[Mortality rate and its statistical properties]

The rate is an epidemiologic measure which has a widespread use in describing the occurrence of diseases. In this paper, with a didactic approach, the definition of the mortality (morbidity) rate is introduced following two ways of reasoning: firstly, in the context of survival analysis, as an instantaneous conditional probability of failure (either disease or death) (instantaneous risk) and, secondly, as a traditional measure of rapidity of change in time. We then proceed to highlight the differences, in terms of definition, interpretation, and application, between the concepts of rate and risk. As a next step the statistical properties of the rate are explored and it is explained why the variability of the measure is simply associated with the numerator (events) and not with the denominator (person-times) of the rate. In this context the Poisson distribution is commonly considered the probability distribution which better describes the statistical variability of the observed events, and examples of such a distribution are presented. When the number of deaths is sufficiently elevated the Poisson distribution can be adequately approximated by the Gauss distribution, which is simpler and in common use in occupational medicine, and formulas are presented to compute mean and variance of the rate in this situation. When the number of deaths is small a suggestion is made of making a log transformation of the rate (or of the deaths) before using the Gauss distribution: formulas are proposed for this situation, too. As a practical application of the statistical properties presented and as a concluding example, a confidence interval for the rate is computed. Numerical and graphical comparisons of the results deriving from the use of different formulas are described.