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

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Abstract
The ratio of a number of "failures" observed in a sample to the total exposure time is a common method of estimating risks of pregnancy, death, or other contingencies. It is known to be a maximum likelihood, consistent estimator of risk if one assumes 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.

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