When calculating the effectiveness of an injury-reducing device from data on injuries with and without the device in operation, one meets the difficulty that injury is merely graded into a few categories from none to fatal, not measured on a continuous scale. This paper discusses the derivation of formulae that overcome this, and of statistical tests for whether the device has any effect.
It is shown that there are two important issues to be considered when deciding on a formula: (i) The distinction between effectiveness relative to (a) the total variability in injury severity in all accidents, or (b) the variability within the particular circumstances of an individual accident. (ii) The shape of the probability distributions of injury severity in the two circumstances, with and without the device.
As to the first issue, effectiveness of type (b) requires each case with the device to be matched with a case without the device, whereas type (a) does not; type (b) leads to a more powerful statistical test, whereas type (a) is more suitable for descriptive purposes.
As to the second issue, it is shown that the logistic distribution has suitable theoretical properties and leads to formulae and statistical tests that are conveniently simple. However, it is demonstrated that formulae that are linear in the probabilities are unlikely to be suitable.