[발췌] Peirce, experiment, cards, subjects

※ 발췌 (excerpts):

출처 0 : https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1743871/pdf/v013p00315.pdf

( ... ... ) Charles Peirce (1839–1914) graduated at the bottom of his class at Harvard (79th out of 90) in 1859 and went on to make major scientific contributions in astronomy, geodetics, psychology, and philosophy (as the founder of ‘‘pragmatism’’). He suffered from repeated bouts of depression and his worldly career was not successful.

In 1879 he was appointed instructor at Johns Hopkins where he carried out the following trial with his colleague Joseph Jastrow. The question addressed was this. Blindfolded people can easily tell the difference between a 1 kg and 2 kg weight. Can a blinded person tell the difference between 1 kg and 1 kg + 2 g? Plus 1 g? What is the difference that is too small to detect? It was a simple question, but very hard to design the experiment to find the ‘‘just noticeable difference’’. The blinded subject sat on one side of a barrier and saw only the end of the balance of a scale. On the other side of the barrier the investigator would add or remove gram weights to the kilogram weight on the scale. This was done randomly using a specially designed deck of cards so the investigator would not impose his own unknown biases in the order of the tests, thereby creating a detectable pattern. The subject let go of the balance. The gram weight was added or subtracted depending on the color of the card drawn from the deck. The subject would press on the balance and say whether the weight was more or less, and his degree of confidence in his choice (0, 1, 2, or 3). Fifty such tests were run at a time and the cards were notched and used as a rapid way of recording the results.

The statistician Stephen Stigler calls this one of the best examples of a carefully developed and explained psychological experiment. The purposeful use of random assignment of experimental changes created a known baseline of error (50/50) which could be used to compare with the actual results.

Stigler considers Peirce to be one of the two greatest American scientific minds of the 19th century. Peirce’s study was published in 1885 but this did not stop him from being
fired from John Hopkins the next year. The reasons are not clear, but, seemingly, were in part due to his divorcing his abused first wife and remarrying. His cocaine addiction
(unproved) probably did not help. It must be said that opiates were legally dispensed at the time.


출처 1: 구글도서


(The title of the chapter is "Experimental science")

( ... ... ) It was lifted off and on by means of a fine India-rubber thread, which was so much stretched by the weight as certainly to avoid any noise or jar from the momentum of the descending pan. A sufficent weight could also be hung on the beam of the balance, so as to take off the entire pressure from the finger at the end of each experiment. This weight could be applied or removed by means of a cam acting upon a lever; and its bearings upon the beam were guarded by India-rubber. It was found that the use of this arrangement, which removed all annoying irregularities of sensation connected with the removal and replace;ent of the greater (initial) pressure, rendered the results more uniform and diminished the probable error. It also shortened the time necessary for performing the experiments, so that a series of 25 experiments was concluded before the effects of fatigue were noticeable. It may be mentioned that certain causes tended to constant decrease of the probable error as the experiments went on, these mainly being an increased skill on the part of the ^operator^ and an education of the sensibility of the ^subject^. The finger was supported in such a way as to be lightly but firmly held in position, all the muscles of the arm being relaxed; and the India-rubber top of the brass enlargement at the end of the beam of the balance was never actually separated from the finger. The projecting arm of a filter-stand (the height of which could be adjusted) with some attachments not necessary to detail, gently prevented by the weight in the pan. In the case of Mr. Peirce as subject (it may be oted that Mr. Peirce is left-handed, Mr. Jastow is strongly right-handed) the tip of forefinger, and in the case of Mr. Jastow of the middle finger, of the left hand were used. In addition, a screen served to prevent the subject from having any indication whatever of the movements of the operator. It is hardly necessary to say that we were fully on guard against unconsicously received indications.

28. The observations were conducted in the following manner: At each sitting three differential weights were employed. At first we always began and ended with the heaviest, but at a later period the plan was to begin on alternate days with the lightest and heaviest. When we began with the heaviest 25 observations were made with that; then 25 with the middle one, and the 25 with the lightest; this constructed one-half of the sitting. It was completed by three more sets of 25, the order of the weights being reversed. When we began with the lightest the heaviest was used for the third and fourth sets. In this way 150 experiments on each of us were taken at one sitting of two hours.

29. A pack of 25 cards were taken, 12 red and 13 black, or vice versa, so that in the 50 experiments made at one sitting with a given differential weight, 25 red and 25 black cards should be used. ( ... ... )


출처 2: 구글도서

(The title of the chapter is "Galtonian Ideas")

( ... ... ) appears in psychophysics well before it appears in education─are we to therefore put physiological psychologists above educational researchers? Of course not.

  I do have an answer to this question, and I will reveal it soon, but in order to make it plausible I have to take a step back and ask: What was it about astronomy other than the astronomers' native intelligence that led them to statistical methods so soon?

Statistical methods for the treatment of astronimical observations evolved over the latter half of the 18th century, and they were reconciled with the mathematical theory of probability early in the 19th century. One key figure in all of this was Pierre Simon Laplace (born 1749, died 1827), another was Carl Friedrich Gauss (born 1777, died 1855). The setting of this work was one of a refined Newtonian theory for the motions of the planets, and the types of problems for which astronomers used statistics involved the determination of the "constants" of that theory. For example, supposing Jupiter traveled about the Sun in an ellipse─what were the coefficients of the equation of that ellipse? And what would they be if we allowed for perturbations owing to the gravitational effect of Saturn? And what if, to help map the stars as a background reference for the motions of planets and comets, five astronomers observed the same star with different results? How to reconcile their answers?

The key point in all this, the anchor for the whole project, was Newtonian theory. When an astronomer resorted to statistics in the 1820s, and the tool he usually reached for was the method of least squares, there was no dount in his mind that he was after something real, definite, objective, something with an independent reality outside of his observations, a genuinely Platonic reality inherited from the then-unshakable edifice of Newtonian theory. The whole of the 19th century theory of errors was keyed to this point─

observation = truth + error.

Without an objective truth, this sort of split would be impossible, for where would error end and truth begin? I might be tempted to refer to such a split as deconstruction, but as I understand the term as now used, a modern deconstructionist would be more likely to identify observation with truth and to deny the possibility of discerning error.

With an objectively defined goal, namely, astronomical or Newtonian "truth," the road was free for probability-based inferences. Probability distributions for errors could be assumed or estimated, observations could be combined by maximum likelihood or least squares, and there was no ambiguity about the nature of the result. I would contrast this with the situation the economists found themselves in a century later. Suppose we wished (as William James Jevons did in the 1860s) to determine the effect to the gold discoveries in California and Australia upon world price levels. You could gather "before" and "after" price data on a number of different commodities important in world trade. But how can you, to use an argument put forth at that time, average pepper and pig iron? How can totally different goods subject to importantly different economic forces be combined into an index that may be considered an estimate of something as real as the position of a star or the shape of the orbit of Jupier? Jevons ^did^ average such quantities, but only with copious apologies, and he did not use probability-based methods to measure the uncertainty in the result. How could he? How would he have defined the object about which he was uncertain? Perhaps Adam Smith was the Newton of economics, but there was no inverse square law for price movements.

Psychologists should feel a particular sympathy for Jevons's problem─for any time groups of people, or examination scores, or, in this day of meta-analysis, a collection of studies of educational interventions, any time a group of different meausre are to be combined in a common analysis the question must be, what is the goal? What is it that I am estimating? There is an answer to this question, but it is not to be found in the stars, and it is not the astronomers' answer. But I am getting ahead of the story. How, I asked, could psychologists bring themselves in the 1860s to use statistical methods, when it took economists another 30-40 years? What was is about the problems the psychophysicists faced that was like the problems the astronomers faced, problems involving the measurement of sensitivity, reaction times, memory? The short answer is, nothing at all; the problems were not intrinsically similar at all. Even the one famous "law" of early psychophysics, Fechner's "Law" relating sensation to the logarithm of intensity, was not a Newtonian law derived from theory, but an empirical constuct that fit only middlingly well, and it is one area of psychophysics where Fechner did not use probability. So what is the answer?

To understand what did happen, and how the psychophysicists managed to creat a surrogate for the law of gravitation, let me look at one the most careful and impressive investigations in the 19th century psychology, one that was performed in the 1880s by the American philosopher Charles S. Peirce, while on the faculty of Johns Hopkins, shortly before the president of that institutin dismissed him, apparently in part because he disapproved of Peirce's handling of his marriages. I would list Peirce as one of the two greatest American scientific minds of that century (the other being physicist J. Willard Gibbs), but Peirce was a strange person, an outlier in any educational theory. He was educated principally at home, by his father, a professor of mathematics at Harvard. The young Peirce went to Harvard, but he gratudated without distinction, 79th in a class of 91. He is said to have had a curious method of self-examination─he was ambidextrous, and he would write out questions with his left hand while simultaneously answering them with his right. He is best know as a philosopher, the father of pragmatism, a word later kidnapped by William Japmes, but he was also a physicist, a cartographer, a mathematician, and a psychologist.

The experiment I describe took place in December 1883 and January 1884 and involved a version of the experiment that Fechner had pioneered, the application of the method of right and wrong cases to the sensation of lifted weights (Peirce and Jastow, 1885; Stigler, 1978c). Indeed, I could use Fechner's own work to make my point, but Peirce's use affords a cleaner and more dramatic example of the idea.

I will briefly describe the method of right and wrong cases, as used here. An experimental subject is confronted with two apparently identical closed boxes; they differ only in that one box containes a single weight D, while the other is heavier─it contains a weight equal to D and a small weight P. The subject lifts one box, then the other, and pronounces a judgment on which is heavier. The numbers of right and wrong cases are tabulated, for various D and P, and with other conditions being varied (left hand versus right hand, heavy-first versus light-first, morning versus evening, and so forth). That, basically, was Fechner's experiment.

Peirce went up a step further, in an attempt to challenge one of Fechner's ideas. Fechner had proposed that despite the fact that sensation increased as stimulation increased, there was a threshold below which there was no sensation at all. He called that threshold the "just noticeable difference." In the context of the lifted weight experiment, Fechner would maintain that as the increment weight P increased, the fraction of "right-cases" would increase, but there was, for each base-weight D and each person, a threshold value for P─a small weight such that if P is less than that value, and there is no sensation, and there is an even chance for a right-case. Weights P that were below the just noticeable difference were indistinguishable from zero. Peirce did not buy this.

So far I have been beating around the bush, evading the question: What was it about problems like this that opened their analysis up to statistical treatment in the manner of the astronomers, and that differed from the problems of the economists? The answer is simple: experimental design. First, the ^possibility^ of experimental design─the ability of the scientist to manipulate the conditions, to sharpen the hypotheses and render limited questions capable of sharp and defitive answers. This alone distanced the experimental psychologists from the economists. And second, the cleverness of the pshcologists in exploiting this advantage to provide a novel surrogate for the anchor of Newtonian law. Let me return to Peirce's work, where this is clearest.

Peirce wanted to measure extremely subtle sensations, the perception of very small incremental weights. And he had a wonderful idea: a blind randomized experiment. In order to eliminate the biases attendant to factors such as which weight was lifted first, how the weights were arranged, or whether the subject knew which was which, Peirce worked with an assistant, Joseph Jastrow, who later had a distinguished career himself in psychology. (Fechner had experimented alone, with himself as both subject and assistant.) And Peirce employed an explicit device for randomizing the order of presentation, the order of placement. He prepared a special deck of cards for this purpose, and Peirce or Jastrow would shuffle and select a card, and prepare the weights, while the other, blind to these preparations, would be experimental subject.

The introduction of randomized experiments is usually associated with Ronald A. Fisher, in work on agricultural experimentation a half-century later, but there is no question that Peirce was clear on what he was doing and why, and his "what and why" were the same as Fisher's. Peirce wrote:

By means of these trifling devices the important object of rapidity was secured, and any possible psychological guessing of what change the operator was likely to select was avoided. A slight disadvantage in this mode of procceding arises from the long runs of one particular kind of change, which would occasionaly be produced by chance and would tend to confuse the mind of the subject. But it seems clear that this advantage was less than that which would have been occasioned by his knowing that there would be no such long runs if any means had been taken to prevent them (Peirce and Jastrow, 1885, p. 80)

Writing elsewhere, in a more philosophical vein, Peirce said that the very possibility of induction depended on such randomization:

The truth is that induction is reasong from a sample taken at random to the whole lot sampled. A sample is a ^random^ one, provided it is drawn by such machinery, artificial or physiological, that in the long run any one individual of the whole lot would get taken as often as any other (Peirce 1957, p. 217)

The point is that Peirce used randomization to create an artificial baseline that was as well understood and as well defined as any of the Platonic constants of Newtonian physincs. If Fechner was correct and the just-noticeable difference existed, then Peirce's scheme woudl guarantee that the probability of a right-case for such a small weight would be 1/2. This was the result of artifice, but it was as real as the position of any star, and it served as the basis for Peirce's subsequent probability calculations of the significance of effects, and of differences between effects.  ( ... ... )


출처 3: 구글도서

( ... ... ) As for the article by Peirce and Jastrow (1885) note the following:

First, only two subjects participated in the whole study, these being the two authors themselves. I.e., they took it in turns to serve as subject or experimenter.

Second, the article ends with the following interesting conclusion:

" The general fact has highly important practical bearings, since it gives new reason for believing that we gather what is passing in one another's minds in large measure from sensations so faint that we are not fairly aware of having them, and can give no account of how we reach our conclusions about such matters. The insignt of femailes as well as certain "telepathic" phenomena may be explained in this way. Such faint sensations ought to be fully studied by the psychological and assiduously cultivated by every man."