Event rates: count time at risk before comparing frequencies
Two datasets can contain the same number of companies and the same number of events while representing very different observation time. A company followed for three months has not contributed the same evidence as one followed for a full year. Before comparing event frequencies, build an exposure ledger that shows when each company actually entered and left the relevant risk set.
The output is an event count divided by summed time at risk, with units attached. That rate is not automatically a probability over one year. First events and recurrent events also need different counting rules; changing only the numerator while leaving the old exposure denominator can produce a misleading result.
Define the event and the clock together
Assume a hypothetical research registry tracks companies until their first verified material reporting correction after enrollment. Eligibility requires no previously qualifying correction within the defined study history. The event date is the prespecified effective date, not whichever date the analyst happened to download a notice.
Use company-days as the exposure unit. Set a common maximum follow-up of 365 days from each company’s enrollment, but allow an earlier first event or loss of observable coverage to end its contribution. The actual observation start and endpoint belong in every row. Calendar entry dates can differ even when the elapsed-time rule is shared.
CDC’s archived lesson on person-time rates, under “Incidence rate or person-time rate,” explains summing observed time until specified endpoints. The company-day ledger here is an original nonclinical application of that denominator logic. It does not imply that the CDC studied these companies or certified their reporting data.
Build the exposure ledger before aggregating
Suppose four eligible companies have the following complete records. Times are elapsed days from enrollment; an event is counted at the end of its contribution.
| Company | Start | Stop | Endpoint | First events | Days at risk |
|---|---|---|---|---|---|
| A | 0 | 365 | administrative end | 0 | 365 |
| B | 0 | 365 | first event | 1 | 365 |
| C | 0 | 180 | first event | 1 | 180 |
| D | 0 | 90 | loss of coverage | 0 | 90 |
The total is 365+365+180+90=1000 company-days and two first events. The observed incidence rate is 2/1000=0.002 events per company-day, or 2 events per 1000 company-days.
C contributes no time after its first event on day 180. D contributes no time after coverage is lost on day 90 because its later event status is unknown, not because an event has already occurred. Adding those 185 and 275 days would introduce ineligible or unobserved exposure. Dividing by four companies gives 2/4, the observed event fraction; D’s unresolved 365-day status prevents that fraction from establishing the cohort’s common 365-day event probability.
In this example, A is known to have no event through day 365, while B has a qualifying event exactly at that endpoint. If all nonevent companies had complete 365-day coverage, the event proportion could indeed describe that common horizon even though event cases stopped contributing risk time earlier. Here D is right censored at 90, so that complete-outcome condition does not hold. Preserve endpoint reasons and any same-day event priority rule.
Compare rates with their own exposure denominators
For a separate illustrative registry E, suppose five events are observed over 5000 comparable company-days under the same definition. Its rate is 1 per 1000 company-days. Comparing the original registry with E gives a rate ratio 2 and rate difference 1 event per 1000 company-days.
Comparing just the counts, two versus five, would suggest the opposite ranking of frequency. The denominator resolves that difference. A rate ratio cancels a consistently used time unit, but the displayed rates and rate difference still require units. Do not divide an events-per-year figure by an events-per-day figure before conversion.
Cochrane Handbook section 6.7 distinguishes counts and rates from binary outcomes and emphasizes extracting event totals together with time at risk. Preserve the number of companies as a description of the study even though the rate denominator is summed company-time. A thousand days concentrated in very few companies can have a different dependence structure from the same total spread across many companies.
Convert a rate into a probability only under an explicit model
Multiplying 0.002 by 365 gives 0.73 events per company-year. It changes the time unit; it does not make the answer a 73% annual probability. A rate can exceed one per time unit, whereas a probability cannot exceed one.
For a purely illustrative constant-hazard first-event model with no competing exits, a daily hazard lambda 0.002 implies 365-day probability 1−exp(−0.002×365)≈0.5181. This differs from 0.73. The exponential conversion introduces assumptions about the hazard over time and the event process; it is not a probability estimate justified merely by observing two events in this four-company ledger.
If the hazard changes with company age or reporting season, one overall rate loses that timing information. A risk calculation then requires an appropriate time-to-event analysis or a justified time-varying model. Likewise, an acquisition that prevents observing the defined event is not interchangeable with a harmless change of data provider; identify competing endpoints rather than treating all stops as equivalent.
Recurrent events require an explicit re-entry rule
Now consider a different outcome: temporary operational incidents that can recur after recovery. In a separate company E, suppose incidents occur on days 90 and 180, and the company is ineligible for another incident until recovery on days 100 and 190. Follow-up ends on day 365.
Its eligible exposure intervals are (0,90], (100,180], and (190,365], with lengths 90+80+175=345 days. The recurrent-event count is two, giving 2/345, about 0.005797 incidents per company-day. The ten-day recovery periods are excluded under this explicitly chosen eligibility rule.
A first-event analysis of the same history would use one event and 90 days, giving 1/90 for that single-company record. Neither numerator can be pasted into the other’s denominator. If the intended process allows risk immediately after an incident, it would use a different exposure rule. State the scientific definition rather than choosing whichever denominator makes a preferred result look smaller.
Repeated events within a company are also dependent. A simple independent Poisson count model is not automatically justified by the presence of a time denominator. Inspect recurrence, concentration and event clustering before using model-based uncertainty intervals. More event rows do not create more independent companies.
Resolve incomplete dates and informative observation
If an annual review establishes only that a correction happened sometime in a year, the exact event date is unknown. A midpoint allocation is an assumption, not a recovered timestamp. Keep the observed interval and test how the exposure total changes under defensible timing alternatives; use a suitable interval-based analysis when exact timing matters.
Loss of coverage can also depend on impending trouble. If difficult companies disappear shortly before corrections, summing their recorded days correctly does not establish unbiased population rates. Investigate the observation mechanism, preserve loss reasons and compare follow-up completeness across groups. An accurate denominator cannot repair unobserved events by itself.
| Step | Input or calculation | Response to missing or conflicting evidence |
|---|---|---|
| Fix eligibility | event definition and first/recurrent target | do not merge incompatible event types |
| Establish the clock | origin, entry, endpoint and units | retain unknown dates as unknown |
| Sum exposure | disjoint eligible intervals per company | remove duplicate or overlapping time |
| Match events | qualifying date and company identifier | reconcile notices describing the same event |
| Compare groups | common definitions and rate units | separate changes of coverage from frequency |
| Assess assumptions | recurrence, censoring and competing exits | qualify interpretation or revise the model |
When two sources disagree on a date, compare legal effective dates, publication dates and discovery dates before selecting the clock. Record the chosen rule and the alternatives; replacing one date silently can change both event order and exposure.
Update the result when a previously missed event is verified, an interval endpoint changes, a company is found ineligible, or the recurrence rule is revised. Keep the count bridge and time bridge separately reproducible. This prevents a later change in observed frequency from being attributed to company behavior when it actually came from longer coverage or a different risk-set definition.
Deduplicate observation time as well as event notices
Two providers can cover the same company over overlapping intervals. Suppose one supplies days 0–180 and another supplies days 90–270, with no interruption of eligible observation. Adding 180+180 gives 360 company-days, while their union is only 270 days; the overlap is 90. This is a separate exposure-ledger example, not an extension of the first-event table after an event.
Choose a documented coverage rule and retain source provenance for the union. The same event may also appear in both feeds and needs a single event identity. Deduplicating only notices while retaining duplicated exposure makes the rate too small; deduplicating only time while counting both notices makes it too large. Reconcile both bridges together before comparing providers or periods.
FAQ
Is two events per 1000 company-days a yearly probability?
No. It is a rate with time in the denominator. A fixed-period probability requires further event-process information or assumptions.
Can every repeat incident be added to a first-event numerator?
No. Recurrent-event counting needs compatible eligible exposure and an appropriate dependence model.