This post follows directly from “Some pharmacological history: an exam from 1959“. In that post, I related how two of my teachers in Leeds, James Dare and George Mogey, had encouraged my interest in statistcs. George Mogey had worked previously at the famous Wellcome Research Labs in Beckenham, Kent. He had been there at the same time as J.W. Trevan, who pioneered accurate methods of biological assay.

Another person who overlapped with Mogey and Trevan at Beckenham was C.L. Oakley. I’m told by Audrey Mogey, George’s widow, that they were good friends of the Oakleys and that probably explains why George Mogey introduced me to Cyril Oakley, who had the chair of bacteriology at Leeds while I was an undergraduate there. Oakley’s Biographical Memoir makes no mention of statistics. The only person I’ve located who knew him is Keith Holland (professor of microbiology at Leeds). He told me

“I was trained by CLO between !961-65 and he inspired me to remain in research into aspects of anaerobic bacteriology and I attended his lectures on statistics, which were highly stimulating and humorous. He frequently used examples of magicians turning lead into gold and I can not recall examples of goats and men.”

The statistical connection stems from an article that was written by Oakley in 1943, Oakley, C. L. (1943). “He-goats into young men: first steps in statistics”, University College Hospital Magazine Vol 28, 16-21. Now you can download a copy of this rather obscure publication.

The action occurs on the Brocken. The paper starts by citing the Illustrated London News (the internet of its age). In 1932 an experiment was done which allegedly dispelled the legend of the Brocken. Here it is.

Oakley uses the Brocken experiment to explain the statistical method known as probit analysis. This was obviously something he’d learned from J.W. Trevan during his time at the Beckenham lab (e.g his classic 1927 paper, The Error of Determination of Toxicity) . And it was my meeting with Oakley, as an undergraduate, that caused me to use his paper as the basis of a section in Lectures on Biostatistics.

It also explains why, ever since the late 50s, I’ve wanted to visit the Brocken. It’s only about 100 km from Göttingen, where I worked often between 1980 and 1985, but at that time the Brocken was in East Germany. I remember looking across the wall at the Harz mountains, when Erwin Neher took us into the country to pick wild bilberries (blueberries, Heidelbeeren). Reunification of Germany occurred while I was working in Heidelberg in 1991 but it was not until a month ago that I got there. We took a rail tour of Germany, and spent four days in the Harz town of Wernigerode, from where we took the Harzer Schmalspurbahn, the steam powered narrow gauge railway, to the Brocken. Here are some pictures of the trip (click first picture for an album)..

All I got was a teapot stand, with a witch on a broomstick (and I don’t even drink the stuff myself).

Interestingly, although there is plenty of tourist tat about the connection with Dr Faust and Goethe, I didn’t find any German who’d heard of the he-goat conversion legend. One of the people involved in the experiment, Harry Price (1881 – 1948) of the National Laboratory of Psychical Research seems to have been behind it, and the history is described by him in the “Bloksberg Tryst” (Blocksberg is another name for the Brocken). Another person who conducted the experiment was Professor Joad (1891 – 1953). I can just remember hearing him on the BBC Home Service (radio) programme, the Brains Trust, which also featured Julian Huxley and Jacob Bronowski (1908 -1974). They were the public intellectuals of the early 1950s. (Much later, I discovered that Bronowski was the father of Lisa Jardine, who now works at UCL).

Oakley (1943) starts by citing the account in the Illustrated London News.

“The legend of the Brocken (the famous peak in the Harz Mountains, noted for its spectre and as the haunt of witches on Walpurgis Night), according to which a virgin he-goat can be converted into “a youth of surpassing beauty” by spells performed in a magic circle at midnight, was tested on June 17 by British and German scientists and investigators, including Professor Joad and Mr. Harry Price, of the National Laboratory of Psychical Research. The object was to expose the fallacy of Black Magic and also to pay a tribute to Goethe, who used the legend in Faust. Some wore evening dress. The goat was anointed with the prescribed compound of scrapings from church bells, bats’ blood, soot and honey. The necessary maiden pure in heart, who removed the white sheet from the goat at the critical moment, was Fräulein Urta Bohn, daughter of one of the German professors taking part in the test. Her mother was a Scotswoman (formerly Miss Gordon). The scene was flood-lit and filmed. As our photographs show, the goat remained a goat and the legend of the Brocken was dispelled”.

Oakley then proposes a biological assay to measure purity in heart.

“It will he observed that the only incompletely controllable variables in the experiment (excluding Iocal variations in the church bells, bat’s blood, soot and honey) are the virgin he-goat and the maiden (virgin?) pure in heart. Virginity may for the present be regarded as an absolute character —purity in heart no doubt varies from person to person.. If, therefore, a reasonably uniform supply of virgin he-goats be obtained, and the percentage of he-goats converted bears
any relation to the purity in heart of the maiden used, we ought appear “>to
be able to measure the degree of purity in heart of the virgins available.”

The argument he uses is based directly on J.W. Trevan. The story reappeared in Chapter 7 (section 7.8, page 111) of Lectures on Biostatistics, where I used it to illustrate confidence intervals for a binomial proportion.

“We shall assume, as Oakley did, that the conversion of he-goats into young men is an all-or-nothing process; either complete conversion or nothing occurs. Oakley supposed, on this basis, that a comparison could be made between, on one hand, the percentage of he-goats converted by maidens of various degrees of purity in heart, and, on the other hand, the sort of pharmacological experiment that involves the measurement of the percentage of individuals showing a specified
effect in response to various doses of a drug. In conformity with the common pharmacological practice he supposed that a plot of percentage he-goat conversion against log purity in heart index (log PHI) would have the sigmoid form shown in Fig. 14.2.4. As explained in Chapter 14, this implies that log PHI required to convert individual he-goats is a normally distributed variable. Furthermore it means that infinite purity in heart is required to produce a population he-goat
conversion rate (HGCR) of 100 per cent..

Although there is a lack of experimental evidence on this point, the present author feels that the assumption of a normal distribution is, as so often happens, without foundation (see § 4.2). The implication of the normality assumption, that there exist he-goats so resistant to conversion that infinite purity in heart is needed to affect them, has
not been (and cannot be) experimentally verified. Furthermore the very idea of infinite purity in heart seems likely to cause despondency in most people, and should therefore be avoided until such time as its necessity may be demonstrated experimentally.”

In the light of these remarks it appears to the present author desirable that the purity in heart index should be redefined simply as the population percentage of he-goats converted. This simple operational definition means that the PHI of all maidens will fall between 0 and 100, and confidence limits for the true PHI can be found easily from the observed conversion rate (which should be binomially distributed, see §§ 3.2-3.5) using Table A2, as explained in §7.7.

For example, if it were observed that a particular maiden caused conversion of r = 2 out of n = 4 he-goats, the estimated PHI would be 100 × 2/4 = 50 per cent, and, from Table A2, confidence limits (P = 0·95) for true PHI are 6.8 – 93.2 per cent. Clearly the information be gained from a sample of only four he-goats is so imprecise that it difficult to conceive what use it could be put to. Oakley recommended that for preliminary experiments at least n = 10 he-goats should be used. If r = 5 (50 per cent) of these were observed to be converted Table A2 would give the confidence limits (P = 0·95) for the true PHI as 18·7 — 81·3 per cent. While the most extreme forms of vice and of virtue appear to be ruled out by this result, there is still considerable uncertainty about the PHI. If a greater degree of confidence were required, as for example, if a potential husband demanded a certain minimum (or, alternatively, a certain maximum) PHI before committing himself, the P = 0.99 confidence limits could found from Table A2. They are 12.8 — 87.2 per cent. The most tolerant suitor might be forgiven for requiring a larger sample.”

The statistics are pretty standard stuff. You can find out more by downloading Lectures on Biostatistics. The binomial distribution in Chapters 3, 7 and 8. Probit analysis is described in Chapter 14.

For some real statistics, please look at “An investigation of the false discovery rate and the misinterpretation of P values“, now available as a preprint on arXiv.

Follow-up