3 YEARS AGO (JUNE 2014): MEMORY LANE

Posted on June 25, 2017 by Mayo

3 years ago…

MONTHLY MEMORY LANE: 3 years ago: June 2014. I mark in red 3-4 posts from each month that seem most apt for general background on key issues in this blog, excluding those reblogged recently[1], and in green up to 4 others of general relevance to philosophy of statistics [2]. Posts that are part of a “unit” or a group count as one.

June 2014

(6/5) Stephen Senn: Blood Simple? The complicated and controversial world of bioequivalence (guest post)
(6/9) “The medical press must become irrelevant to publication of clinical trials.”
(6/11) A. Spanos: “Recurring controversies about P values and confidence intervals revisited”
(6/14) “Statistical Science and Philosophy of Science: where should they meet?”
(6/21) Big Bayes Stories? (draft ii)
(6/25) Blog Contents: May 2014
(6/28) Sir David Hendry Gets Lifetime Achievement Award
(6/30) Some ironies in the ‘replication crisis’ in social psychology (4^th and final installment)

[1] Monthly memory lanes began at the blog’s 3-year anniversary in Sept, 2014.

[2] New Rule, July 30,2016, March 30,2017 (moved to 4) -very convenient way to allow data-dependent choices.

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Can You Change Your Bayesian Prior? The one post whose comments (some of them) will appear in my new book

Posted on June 18, 2017 by Mayo

I blogged this exactly 2 years ago here, seeking insight for my new book (Mayo 2017). Over 100 (rather varied) interesting comments ensued. This is the first time I’m incorporating blog comments into published work. You might be interested to follow the nooks and crannies from back then, or add a new comment to this.

This is one of the questions high on the “To Do” list I’ve been keeping for this blog. The question grew out of discussions of “updating and downdating” in relation to papers by Stephen Senn (2011) and Andrew Gelman (2011) in Rationality, Markets, and Morals.[i]

“As an exercise in mathematics [computing a posterior based on the client’s prior probabilities] is not superior to showing the client the data, eliciting a posterior distribution and then calculating the prior distribution; as an exercise in inference Bayesian updating does not appear to have greater claims than ‘downdating’.” (Senn, 2011, p. 59)

“If you could really express your uncertainty as a prior distribution, then you could just as well observe data and directly write your subjective posterior distribution, and there would be no need for statistical analysis at all.” (Gelman, 2011, p. 77)

But if uncertainty is not expressible as a prior, then a major lynchpin for Bayesian updating seems questionable. If you can go from the posterior to the prior, on the other hand, perhaps it can also lead you to come back and change it.

Is it legitimate to change one’s prior based on the data?

I don’t mean update it, but reject the one you had and replace it with another. My question may yield different answers depending on the particular Bayesian view. I am prepared to restrict the entire question of changing priors to Bayesian “probabilisms”, meaning the inference takes the form of updating priors to yield posteriors, or to report a comparative Bayes factor. Interpretations can vary. In many Bayesian accounts the prior probability distribution is a way of introducing prior beliefs into the analysis (as with subjective Bayesians) or, conversely, to avoid introducing prior beliefs (as with reference or conventional priors). Empirical Bayesians employ frequentist priors based on similar studies or well established theory. There are many other variants.

S. SENN: According to Senn, one test of whether an approach is Bayesian is that while “arrival of new data will, of course, require you to update your prior distribution to being a posterior distribution, no conceivable possible constellation of results can cause you to wish to change your prior distribution. If it does, you had the wrong prior distribution and this prior distribution would therefore have been wrong even for cases that did not leave you wishing to change it.” (Senn, 2011, 63)

“If you cannot go back to the drawing board, one seems stuck with priors one now regards as wrong; if one does change them, then what was the meaning of the prior as carrying prior information?” (Senn, 2011, p. 58)

I take it that Senn is referring to a Bayesian prior expressing belief. (He will correct me if I’m wrong.)[ii] Senn takes the upshot to be that priors cannot be changed based on data. Is there a principled ground for blocking such moves?

I.J. GOOD: The traditional idea was that one would have thought very hard about one’s prior before proceeding—that’s what Jack Good always said. Good advocated his device of “imaginary results” whereby one would envisage all possible results in advance (1971, p. 431) and choose a prior that you can live with whatever happens. This could take a long time! Given how difficult this would be, in practice, Good allowed

“that it is possible after all to change a prior in the light of actual experimental results” [but] rationality of type II has to be used.” (Good 1971, p. 431)

Maybe this is an example of what Senn calls requiring the informal to come to the rescue of the formal? Good was commenting on D. J. Bartholomew [iii] in the same wonderful volume (edited by Godambe and Sprott).

D. LINDLEY: According to subjective Bayesian Dennis Lindley:

“[I]f a prior leads to an unacceptable posterior then I modify it to cohere with properties that seem desirable in the inference.”(Lindley 1971, p. 436)

This would seem to open the door to all kinds of verification biases, wouldn’t it? This is the same Lindley who famously declared:

“I am often asked if the method gives the right answer: or, more particularly, how do you know if you have got the right prior. My reply is that I don’t know what is meant by “right” in this context. The Bayesian theory is about coherence, not about right or wrong.” (1976, p. 359)

H. KYBURG: Philosopher Henry Kyburg (who wrote a book on subjective probability, but was or became a frequentist) gives what I took to be the standard line (for subjective Bayesians at least):

There is no way I can be in error in my prior distribution for μ ––unless I make a logical error–… . It is that very fact that makes this prior distribution perniciously subjective. It represents an assumption that has consequences, but cannot be corrected by criticism or further evidence.” (Kyburg 1993, p. 147)

It can be updated of course via Bayes rule.

D.R. COX: While recognizing the serious problem of “temporal incoherence”, (a violation of diachronic Bayes updating), David Cox writes:

“On the other hand [temporal coherency] is not inevitable and there is nothing intrinsically inconsistent in changing prior assessments” in the light of data; however, the danger is that “even initially very surprising effects can post hoc be made to seem plausible.” (Cox 2006, p. 78)

An analogous worry would arise, Cox notes, if frequentists permit data dependent selections of hypotheses (significance seeking, cherry picking, etc). However, frequentists (if they are not to be guilty of cheating) would need to take into account any adjustments to the overall error probabilities of the test. But the Bayesian is not in the business of computing error probabilities associated with a method for reaching posteriors. At least not traditionally. Would Bayesians even be required to report such shifts of priors? (A principle is needed.)

What if the proposed adjustment of prior is based on the data and resulting likelihoods, rather than an impetus to ensure one’s favorite hypothesis gets a desirable posterior? After all, Jim Berger says that prior elicitation typically takes place after “the expert has already seen the data” (2006, p. 392). Do they instruct them to try not to take the data into account? Anyway, if the prior is determined post-data, then one wonders how it can be seen to reflect information distinct from the data under analysis. All the work to obtain posteriors would have been accomplished by the likelihoods. There’s also the issue of using data twice.

So what do you think is the answer? Does it differ for subjective vs conventional vs other stripes of Bayesian?

[i]Both were contributions to the RMM (2011) volume: Special Topic: Statistical Science and Philosophy of Science: Where Do (Should) They Meet in 2011 and Beyond? (edited by D. Mayo, A. Spanos, and K. Staley). The volume was an outgrowth of a 2010 conference that Spanos and I (and others) ran in London (LSE), and conversations that emerged soon after. See full list of participants, talks and sponsors here.

[ii] Senn and I had a published exchange on his paper that was based on my “deconstruction” of him on this blog, followed by his response! The published comments are here (Mayo) and here (Senn).

[iii] At first I thought Good was commenting on Lindley. Bartholomew came up in this blog in discussing when Bayesians and frequentists can agree on numbers.

WEEKEND READING

Gelman, A. 2011. “Induction and Deduction in Bayesian Data Analysis.”
Senn, S. 2011. “You May Believe You Are a Bayesian But You Are Probably Wrong.”
Berger, J. O. 2006. “The Case for Objective Bayesian Analysis.”

Discussions and Responses on Senn and Gelman can be found searching this blog:

Commentary on Berger & Goldstein: Christen, Draper, Fienberg, Kadane, Kass, Wasserman,
Rejoinders: Berger, Goldstein,

REFERENCES

Berger, J. O. 2006. “The Case for Objective Bayesian Analysis.” Bayesian Analysis 1 (3): 385–402.

Cox, D. R. 2006. Principles of Statistical Inference. Cambridge, UK: Cambridge University Press.

Mayo, D. G. 2017. Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars. Cambridge.

Categories: Bayesian priors, Bayesian/frequentist | 11 Comments

Performance or Probativeness? E.S. Pearson’s Statistical Philosophy

Posted on June 12, 2017 by Mayo

E.S. Pearson (11 Aug, 1895-12 June, 1980)

E.S. Pearson died on this day in 1980. Aside from being co-developer of Neyman-Pearson statistics, Pearson was interested in philosophical aspects of statistical inference. A question he asked is this: Are methods with good error probabilities of use mainly to supply procedures which will not err too frequently in some long run? (performance). Or is it the other way round: that the control of long run error properties are of crucial importance for probing the causes of the data at hand? (probativeness). I say no to the former and yes to the latter. But how exactly does it work? It’s not just the frequentist error statistician who faces this question, but also some contemporary Bayesians who aver that the performance or calibration of their methods supplies an evidential (or inferential or epistemic) justification (e.g., Robert Kass 2011). The latter generally ties the reliability of the method that produces the particular inference C to degrees of belief in C. The inference takes the form of a probabilism, e.g., Pr(C|x), equated, presumably, to the reliability (or coverage probability) of the method. But why? The frequentist inference is C, which is qualified by the reliability of the method, but there’s no posterior assigned C. Again, what’s the rationale? I think existing answers (from both tribes) come up short in non-trivial ways.

I’ve recently become clear (or clearer) on a view I’ve been entertaining for a long time. There’s more than one goal in using probability, but when it comes to statistical inference in science, I say, the goal is not to infer highly probable claims (in the formal sense)* but claims which have been highly probed and have passed severe probes. Even highly plausible claims can be poorly tested (and I require a bit more of a test than informal uses of the word.) The frequency properties of a method are relevant in those contexts where they provide assessments of a method’s capabilities and shortcomings in uncovering ways C may be wrong. Knowledge of the methods capabilities are used, in turn, to ascertain how well or severely C has been probed. C is warranted only to the extent that it survived a severe probe of ways it can be incorrect. There’s poor evidence for C when little has been done to rule out C’s flaws. The most important role of error probabilities is in blocking inferences to claims that have not passed severe tests, but also to falsify (statistically) claims whose denials pass severely. This view is in the spirit of E.S. Pearson, Peirce, and Popper–though none fully worked it out. That’s one of the things I do or try to in my latest work. Each supplied important hints. The following remarks of Pearson, earlier blogged here, contains some of his hints.

*Nor to give a comparative assessment of the probability of claims

From Pearson, E. S. (1947)

“How far then, can one go in giving precision to a philosophy of statistical inference?” (Pearson 1947, 172)

Pearson considers the rationale that might be given to N-P tests in two types of cases, A and B:

“(A) At one extreme we have the case where repeated decisions must be made on results obtained from some routine procedure…

(B) At the other is the situation where statistical tools are applied to an isolated investigation of considerable importance…?” (ibid., 170)

In cases of type A, long-run results are clearly of interest, while in cases of type B, repetition is impossible and may be irrelevant:

“In other and, no doubt, more numerous cases there is no repetition of the same type of trial or experiment, but all the same we can and many of us do use the same test rules to guide our decision, following the analysis of an isolated set of numerical data. Why do we do this? What are the springs of decision? Is it because the formulation of the case in terms of hypothetical repetition helps to that clarity of view needed for sound judgment?

Or is it because we are content that the application of a rule, now in this investigation, now in that, should result in a long-run frequency of errors in judgment which we control at a low figure?” (Ibid., 173)

Although Pearson leaves this tantalizing question unanswered, claiming, “On this I should not care to dogmatize”, in studying how Pearson treats cases of type B, it is evident that in his view, “the formulation of the case in terms of hypothetical repetition helps to that clarity of view needed for sound judgment” in learning about the particular case at hand.

“Whereas when tackling problem A it is easy to convince the practical man of the value of a probability construct related to frequency of occurrence, in problem B the argument that ‘if we were to repeatedly do so and so, such and such result would follow in the long run’ is at once met by the commonsense answer that we never should carry out a precisely similar trial again.

Nevertheless, it is clear that the scientist with a knowledge of statistical method behind him can make his contribution to a round-table discussion…” (Ibid., 171).

Pearson gives the following example of a case of type B (from his wartime work), where he claims no repetition is intended:

“Example of type B. Two types of heavy armour-piercing naval shell of the same caliber are under consideration; they may be of different design or made by different firms…. Twelve shells of one kind and eight of the other have been fired; two of the former and five of the latter failed to perforate the plate….”(Pearson 1947, 171)

“Starting from the basis that, individual shells will never be identical in armour-piercing qualities, however good the control of production, he has to consider how much of the difference between (i) two failures out of twelve and (ii) five failures out of eight is likely to be due to this inevitable variability. ..”(Ibid.,)

We’re interested in considering what other outcomes could have occurred, and how readily, in order to learn what variability alone is capable of producing. As a noteworthy aside, Pearson shows that treating the observed difference (between the two proportions) in one way yields an observed significance level of 0.052; treating it differently (along Barnard’s lines), he gets 0.025 as the (upper) significance level. But in scientific cases, Pearson insists, the difference in error probabilities makes no real difference to substantive judgments in interpreting the results. Only in an unthinking, automatic, routine use of tests would it matter:

“Were the action taken to be decided automatically by the side of the 5% level on which the observation point fell, it is clear that the method of analysis used would here be of vital importance. But no responsible statistician, faced with an investigation of this character, would follow an automatic probability rule.” (ibid., 192)

The two analyses correspond to the tests effectively asking different questions, and if we recognize this, says Pearson, different meanings may be appropriately attached.

Three Steps in the Original Construction of Tests

After setting up the test (or null) hypothesis, and the alternative hypotheses against which “we wish the test to have maximum discriminating power” (Pearson 1947, 173), Pearson defines three steps in specifying tests:

“Step 1. We must specify the experimental probability set, the set of results which could follow on repeated application of the random process used in the collection of the data…

Step 2. We then divide this set [of possible results] by a system of ordered boundaries…such that as we pass across one boundary and proceed to the next, we come to a class of results which makes us more and more inclined on the Information available, to reject the hypothesis tested in favour of alternatives which differ from it by increasing amounts” (Pearson 1966a, 173).

“Step 3. We then, if possible[i], associate with each contour level the chance that, if [the null] is true, a result will occur in random sampling lying beyond that level” (ibid.).

Pearson warns that:

“Although the mathematical procedure may put Step 3 before 2, we cannot put this into operation before we have decided, under Step 2, on the guiding principle to be used in choosing the contour system. That is why I have numbered the steps in this order.” (Ibid. 173).

Strict behavioristic formulations jump from step 1 to step 3, after which one may calculate how the test has in effect accomplished step 2. However, the resulting test, while having adequate error probabilities, may have an inadequate distance measure and may even be irrelevant to the hypothesis of interest. This is one reason critics can construct howlers that appear to be licensed by N-P methods, and which make their way from time to time into this blog.

So step 3 remains crucial, even for cases of type [B]. There are two reasons: pre-data planning—that’s familiar enough—but secondly, for post-data scrutiny. Post data, step 3 enables determining the capability of the test to have detected various discrepancies, departures, and errors, on which a critical scrutiny of the inferences are based. More specifically, the error probabilities are used to determine how well/poorly corroborated, or how severely tested, various claims are, post-data.

If we can readily bring about statistically significantly higher rates of success with the first type of armour-piercing naval shell than with the second (in the above example), we have evidence the first is superior. Or, as Pearson modestly puts it: the results “raise considerable doubts as to whether the performance of the [second] type of shell was as good as that of the [first]….” (Ibid., 192)[ii]

Still, while error rates of procedures may be used to determine how severely claims have/have not passed they do not automatically do so—hence, again, opening the door to potential howlers that neither Egon nor Jerzy for that matter would have countenanced.

Neyman Was the More Behavioristic of the Two

Pearson was (rightly) considered to have rejected the more behaviorist leanings of Neyman.

Here’s a snippet from an unpublished letter he wrote to Birnbaum (1974) about the idea that the N-P theory admits of two interpretations: behavioral and evidential:

“I think you will pick up here and there in my own papers signs of evidentiality, and you can say now that we or I should have stated clearly the difference between the behavioral and evidential interpretations. Certainly we have suffered since in the way the people have concentrated (to an absurd extent often) on behavioral interpretations”.

In Pearson’s (1955) response to Fisher (blogged here):

“To dispel the picture of the Russian technological bogey, I might recall how certain early ideas came into my head as I sat on a gate overlooking an experimental blackcurrant plot….!” (Pearson 1955, 204)

“To the best of my ability I was searching for a way of expressing in mathematical terms what appeared to me to be the requirements of the scientist in applying statistical tests to his data. After contact was made with Neyman in 1926, the development of a joint mathematical theory proceeded much more surely; it was not till after the main lines of this theory had taken shape with its necessary formalization in terms of critical regions, the class of admissible hypotheses, the two sources of error, the power function, etc., that the fact that there was a remarkable parallelism of ideas in the field of acceptance sampling became apparent. Abraham Wald’s contributions to decision theory of ten to fifteen years later were perhaps strongly influenced by acceptance sampling problems, but that is another story.“ (ibid., 204-5).

“It may be readily agreed that in the first Neyman and Pearson paper of 1928, more space might have been given to discussing how the scientific worker’s attitude of mind could be related to the formal structure of the mathematical probability theory….Nevertheless it should be clear from the first paragraph of this paper that we were not speaking of the final acceptance or rejection of a scientific hypothesis on the basis of statistical analysis…. Indeed, from the start we shared Professor Fisher’s view that in scientific enquiry, a statistical test is ‘a means of learning”… (Ibid., 206)

“Professor Fisher’s final criticism concerns the use of the term ‘inductive behavior’; this is Professor Neyman’s field rather than mine.” (Ibid., 207)

__________________________

References:

Kass, R. (2011), “Statistical Inference: The Big Picture,” Statistical Science 26, No. 1, 1–9.

Pearson, E. S. (1935), The Application of Statistical Methods to Industrial Standardization and Quality Control, London: British Standards Institution.

Pearson, E. S. (1947), “The Choice of Statistical Tests illustrated on the Interpretation of Data Classed in a 2×2 Table,” Biometrika 34(1/2): 139-167.

Pearson, E. S. (1955), “Statistical Concepts and Their Relationship to Reality” Journal of the Royal Statistical Society, Series B, (Methodological), 17(2): 204-207.

Neyman, J. and Pearson, E. S. (1928), “On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference, Part I.” Biometrika 20(A): 175-240.

[i] In some cases only an upper limit to this error probability may be found.

[ii] Pearson inadvertently switches from number of failures to number of successes in the conclusion of this paper.

Categories: E.S. Pearson, highly probable vs highly probed, phil/history of stat | Leave a comment

3 YEARS AGO (May 2014): MEMORY LANE

Posted on June 4, 2017 by Mayo

3 years ago…

MONTHLY MEMORY LANE: 3 years ago: May 2014. I leave them unmarked this month, read whatever looks interesting.

May 2014

(5/1) Putting the brakes on the breakthrough: An informal look at the argument for the Likelihood Principle
(5/3) You can only become coherent by ‘converting’ non-Bayesianly
(5/6) Winner of April Palindrome contest: Lori Wike
(5/7) A. Spanos: Talking back to the critics using error statistics (Phil6334)
(5/10) Who ya gonna call for statistical Fraudbusting? R.A. Fisher, P-values, and error statistics (again)
(5/15) Scientism and Statisticism: a conference* (i)
(5/17) Deconstructing Andrew Gelman: “A Bayesian wants everybody else to be a non-Bayesian.”
(5/20) The Science Wars & the Statistics Wars: More from the Scientism workshop
(5/25) Blog Table of Contents: March and April 2014
(5/27) Allan Birnbaum, Philosophical Error Statistician: 27 May 1923 – 1 July 1976
(5/31) What have we learned from the Anil Potti training and test data frameworks? Part 1 (draft 2)

[1] Monthly memory lanes began at the blog’s 3-year anniversary in Sept, 2014.

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Allan Birnbaum: Foundations of Probability and Statistics (27 May 1923 – 1 July 1976)

Posted on May 27, 2017 by Mayo

27 May 1923-1 July 1976

Today is Allan Birnbaum’s birthday. In honor of his birthday, I’m posting the articles in the Synthese volume that was dedicated to his memory in 1977. The editors describe it as their way of “paying homage to Professor Birnbaum’s penetrating and stimulating work on the foundations of statistics”. I paste a few snippets from the articles by Giere and Birnbaum. If you’re interested in statistical foundations, and are unfamiliar with Birnbaum, here’s a chance to catch up. (Even if you are,you may be unaware of some of these key papers.)

HAPPY BIRTHDAY ALLAN!

Synthese Volume 36, No. 1 Sept 1977: Foundations of Probability and Statistics, Part I

Editorial Introduction:

This special issue of Synthese on the foundations of probability and statistics is dedicated to the memory of Professor Allan Birnbaum. Professor Birnbaum’s essay ‘The Neyman-Pearson Theory as Decision Theory; and as Inference Theory; with a Criticism of the Lindley-Savage Argument for Bayesian Theory’ was received by the editors of Synthese in October, 1975, and a decision was made to publish a special symposium consisting of this paper together with several invited comments and related papers. The sad news about Professor Birnbaum’s death reached us in the summer of 1976, but the editorial project could nevertheless be completed according to the original plan. By publishing this special issue we wish to pay homage to Professor Birnbaum’s penetrating and stimulating work on the foundations of statistics. We are grateful to Professor Ronald Giere who wrote an introductory essay on Professor Birnbaum’s concept of statistical evidence and who compiled a list of Professor Birnbaum’s publications.

THE EDITORS

Table of Contents

Editorial Introduction. (1977). Synthese, 36(1), 3-3.

Giere, R. (1977). Allan Birnbaum’s Conception of Statistical Evidence. Synthese, 36(1), 5-13.

SUFFICIENCY, CONDITIONALLY AND LIKELIHOOD In December of 1961 Birnbaum presented the paper ‘On the Foundations, of Statistical Inference’ (Birnbaum [19]) at a special discussion meeting of the American Statistical Association. Among the discussants was L. J. Savage who pronounced it “a landmark in statistics”. Explicitly denying any “intent to speak with exaggeration or rhetorically”, Savage described the occasion as “momentous in the history of statistics”. “It would be hard”, he said, “to point to even a handful of comparable events” (Birnbaum [19], pp. 307-8). The reasons for Savage’s enthusiasm are obvious. Birnbaum claimed to have shown that two principles widely held by non-Bayesian statisticians (sufficiency and conditionality) jointly imply an important consequence of Bayesian statistics (likelihood).”[1]

Giere, R. (1977). Publications by Allan Birnbaum. Synthese, 36(1), 15-17.

Birnbaum, A. (1977). The Neyman-Pearson Theory as Decision Theory, and as Inference Theory; With a Criticism of the Lindley-Savage Argument for Bayesian Theory. Synthese, 36(1), 19-49.

INTRODUCTION AND SUMMARY ….Two contrasting interpretations of the decision concept are formulated: behavioral, applicable to ‘decisions’ in a concrete literal sense as in acceptance sampling; and evidential, applicable to ‘decisions’ such as ‘reject H in a research context, where the pattern and strength of statistical evidence concerning statistical hypotheses is of central interest. Typical standard practice is characterized as based on the confidence concept of statistical evidence, which is defined in terms of evidential interpretations of the ‘decisions’ of decision theory. These concepts are illustrated by simple formal examples with interpretations in genetic research, and are traced in the writings of Neyman, Pearson, and other writers. The Lindley-Savage argument for Bayesian theory is shown to have no direct cogency as a criticism of typical standard practice, since it is based on a behavioral, not an evidential, interpretation of decisions.

Lindley, D. (1977). The Distinction between Inference and Decision. Synthese, 36(1), 51-58.

Pratt, J. (1977). ‘Decisions’ as Statistical Evidence and Birnbaum’s ‘Confidence Concept’Synthese, 36(1), 59-69.

Smith, C. (1977). The Analogy between Decision and Inference. Synthese, 36(1), 71-85.

Kyburg, H. (1977). Decisions, Conclusions, and Utilities. Synthese, 36(1), 87-96.

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

Lecam, L. (1977).A Note on Metastatistics or ‘An Essay toward Stating a Problem in the Doctrine of Chances’. Synthese, 36(1), 133-160.

Kiefer, J. (1977). The Foundations of Statistics Are There Any? Synthese, 36(1), 161-176.

[1]By “likelihood” here, Giere means the (strong) Likelihood Principle (SLP). Dotted through the first 3 years of this blog are a number of (formal and informal) posts on his SLP result, and my argument as to why it is unsound. I wrote a paper on this that appeared in Statistical Science 2014. You can find it along with a number of comments and my rejoinder in this post: Statistical Science: The Likelihood Principle Issue is Out.The consequences of having found his proof unsound gives a new lease on life to statistical foundations, or so I argue in my rejoinder.

Categories: Birnbaum, Likelihood Principle, Statistics, strong likelihood principle | Tags: Birnbaum | 1 Comment

Frequentstein’s Bride: What’s wrong with using (1 – β)/α as a measure of evidence against the null?

Posted on May 23, 2017 by Mayo

ONE YEAR AGO: …and growing more relevant all the time. Rather than leak any of my new book*, I reblog some earlier posts, even if they’re a bit scruffy. This was first blogged here (with a slightly different title). It’s married to posts on “the P-values overstate the evidence against the null fallacy”, such as this, and is wedded to this one on “How to Tell What’s True About Power if You’re Practicing within the Frequentist Tribe”.

In their “Comment: A Simple Alternative to p-values,” (on the ASA P-value document), Benjamin and Berger (2016) recommend researchers report a pre-data Rejection Ratio:

It is the probability of rejection when the alternative hypothesis is true, divided by the probability of rejection when the null hypothesis is true, i.e., the ratio of the power of the experiment to the Type I error of the experiment. The rejection ratio has a straightforward interpretation as quantifying the strength of evidence about the alternative hypothesis relative to the null hypothesis conveyed by the experimental result being statistically significant. (Benjamin and Berger 2016, p. 1)

The recommendation is much more fully fleshed out in a 2016 paper by Bayarri, Benjamin, Berger, and Sellke (BBBS 2016): Rejection Odds and Rejection Ratios: A Proposal for Statistical Practice in Testing Hypotheses. Their recommendation is:

…that researchers should report the ‘pre-experimental rejection ratio’ when presenting their experimental design and researchers should report the ‘post-experimental rejection ratio’ (or Bayes factor) when presenting their experimental results. (BBBS 2016, p. 3)….

The (pre-experimental) ‘rejection ratio’ R_pre , the ratio of statistical power to significance threshold (i.e., the ratio of the probability of rejecting under H₁ and H₀ respectively), is shown to capture the strength of evidence in the experiment for H₁over H₀. (ibid., p. 2)

But it does no such thing! [See my post from the FUSION 2016 conference here.] J. Berger, and his co-authors, will tell you the rejection ratio (and a variety of other measures created over the years) are entirely frequentist because they are created out of frequentist error statistical measures. But a creation built on frequentist measures doesn’t mean the resulting animal captures frequentist error statistical reasoning. It might be a kind of Frequentstein monster! [1]

~~~~~~~~~~~~~~

The Law of Comparative Support

It comes from a comparativist support position which has intrinsic plausibility, although I do not hold to it. It is akin to what some likelihoodists call “the law of support”: if H₁ make the observed results probable, while H₀ make them improbable, then the results are strong (or at least better) evidence for H₁ compared to H₀ . It appears to be saying (sensibly) that you have better evidence for a hypothesis that best “explains” the data, only this is not a good measure of explanation. It is not generally required H₀ and H₁ be exhaustive. Even if you hold a comparative support position, the “ratio of statistical power to significance threshold” isn’t a plausible measure for this. Now BBBS also object to the Rejection Ratio, but ~~only~~ largely because it’s not sensitive to the actual outcome; so they recommend the Bayes Factor post data. My criticism is much, much deeper. To get around the data-dependent part, let’s assume throughout that we’re dealing with a result just statistically significant at the α level.

~~~~~~~~~~~~~~

Take a one-sided Normal test T+: with n iid samples:

H₀: µ ≤ 0 against H₁: µ > 0

σ = 10, n = 100, σ/√n =σ_x= 1, α = .025.

So the test would reject H₀ iff Z > c_.025 =1.96. (1.96. is the “cut-off”.)

People often talk of a test “having a power” but the test actually specifies a power function that varies with different point values in the alternative H₁ . The power of test T+ in relation to point alternative µ’ is

Pr(Z > 1.96; µ = µ’).

We can abbreviate this as POW(T+,µ’).

~~~~~~~~~~~~~~

Jacob Cohen’s slips

By the way, Jacob Cohen, a founder of power analysis, makes a few slips in introducing power, even though he correctly computes power through the book (so far as I know). [2] Someone recently reminded me of this, and given the confusion about power, maybe it’s had more of an ill effect than I assumed.

In the first sentence on p. 1 of Statistical Power Analysis for the Behavioral Sciences, Cohen says “The power of a statistical test is the probability it will yield statistically significant results.” Also faulty, and for two reasons, is what he says on p. 4: “The power of a statistical test of a null hypothesis is the probability that it will lead to the rejection of the null hypothesis, i.e., the probability that it will result in the conclusion that the phenomenon exists.”

Do you see the two mistakes?

~~~~~~~~~~~~~~

Examples of alternatives against which T+ has high power:

If we add σ_x(i.e.,σ/√n) to the cut-off (1.96) we are at an alternative value for µ that test T+ has .84 power to detect. In this example, σ_x = 1.
If we add 3σ_xto the cut-off we are at an alternative value for µ that test T+ has ~ .999 power to detect. This value, which we can write as µ^.⁹⁹⁹ = 4.96

Let the observed outcome just reach the cut-off to reject the null, z₀= 1.96.

If we were to form a “rejection ratio” or a “likelihood ratio” of μ = 4.96 compared to μ₀ = 0 using

[POW(T+, 4.96)]/α,

it would be 40. (.999/.025).

It is absurd to say the alternative 4.96 is supported 40 times as much as the null, even understanding support as comparative likelihood or something akin. The data 1.96 are even closer to 0 than to 4.96. The same point can be made with less extreme cases.) What is commonly done next is to assign priors of .5 to the two hypotheses, yielding

Pr(H₀|z₀) = 1/(1 + 40) = .024, so Pr(H₁|z₀) = .976.

Such an inference is highly unwarranted and would almost always be wrong. Back to our question:

Here’s my explanation for why some think it’s plausible to compute comparative evidence this way:

I presume it stems comes from the comparativist support position noted above. I’m guessing they’re reasoning as follows:

The probability is very high that z > 1.96 under the assumption that μ = 4.96.

The probability is low that z > 1.96 under the assumption that μ = μ₀ = 0.

We’ve observed z₀= 1.96 (so you’ve observed z > 1.96).

Therefore, μ = 4.96 makes the observation more probable than does μ = 0.

Therefore the outcome is (comparatively) better evidence for μ= 4.96 than for μ = 0.

But the “outcome” for a likelihood is to be the specific outcome, and the comparative appraisal of which hypothesis accords better with the data only makes sense when one keeps to this.

I can pick any far away alternative I like for purposes of getting high power, and we wouldn’t want to say that just reaching the cut-off (1.96) is good evidence for it! Power works in the reverse. That is,

If POW(T+,µ’) is high, then z₀= 1.96 is poor evidence that μ > μ’.

That’s because were μ as great as μ’, with high probability we would have observed a larger z value (smaller p-value) than we did. Power may, if one wishes, be seen as a kind of distance measure, but (just like α) it is inverted.

(Note that our inferences take the form μ > μ’, μ < μ’, etc. rather than to a point value.)

In fact:

if Pr(Z > z₀;μ =μ’) = high , then Z = z₀ is strong evidence that μ < μ’!

Rather than being evidence for μ’, the statistically significant result is evidence against μ being as high as μ’.
~~~~~~~~~~~~~~

A post by Stephen Senn:

In my favorite guest post by Stephen Senn here, Senn strengthens a point from his 2008 book (p. 201), namely, that the following is “nonsense”:

[U]pon rejecting the null hypothesis, not only may we conclude that the treatment is effective but also that it has a clinically relevant effect. (Senn 2008, p. 201)

Now the test is designed to have high power to detect a clinically relevant effect (usually .8 or .9). I happen to have chosen an extremely high power (.999) but the claim holds for any alternative that the test has high power to detect. The clinically relevant discrepancy, as he describes it, is one “we should not like to miss”, but obtaining a statistically significant result is not evidence we’ve found a discrepancy that big.

Supposing that it is, is essentially to treat the test as if it were:

H₀: μ < 0 vs H₁: μ > 4.96

This, he says, is “ludicrous”as it:

would imply that we knew, before conducting the trial, that the treatment effect is either zero or at least equal to the clinically relevant difference. But where we are unsure whether a drug works or not, it would be ludicrous to maintain that it cannot have an effect which, while greater than nothing, is less than the clinically relevant difference. (Senn, 2008, p. 201)

The same holds with H₀: μ = 0 as null.

If anything, it is the lower confidence limit that we would look at to see what discrepancies from 0 are warranted. The lower .975 limit (if one-sided) or .95 (if two-sided) would be 0 and .3, respectively. So we would be warranted in inferring from z:

μ > 0 or μ > .3.

~~~~~~~~~~~~~~

What does the severe tester say?

In sync with the confidence interval, she would say SEV(μ > 0) = .975 (if one-sided), and would also note some other benchmarks, e.g., SEV(μ > .96) = .84.

Equally important for her is a report of what is poorly warranted. In particular the claim that the data indicate

μ > 4.96

would be wrong over 99% of the time!

Of course, I would want to use the actual result, rather than the cut-off for rejection (as with power) but the reasoning is the same, and here I deliberately let the outcome just hit the cut-off for rejection.

~~~~~~~~~~~~~~

The (Type 1, 2 error probability) trade-off vanishes

Notice what happens if we consider the “real Type 1 error” as Pr(H₀|z₀)

Since Pr(H₀|z₀) decreases with increasing power, it decreases with decreasing Type 2 error. So we know that to identify “Type 1 error” and Pr(H₀|z₀) is to use language in a completely different way than the one in which power is defined. For there we must have a trade-off between Type 1 and 2 error probabilities.

Upshot

Using size/ power as a likelihood ratio, or even as a preregistrated estimate of expected strength of evidence (with which to accord a rejection) is problematic. The error statistician is not in the business of making inferences to point values, nor to comparative appraisals of different point hypotheses. It’s not unusual for criticisms to start out forming these ratios, and then blame the “tail areas” for exaggerating the evidence against the test hypothesis. We don’t form those ratios. But the pre-data Rejection Ratio is also misleading as an assessment alleged to be akin to a Bayes ratio or likelihood assessment. You can marry frequentist components and end up with something frequentsteinian.

REFERENCES

Bayarri, M., Benjamin, D., Berger, J., & Sellke, T. (2016, in press). “Rejection Odds and Rejection Ratios: A Proposal for Statistical Practice in Testing Hypotheses“, Journal of Mathematical Psychology

Benjamin, D. & Berger J. 2016. “Comment: A Simple Alternative to P-values,” The American Statistician (online March 7, 2016).

Cohen, J. 1988. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Erlbaum.

Mayo, D. 2016. “Don’t throw out the Error Control Baby with the Error Statistical Bathwater“. (My comment on the ASA document)

Mayo, D. 2003. Comments on J. Berger’s, “Could Jeffreys, Fisher and Neyman have Agreed on Testing? (pp. 19-24)

*Mayo, D. Statistical Inference as Severe Testing, forthcoming (2017) CUP.

Senn, S. 2008. Statistical Issues in Drug Development, 2^nd ed. Chichster, New Sussex: Wiley Interscience, John Wiley & Sons.

Wasserstein, R. & Lazar, N. 2016. “The ASA’s Statement on P-values: Context, Process and Purpose”, The American Statistician (online March 7, 2016).

[1] I don’t say there’s no context where the Rejection Ratio has a frequentist role. It may arise in a diagnostic screening or empirical Bayesian context where one has to deal with a dichotomy. See, for example, this post (“Beware of questionable front page articles telling you to beware…”)

[2] It may also be found in Neyman! (Search this blog under Neyman’s Nursery.) However, Cohen uniquely provides massive power computations, before it was all computerized.

Categories: Bayesian/frequentist, fallacy of rejection, J. Berger, power, S. Senn | 17 Comments

3 YEARS AGO (APRIL 2014): MEMORY LANE

Posted on May 15, 2017 by Mayo

3 years ago…

MONTHLY MEMORY LANE: 3 years ago: April 2014. I mark in red three posts from each month that seem most apt for general background on key issues in this blog, excluding those reblogged recently[1], and in green up to 4 others I’d recommend[2]. Posts that are part of a “unit” or a group count as one. For this month, I’ll include all the 6334 seminars as “one”.

April 2014

(4/1) April Fool’s. Skeptical and enthusiastic Bayesian priors for beliefs about insane asylum renovations at Dept of Homeland Security: I’m skeptical and unenthusiastic
(4/3) Self-referential blogpost (conditionally accepted*)
(4/5) Who is allowed to cheat? I.J. Good and that after dinner comedy hour. . ..
(4/6) Phil6334: Duhem’s Problem, highly probable vs highly probed; Day #9 Slides
(4/8) “Out Damned Pseudoscience: Non-significant results are the new ‘Significant’ results!” (update)
(4/12) “Murder or Coincidence?” Statistical Error in Court: Richard Gill (TEDx video)
(4/14) Phil6334: Notes on Bayesian Inference: Day #11 Slides
(4/16) A. Spanos: Jerzy Neyman and his Enduring Legacy
(4/17) Duality: Confidence intervals and the severity of tests
(4/19) Getting Credit (or blame) for Something You Didn’t Do (BP oil spill)
(4/21) Phil 6334: Foundations of statistics and its consequences: Day#12
(4/23) Phil 6334 Visitor: S. Stanley Young, “Statistics and Scientific Integrity”
(4/26) Reliability and Reproducibility: Fraudulent p-values through multiple testing (and other biases): S. Stanley Young (Phil 6334: Day #13)
(4/30) Able Stats Elba: 3 Palindrome nominees for April! (rejected post)

[1] Monthly memory lanes began at the blog’s 3-year anniversary in Sept, 2014.

[2] New Rule, July 30,2016, March 30,2017 (moved to 4) -very convenient way to allow data-dependent choices.

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Categories: 3-year memory lane, Statistics | Leave a comment

How to tell what’s true about power if you’re practicing within the error-statistical tribe

Posted on May 8, 2017 by Mayo

This is a modified reblog of an earlier post, since I keep seeing papers that confuse this.

Suppose you are reading about a result x that is just statistically significant at level α (i.e., P-value = α) in a one-sided test T+ of the mean of a Normal distribution with n iid samples, and (for simplicity) known σ: H₀: µ ≤ 0 against H₁: µ > 0.

I have heard some people say:

A. If the test’s power to detect alternative µ’ is very low, then the just statistically significant x is poor evidence of a discrepancy (from the null) corresponding to µ’. (i.e., there’s poor evidence that µ > µ’ ).*See point on language in notes.

They will generally also hold that if POW(µ’) is reasonably high (at least .5), then the inference to µ > µ’ is warranted, or at least not problematic.

I have heard other people say:

B. If the test’s power to detect alternative µ’ is very low, then the just statistically significant x is good evidence of a discrepancy (from the null) corresponding to µ’ (i.e., there’s good evidence that µ > µ’).

They will generally also hold that if POW(µ’) is reasonably high (at least .5), then the inference to µ > µ’ is unwarranted.

Which is correct, from the perspective of the (error statistical) philosophy, within which power and associated tests are defined? Continue reading →

Categories: power, reforming the reformers | 17 Comments

“Fusion-Confusion?” My Discussion of Nancy Reid: “BFF Four- Are we Converging?”

Posted on May 2, 2017 by Mayo

Here are the slides from my discussion of Nancy Reid today at BFF4: The Fourth Bayesian, Fiducial, and Frequentist Workshop: May 1-3, 2017 (hosted by Harvard University)

Categories: Bayesian/frequentist, C.S. Peirce, confirmation theory, fiducial probability, Fisher, law of likelihood, Popper | Tags: Hacking | 1 Comment

S. Senn: “Automatic for the people? Not quite” (Guest post)

Posted on April 23, 2017 by Mayo

Stephen Senn

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

Automatic for the people? Not quite

What caught my eye was the estimable (in its non-statistical meaning) Richard Lehman tweeting about the equally estimable John Ioannidis. For those who don’t know them, the former is a veteran blogger who keeps a very cool and shrewd eye on the latest medical ‘breakthroughs’ and the latter a serial iconoclast of idols of scientific method. This is what Lehman wrote

Ioannidis hits 8 on the Richter scale: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0173184 … Bayes factors consistently quantify strength of evidence, p is valueless.

Since Ioannidis works at Stanford, which is located in the San Francisco Bay Area, he has every right to be interested in earthquakes but on looking up the paper in question, a faint tremor is the best that I can afford it. I shall now try and explain why, but before I do, it is only fair that I acknowledge the very generous, prompt and extensive help I have been given to understand the paper[1] in question by its two authors Don van Ravenzwaaij and Ioannidis himself. Continue reading →

Categories: Bayesian/frequentist, Error Statistics, S. Senn | 18 Comments

The Fourth Bayesian, Fiducial and Frequentist Workshop (BFF4): Harvard U

Posted on April 21, 2017 by Mayo

May 1-3, 2017
Hilles Event Hall, 59 Shepard St. MA

The Department of Statistics is pleased to announce the 4th Bayesian, Fiducial and Frequentist Workshop (BFF4), to be held on May 1-3, 2017 at Harvard University. The BFF workshop series celebrates foundational thinking in statistics and inference under uncertainty. The three-day event will present talks, discussions and panels that feature statisticians and philosophers whose research interests synergize at the interface of their respective disciplines. Confirmed featured speakers include Sir David Cox and Stephen Stigler.

The program will open with a featured talk by Art Dempster and discussion by Glenn Shafer. The featured banquet speaker will be Stephen Stigler. Confirmed speakers include:

Featured Speakers and Discussants: Arthur Dempster (Harvard); Cynthia Dwork (Harvard); Andrew Gelman (Columbia); Ned Hall (Harvard); Deborah Mayo (Virginia Tech); Nancy Reid (Toronto); Susanna Rinard (Harvard); Christian Robert (Paris-Dauphine/Warwick); Teddy Seidenfeld (CMU); Glenn Shafer (Rutgers); Stephen Senn (LIH); Stephen Stigler (Chicago); Sandy Zabell (Northwestern)

Invited Speakers and Panelists: Jim Berger (Duke); Emery Brown (MIT/MGH); Larry Brown (Wharton); David Cox (Oxford; remote participation); Paul Edlefsen (Hutch); Don Fraser (Toronto); Ruobin Gong (Harvard); Jan Hannig (UNC); Alfred Hero (Michigan); Nils Hjort (Oslo); Pierre Jacob (Harvard); Keli Liu (Stanford); Regina Liu (Rutgers); Antonietta Mira (USI); Ryan Martin (NC State); Vijay Nair (Michigan); James Robins (Harvard); Daniel Roy (Toronto); Donald B. Rubin (Harvard); Peter XK Song (Michigan); Gunnar Taraldsen (NUST); Tyler VanderWeele (HSPH); Vladimir Vovk (London); Nanny Wermuth (Chalmers/Gutenberg); Min-ge Xie (Rutgers)

Continue reading →

Categories: Announcement, Bayesian/frequentist | 2 Comments

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

Posted on April 18, 2017 by Mayo

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”. Continue reading →

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

Posted on April 17, 2017 by Mayo

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: behavioristic vs evidential | 2 Comments

A. Spanos: Jerzy Neyman and his Enduring Legacy

Posted on April 16, 2017 by Mayo

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 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 M_θ(x) was based on the idea of ‘a hypothetical infinite population’, chosen so as to ensure that the observed data x₀:=(x₁,x₂,…,x_n) 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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If you’re seeing limb-sawing in P-value logic, you’re sawing off the limbs of reductio arguments

Posted on April 15, 2017 by Mayo

I was just reading a paper by Martin and Liu (2014) in which they allude to the “questionable logic of proving H₀ false by using a calculation that assumes it is true”(p. 1704). They say they seek to define a notion of “plausibility” that

“fits the way practitioners use and interpret p-values: a small p-value means H₀ is implausible, given the observed data,” but they seek “a probability calculation that does not require one to assume that H₀ is true, so one avoids the questionable logic of proving H₀ false by using a calculation that assumes it is true“(Martin and Liu 2014, p. 1704).

Questionable? A very standard form of argument is a reductio (ad absurdum) wherein a claim C is inferred (i.e., detached) by falsifying ~C, that is, by showing that assuming ~C entails something in conflict with (if not logically contradicting) known results or known truths [i]. Actual falsification in science is generally a statistical variant of this argument. Supposing H₀in p-value reasoning plays the role of ~C. Yet some aver it thereby “saws off its own limb”! Continue reading →

Categories: P-values, reforming the reformers, Statistics | 13 Comments

3 YEARS AGO (MARCH 2014): MEMORY LANE

Posted on April 9, 2017 by Mayo

3 years ago…

MONTHLY MEMORY LANE: 3 years ago: March 2014. I mark in red three posts from each month that seem most apt for general background on key issues in this blog, excluding those reblogged recently[1], and in green up to 4 others I’d recommend[2]. Posts that are part of a “unit” or a group count as one. 3/19 and 3/17 are one, as are 3/19, 3/12 and 3/4, and the 6334 items 3/11, 3/22 and 3/26. So that covers nearly all the posts!

March 2014

(3/1) Cosma Shalizi gets tenure (at last!) (metastat announcement)
(3/2) Significance tests and frequentist principles of evidence: Phil6334 Day #6
(3/3) Capitalizing on Chance (ii)
(3/4) Power, power everywhere–(it) may not be what you think! [illustration]
(3/8) Msc kvetch: You are fully dressed (even under you clothes)?
(3/8) Fallacy of Rejection and the Fallacy of Nouvelle Cuisine
(3/11) Phil6334 Day #7: Selection effects, the Higgs and 5 sigma, Power
(3/12) Get empowered to detect power howlers
(3/15) New SEV calculator (guest app: Durvasula)
(3/17) Stephen Senn: “Delta Force: To what extent is clinical relevance relevant?” (Guest Post)
(3/19) Power taboos: Statue of Liberty, Senn, Neyman, Carnap, Severity
(3/22) Fallacies of statistics & statistics journalism, and how to avoid them: Summary & Slides Day #8 (Phil 6334)
(3/25) The Unexpected Way Philosophy Majors Are Changing The World Of Business
(3/26) Phil6334:Misspecification Testing: Ordering From A Full Diagnostic Menu (part 1)
(3/28) Severe osteometric probing of skeletal remains: John Byrd
(3/29) Winner of the March 2014 palindrome contest (rejected post)
(3/30) Phil6334: March 26, philosophy of misspecification testing (Day #9 slides)

[1] Monthly memory lanes began at the blog’s 3-year anniversary in Sept, 2014.

[2] New Rule, July 30,2016, March 30,2017 (moved to 4) -very convenient way to allow data-dependent choices.

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Categories: 3-year memory lane, Error Statistics, Statistics | Leave a comment

Announcement: Columbia Workshop on Probability and Learning (April 8)

Posted on April 3, 2017 by Mayo

I’m speaking on “Probing with Severity” at the “Columbia Workshop on Probability and Learning” On April 8:

Meetings of the Formal Philosophy Group at Columbia

April 8, 2017

Department of Philosophy, Columbia University

Room 716
Philosophy Hall, 1150 Amsterdam Avenue
New York 10027
United States

Sponsor(s):

The Formal Philosophy Group (Columbia)

Main speakers:

Gordon Belot (University of Michigan, Ann Arbor)

Simon Huttegger (University of California, Irvine)

Deborah Mayo (Virginia Tech)

Teddy Seidenfeld (Carnegie Mellon University)

Organisers:

Michael Nielsen (Columbia University)

Rush Stewart (Columbia University)

Details

Unfortunately, access to Philosophy Hall is by swipe access on the weekends. However, students and faculty will be entering and exiting the building throughout the day (with relateively high frequency since there is a popular cafe on the main floor).

https://www.facebook.com/events/1869417706613583/

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Er, about those other approaches, hold off until a balanced appraisal is in

Posted on April 1, 2017 by Mayo

I could have told them that the degree of accordance enabling the ASA’s “6 principles” on p-values was unlikely to be replicated when it came to most of the “other approaches” with which some would supplement or replace significance tests– notably Bayesian updating, Bayes factors, or likelihood ratios (confidence intervals are dual to hypotheses tests). [My commentary is here.] So now they may be advising a “hold off” or “go slow” approach until some consilience is achieved. Is that it? I don’t know. I was tweeted an article about the background chatter taking place behind the scenes; I wasn’t one of people interviewed for this. Here are some excerpts, I may add more later after it has had time to sink in. (check back later)

“Reaching for Best Practices in Statistics: Proceed with Caution Until a Balanced Critique Is In”

J. Hossiason

“[A]ll of the other approaches*, as well as most statistical tools, may suffer from many of the same problems as the p-values do. What level of likelihood ratio in favor of the research hypothesis will be acceptable to the journal? Should scientific discoveries be based on whether posterior odds pass a specific threshold (P3)? Does either measure the size of an effect (P5)?…How can we decide about the sample size needed for a clinical trial—however analyzed—if we do not set a specific bright-line decision rule? 95% confidence intervals or credence intervals…offer no protection against selection when only those that do not cover 0, are selected into the abstract (P4). (Benjamini, ASA commentary, pp. 3-4)

What’s sauce for the goose is sauce for the gander right? Many statisticians seconded George Cobb who urged “the board to set aside time at least once every year to consider the potential value of similar statements” to the recent ASA p-value report. Disappointingly, a preliminary survey of leaders in statistics, many from the original p-value group, aired striking disagreements on best and worst practices with respect to these other approaches. The Executive Board is contemplating a variety of recommendations, minimally, that practitioners move with caution until they can put forward at least a few agreed upon principles for interpreting and applying Bayesian inference methods. The words we heard ranged from “go slow” to “moratorium“ [emphasis mine]. Having been privy to some of the results of this survey, we at Stat Report Watch decided to contact some of the individuals involved. Continue reading →

Categories: P-values, reforming the reformers, Statistics | 6 Comments

Slides from the Boston Colloquium for Philosophy of Science: “Severe Testing: The Key to Error Correction”

Posted on March 26, 2017 by Mayo

Slides from my March 17 presentation on “Severe Testing: The Key to Error Correction” given at the Boston Colloquium for Philosophy of Science Alfred I.Taub forum on “Understanding Reproducibility and Error Correction in Science.”

Categories: fallacy of rejection, Fisher, fraud, frequentist/Bayesian, Likelihood Principle, reforming the reformers | 16 Comments