Error Statistics

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:

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. Particularly with reference to applied statistical work in a variety of domains of Science, my own thoughts of tests of significance, or EQUIVALENTLY of tests of statistical hypotheses, are that they are tools to reduce the frequency of errors….

(iv) A similar remark applies to the use of the words “decision” or “conclusion”. It seem to me that at our discussion, these particular words were used to designate only something like a final outcome of complicated analysis involving several tests of different hypotheses. In my own way of speaking, I do not hesitate to use the words ‘decision’ or “conclusion” every time they come handy. For example, in the analysis of the follow-up data for the [home ownership] experiment, Mark Eudey and I started by considering the importance of bias in forming the experimental and control groups of families. As a result of the tests we applied, we decided to act on the assumption (or concluded) that the two groups are not random samples from the same population. Acting on this assumption (or having reached this conclusions), we sought for ways to analyze that data other than by comparing the experimental and the control groups. The analyses we performed led us to “conclude” or “decide” that the hypotheses tested could be rejected without excessive risk of error. In other words, after considering the probability of error (that is, after considering how frequently we would be in error if in conditions of our data we rejected the hypotheses tested), we decided to act on the assumption that “high” scores on “potential” and on “education” are indicative of better chances of success in the drive to home ownership. (750-1; the emphasis is Neyman’s)

To read the full (short) paper: Tests of Statistical Hypotheses and Their Use in Studies of Natural Phenomena.

Following Neyman, I’ve “decided” to use the terms ‘tests of hypotheses’ and ‘tests of significance’ interchangeably in my book.[1] Now it’s true that Neyman was more behavioristic than Pearson, and it’s also true that tests of statistical hypotheses or tests of significance need an explicit reformulation and statistical philosophy to explicate the role of error probabilities in inference. My way of providing this has been in terms of severe tests. However, in Neyman-Pearson applications, more than in their theory, you can find many examples as well. Recall Neyman’s paper, “The Problem of Inductive Inference” (Neyman 1955) wherein Neyman is talking to none other than the logical positivist philosopher of confirmation, Rudolf Carnap:

I am concerned with the term “degree of confirmation” introduced by Carnap.  …We have seen that the application of the locally best one-sided test to the data … failed to reject the hypothesis [that the n observations come from a source in which the null hypothesis is true].  The question is: does this result “confirm” the hypothesis that H0 is true of the particular data set? (Neyman, pp 40-41).

Neyman continues:

The answer … depends very much on the exact meaning given to the words “confirmation,” “confidence,” etc.  If one uses these words to describe one’s intuitive feeling of confidence in the hypothesis tested H0, then…. the attitude described is dangerous.… [T]he chance of detecting the presence [of discrepancy from the null], when only [n] observations are available, is extremely slim, even if [the discrepancy is present].  Therefore, the failure of the test to reject H0 cannot be reasonably considered as anything like a confirmation of H0.  The situation would have been radically different if the power function [corresponding to a discrepancy of interest] were, for example, greater than 0.95. (ibid.)

The general conclusion is that it is a little rash to base one’s intuitive confidence in a given hypothesis on the fact that a test failed to reject this hypothesis. A more cautious attitude would be to form one’s intuitive opinion only after studying the power function of the test applied.

I’m adding another paper of Neyman’s that echoes these same sentiments on the use of power, post data to evaluate what is “confirmed” ‘The Use of the Concept of Power in Agricultural Experimentation’.

Neyman, like Peirce, Popper and many others, hold that the only “logic” is deductive logic. “Confirmation” for Neyman is akin to Popperian “corroboration”–you could corroborate a hypothesis H only to the extent that it passed a severe test–one with a high probability of having found flaws in H, if they existed.  Of course, Neyman puts this in terms of having high power to reject H, if H is false, and high probability of finding no evidence against H if true, but it’s the same idea. But the use of power post-data is to interpret the discrepancies warranted in the given test. (This third use of power is also in Neyman 1956, responding to Fisher, the Triad).Unlike Popper, however, Neyman actually provides a methodology that can be shown to accomplish the task reliably.

Still, Fisher was correct to claim that Neyman is merely recording his preferred way of speaking. One could choose a different way. For example, Peirce defined induction as passing a severe test, and Popper said you could define it that way if you wanted to. But the main thing is that Neyman is attempting to distinguish the “inductive” or “evidence transcending” conclusions that statistics affords, on his approach,[2] from assigning to hypotheses degrees of belief, probability, support, plausibility or the like.

De Finetti gets it right when he says that the expression “inductive behavior…that was for Neyman simply a slogan underlining and explaining the difference between his own, the Bayesian and the Fisherian formulations” became, with Wald’s work, “something much more substantial” (de Finetti 1972, p.176). De Finetti called this “the involuntarily destructive aspect of Wald’s work” (ibid.).

Related papers on tests:

[1] That really is a decision, though it’s based on evidence that doing so is in sync with what both Neyman and Pearson thought. There are plenty of times, by the way, where Fisher is more behavioristic and less evidential than is Neyman, and certainly less than E. Pearson. I think this “he said/she said” route to understanding statistical methods is a huge mistake. I keep saying, “It’s the method’s stupid!” This is now the title of Excursion 3 Tour II of my book Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (2018, CUP).

[2] And, Neyman rightly assumed at first, from Fisher’s approach. Fisher’s loud rants, later on, that Neyman turned his tests into crude acceptance sampling affairs akin to Russian 5 year-plans, and money-making goals of U.S. commercialism, all occurred after the break in 1935 which registered a conflict of egos, not statistical philosophies. Look up “anger management” on this blog.

Fisher is the arch anti-Bayesian; whereas, Neyman experimented with using priors at the start. The problem wasn’t so much viewing parameters as random variables, but lacking knowledge of what their frequentist distributions could possibly be. Thus he sought methods whose validity held up regardless of priors.  Here E. Pearson was closer to Fisher, but unlike the two others, he was a really nice guy. (I hope everyone knows I’m talking of Egon here, not his mean daddy.) See chapter 11 of EGEK (1996):

[3] Who drew the picture of Neyman above? Anyone know?

References

de Finetti, B. 1972. Probability, Induction and Statistics: The Art of Guessing. Wiley.

Neyman, J. 1957. “The Use of the Concept of Power in Agricultural Experimentation, Journal of the Indian Society of Agricultural Statistics, 9(1): 9–17.

Neyman, J. 1976. “Tests of Statistical Hypotheses and Their Use in Studies of Natural Phenomena.Commun. Statist. Theor. Meth. A5(8), 737-751.

Reader: This and other Neyman blogposts have been incorporated into my book, Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (2018, CUP). Several excerpts can be found on this blog. Look up excerpts and mementos.

 

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Neyman vs the ‘Inferential’ Probabilists

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We celebrated Jerzy Neyman’s Birthday (April 16, 1894) last night in our seminar: here’s a pic of the cake.  My entry today is a brief excerpt and a link to a paper of his that we haven’t discussed much on this blog: Neyman, J. (1962), ‘Two Breakthroughs in the Theory of Statistical Decision Making‘ [i] It’s chock full of ideas and arguments, but the one that interests me at the moment is Neyman’s conception of “his breakthrough”, in relation to a certain concept of “inference”.  “In the present paper” he tells us, “the term ‘inferential theory’…will be used to describe the attempts to solve the Bayes’ problem with a reference to confidence, beliefs, etc., through some supplementation …either a substitute a priori distribution [exemplified by the so called principle of insufficient reason] or a new measure of uncertainty” such as Fisher’s fiducial probability. So if you hear Neyman rejecting “inferential accounts” you have to understand it in this very specific way: he’s rejecting “new measures of confidence or diffidence”. Here he alludes to them as “easy ways out”. Now Neyman always distinguishes his error statistical performance conception from Bayesian and Fiducial probabilisms [ii]. The surprising twist here is semantical and the culprit is none other than…Allan Birnbaum. Yet Birnbaum gets short shrift, and no mention is made of our favorite “breakthrough” (or did I miss it?).

drawn by his wife,Olga

Note: In this article,”attacks” on various statistical “fronts” refers to ways of attacking problems in one or another statistical research program.
HAPPY BIRTHDAY WEEK FOR NEYMAN! Continue reading

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Several reviews of Deborah Mayo’s new book, Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars « Statistical Modeling, Causal Inference, and Social Science

Source: Several reviews of Deborah Mayo’s new book, Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars « Statistical Modeling, Causal Inference, and Social Science

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Excursion 1 Tour II: Error Probing Tools versus Logics of Evidence-Excerpt

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For the first time, I’m excerpting all of Excursion 1 Tour II from SIST (2018, CUP).

1.4 The Law of Likelihood and Error Statistics

If you want to understand what’s true about statistical inference, you should begin with what has long been a holy grail–to use probability to arrive at a type of logic of evidential support–and in the first instance you should look not at full-blown Bayesian probabilism, but at comparative accounts that sidestep prior probabilities in hypotheses. An intuitively plausible logic of comparative support was given by the philosopher Ian Hacking (1965)–the Law of Likelihood. Fortunately, the Museum of Statistics is organized by theme, and the Law of Likelihood and the related Likelihood Principle is a big one. Continue reading

Categories: Error Statistics, law of likelihood, SIST | 2 Comments

American Phil Assoc Blog: The Stat Crisis of Science: Where are the Philosophers?

Ship StatInfasST

The Statistical Crisis of Science: Where are the Philosophers?

This was published today on the American Philosophical Association blog. 

“[C]onfusion about the foundations of the subject is responsible, in my opinion, for much of the misuse of the statistics that one meets in fields of application such as medicine, psychology, sociology, economics, and so forth.” (George Barnard 1985, p. 2)

“Relevant clarifications of the nature and roles of statistical evidence in scientific research may well be achieved by bringing to bear in systematic concert the scholarly methods of statisticians, philosophers and historians of science, and substantive scientists…” (Allan Birnbaum 1972, p. 861).

“In the training program for PhD students, the relevant basic principles of philosophy of science, methodology, ethics and statistics that enable the responsible practice of science must be covered.” (p. 57, Committee Investigating fraudulent research practices of social psychologist Diederik Stapel)

I was the lone philosophical observer at a special meeting convened by the American Statistical Association (ASA) in 2015 to construct a non-technical document to guide users of statistical significance tests–one of the most common methods used to distinguish genuine effects from chance variability across a landscape of social, physical and biological sciences.

It was, by the ASA Director’s own description, “historical”, but it was also highly philosophical, and its ramifications are only now being discussed and debated. Today, introspection on statistical methods is rather common due to the “statistical crisis in science”. What is it? In a nutshell: high powered computer methods make it easy to arrive at impressive-looking ‘findings’ that too often disappear when others try to replicate them when hypotheses and data analysis protocols are required to be fixed in advance.

Continue reading

Categories: Error Statistics, Philosophy of Statistics, Summer Seminar in PhilStat | 2 Comments

Little Bit of Logic (5 mini problems for the reader)

Little bit of logic (5 little problems for you)[i]

Deductively valid arguments can readily have false conclusions! Yes, deductively valid arguments allow drawing their conclusions with 100% reliability but only if all their premises are true. For an argument to be deductively valid means simply that if the premises of the argument are all true, then the conclusion is true. For a valid argument to entail  the truth of its conclusion, all of its premises must be true.  In that case the argument is said to be (deductively) sound.

Equivalently, using the definition of deductive validity that I prefer: A deductively valid argument is one where, the truth of all its premises together with the falsity of its conclusion, leads to a logical contradiction (A & ~A).

Show that an argument with the form of disjunctive syllogism can have a false conclusion. Such an argument take the form (where A, B are statements): Continue reading

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Mayo-Spanos Summer Seminar PhilStat: July 28-Aug 11, 2019: Instructions for Applying Now Available

INSTRUCTIONS FOR APPLYING ARE NOW AVAILABLE

See the Blog at SummerSeminarPhilStat

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You Should Be Binge Reading the (Strong) Likelihood Principle

 

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An essential component of inference based on familiar frequentist notions: p-values, significance and confidence levels, is the relevant sampling distribution (hence the term sampling theory, or my preferred error statistics, as we get error probabilities from the sampling distribution). This feature results in violations of a principle known as the strong likelihood principle (SLP). To state the SLP roughly, it asserts that all the evidential import in the data (for parametric inference within a model) resides in the likelihoods. If accepted, it would render error probabilities irrelevant post data.

SLP (We often drop the “strong” and just call it the LP. The “weak” LP just boils down to sufficiency)

For any two experiments E1 and E2 with different probability models f1, f2, but with the same unknown parameter θ, if outcomes x* and y* (from E1 and E2 respectively) determine the same (i.e., proportional) likelihood function (f1(x*; θ) = cf2(y*; θ) for all θ), then x* and y* are inferentially equivalent (for an inference about θ).

(What differentiates the weak and the strong LP is that the weak refers to a single experiment.)
Continue reading

Categories: Error Statistics, Statistics, strong likelihood principle | 1 Comment

Excerpt from Excursion 4 Tour I: The Myth of “The Myth of Objectivity” (Mayo 2018, CUP)

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Tour I The Myth of “The Myth of Objectivity”*

 

Objectivity in statistics, as in science more generally, is a matter of both aims and methods. Objective science, in our view, aims to find out what is the case as regards aspects of the world [that hold] independently of our beliefs, biases and interests; thus objective methods aim for the critical control of inferences and hypotheses, constraining them by evidence and checks of error. (Cox and Mayo 2010, p. 276)

Whenever you come up against blanket slogans such as “no methods are objective” or “all methods are equally objective and subjective” it is a good guess that the problem is being trivialized into oblivion. Yes, there are judgments, disagreements, and values in any human activity, which alone makes it too trivial an observation to distinguish among very different ways that threats of bias and unwarranted inferences may be controlled. Is the objectivity–subjectivity distinction really toothless, as many will have you believe? I say no. I know it’s a meme promulgated by statistical high priests, but you agreed, did you not, to use a bit of chutzpah on this excursion? Besides, cavalier attitudes toward objectivity are at odds with even more widely endorsed grass roots movements to promote replication, reproducibility, and to come clean on a number of sources behind illicit results: multiple testing, cherry picking, failed assumptions, researcher latitude, publication bias and so on. The moves to take back science are rooted in the supposition that we can more objectively scrutinize results – even if it’s only to point out those that are BENT. The fact that these terms are used equivocally should not be taken as grounds to oust them but rather to engage in the difficult work of identifying what there is in “objectivity” that we won’t give up, and shouldn’t. Continue reading

Categories: Error Statistics, SIST, Statistical Inference as Severe Testing | 4 Comments

Summer Seminar PhilStat: July 28-Aug 11, 2019 (ii)

First draft of PhilStat Announcement

 

Categories: Announcement, Error Statistics | 5 Comments

It’s the Methods, Stupid: Excerpt from Excursion 3 Tour II (Mayo 2018, CUP)

Tour II It’s the Methods, Stupid

There is perhaps in current literature a tendency to speak of the Neyman–Pearson contributions as some static system, rather than as part of the historical process of development of thought on statistical theory which is and will always go on. (Pearson 1962, 276)

This goes for Fisherian contributions as well. Unlike museums, we won’ t remain static. The lesson from Tour I of this Excursion is that Fisherian and Neyman– Pearsonian tests may be seen as offering clusters of methods appropriate for different contexts within the large taxonomy of statistical inquiries. There is an overarching pattern: Continue reading

Categories: Error Statistics, Statistical Inference as Severe Testing | 4 Comments

Memento & Quiz (on SEV): Excursion 3, Tour I

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As you enjoy the weekend discussion & concert in the Captain’s Central Limit Library & Lounge, your Tour Guide has prepared a brief overview of Excursion 3 Tour I, and a short (semi-severe) quiz on severity, based on exhibit (i).*

 

We move from Popper through a gallery on “Data Analysis in the 1919 Eclipse tests of the General Theory of Relativity (GTR)” (3.1) which leads to the main gallery on the origin of statistical tests (3.2) by way of a look at where the main members of our statistical cast are in 1919: Fisher, Neyman and Pearson. From the GTR episode, we identify the key elements of a statistical test–the steps in E.S. Pearson’s opening description of tests in 3.2. The classical testing notions–type I and II errors, power, consistent tests–are shown to grow out of requiring probative tests. The typical (behavioristic) formulation of N-P tests came later. The severe tester breaks out of the behavioristic prison. A first look at the severity construal of N-P tests is in Exhibit (i). Viewing statistical inference as severe testing shows how to do all N-P tests do (and more) while a member of the Fisherian Tribe (3.3). We consider the frequentist principle of evidence FEV and the divergent interpretations that are called for by Cox’s taxonomy of null hypotheses. The last member of the taxonomy–substantively based null hypotheses–returns us to the opening episode of GTR. Continue reading

Categories: Severity, Statistical Inference as Severe Testing | 16 Comments

First Look at N-P Methods as Severe Tests: Water plant accident [Exhibit (i) from Excursion 3]

Excursion 3 Exhibit (i)

Exhibit (i) N-P Methods as Severe Tests: First Look (Water Plant Accident)

There’s been an accident at a water plant where our ship is docked, and the cooling system had to be repaired.  It is meant to ensure that the mean temperature of discharged water stays below the temperature that threatens the ecosystem, perhaps not much beyond 150 degrees Fahrenheit. There were 100 water measurements taken at randomly selected times and the sample mean x computed, each with a known standard deviation σ = 10.  When the cooling system is effective, each measurement is like observing X ~ N(150, 102). Because of this variability, we expect different 100-fold water samples to lead to different values of X, but we can deduce its distribution. If each X ~N(μ = 150, 102) then X is also Normal with μ = 150, but the standard deviation of X is only σ/√n = 10/√100 = 1. So X ~ N(μ = 150, 1). Continue reading

Categories: Error Statistics, Severity, Statistical Inference as Severe Testing | 44 Comments

Stephen Senn: Rothamsted Statistics meets Lord’s Paradox (Guest Post)

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Stephen Senn
Consultant Statistician
Edinburgh

The Rothamsted School

I never worked at Rothamsted but during the eight years I was at University College London (1995-2003) I frequently shared a train journey to London from Harpenden (the village in which Rothamsted is situated) with John Nelder, as a result of which we became friends and I acquired an interest in the software package Genstat®.

That in turn got me interested in John Nelder’s approach to analysis of variance, which is a powerful formalisation of ideas present in the work of others associated with Rothamsted. Nelder’s important predecessors in this respect include, at least, RA Fisher (of course) and Frank Yates and others such as David Finney and Frank Anscombe. John died in 2010 and I regard Rosemary Bailey, who has done deep and powerful work on randomisation and the representation of experiments through Hasse diagrams, as being the greatest living proponent of the Rothamsted School. Another key figure is Roger Payne who turned many of John’s ideas into code in Genstat®. Continue reading

Categories: Error Statistics | 11 Comments

The Replication Crises and its Constructive Role in the Philosophy of Statistics-PSA2018

Below are my slides from a session on replication at the recent Philosophy of Science Association meetings in Seattle.

 

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Tour Guide Mementos (Excursion 1, Tour I of How to Get Beyond the Statistics Wars)

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Tour guides in your travels jot down Mementos and Keepsakes from each Tour[i] of my new book: Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (CUP 2018). Their scribblings, which may at times include details, at other times just a word or two, may be modified through the Tour, and in response to questions from travelers (so please check back). Since these are just mementos, they should not be seen as replacements for the more careful notions given in the journey (i.e., book) itself. Still, you’re apt to flesh out your notes in greater detail, so please share yours (along with errors you’re bound to spot), and we’ll create Meta-Mementos. Continue reading

Categories: Error Statistics, Statistical Inference as Severe Testing | 8 Comments

Philosophy of Statistics & the Replication Crisis in Science: A philosophical intro to my book (slides)

a road through the jungle

In my talk yesterday at the Philosophy Department at Virginia Tech, I introduced my new book: Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (Cambridge 2018). I began with my preface (explaining the meaning of my title), and turned to the Statistics Wars, largely from Excursion 1 of the book. After the sum-up at the end, I snuck in an example from the replication crisis in psychology. Here are the slides.

 

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RSS 2018 – Significance Tests: Rethinking the Controversy

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Day 2, Wednesday 05/09/2018

11:20 – 13:20

Keynote 4 – Significance Tests: Rethinking the Controversy Assembly Room

Speakers:
Sir David Cox, Nuffield College, Oxford
Deborah Mayo, Virginia Tech
Richard Morey, Cardiff University
Aris Spanos, Virginia Tech

Intermingled in today’s statistical controversies are some long-standing, but unresolved, disagreements on the nature and principles of statistical methods and the roles for probability in statistical inference and modelling. In reaction to the so-called “replication crisis” in the sciences, some reformers suggest significance tests as a major culprit. To understand the ramifications of the proposed reforms, there is a pressing need for a deeper understanding of the source of the problems in the sciences and a balanced critique of the alternative methods being proposed to supplant significance tests. In this session speakers offer perspectives on significance tests from statistical science, econometrics, experimental psychology and philosophy of science. There will be also be panel discussion.

Categories: Error Statistics | 2 Comments

Neyman vs the ‘Inferential’ Probabilists continued (a)

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Today is Jerzy Neyman’s Birthday (April 16, 1894 – August 5, 1981).  I am posting a brief excerpt and a link to a paper of his that I hadn’t posted before: Neyman, J. (1962), ‘Two Breakthroughs in the Theory of Statistical Decision Making‘ [i] It’s chock full of ideas and arguments, but the one that interests me at the moment is Neyman’s conception of “his breakthrough”, in relation to a certain concept of “inference”.  “In the present paper” he tells us, “the term ‘inferential theory’…will be used to describe the attempts to solve the Bayes’ problem with a reference to confidence, beliefs, etc., through some supplementation …either a substitute a priori distribution [exemplified by the so called principle of insufficient reason] or a new measure of uncertainty” such as Fisher’s fiducial probability. Now Neyman always distinguishes his error statistical performance conception from Bayesian and Fiducial probabilisms [ii]. The surprising twist here is semantical and the culprit is none other than…Allan Birnbaum. Yet Birnbaum gets short shrift, and no mention is made of our favorite “breakthrough” (or did I miss it?). [iii] I’ll explain in later stages of this post & in comments…(so please check back); I don’t want to miss the start of the birthday party in honor of Neyman, and it’s already 8:30 p.m in Berkeley!

Note: In this article,”attacks” on various statistical “fronts” refers to ways of attacking problems in one or another statistical research program. HAPPY BIRTHDAY NEYMAN! Continue reading

Categories: Bayesian/frequentist, Error Statistics, Neyman, Statistics | Leave a comment

S. Senn: Being a statistician means never having to say you are certain (Guest Post)

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Stephen Senn
Head of  Competence Center
for Methodology and Statistics (CCMS)
Luxembourg Institute of Health
Twitter @stephensenn

Being a statistician means never having to say you are certain

A recent discussion of randomised controlled trials[1] by Angus Deaton and Nancy Cartwright (D&C) contains much interesting analysis but also, in my opinion, does not escape rehashing some of the invalid criticisms of randomisation with which the literatures seems to be littered. The paper has two major sections. The latter, which deals with generalisation of results, or what is sometime called external validity, I like much more than the former which deals with internal validity. It is the former I propose to discuss.

Continue reading

Categories: Error Statistics, RCTs, Statistics | 26 Comments

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