Intermittent Missingness Versus Monotone Dropout

Two Different Observation Patterns

A participant who leaves follow-up is not the same as a participant whose measurement is missed and later recorded again. The distinction matters because the usual inverse-probability-of-censoring construction represents loss to follow-up as an absorbing event: after censoring, later outcomes are unavailable. Intermittent missingness does not have that structure. An analysis that treats every missing visit as permanent dropout can discard valid later measurements, while a dropout weight applied to arbitrary gaps may assign the wrong observation probabilities.

Let \(Y_t\) be the outcome at visit \(t\), for \(t=1,\ldots,K\), and let \(R_t=1\) if \(Y_t\) is observed and \(R_t=0\) otherwise. The complete outcome vector is \((Y_1,\ldots,Y_K)\), while the observation pattern is \((R_1,\ldots,R_K)\). Assume for now that \(R_0=1\), so everyone is observed at baseline. The observation pattern \(1,1,0,0\) is monotone dropout; \(1,0,1,0\) is intermittent missingness.

These are properties of the recorded observation process, not of the outcome distribution. A missed visit may be unrelated to prognosis or highly informative; a permanent dropout may also be informative. In either case, weighting requires assumptions about why measurements are observed, as well as sufficient probability of observation for the histories relevant to the target.

What Monotone Censoring Weights Represent

Under monotone observation, define the conditional probability of remaining observed at visit \(j\), among those observed through visit \(j-1\), as \(q_j(H_{j-1})=P(R_j=1\mid R_{j-1}=1,H_{j-1})\). Here \(H_{j-1}\) denotes the history available before the observation decision at \(j\), such as baseline variables and previously observed outcomes and covariates. The probability of being observed through visit \(t\), conditional on the observed history along the way, is represented by the product of these conditional probabilities.

For the standard weighting identity, suppose \(Z\) is a full-data quantity whose mean is the target. Assume: (1) \(Z\) is integrable; (2) each \(q_j(H_{j-1})\) is the correctly specified conditional probability of observation among those at risk and is positive wherever the target population has support; and (3) at each decision, conditional on the observed history among those still observed, observation is independent of \(Z\). In applications, this is a sequential independent-censoring assumption: the history must contain the predictors needed to account for informative censoring. It is not established merely by fitting a model to the observed data.

Proof. Write \(G_j=\prod_{m=1}^{j}q_m(H_{m-1})\) and \(G_0=1\). Conditional on the history through \(j-1\) for a participant still observed, the observation decision has probability \(q_j(H_{j-1})\). By the sequential independence assumption, weighting the observed cases by \(1/q_j(H_{j-1})\) reverses that selection for \(Z\). In particular, the conditional expectation of the contribution at step \(j\), given the at-risk history and \(Z\), is \(E[R_j Z/q_j(H_{j-1})\mid R_{j-1}=1,H_{j-1},Z]=Z\). Because observation is monotone, \(R_t=1\) requires every earlier observation decision to be successful. Applying the conditional expectation identity successively at visits \(1,\ldots,t\) therefore gives \(E[R_t Z/G_t]=E[Z]\). Positivity ensures that each division is defined on the target support; integrability ensures the expectations exist. \(\square\)

Intermittent Observation Requires a Different Probability

When observation can resume, the probability \(G_t\) above is not the probability of observing visit \(t\). For a participant missing visit \(t-1\), the chance of being observed at \(t\) may be positive. The relevant probability depends on the observation process and the information available before that visit. To estimate one marginal mean \(E[Y_t]\), a direct observation-probability argument can sometimes suffice, but it is not automatically a longitudinal censoring-weight analysis.

Derivation. Since \(\pi_t(H_t)\) is determined by \(H_t\), conditional independence gives \(E[R_tY_t/\pi_t(H_t)\mid Y_t,H_t]=Y_t E[R_t\mid Y_t,H_t]/\pi_t(H_t)=Y_t\). Taking expectations on both sides yields the identity. Positivity makes the ratio defined, and integrability is needed for the expectations. This result concerns a single-time marginal mean. If missingness depends on unobserved information even after conditioning on \(H_t\), or if positivity fails, the identity does not follow.

Why a Dropout Weight Does Not Fill an Observation Gap

A monotone dropout model treats \(R_{j-1}=0\) as being out of the risk set for later observation. Under that model, \(P(R_j=1\mid R_{j-1}=0)=0\). In an intermittent process this probability can be positive. Therefore a product of “remaining uncensored” probabilities either assigns zero probability to a resumed record or misrepresents the chance that its measurement was observed. It cannot recover an unrecorded outcome from a gap, and it does not by itself make a partially observed history equivalent to a complete one.

The distinction is also important for analyses involving multiple visits together. A weight that identifies \(E[Y_t]\) at each visit separately does not automatically identify a joint quantity involving \((Y_1,\ldots,Y_K)\), such as a function requiring all visits to be observed. The joint observation pattern and the assumptions linking it to the full outcome history must match the target. A participant with pattern \(1,0,1\) contributes observed information at visits 1 and 3, but not the missing value at visit 2. Multiplying marginal visit-specific weights does not turn that record into a fully observed trajectory unless a suitable joint derivation justifies the construction.

Diagnostics, Alternatives, and Limits

Begin by tabulating observation patterns across visits. A pattern with a 0 followed by a 1 is direct evidence that a single absorbing censoring time is inadequate for the recorded process. Also inspect estimated observation probabilities by visit and history. Probabilities near zero imply large inverse weights; an estimated zero makes the corresponding ratio undefined and signals a positivity problem, a sparse history, or model failure. These checks can reveal structural mismatch and unstable estimation, but they cannot establish conditional independence between observation and missing outcomes.

The main alternative depends on the target. For an individual visit mean, an observation-probability model for \(R_t\) conditional on observed predictors may support the single-time identity above. For a longitudinal target involving several outcomes, one may need a model for the joint or sequential observation process that explicitly allows missed visits and return, or an analysis that models the outcome distribution under stated missing-data assumptions. These approaches require their own assumptions and do not make arbitrary missingness ignorable. In particular, neither inverse weighting nor a pattern model repairs missing-not-at-random dependence on unobserved outcomes without additional identifying assumptions or sensitivity analysis.