The Goldsboro Incident really happened! Nonlinearity, mathematical and statistical models


Opening up the paper today, I was pleased to see this story on the front page of the Guardian, about the Goldsboro incident in November 1961. Why pleased? Well for years the Goldsboro incident has been my analogy of choice for explaining the difference between linearity and nonlinearity, based on an interpretation of nonlinearity inspired by George Sugihara on physical vs. biological noise. I’ve always prefaced this analogy by saying that it was unconfirmed but useful – and now it appears to be true! So what happened in Goldsboro? From the companion piece in the Guardian:

The document, obtained by the investigative journalist Eric Schlosser under the Freedom of Information Act, gives the first conclusive evidence that the US was narrowly spared a disaster of monumental proportions when two Mark 39 hydrogen bombs were accidentally dropped over Goldsboro, North Carolina on 23 January 1961. The bombs fell to earth after a B-52 bomber broke up in mid-air, and one of the devices behaved precisely as a nuclear weapon was designed to behave in warfare: its parachute opened, its trigger mechanisms engaged, and only one low-voltage switch prevented untold carnage.


The conventional interpretation of nonlinearity. Doubling the input either more than doubles (e.g. oversteer in a car) or less than doubles (e.g. understeer) the response.

Our formal understanding of nonlinearity is based on the idea that, if we consider a response to an input, doubling the input will result, if there is a linear response, in a doubling of the response. Thus if I press the accelerator on my car twice as hard, I might expect to travel (approximately) twice as fast. In a nonlinear response, the return is either more than or less than twice.   However, an alternative understanding of nonlinearity is illustrated by the Goldsboro Incident, where the difference between 5 of 6 safeties failing, and 6 of 6, is the difference between an incident quietly swept under the rug for 50 years, and a monumental disaster.

The difference between 5 switches being triggered and 6 is the difference between a hole in the ground and a nuclear explosion.

The difference between 5 switches being triggered and 6 is the difference between a hole in the ground and a nuclear explosion.

This interpretation of nonlinearity can be viewed in terms of the difference between multiplication and addition. We are quite good at predicting additive phenomena; the problem is, we are are less proficient when it comes to multiplication. The recent story of the death of four year old Daniel Pelka (and this is a type of story repeated with tragic Sisyphean regularity) is a case in point. How could this happen? How could so many safety checks fail? How could so many people miss the warning signs? The truth of the matter is likely to be that there are many, many more cases where “the system” almost fails, but with no observable consequence. Overburdened, pressurised staff, sometimes under motivated or under pressure not to raise alarms unnecessarily, may cut corners or make mistakes far more often than we are aware. It is also likely true that because there is no immediate consequence to these actions (the effect of nonlinearity) the potential for disaster is missed. The question may in fact not be, why does this happen, but why does it not happen more often?

And this leads us to the concept of extrapolation and mathematical and statistical models. Statistical models are fantastically valuable tools for rigorously describing relationships in data. However they are fundamentally ontological in nature; that is, built to classify rather than to explain mechanisms, and thus the ultimate arbiter of the quality of a statistical model is the fit to the data. Of course, in designing the statistical model and in interpreting it, a good scientist will be aware of the existence of these underlying mechanisms. This awareness will drive both experimental design and observation, and the interpretation of the statistics. However, these considerations lie outside the statistical model itself. In contrast, mathematical models should be phenomenological, i.e. built to directly describe the often nonlinear relationships between variables, and therefore they are better suited to extrapolate or predict away from the data, rather than interpolate. What is often not understood, is that even very good mathematical models may give an inferior fit to the statistical within close bounds of the data – the aim is not to develop the best fit to the data, but to better be able to predict what may occur, when moving farther away from known data.

Mathematical models can often provide a poor fit the data, but, if formulated to appropriately describe a fundamental aspect of the data, can provide insight into possible trends as we move away from the known data.

Mathematical models can often provide a poor fit the data, but, if formulated to appropriately describe a fundamental aspect of the data, can provide insight into possible trends as we move away from the known data.

Of course, this is at best a caricature of both mathematical and statistical models, with modern quantitative sciences using in various ways combinations of both of them. Nevertheless there is a fundamental difference in models that aim to describe, and models that aim to explain, a difference that must be considered when evaluating the interpretation of any model.

Nonlinearity is a critical concept in ecology, evolution and epidemiology. The emergence of new pathogens is one example of this. For example, in a paper a few years ago, Nim Pathy and Angela McLean used a theoretical model, to ask whether or not a pathogen (in this case, avian influenza) that has caused hundreds of cases but with little transmission indicates that the species barrier cannot be crossed. Another way of looking at this question is to ask which is worse, 4 introductions of avian flu into humans from birds, or a single introduction, where a chain of 4 infections in humans occurs but the disease then fails? Extrapolation from currently observed data requires an insight into the underlying mechanisms that drive the phenomenon to be understood (in this case, the emergence of a new human pathogen). What Pathy and McLean showed using nonlinear mathematical models, was that a lack of demonstrated transmission cannot rule the possibility of adaptability, regardless of how many zoonoses have occurred – thus even when we think we are safe, we are not necessarily so.

Of course, while I am (unsurprisingly) a keen proponent of the use of mathematical models, it must always be kept in mind that prophecy is difficult, and the biblical admonition against following false prophets reflects the popularity of trying to predict the future, the frequency of our failures, and the ease with which we can be led into following those predictions, especially when espoused by recognised experts.


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