Signature
lambda = D / PT; T_mean = 1 / lambda
| Inputs | Definition | Unit |
|---|---|---|
D | Events among the records, each after its entry time | count |
PT | Sum over records of exit time minus entry time | person-months |
lambda | Event rate per unit of person-time under an exponential model | events per person-month |
|---|---|---|
T_mean | Mean of the fitted exponential distribution, 1 / lambda | months |
Function
Survival estimation from left-truncated records with entry-restricted risk sets
Maps records with an entry time, an exit time and an event indicator to survival from the time origin when people come under observation only after the origin. Each person informs survival only beyond entry, conditional on surviving to entry, so risk sets and likelihood contributions start at entry. The notation follows the Left Truncation article and its six registry patients.
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Implementations
Excel
Exponential rate and mean from named entry, exit and status ranges
With EntryTimes, ExitTimes and Status named, the first formula returns the rate, held in ExpRate, and the second the mean, held in ExpMeanTime.
=SUM(Status)/(SUM(ExitTimes)-SUM(EntryTimes)); =1/ExpRate
Assumptions
Constant hazard from the time origin
The hazard is the same at every time since the origin, so time before entry carries no information once the entry term is included; with a changing hazard the Weibull form of HE-FM-LTR-004 or another distribution is needed.
Independent entry for the exponential fit
Entry times are independent of event times given that events follow entry, as for the risk-set correction.
Worked examples
Exponential rate for the six registry patients
Four deaths over 103 person-months from entry give a rate of about 0.0388 per month, about 0.47 per person-year as in the article, and a mean of 25.75 months (computed here for illustration).
D = 4; PT = 103; lambda = 0.0388; T_mean = 25.75
Exponential rate when person-time is counted from diagnosis
Counting from diagnosis gives 147 person-months, a rate of about 0.0272 per month (0.33 per person-year, as in the article) and a mean of 36.75 months, 11 months longer (computed here for illustration).
D = 4; PT = 147; lambda = 0.0272; T_mean = 36.75
Common errors
Counting person-time from the origin for late entrants
The 44 months before entry for patients D, E and F could not end in a recorded death; including them lowers the rate from about 0.47 to 0.33 per person-year and lengthens the fitted mean by 11 months, and cost or event rates per person-year are diluted the same way.
Fitting a parametric curve without entry times
Stata's streg manual states that each subject contributes its density or survival conditional on the entry time; a fit that leaves out the entry term treats late entrants as observed from the origin and carries the bias into the extrapolated tail.
Sources
Likelihood contributions conditional on the entry time in parametric survival models
StataCorp. Stata Survival Analysis Reference Manual, Release 19. College Station, TX: Stata Press; 2025. Entry streg, Methods and formulas: a subject known to fail at time t contributes the density at t conditional on the entry time t0, f(t)/S(t0), and a censored observation contributes S(t)/S(t0); Remarks: streg is appropriate for data exhibiting delayed entry. With the exponential density these contributions give the rate in this record.
Counting-type survival records treated as left-truncated in flexsurvreg
Jackson C. flexsurv: Flexible parametric survival and multi-state models. R package version 2.3. Help page flexsurvreg. Argument formula: Surv objects of type right, counting, interval1 or interval2 are supported, corresponding to right-censored, left-truncated or interval-censored observations.
Canonical Identity
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