A Scale for Measuring Very Rare Events

There has long been a need for a better system for describing the probabilities of very rare and nearly certain events. The problem is not just that the public has great difficulty in comprehending the risks of events such as cancer or nuclear power plant meltdown, to name just two examples, but also that scientists themselves often find probability calculations to be something of a black art, to be left strictly in the hands of statisticians. A system is needed for describing and calculating probabilites on an "order of magnitude" basis, so that very rare or nearly certain events can be compared and comprehended in a more intuitive way. A new scale is proposed here, which will help the scientific and technical community describe and compare these probabilities. It is hoped that this scale will provide a uniform standard with which the probabilities of both very rare or nearly certain events can be communicated and intelligently compared.

The new scale can be referred to as the log-odds scale. It is logarithmic, and similar in spirit to the Richter Scale. To convert a probability to log-odds, first convert to odds, then take the logarithm (base 10). In algebraic terms, the scale value r for a probability of p is given by:

r = log₁₀[ p/(1–p) ].

Table 1 exhibits the correspondence in terms of metric name, log-odds, odds, and probability. As will be demonstrated below, this scale is ideally suited for the kinds of rough calculations that are routinely required in risk assessment in all disciplines of science. Table 1 also shows the approximate probabilities corresponding to the points on the scale. The tabled probability for a scale value of r is correct to within 10^–2|r|.

An event that registers 6 on the log-odds scale is nearly certain to happen — there is a one-in-a-million chance that it will not occur. An event that registers a 5 is less certain — the odds that it will not happen are only one in 100,000. For every one point reduction in the log-odds scale, the odds against are reduced by a factor of ten. Rare events have negative scale values: an event that registers a –9 on the log-odds scale has a one-in-a-billion chance of occurring.

Log-odds have, of course, been used by statisticians for many decades⁽¹⁾, but only for technical purposes and never as a communication tool. The proposal here is, in essence, to organize the log-odds concept into a useful measurement system, complete with metric terminology and rules for order-of-magnitude calculations.

Several examples of contemporary interest may serve to demonstrate the use of the proposed scale:

1. Shortly after the nuclear accident at Chernobyl, it was reported in the press that the Russian assessment of the chances of a serious accident in this class of plant had been placed at roughly one in 10,000 plant-years. Thus, by their estimation, the explosion at Chernobyl was an event that registered –4 on the log-odds scale (i.e. 100 microchances per plant per year).
2. The U.S. probability of death due to lung cancer for smokers is about one millichance per year (1 per 1000 per annum). Thus death due to lung cancer is an event that registers –3 on the annual log-odds scale for each smoker. For non-smokers this event is much less likely: it registers –4.15 on the annual log-odds scale. The difference in scale values (1.15) indicates that the increase in risk is greater than an order of magnitude.
3. The probability of the decay of a single proton in a vacuum is a rare event with an annual log-odds scale value of –32, corresponding to an expected lifetime of about 10³² years.

The utility of the log-odds scale for describing and comparing rare events becomes evident when risk calculations are to be performed. Four rules for approximate risk calculations can be simply stated:

1. The log-odds for the non-occurrence of an event is the negative of the log-odds for its occurrence.
2. The log-odds for the simultaneous occurrence of two independent rare events is the sum of their individual log-odds.
3. The log-odds for rare event A given rare event B is the log-odds of the simultaneous occurence of A and B minus the log-odds for B.
4. The log-odds for the occurrence of at least one of 10^k rare independent events, each with log-odds r, is r+k.

Rule 1 is exact, while Rules 2–4 are approximate (the quality of the approximation is given in Table 1). For now it will suffice to say that these rules are quite adequate for rough calculations with events whose scale values are in the millichance range, and that the quality improves exponentially as the events become more rare. Calculating with log-odds rather than probabilities is reasonable when, as is usually the case, the probabilities are known only to within an order of magnitude.

Rule 1 can be used to show, for example, that if a Chernobyl-class accident is a 100 microchance per year event (–4 on the log-odds scale), then the log-odds for such an accident not occurring is +4. Stated in metric terminology, there are 10 kilochances per year that any particular Russian nuclear power-plant will not have a Chernobyl-class accident.

Rule 2 can be used to think about the components of risk. Suppose that the annual log-odds that the operators of a nuclear plant operators will sleep at least one night on the job is –3, and that the annual log-odds for a nighttime serious equipment failure at the plant is –2. Then Rule 1 suggests that the annual log-odds for an unobserved serious equipment failure at the plant is –3 – 2 = –5, i.e. a 10 microchance per year event.

Rule 4 can be used for repetitions of similar independent events. For example, suppose there are 1000 nuclear reactors operating worldwide with roughly the same log-odds scale value as the Chernobyl plant (i.e. –4), then the value for at least one serious accident in any given year is –4 + 3 = –1. This suggests that we can expect, on average, one Chernobyl-class accident per decade, using the Russian's own estimates of odds.

Rule 4 also provides a justification for the metric system nomenclature presented in Table 1 to describe the points on the log-odds scale. Consider a repeatable experiment that only has two outcomes: success or failure. It should be clear that the occurrence of at least one success in 10 independent repetitions of the experiment is an event that is a full point higher on the scale than the occurrence of one success in just one trial. Similarly, one can say that the occurrence of at least one microchance event in 1000 independent trials constitutes a millichance event, etc. In the extreme example of proton decay, in order to have merely an even chance of seeing at least one proton decay, one would have to continuously observe an ensemble of 10³² protons for a full year (over 150,000 liters of water).

When the probability of an event E depends upon a quantitative variable, the relationship frequently has a logistic shape. The mathematical form for this relationship is usually assumed to have this form:

Odds{ E | x } = 10^a+bx : 1
Prob{ E | x } = 1 / (1 + 10^–(a+bx)).

Under this assumption, the log-odds for event E given x is simply a+bx. This linear relationship, well-known to statisticians as the Cox model⁽²⁾, provides perhaps the strongest reason for using the log-odds scale instead of the probability scale. When the log-odds scale is used in this context, the slope coefficient b is immediately interpretable as the change in log-odds per unit change in x. In the smoking example, the contribution of smoking to the annual log-odds for death due to lung cancer is about 0.06 per daily cigarette [ 1.15/20 = 0.0575 = 0.06 ].

To summarize, there are many reasons why the probabilities of very rare events should be reported using the log-odds scale instead of the probability scale. Mathematically the two are equivalent, but the edge goes to log-odds for intuitive appeal and simplicity of calculation. Further, the odds scale is easily integrated into the metric system, with natural terminology and great expressive power. Historically, the odds scale was the preferred system for measuring probabilities by professional gamblers; it was not until recent times that probabilities were used at all. For the sake of better scientific communication and understanding, the log-odds scale should be used when discussing either very rare or nearly certain events.

Footnotes
(1) Fleiss, JL. Statistical Methods for Rates and Proportions. Wiley, 1981.
(2) Cox, DR. Journal of the Royal Statistical Society, B-34, 187-220, 1972.

Copyright © 1987 by Loren Cobb. All rights reserved. Latest revision: April, 1998.
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