Delayed Entry

Concept

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Theoretically, Delayed Entry is a survival analysis concept in which individuals enter the risk set after the formal origin of the study time scale rather than at time zero. Also known as left truncation, delayed entry occurs when participants become observable only after surviving to a later entry time. The concept is founded on survival analysis and counting process theory. It exists to ensure that only individuals who are genuinely at risk and under observation contribute to estimation of survival probabilities and hazard functions.

Mathematically, Delayed Entry is represented by defining separate entry and exit times for each individual. An individual contributes to the likelihood only after their entry time and remains in the risk set until the event of interest or censoring. Likelihood functions, Kaplan?Meier estimators and Cox proportional hazards models are modified so that risk sets include only individuals satisfying the delayed entry criterion at each event time.

In practice, Delayed Entry is incorporated into survival datasets by recording both the time of study entry and the time of event or censoring. Statistical software automatically constructs the appropriate risk sets for estimation of survival functions and hazard ratios. Delayed entry commonly occurs in registry studies, longitudinal cohort studies and health technology assessments where participants enter observation after diagnosis, treatment initiation or another qualifying event.

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Purpose

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Used to account for participants entering observation after the start of the study time scale, ensuring valid estimation of survival functions and treatment effects by constructing appropriate risk sets.

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Mathematical Formulae

Primary Formula

Observed follow-up interval:

L? < T? � C?

where:

• L? = delayed entry (left truncation) time
• T? = event time
• C? = censoring time

Supporting Formulae

Risk set at time t:

R(t) = {i : L? � t < T?}

Partial likelihood for the Cox proportional hazards model:

L(?) = ? exp(??X?) / ???R(t?) exp(??X?)

where R(t?) is the delayed-entry risk set at event time t?.

Related Mathematical Methods

• Left Truncation
• Survival Analysis
• Kaplan?Meier Estimation
• Cox Proportional Hazards Model
• Partial Likelihood Estimation
• Counting Process Models
• Maximum Likelihood Estimation

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Example

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A cancer registry follows patients from diagnosis, but only those alive when the registry was established are enrolled. One patient is diagnosed at month 0, enters the registry at month 18 and dies at month 42. The participant therefore enters the risk set at 18 months and contributes survival information only between months 18 and 42. Failure to account for delayed entry would overestimate survival because individuals who died before registry enrolment would be excluded.

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Excel Implementation

Function	Example Formula	Health Economics Application
MAX	=MAX(Entry_Time,Analysis_Start)	Determine the effective start of follow-up.
IF	=IF(Event_Time>=Entry_Time,1,0)	Verify valid delayed-entry observations.
MIN	=MIN(Event_Time,Censor_Time)	Calculate observed exit time.
DAYS	=DAYS(Exit_Date,Entry_Date)	Calculate follow-up duration after delayed entry.
FILTER	=FILTER(A2:E1000,B2:B1000<=Analysis_Time)	Identify individuals eligible for the risk set at a specified time.

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VBA (Optional)

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VBA can automate construction of delayed-entry survival datasets by generating entry times, exit times and time-specific risk sets for survival analyses.

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Sources

• Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
• Therneau TM, Grambsch PM. Modeling Survival Data: Extending the Cox Model.
• Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data.
• Andersen PK, Borgan ?, Gill RD, Keiding N. Statistical Models Based on Counting Processes.
• NICE. Health Technology Evaluation Manual.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.