulate plausible differences in post-censoring outcomes, while weighting is often attractive when censoring can be adequately explained by measured history. Neither method removes the need to assess assumptions, and estimates can be unstable when patterns or weights have little support.

Worked Example: A Cause-Specific Departure

In one hypothetical interval, let the reference probabilities be \(q_1=0.08\), \(q_2=0.04\), and \(q_0=0.88\). Suppose the sensitivity scenario sets \(\delta_1=\log(3)\) and \(\delta_2=0\). This represents tripling cause 1’s odds relative to no event, while leaving cause 2’s corresponding odds unchanged before renormalization.

The denominator is \(0.88+3(0.08)+1(0.04)=1.16\). The tilted probabilities are \(q_1^{(\delta)}=0.24/1.16=0.2069\), \(q_2^{(\delta)}=0.04/1.16=0.0345\), and \(q_0^{(\delta)}=0.88/1.16=0.7586\). They sum to 1.0000 after rounding. Cause 1’s probability rises from 8% to about 20.7%, but cause 2’s probability falls from 4% to about 3.45% because the model renormalizes all outcomes. It would be incorrect to say that the cause 1 probability itself was tripled: the parameter triples cause 1’s odds relative to no event.

Common Pitfall and Limitations

A frequent mistake is to treat the independent-censoring benchmark as a finding from the data: “the estimated sensitivity parameter is zero, so censoring is noninformative.” The post-censoring outcomes are missing, so the data generally cannot determine whether the sensitivity parameter is zero. The correction is to label \(\delta=0\) as a reference assumption, examine a scientifically plausible range of nonzero values, and report how the target estimate changes across that range.

Other limitations follow from the same lack of identification. Results depend on the chosen pattern definition, reference model, covariates, and sensitivity scale. Patterns that are too broad can hide meaningful differences; patterns that are too fine can leave little data to estimate a reference distribution. Moreover, sensitivity parameters that are easy to interpret as conditional odds tilts may not correspond to equally plausible departures across all times or covariate groups. Report the patterns, reference model, sensitivity values, and resulting estimates so readers can see exactly which assumptions drive the conclusions.

Key takeaway: A pattern-mixture model makes post-censoring event distributions explicit by censoring pattern. Its sensitivity parameters cannot be learned from the missing outcomes themselves; their role is to show how competing-risk conclusions vary under defensible departures from a reference distribution.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions.

In one interval, \(q_1=0.10\), \(q_2=0.10\), and \(q_0=0.80\). If \(\delta_1=\log(2)\) and \(\delta_2=0\), calculate all three tilted probabilities.
Why does \(\delta=0\) not demonstrate from the observed data that censoring is noninformative?
A sample has 50 participants with observed cause 1 events by the horizon, 20 censored participants with pattern A, and 30 with pattern B. Their modeled post-censoring cause 1 probabilities are 0.20 and 0.10. With \(n=200\), calculate the estimated overall cause 1 risk if there are no other cause 1 contributions.
What does a lack of comparable observed participants after a particular censoring time imply for the pattern-mixture analysis?

Solutions. (1) The denominator is \(0.80+2(0.10)+0.10=1.10\). The probabilities are \(q_1^{(\delta)}=0.20/1.10=0.1818\), \(q_2^{(\delta)}=0.10/1.10=0.0909\), and \(q_0^{(\delta)}=0.80/1.10=0.7273\). (2) Post-censoring event outcomes are unobserved, so \(\delta=0\) is a reference assumption, not an empirically verified conclusion. (3) The estimate is \((50+20(0.20)+30(0.10))/200=(50+4+3)/200=0.285\), or 28.5%. (4) The reference distribution is weakly supported in that region, so the analysis relies more heavily on extrapolation and should be reported as correspondingly uncertain and assumption-dependent.