Characteristics of a Ratio Used to Estimate Failure Rates: Occurrences Per Person Year of Exposure

The ratio of the number of `failures' observed in a sample to the total exposure time is known to be a maximum likelihood, consistent estimator of risk in a homogeneous population with constant risk per unit time. Two models are here presented for both homogeneous and heterogeneous populations...

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Bibliographic Details
Published in:Biometrics 1966-06, Vol.22 (2), p.310-321
Main Author: Sheps, Mindel C.
Format: Article
Language:English
Subjects:
Biometrics
Biometry
Conditional probabilities
Estimators
Fertilization
Health hazards
Humans
Life tables
Mathematical constants
Maximum likelihood estimation
Models, Theoretical
Population estimates
Sample size
Statistical variance
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Summary:The ratio of the number of `failures' observed in a sample to the total exposure time is known to be a maximum likelihood, consistent estimator of risk in a homogeneous population with constant risk per unit time. Two models are here presented for both homogeneous and heterogeneous populations that are subject to risks of being lost to observation as well as of failure. The exact distribution of the ratio is derived and approximations are used to study its small sample properties in these populations. For homogeneous populations, the bias in the ratio (which is not large) and its considerable skewness are diminished with increasing sample size but tend to be aggravated with increasing duration of observations. For heterogeneous populations, the expected value of the ratio is a decreasing function of the duration of the observations. Particularly when losses are included, the ratio may be so erratic that it cannot be said to estimate any quantity of interest or to be a meaningful approach to the comparison of several groups.
ISSN:0006-341X
1541-0420
DOI:10.2307/2528521