Neyman

Jerzy Neyman and “Les Miserables Citations” (statistical theater in honor of his birthday)

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Neyman April 16, 1894 – August 5, 1981

For my final Jerzy Neyman item, here’s the post I wrote for his birthday last year: 

A local acting group is putting on a short theater production based on a screenplay I wrote:  “Les Miserables Citations” (“Those Miserable Quotes”) [1]. The “miserable” citations are those everyone loves to cite, from their early joint 1933 paper:

We are inclined to think that as far as a particular hypothesis is concerned, no test based upon the theory of probability can by itself provide any valuable evidence of the truth or falsehood of that hypothesis.

But we may look at the purpose of tests from another viewpoint. Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behavior with regard to them, in following which we insure that, in the long run of experience, we shall not be too often wrong. (Neyman and Pearson 1933, pp. 290-1).

In this early paper, Neyman and Pearson were still groping toward the basic concepts of tests–for example, “power” had yet to be coined. Taken out of context, these quotes have led to knee-jerk (behavioristic) interpretations which neither Neyman nor Pearson would have accepted. What was the real context of those passages? Well, the paper opens, just five paragraphs earlier, with a discussion of a debate between two French probabilists—Joseph Bertrand, author of “Calculus of Probabilities” (1907), and Emile Borel, author of “Le Hasard” (1914)! According to Neyman, what served “as an inspiration to Egon S. Pearson and myself in our effort to build a frequentist theory of testing hypotheses”(1977, p. 103) initially grew out of remarks of Borel, whose lectures Neyman had attended in Paris. He returns to the Bertrand-Borel debate in four different papers, and circles back to it often in his talks with his biographer, Constance Reid. His student Erich Lehmann (1993), regarded as the authority on Neyman, wrote an entire paper on the topic: “The Bertrand-Borel Debate and the Origins of the Neyman Pearson Theory”.

Since it’s Saturday night, let’s listen in on this one act play, just about to begin at the Elba Dinner Theater. Don’t worry, food and drink are allowed to be taken in. (I’ve also included, in the References, several links to papers for your weekend reading enjoyment!)  There go les trois coups–the curtain’s about to open!

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The curtain opens with a young Neyman and Pearson (from 1933) standing mid-stage, lit by a spotlight. (Neyman does the talking, since its his birthday).

Neyman: “Bertrand put into statistical form a variety of hypotheses, as for example the hypothesis that a given group of stars…form a ‘system.’ His method of attack, which is that in common use, consisted essentially in calculating the probability, P, that a certain character, x, of the observed facts would arise if the hypothesis tested were true. If P were very small, this would generally be considered as an indication that…H was probably false, and vice versa. Bertrand expressed the pessimistic view that no test of this kind could give reliable results.

Borel, however, considered…that the method described could be applied with success provided that the character, x, of the observed facts were properly chosen—were, in fact, a character which he terms ‘en quelque sorte remarquable’” (Neyman and Pearson 1933, p.141/290).

The stage fades to black, then a spotlight shines on Bertrand, stage right.

Bertrand: “How can we decide on the unusual results that chance is incapable of producing?…The Pleiades appear closer to each other than one would naturally expect…In order to make the vague idea of closeness more precise, should we look for the smallest circle that contains the group? the largest of the angular distances? the sum of squares of all the distances?…Each of these quantities is smaller for the group of the Pleiades than seems plausible. Which of them should provide the measure of implausibility. …

[He turns to the audience, shaking his head.]

The application of such calculations to questions of this kind is a delusion and an abuse.” (Bertrand, 1907, p. 166; Lehmann 1993, p. 963).

The stage fades to black, then a spotlight appears on Borel, stage left.

Borel: “The particular form that problems of causes often take…is the following: Is such and such a result due to chance or does it have a cause? It has often been observed how much this statement lacks in precision. Bertrand has strongly emphasized this point. But …to refuse to answer under the pretext that the answer cannot be absolutely precise, is to… misunderstand the essential nature of the application of mathematics.” (ibid. p. 964) Bertrand considers the Pleiades. ‘If one has observed a [precise angle between the stars]…in tenths of seconds…one would not think of asking to know the probability [of observing exactly this observed angle under chance] because one would never have asked that precise question before having measured the angle’… (ibid.)

The question is whether one has the same reservations in the case in which one states that one of the angles of the triangle formed by three stars has “une valeur remarquable” [a striking or noteworthy value], and is for example equal to the angle of the equilateral triangle…. (ibid.)

Here is what one can say on this subject: One should carefully guard against the tendency to consider as striking an event that one has not specified beforehand, because the number of such events that may appear striking, from different points of view, is very substantial” (ibid. p. 964).J. Neyman and E. Pearson

The stage fades to black, then a spotlight beams on Neyman and Pearson mid-stage. (Neyman does the talking)

Neyman: “We appear to find disagreement here, but are inclined to think that…the two writers [Bertrand and Borel] are not really considering precisely the same problem. In general terms the problem is this: Is it possible that there are any efficient tests of hypotheses based upon the theory of probability, and if so, what is their nature. …What is the precise meaning of the words ‘an efficient test of a hypothesis’?” (1933, p. 140/290)

“[W]e may consider some specified hypothesis, as that concerning the group of stars, and look for a method which we should hope to tell us, with regard to a particular group of stars, whether they form a system, or are grouped ‘by chance,’…their relative movements unrelated.” (ibid.)
“If this were what is required of ‘an efficient test’, we should agree with Bertrand in his pessimistic view. For however small be the probability that a particular grouping of a number of stars is due to ‘chance’, does this in itself provide any evidence of another ‘cause’ for this grouping but ‘chance’? …Indeed, if x is a continuous variable—as for example is the angular distance between two stars—then any value of x is a singularity of relative probability equal to zero. We are inclined to think that as far as a particular hypothesis is concerned, no test based upon the theory of probability can by itself provide any valuable evidence of the truth or falsehood of that hypothesis. But we may look at the purpose of tests from another view-point.” (Emphasis added; ibid. pp. 141-2; 290-1)

Fade to black, spot on narrator mid-stage:

Narrator: We all know our famous (miserable) lines are about to come. But let’s linger on the “as far as a particular hypothesis is concerned” portion. For any particular case, one may identify a data dependent feature x that would be highly improbable “under the particular hypothesis of chance”. We must “carefully guard,” Borel warns, “against the tendency to consider as striking an event that one has not specified beforehand”. But if you are required to set the test’s capabilities ahead of time then you need to specify the type of falsity of Ho, the distance measure or test statistic beforehand. An efficient test should capture Fisher’s concern with tests sensitive to departures of interest. Listen to Neyman over 40 years later, reflecting on the relevance of Borel’s position in 1977.

Fade to black. Spotlight on an older Neyman, stage right.

Neyman April 16, 1894 – August 5, 1981

Neyman: “The question (what is an efficient test of a statistical hypothesis) is about an intelligible methodology for deciding whether the observed difference…contradicts the stochastic model….

This question was the subject of a lively discussion by Borel and others. Borel was optimistic but insisted that: (a) the criterion to test a hypothesis (a ‘statistical hypothesis’) using some observations must be selected not after the examination of the results of observation, but before, and (b) this criterion should be a function of the observations (of some sort remarkable) (Neyman 1977, pp. 102-103).
It is these remarks of Borel that served as an inspiration to Egon S. Pearson and myself in our effort to build a frequentist theory of testing hypotheses.”(ibid. p. 103)

Fade to back. Spotlight on an older Egon Pearson writing a letter to Neyman about the preprint Neyman sent of his 1977 paper. (The letter is unpublished, but I cite Lehmann 1993).

Pearson: “I remember that you produced this quotation [from Borel] when we began to get our [1933] paper into shape… The above stages [wherein he had been asking ‘Why use that particular test statistic?’] led up to Borel’s requirement of finding…a criterion which was ‘a function of the observations ‘en quelque sorte remarquable’. Now my point is that you and I (perhaps my first leading) had ourselves reached the Borel requirement independently of Borel, because we were serious humane thinkers; Borel’s expression neatly capped our own.”

Fade to black. End Play

Egon has the habit of leaving the most tantalizing claims unpacked, and this is no exception: What exactly is the Borel requirement already reached due to their being “serious humane thinkers”? I can well imagine growing this one act play into something like the expressionist play of Michael Fraylin, Copenhagen, wherein a variety of alternative interpretations are entertained based on subsequent work and remarks. I don’t say that it would enjoy a long life on Broadway, but a small handful of us would relish it.

As with my previous attempts at “statistical theatre of the absurd, (e.g., “Stat on a hot-tin roof”) there would be no attempt at all to popularize—only published quotes and closely remembered conversations would be included.

Deconstructions on the Meaning of the Play by Theater Critics

It’s not hard to see that “as far as a particular” star grouping is concerned, we cannot expect a reliable inference to just any non-chance effect discovered in the data. The more specific the feature is to these particular observations, the more improbable. What’s the probability of 3 hurricanes followed by 2 plane crashes (as occurred last month, say)? Harold Jeffreys put it this way: any sample is improbable in some respect;to cope with this fact statistical method does one of two things: appeals to prior probabilities of a hypothesis or to error probabilities of a procedure. The former can check our tendency to find a more likely explanation H’ than chance by an appropriately low prior weight to H’. What does the latter approach do? It says, we need to consider the problem as of a general type. It’s a general rule, from a test statistic to some assertion about alternative hypotheses, expressing the non-chance effect. Such assertions may be in error but we can control such erroneous interpretations. We deliberately move away from the particularity of the case at hand, to the general type of mistake that could be made.

Isn’t this taken care of by Fisher’s requirement that Pr(P < p0; Ho) = p—that the test rarely rejects the null if true?   It may be, in practice, Neyman and Pearson thought, but only with certain conditions that were not explicitly codified by Fisher’s simple significance tests. With just the null hypothesis, it is unwarranted to take low P-values as evidence for a specific “cause” or non-chance explanation. Many could be erected post data, but the ways these could be in error would not have been probed. Fisher (1947, p. 182) is well aware that “the same data may contradict the hypothesis in any of a number of different ways,” and that different corresponding tests would be used.

The notion that different tests of significance are appropriate to test different features of the same null hypothesis presents no difficulty to workers engaged in practical experimentation. [T]he experimenter is aware of what observational discrepancy it is which interests him, and which he thinks may be statistically significant, before he inquires what test of significance, if any, is available appropriate to his needs (ibid., p. 185).

Even if “an experienced experimenter” knows the appropriate test, this doesn’t lessen the importance of NP’s interest in seeking to identify a statistical rationale for the choices made on informal grounds. In today’s world, if not in Fisher’s day, there’s legitimate concern about selecting the alternative that gives the more impressive P-value.

Here’s Egon Pearson writing with Chandra Sekar: In testing if a sample has been drawn from a single normal population, “it is not possible to devise an efficient test if we only bring into the picture this single normal probability distribution with its two unknown parameters. We must also ask how sensitive the test is in detecting failure of the data to comply with the hypotheses tested, and to deal with this question effectively we must be able to specify the directions in which the hypothesis may fail” ( p. 121). “It is sometimes held that the criterion for a test can be selected after the data, but it will be hard to be unprejudiced at this point” (Pearson & Chandra Sekar, 1936, p. 129).

To base the choice of the test of a statistical hypothesis upon an inspection of the observations is a dangerous practice; a study of the configuration of a sample is almost certain to reveal some feature, or features, which are exceptions if the hypothesis is true….By choosing the feature most unfavourable to Ho out of a very large number of features examined it will usually be possible to find some reason for rejecting the hypothesis. It must be remembered, however, that the point now at issue will not be whether it is exceptional to find a given criterion with so unfavourable a value. We shall need to find an answer to the more difficult question. Is it exceptional that the most unfavourable criterion of the n, say, examined should have as unfavourable a value as this? (ibid., p. 127).

Notice, the goal is not behavioristic; it’s a matter of avoiding the glaring fallacies in the test at hand, fallacies we know all too well.

“The statistician who does not know in advance with which type of alternative to H0 he may be faced, is in the position of a carpenter who is summoned to a house to undertake a job of an unknown kind and is only able to take one tool with him! Which shall it be? Even if there is an omnibus tool, it is likely to be far less sensitive at any particular job than a specialized one; but the specialized tool will be quite useless under the wrong conditions” (ibid., p. 126).

In a famous result, Neyman (1952) demonstrates that by dint of a post-data choice of hypothesis, a result that leads to rejection in one test yields the opposite conclusion in another, both adhering to a fixed significance level. [Fisher concedes this as well.] If you are keen to ensure the test is capable of teaching about discrepancies of interest, you should prespecify an alternative hypothesis, where the null and alternative hypothesis exhaust the space, relative to a given question. We can infer discrepancies from the null, as well as corroborate their absence by considering those the test had high power to detect.

Playbill Souvenir

Let’s flesh out Neyman’s conclusion to the Borel-Bertrand debate: if we accept the words, “an efficient test of the hypothesis H” to mean a statistical (methodological) falsification rule that controls the probabilities of erroneous interpretations of data, and ensures the rejection was because of the underlying cause (as modeled), then we agree with Borel that efficient tests are possible. This requires (a) a prespecified test criterion to avoid verification biases while ensuring power (efficiency), and (b) consideration of alternative hypotheses to avoid fallacies of acceptance and rejection. We must steer clear of isolated or particular curiosities to find indications that we are tracking genuine effects.

“Fisher’s the one to be credited,” Pearson remarks, “for his emphasis on planning an experiment, which led naturally to the examination of the power function, both in choosing the size of sample so as to enable worthwhile results to be achieved, and in determining the most appropriate tests” (Pearson 1962, p. 277). If you’re planning, you’re prespecifying, perhaps, nowadays, by means of explicit preregistration.

Nevertheless prespecifying the question (or test statistic) is distinct from predesignating a cut-off P-value for significance. Discussions of tests often suppose one is somehow cheating if the attained P-value is reported, as if it loses its error probability status. It doesn’t.[2] I claim they are confusing prespecifying the question or hypothesis, with fixing the P-value in advance–a confusion whose origin stems from failing to identify the rationale behind conventions of tests, or so I argue. Nor is it even that the predesignation is essential, rather than an excellent way to promote valid error probabilities.

But not just any characteristic of the data affords the relevant error probability assessment. It has got to be pretty remarkable!

Enter those pivotal statistics called upon in Fisher’s Fiducial inference. In fact, the story could well be seen to continue in the following two posts: “You can’t take the Fiducial out of Fisher if you want to understand the N-P performance philosophy“, and ” Deconstructing the Fisher-Neyman conflict wearing fiducial glasses”.

[1] Or, it might have been titled, “A Polish Statistician in Paris”, given the remake of “An American in Paris” is still going strong on Broadway, last time I checked.

[2] We know that Lehmann insisted people report the attained p-value so that others could apply their own preferred error probabilities. N-P felt the same way. (I may add some links to relevant posts later on.)

REFERENCES

Bertrand, J. (1888/1907). Calcul des Probabilités. Paris: Gauthier-Villars.

Borel, E. 1914. Le Hasard. Paris: Alcan.

Fisher, R. A. 1947. The Design of Experiments (4th ed.). Edinburgh: Oliver and Boyd.

Lehmann, E.L. 2012. “The Bertrand-Borel Debate and the Origins of the Neyman-Pearson Theory” in J. Rojo (ed.), Selected Works of E. L. Lehmann, 2012, Springer US, Boston, MA, pp. 965-974.

Neyman, J. 1952. Lectures and Conferences on Mathematical Statistics and Probability. 2nd ed. Washington, DC: Graduate School of U.S. Dept. of Agriculture.

Neyman, J. 1977. “Frequentist Probability and Frequentist Statistics“, Synthese 36(1): 97–131.

Neyman, J. & Pearson, E. 1933. “On the Problem of the Most Efficient Tests of Statistical Hypotheses“, Philosophical Transactions of the Royal Society of London 231. Series A, Containing Papers of a Mathematical or Physical Character: 289–337.

Pearson, E. S. 1962. “Some Thoughts on Statistical Inference”, The Annals of Mathematical Statistics, 33(2): 394-403.

Pearson, E. S. & Sekar, C. C. 1936. “The Efficiency of Statistical Tools and a Criterion for the Rejection of Outlying Observations“, Biometrika 28(3/4): 308-320. Reprinted (1966) in The Selected Papers of E. S. Pearson, (pp. 118-130). Berkeley: University of California Press.

Reid, Constance (1982). Neyman–from life

 

 

Categories: E.S. Pearson, Neyman, Statistics | Leave a comment

Neyman: Distinguishing tests of statistical hypotheses and tests of significance might have been a lapse of someone’s pen

neyman

April 16, 1894 – August 5, 1981

I’ll continue to post Neyman-related items this week in honor of his birthday. This isn’t the only paper in which Neyman makes it clear he denies a distinction between a test of  statistical hypotheses and significance tests. He and E. Pearson also discredit the myth that the former is only allowed to report pre-data, fixed error probabilities, and are justified only by dint of long-run error control. Controlling the “frequency of misdirected activities” in the midst of finding something out, or solving a problem of inquiry, on the other hand, are epistemological goals. What do you think?

Tests of Statistical Hypotheses and Their Use in Studies of Natural Phenomena
by Jerzy Neyman

ABSTRACT. Contrary to ideas suggested by the title of the conference at which the present paper was presented, the author is not aware of a conceptual difference between a “test of a statistical hypothesis” and a “test of significance” and uses these terms interchangeably. A study of any serious substantive problem involves a sequence of incidents at which one is forced to pause and consider what to do next. In an effort to reduce the frequency of misdirected activities one uses statistical tests. The procedure is illustrated on two examples: (i) Le Cam’s (and associates’) study of immunotherapy of cancer and (ii) a socio-economic experiment relating to low-income homeownership problems.

I recommend, especially, the example on home ownership. Here are two snippets: Continue reading

Categories: Error Statistics, Neyman, Statistics | Tags: | 2 Comments

A. Spanos: Jerzy Neyman and his Enduring Legacy

Today is Jerzy Neyman’s birthday. I’ll post various Neyman items this week in honor of it, starting with a guest post by Aris Spanos. Happy Birthday Neyman!

A. Spanos

A Statistical Model as a Chance Mechanism
Aris Spanos 

Jerzy Neyman (April 16, 1894 – August 5, 1981), was a Polish/American statistician[i] who spent most of his professional career at the University of California, Berkeley. Neyman is best known in statistics for his pioneering contributions in framing the Neyman-Pearson (N-P) optimal theory of hypothesis testing and his theory of Confidence Intervals. (This article was first posted here.)

Neyman: 16 April

Neyman: 16 April 1894 – 5 Aug 1981

One of Neyman’s most remarkable, but least recognized, achievements was his adapting of Fisher’s (1922) notion of a statistical model to render it pertinent for  non-random samples. Fisher’s original parametric statistical model Mθ(x) was based on the idea of ‘a hypothetical infinite population’, chosen so as to ensure that the observed data x0:=(x1,x2,…,xn) can be viewed as a ‘truly representative sample’ from that ‘population’:

“The postulate of randomness thus resolves itself into the question, Of what population is this a random sample? (ibid., p. 313), underscoring that: the adequacy of our choice may be tested a posteriori.’’ (p. 314) Continue reading

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R.A. Fisher: “Statistical methods and Scientific Induction”

I continue a week of Fisherian posts in honor of his birthday (Feb 17). This is his contribution to the “Triad”–an exchange between  Fisher, Neyman and Pearson 20 years after the Fisher-Neyman break-up. They are each very short.

17 February 1890 — 29 July 1962

“Statistical Methods and Scientific Induction”

by Sir Ronald Fisher (1955)

SUMMARY

The attempt to reinterpret the common tests of significance used in scientific research as though they constituted some kind of  acceptance procedure and led to “decisions” in Wald’s sense, originated in several misapprehensions and has led, apparently, to several more.

The three phrases examined here, with a view to elucidating they fallacies they embody, are:

  1. “Repeated sampling from the same population”,
  2. Errors of the “second kind”,
  3. “Inductive behavior”.

Mathematicians without personal contact with the Natural Sciences have often been misled by such phrases. The errors to which they lead are not only numerical.

To continue reading Fisher’s paper.

The most noteworthy feature is Fisher’s position on Fiducial inference, typically downplayed. I’m placing a summary and link to Neyman’s response below–it’s that interesting. Continue reading

Categories: fiducial probability, Fisher, Neyman, phil/history of stat | 6 Comments

Jerzy Neyman and “Les Miserables Citations” (statistical theater in honor of his birthday)

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Neyman April 16, 1894 – August 5, 1981

In honor of Jerzy Neyman’s birthday today, a local acting group is putting on a short theater production based on a screenplay I wrote:  “Les Miserables Citations” (“Those Miserable Quotes”) [1]. The “miserable” citations are those everyone loves to cite, from their early joint 1933 paper:

We are inclined to think that as far as a particular hypothesis is concerned, no test based upon the theory of probability can by itself provide any valuable evidence of the truth or falsehood of that hypothesis.

But we may look at the purpose of tests from another viewpoint. Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behavior with regard to them, in following which we insure that, in the long run of experience, we shall not be too often wrong. (Neyman and Pearson 1933, pp. 290-1).

Continue reading

Categories: E.S. Pearson, Neyman, Statistics | 7 Comments

Deconstructing the Fisher-Neyman conflict wearing fiducial glasses (continued)

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Fisher/ Neyman

This continues my previous post: “Can’t take the fiducial out of Fisher…” in recognition of Fisher’s birthday, February 17. I supply a few more intriguing articles you may find enlightening to read and/or reread on a Saturday night

Move up 20 years to the famous 1955/56 exchange between Fisher and Neyman. Fisher clearly connects Neyman’s adoption of a behavioristic-performance formulation to his denying the soundness of fiducial inference. When “Neyman denies the existence of inductive reasoning, he is merely expressing a verbal preference. For him ‘reasoning’ means what ‘deductive reasoning’ means to others.” (Fisher 1955, p. 74).

Fisher was right that Neyman’s calling the outputs of statistical inferences “actions” merely expressed Neyman’s preferred way of talking. Nothing earth-shaking turns on the choice to dub every inference “an act of making an inference”.[i] The “rationality” or “merit” goes into the rule. Neyman, much like Popper, had a good reason for drawing a bright red line between his use of probability (for corroboration or probativeness) and its use by ‘probabilists’ (who assign probability to hypotheses). Fisher’s Fiducial probability was in danger of blurring this very distinction. Popper said, and Neyman would have agreed, that he had no problem with our using the word induction so long it was kept clear it meant testing hypotheses severely. Continue reading

Categories: fiducial probability, Fisher, Neyman, Statistics | 55 Comments

Erich Lehmann: Neyman-Pearson & Fisher on P-values

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lone book on table

Today is Erich Lehmann’s birthday (20 November 1917 – 12 September 2009). Lehmann was Neyman’s first student at Berkeley (Ph.D 1942), and his framing of Neyman-Pearson (NP) methods has had an enormous influence on the way we typically view them.

I got to know Erich in 1997, shortly after publication of EGEK (1996). One day, I received a bulging, six-page, handwritten letter from him in tiny, extremely neat scrawl (and many more after that).  He began by telling me that he was sitting in a very large room at an ASA (American Statistical Association) meeting where they were shutting down the conference book display (or maybe they were setting it up), and on a very long, wood table sat just one book, all alone, shiny red.  He said he wondered if it might be of interest to him!  So he walked up to it….  It turned out to be my Error and the Growth of Experimental Knowledge (1996, Chicago), which he reviewed soon after[0]. (What are the chances?) Some related posts on Lehmann’s letter are here and here.

One of Lehmann’s more philosophical papers is Lehmann (1993), “The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two?” We haven’t discussed it before on this blog. Here are some excerpts (blue), and remarks (black)

Erich Lehmann 20 November 1917 – 12 September 2009

Erich Lehmann 20 November 1917 – 12 September 2009

…A distinction frequently made between the approaches of Fisher and Neyman-Pearson is that in the latter the test is carried out at a fixed level, whereas the principal outcome of the former is the statement of a p value that may or may not be followed by a pronouncement concerning significance of the result [p.1243].

The history of this distinction is curious. Throughout the 19th century, testing was carried out rather informally. It was roughly equivalent to calculating an (approximate) p value and rejecting the hypothesis if this value appeared to be sufficiently small. … Fisher, in his 1925 book and later, greatly reduced the needed tabulations by providing tables not of the distributions themselves but of selected quantiles. … These tables allow the calculation only of ranges for the p values; however, they are exactly suited for determining the critical values at which the statistic under consideration becomes significant at a given level. As Fisher wrote in explaining the use of his [chi square] table (1946, p. 80):

In preparing this table we have borne in mind that in practice we do not want to know the exact value of P for any observed [chi square], but, in the first place, whether or not the observed value is open to suspicion. If P is between .1 and .9, there is certainly no reason to suspect the hypothesis tested. If it is below .02, it is strongly indicated that the hypothesis fails to account for the whole of the facts. We shall not often be astray if we draw a conventional line at .05 and consider that higher values of [chi square] indicate a real discrepancy.

Similarly, he also wrote (1935, p. 13) that “it is usual and convenient for experimenters to take 5 percent as a standard level of significance, in the sense that they are prepared to ignore all results which fail to reach this standard .. .” …. Continue reading

Categories: Neyman, P-values, phil/history of stat, Statistics | Tags: , | 4 Comments

Neyman: Distinguishing tests of statistical hypotheses and tests of significance might have been a lapse of someone’s pen

Neyman April 16, 1894 – August 5, 1981

April 16, 1894 – August 5, 1981

Tests of Statistical Hypotheses and Their Use in Studies of Natural Phenomena” by Jerzy Neyman

ABSTRACT. Contrary to ideas suggested by the title of the conference at which the present paper was presented, the author is not aware of a conceptual difference between a “test of a statistical hypothesis” and a “test of significance” and uses these terms interchangeably. A study of any serious substantive problem involves a sequence of incidents at which one is forced to pause and consider what to do next. In an effort to reduce the frequency of misdirected activities one uses statistical tests. The procedure is illustrated on two examples: (i) Le Cam’s (and associates’) study of immunotherapy of cancer and (ii) a socio-economic experiment relating to low-income homeownership problems.

Neyman died on August 5, 1981. Here’s an unusual paper of his, “Tests of Statistical Hypotheses and Their Use in Studies of Natural Phenomena.” I have been reading a fair amount by Neyman this summer in writing about the origins of his philosophy, and have found further corroboration of the position that the behavioristic view attributed to him, while not entirely without substance*, is largely a fable that has been steadily built up and accepted as gospel. This has justified ignoring Neyman-Pearson statistics (as resting solely on long-run performance and irrelevant to scientific inference) and turning to crude variations of significance tests, that Fisher wouldn’t have countenanced for a moment (so-called NHSTs), lacking alternatives, incapable of learning from negative results, and permitting all sorts of P-value abuses–notably going from a small p-value to claiming evidence for a substantive research hypothesis. The upshot is to reject all of frequentist statistics, even though P-values are a teeny tiny part. *This represents a change in my perception of Neyman’s philosophy since EGEK (Mayo 1996).  I still say that that for our uses of method, it doesn’t matter what anybody thought, that “it’s the methods, stupid!” Anyway, I recommend, in this very short paper, the general comments and the example on home ownership. Here are two snippets: Continue reading

Categories: Error Statistics, Neyman, Statistics | Tags: | 19 Comments

NEYMAN: “Note on an Article by Sir Ronald Fisher” (3 uses for power, Fisher’s fiducial argument)

Note on an Article by Sir Ronald Fisher

By Jerzy Neyman (1956)

Summary

(1) FISHER’S allegation that, contrary to some passages in the introduction and on the cover of the book by Wald, this book does not really deal with experimental design is unfounded. In actual fact, the book is permeated with problems of experimentation.  (2) Without consideration of hypotheses alternative to the one under test and without the study of probabilities of the two kinds, no purely probabilistic theory of tests is possible.  (3) The conceptual fallacy of the notion of fiducial distribution rests upon the lack of recognition that valid probability statements about random variables usually cease to be valid if the random variables are replaced by their particular values.  The notorious multitude of “paradoxes” of fiducial theory is a consequence of this oversight.  (4)  The idea of a “cost function for faulty judgments” appears to be due to Laplace, followed by Gauss.

1. Introduction

In a recent article (Fisher, 1955), Sir Ronald Fisher delivered an attack on a a substantial part of the research workers in mathematical statistics. My name is mentioned more frequently than any other and is accompanied by the more expressive invectives. Of the scientific questions raised by Fisher many were sufficiently discussed before (Neyman and Pearson, 1933; Neyman, 1937; Neyman, 1952). In the present note only the following points will be considered: (i) Fisher’s attack on the concept of errors of the second kind; (ii) Fisher’s reference to my objections to fiducial probability; (iii) Fisher’s reference to the origin of the concept of loss function and, before all, (iv) Fisher’s attack on Abraham Wald.

THIS SHORT (5 page) NOTE IS NEYMAN’S PORTION OF WHAT I CALL THE “TRIAD”. LET ME POINT YOU TO THE TOP HALF OF p. 291, AND THE DISCUSSION OF FIDUCIAL INFERENCE ON p. 292 HERE.


Categories: Fisher, Neyman, phil/history of stat, Statistics | Tags: , , | 2 Comments

Neyman: Distinguishing tests of statistical hypotheses and tests of significance might have been a lapse of someone’s pen

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Neyman, drawn by ?

Tests of Statistical Hypotheses and Their Use in Studies of Natural Phenomena” by Jerzy Neyman

ABSTRACT. Contrary to ideas suggested by the title of the conference at which the present paper was presented, the author is not aware of a conceptual difference between a “test of a statistical hypothesis” and a “test of significance” and uses these terms interchangeably. A study of any serious substantive problem involves a sequence of incidents at which one is forced to pause and consider what to do next. In an effort to reduce the frequency of misdirected activities one uses statistical tests. The procedure is illustrated on two examples: (i) Le Cam’s (and associates’) study of immunotherapy of cancer and (ii) a socio-economic experiment relating to low-income homeownership problems.

I hadn’t posted this paper of Neyman’s before, so here’s something for your weekend reading:  “Tests of Statistical Hypotheses and Their Use in Studies of Natural Phenomena.”  I recommend, especially, the example on home ownership. Here are two snippets:

1. INTRODUCTION

The title of the present session involves an element that appears mysterious to me. This element is the apparent distinction between tests of statistical hypotheses, on the one hand, and tests of significance, on the other. If this is not a lapse of someone’s pen, then I hope to learn the conceptual distinction. Continue reading

Categories: Error Statistics, Neyman, Statistics | Tags: | 18 Comments

A. Spanos: Jerzy Neyman and his Enduring Legacy

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A Statistical Model as a Chance Mechanism
Aris Spanos 

Today is the birthday of Jerzy Neyman (April 16, 1894 – August 5, 1981). Neyman was a Polish/American statistician[i] who spent most of his professional career at the University of California, Berkeley. Neyman is best known in statistics for his pioneering contributions in framing the Neyman-Pearson (N-P) optimal theory of hypothesis testing and his theory of Confidence Intervals. (This article was first posted here.)

Neyman: 16 April

Neyman: 16 April 1894 – 5 Aug 1981

One of Neyman’s most remarkable, but least recognized, achievements was his adapting of Fisher’s (1922) notion of a statistical model to render it pertinent for  non-random samples. Fisher’s original parametric statistical model Mθ(x) was based on the idea of ‘a hypothetical infinite population’, chosen so as to ensure that the observed data x0:=(x1,x2,…,xn) can be viewed as a ‘truly representative sample’ from that ‘population’:

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Fisher and Neyman

“The postulate of randomness thus resolves itself into the question, Of what population is this a random sample? (ibid., p. 313), underscoring that: the adequacy of our choice may be tested a posteriori.’’ (p. 314)

In cases where data x0 come from sample surveys or it can be viewed as a typical realization of a random sample X:=(X1,X2,…,Xn), i.e. Independent and Identically Distributed (IID) random variables, the ‘population’ metaphor can be helpful in adding some intuitive appeal to the inductive dimension of statistical inference, because one can imagine using a subset of a population (the sample) to draw inferences pertaining to the whole population. Continue reading

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Neyman, Power, and Severity

April 16, 1894 – August 5, 1981

NEYMAN: April 16, 1894 – August 5, 1981

Jerzy Neyman: April 16, 1894-August 5, 1981. This reblogs posts under “The Will to Understand Power” & “Neyman’s Nursery” here & here.

Way back when, although I’d never met him, I sent my doctoral dissertation, Philosophy of Statistics, to one person only: Professor Ronald Giere. (And he would read it, too!) I knew from his publications that he was a leading defender of frequentist statistical methods in philosophy of science, and that he’d worked for at time with Birnbaum in NYC.

Some ten 15 years ago, Giere decided to quit philosophy of statistics (while remaining in philosophy of science): I think it had to do with a certain form of statistical exile (in philosophy). He asked me if I wanted his papers—a mass of work on statistics and statistical foundations gathered over many years. Could I make a home for them? I said yes. Then came his caveat: there would be a lot of them.

As it happened, we were building a new house at the time, Thebes, and I designed a special room on the top floor that could house a dozen or so file cabinets. (I painted it pale rose, with white lacquered book shelves up to the ceiling.) Then, for more than 9 months (same as my son!), I waited . . . Several boxes finally arrived, containing hundreds of files—each meticulously labeled with titles and dates.  More than that, the labels were hand-typed!  I thought, If Ron knew what a slob I was, he likely would not have entrusted me with these treasures. (Perhaps he knew of no one else who would  actually want them!) Continue reading

Categories: Neyman, phil/history of stat, power, Statistics | Tags: , , , | 5 Comments

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