Imputation With Delayed Entry

Why Delayed Entry Changes the Imputation Data

In survival studies, time zero is a defined origin on the analysis time scale. It might be diagnosis, birth, or a common calendar date. Some participants are not observed from that origin: they join the study later, after having remained event-free long enough to enter. This is delayed entry, also called left truncation. It changes not only how much follow-up a participant contributes, but also which covariate history the imputation model can legitimately use.

For example, a participant may enter a study five years after diagnosis. If the event has not occurred by entry, the study observes that participant from year five onward. It does not follow that the participant was observed, or that their covariates were measured, during years zero through five. Coding them as at risk and observed from time zero creates a history that the data do not support.

The time scale and its origin matter. If follow-up is measured from study enrollment, a participant enrolled at enrollment time has entry time zero by definition. If follow-up is measured from an earlier origin, such as diagnosis, enrollment may occur at a positive time. Entry is not a new event time; it marks when this participant becomes observable and at risk in the study.

Representing Entry and Covariate History

For an individual with one analysis interval, the start time is \(L_i\), not zero. Their observed time at risk is \(T_i-L_i\). With time-varying covariates, split the observed follow-up at the relevant measurement or update times, following the start-stop convention. The first interval starts at entry; later intervals begin when the recorded covariate value changes or a new interval is otherwise required. No interval should be created before entry merely to make records start at zero.

The half-open interval is a convenient bookkeeping convention: an interval ends where the next one starts, so adjacent records do not overlap. Whether a terminal event is assigned to the endpoint of the final interval is handled by the event-status convention in the analysis data. That endpoint convention does not add pre-entry time at risk.

A covariate value can be missing during observed follow-up, or it can be unavailable because the participant had not yet entered the study. These are different situations. MICE can impute a missing value at an observed visit using an appropriate conditional model. It cannot turn a period with no observation into an observed covariate history without additional data and assumptions. If reliable pre-entry records exist, they may be incorporated as data; otherwise, do not label invented pre-entry values as measured history.

What the Imputation Model Must Respect

MICE fits a sequence of conditional models for variables with missing values. With delayed entry, the conditional model for a missing covariate must describe the data among people who entered observation, using their observed history and relevant survival information. The model should include entry time, or a suitable representation of it, when entry timing is informative about the covariate or its distribution. It should also use survival predictors and covariate history aligned to the participant’s observed times, as appropriate for the variable being imputed.

The information available for imputing a covariate measured at time \(t\) must not include covariate values from later times as if they were known at \(t\). A missing value at entry can use information measured at entry and other suitable observed variables; a missing value at a later visit can use the recorded history available up to that visit, along with properly constructed outcome information used in the imputation model. The earlier material on aligning covariate values with event and censoring times is relevant here: time labels and event status must correspond to the records being imputed.

This expression describes the conditioning information, not a guarantee that any chosen set of predictors is adequate. In particular, including \(L_i\) as a predictor does not by itself correct a misspecified imputation model or undo selection into the observed cohort. The imputation model must represent the relevant association between entry, covariates, and survival among those who entered. As with other MICE applications, a missing-at-random analysis requires that, conditional on variables in the imputation model, the probability a value is missing does not depend on its unseen value. If this is implausible, the usual MICE analysis alone does not resolve that uncertainty.

The survival information supplied to an imputation model must also respect delayed entry. For example, an outcome summary based on event status and cumulative hazard should be constructed consistently with the observed risk periods. The earlier material on survival predictors in covariate imputation gives the relevant predictor context; the specific calculation of cumulative-hazard quantities with left truncation is a separate issue. Do not use a summary computed as though a participant had been under observation from time zero.

A Short Justification: Why Time Before Entry Is Excluded

An event-free participant cannot contribute observed risk time before entering the study: the study has no record of their being under observation then. Define the at-risk indicator \(Y_i(t)\) to equal 1 when participant \(i\) is under observation and event-free immediately before time \(t\), and 0 otherwise. Apart from endpoint conventions,

The indicator is zero for \(t<L_i\), so those times contribute neither observed risk time nor observed covariate history. Integrating the indicator over the interval from entry to the terminal time gives \(T_i-L_i\); integrating from zero would incorrectly add \(L_i\) units of unobserved time. This elementary representation explains why the first start time in the imputation data must be the participant’s entry time.

Assumptions, Diagnostics, and Alternatives

A delayed-entry MICE analysis depends first on correct data representation: the time origin, entry time, terminal time, event indicator, and covariate measurement times must refer to compatible scales. It also depends on suitable conditional imputation models and on an adequate account of the observed survival information. Under a missing-at-random analysis, missingness must be conditionally independent of the unseen values after accounting for the included information. These assumptions are substantive, not consequences of running chained equations.

Useful checks include verifying that every participant has \(T_i\geq L_i\), that no start-stop row begins before entry, that interval boundaries meet without unintended overlap or gaps, and that terminal event status is attached to the correct final record. Compare entry-time distributions across missingness patterns and inspect imputed covariate distributions by entry time and follow-up interval. Implausible jumps at entry, imputed values outside permissible ranges, or systematically different imputations for otherwise similar participants at different entry times may indicate a poorly specified model. These checks can reveal problems but cannot prove that missingness assumptions hold.

MICE is not automatically superior in every application. If missingness is rare and the analysis assumptions justify it, a simpler analysis may be reasonable; if the conditional models are badly specified, multiple imputations can still be misleading. Conversely, complete-case analysis and single-value filling do not solve delayed-entry representation. Regardless of the missing-data strategy, time at risk and the observed covariate history must begin at the actual entry time.

Common Mistakes

• Wrong: “The time origin is diagnosis, so every participant’s record starts at diagnosis.” Correction: A participant who enters later contributes observed follow-up beginning at \(L_i\), not at the time origin.
• Wrong: “A missing covariate at entry means the participant’s whole pre-entry history is missing and should be imputed.” Correction: Impute the missing measurement at entry; do not invent earlier measurements unless the data and a defensible model support them.
• Wrong: “Including entry time guarantees that delayed-entry selection is handled.” Correction: Entry time can be important, but model adequacy and selection assumptions still require assessment.
• Wrong: “An imputed value can use any value observed anywhere in the participant’s record.” Correction: Preserve the temporal ordering and avoid using future covariate information as though it were known at the imputed time.