Statistics

Phil/Stat/Law: What Bayesian prior should a jury have? (Schachtman)

Posted on July 17, 2013 by Mayo

wavy capitalNathan Schachtman, Esq., PC* emailed me the following interesting query a while ago:

NAS-3When I was working through some of the Bayesian in the law issues with my class, I raised the problem of priors of 0 and 1 being off “out of bounds” for a Bayesian analyst.  I didn’t realize then that the problem had a name:  Cromwell’s Rule.

My point was then, and more so now, what is the appropriate prior the jury should have when it is sworn?  When it hears opening statements?  Just before the first piece of evidence is received?

Do we tell the jury that the defendant is presumed innocent, which means that it’s ok to entertain a very, very small prior probability of guilt, say no more than 1/N, where N is the total population of people? This seems wrong as a matter of legal theory.  But if the prior = 0, then no amount of evidence can move the jury off its prior.

*Schachtman’s legal practice focuses on the defense of product liability suits, with an emphasis on the scientific and medico-legal issues.  He teaches a course in statistics in the law at the Columbia Law School, NYC. He also has a legal blog here.

Categories: PhilStatLaw, Statistics | Tags: | 27 Comments

Stephen Senn: Indefinite irrelevance

Posted on July 14, 2013 by Mayo
Stephen SennStephen Senn
Head, Methodology and Statistics Group,
Competence Center for Methodology and Statistics (CCMS),
Luxembourg

At a workshop on randomisation I attended recently I was depressed to hear what I regard as hackneyed untruths treated as if they were important objections. One of these is that of indefinitely many confounders. The argument goes that although randomisation may make it probable that some confounders are reasonably balanced between the arms, since there are indefinitely many of these, the chance that at least some are badly confounded is so great as to make the procedure useless.

This argument is wrong for several related reasons. The first is to do with the fact that the total effect of these indefinitely many confounders is bounded. This means that the argument put forward is analogously false to one in which it were claimed that the infinite series ½, ¼,⅛ …. did not sum to a limit because there were infinitely many terms. The fact is that the outcome value one wishes to analyse poses a limit on the possible influence of the covariates. Suppose that we were able to measure a number of covariates on a set of patients prior to randomisation (in fact this is usually not possible but that does not matter here). Now construct principle components, C1, C2… .. based on these covariates. We suppose that each of these predict to a greater or lesser extent the outcome, Y  (say).  In a linear model we could put coefficients on these components, k1, k2… (say). However one is not free to postulate anything at all by way of values for these coefficients, since it has to be the case for any set of m such coefficients that inequality (2)where  V(  ) indicates variance of. Thus variation in outcome bounds variation in prediction. This total variation in outcome has to be shared between the predictors and the more predictors you postulate there are, the less on average the influence per predictor.

The second error is to ignore the fact that statistical inference does not proceed on the basis of signal alone but also on noise. It is the ratio of these that is important. If there are indefinitely many predictors then there is no reason to suppose that their influence on the variation between treatment groups will be bigger than their variation within groups and both of these are used to make the inference. Continue reading →

Categories: RCTs, Statistics, Stephen Senn | 15 Comments

Is Particle Physics Bad Science? (memory lane)

Posted on July 11, 2013 by Mayo

Memory Lane: reblog July 11, 2012 (+ updates at the end). 

I suppose[ed] this was somewhat of a joke from the ISBA, prompted by Dennis Lindley, but as I [now] accord the actual extent of jokiness to be only ~10%, I’m sharing it on the blog [i].  Lindley (according to O’Hagan) wonders why scientists require so high a level of statistical significance before claiming to have evidence of a Higgs boson.  It is asked: “Are the particle physics community completely wedded to frequentist analysis?  If so, has anyone tried to explain what bad science that is?”

Bad science?   I’d really like to understand what these representatives from the ISBA would recommend, if there is even a shred of seriousness here (or is Lindley just peeved that significance levels are getting so much press in connection with so important a discovery in particle physics?)

Well, read the letter and see what you think.

On Jul 10, 2012, at 9:46 PM, ISBA Webmaster wrote:

Dear Bayesians,

A question from Dennis Lindley prompts me to consult this list in search of answers.

We’ve heard a lot about the Higgs boson.  The news reports say that the LHC needed convincing evidence before they would announce that a particle had been found that looks like (in the sense of having some of the right characteristics of) the elusive Higgs boson.  Specifically, the news referred to a confidence interval with 5-sigma limits.

Now this appears to correspond to a frequentist significance test with an extreme significance level.  Five standard deviations, assuming normality, means a p-value of around 0.0000005.  A number of questions spring to mind.

1.  Why such an extreme evidence requirement?  We know from a Bayesian  perspective that this only makes sense if (a) the existence of the Higgs  boson (or some other particle sharing some of its properties) has extremely small prior probability and/or (b) the consequences of erroneously announcing its discovery are dire in the extreme.  Neither seems to be the case, so why  5-sigma?

2.  Rather than ad hoc justification of a p-value, it is of course better to do a proper Bayesian analysis.  Are the particle physics community completely wedded to frequentist analysis?  If so, has anyone tried to explain what bad science that is? Continue reading →

Categories: philosophy of science, Statistics | Tags: , , , , , | Leave a comment

Bad news bears: ‘Bayesian bear’ rejoinder-reblog mashup

Posted on July 6, 2013 by Mayo

Oh No! It’s those mutant bears again. To my dismay, I’ve been sent, for the third time, that silly, snarky, adolescent, clip of those naughty “what the p-value” bears (first posted on Aug 5, 2012), who cannot seem to get a proper understanding of significance tests into their little bear brains. So apparently some people haven’t seen my rejoinder which, as I said then, practically wrote itself. So since it’s Saturday night here at the Elbar Room, let’s listen in to a mashup of both the clip and my original rejoinder (in which p-value bears are replaced with hypothetical Bayesian bears). 

These stilted bear figures and their voices are sufficiently obnoxious in their own right, even without the tedious lampooning of p-values and the feigned horror at learning they should not be reported as posterior probabilities.

Mayo’s Rejoinder:

Bear #1: Do you have the results of the study?

Bear #2:Yes. The good news is there is a .996 probability of a positive difference in the main comparison.

Bear #1: Great. So I can be well assured that there is just a .004 probability that such positive results would occur if they were merely due to chance.

Bear #2: Not really, that would be an incorrect interpretation. Continue reading →

Categories: Bayesian/frequentist, Comedy, P-values, Statistics | Tags: , , , | 13 Comments

Phil/Stat/Law: 50 Shades of gray between error and fraud

Posted on July 3, 2013 by Mayo

500x307-embo-reports-vol-73-meeting-report-fig-1-abcAn update on the Diederik Stapel case: July 2, 2013, The Scientist, “Dutch Fraudster Scientist Avoids Jail”.

Two years after being exposed by colleagues for making up data in at least 30 published journal articles, former Tilburg University professor Diederik Stapel will avoid a trial for fraud. Once one of the Netherlands’ leading social psychologists, Stapel has agreed to a pre-trial settlement with Dutch prosecutors to perform 120 hours of community service.

According to Dutch newspaper NRC Handeslblad, the Dutch Organization for Scientific Research awarded Stapel $2.8 million in grants for research that was ultimately tarnished by misconduct. However, the Dutch Public Prosecution Service and the Fiscal Information and Investigation Service said on Friday (June 28) that because Stapel used the grant money for student and staff salaries to perform research, he had not misused public funds. …

In addition to the community service he will perform, Stapel has agreed not to make a claim on 18 months’ worth of illness and disability compensation that he was due under his terms of employment with Tilburg University. Stapel also voluntarily returned his doctorate from the University of Amsterdam and, according to Retraction Watch, retracted 53 of the more than 150 papers he has co-authored.

“I very much regret the mistakes I have made,” Stapel told ScienceInsider. “I am happy for my colleagues as well as for my family that with this settlement, a court case has been avoided.”

No surprise he’s not doing jail time, but 120 hours of community service?  After over a decade of fraud, and tainting 14 of 21 of the PhD theses he supervised?  Perhaps the “community service” should be to actually run the experiments he had designed?  What about his innocence of misusing public funds? Continue reading →

Categories: PhilStatLaw, spurious p values, Statistics | 13 Comments

Blog Contents: mid-year

Posted on June 30, 2013 by Mayo

Error Statistics Philosophy BLOG: Table of Contents 2013 (January-June)*

img_02443January 2013

(1/2) Severity as a ‘Metastatistical’ Assessment
(1/4) Severity Calculator
(1/6) Guest post: Bad Pharma? (S. Senn)
(1/9) RCTs, skeptics, and evidence-based policy
(1/10) James M. Buchanan
(1/11) Aris Spanos: James M. Buchanan: a scholar, teacher and friend
(1/12) Error Statistics Blog: Table of Contents
(1/15) Ontology & Methodology: Second call for Abstracts, Papers
(1/18) New Kvetch/PhilStock
(1/19) Saturday Night Brainstorming and Task Forces: (2013) TFSI on NHST
(1/22) New PhilStock
(1/23) P-values as posterior odds?
(1/26) Coming up: December U-Phil Contributions….
(1/27) U-Phil: S. Fletcher & N. Jinn
(1/30) U-Phil: J. A. Miller: Blogging the SLP Continue reading →

Categories: Metablog, Statistics | Leave a comment

Why I am not a “dualist” in the sense of Sander Greenland

Posted on June 26, 2013 by Mayo

Janus--2face

This post picks up, and continues, an exchange that began with comments on my June 14 blogpost (between Sander Greenland, Nicole Jinn, and I). My new response is at the end. The concern is how to expose and ideally avoid some of the well known flaws and foibles in statistical inference, thanks to gaps between data and statistical inference, and between statistical inference and substantive claims. I am not rejecting the use of multiple methods in the least (they are highly valuable when one method is capable of detecting or reducing flaws in one or more others). Nor am I speaking of classical dualism in metaphysics (which I also do not espouse). I begin with Greenland’s introduction of this idea in his comment… (For various earlier comments, see the post.)

Sander Greenland 

…. I sense some confusion of criticism of the value of tests as popular tools vs. criticism of their logical foundation. I am a critic in the first, practical category, who regards the adoption of testing outside of narrow experimental programs as an unmitigated disaster, resulting in publication bias, prosecutor-type fallacies, and affirming the consequent fallacies throughout the health and social science literature. Even though testing can in theory be used soundly, it just hasn’t done well in practice in these fields. This could be ascribed to human failings rather than failings of received testing theories, but I would require any theory of applied statistics to deal with human limitations, just as safety engineering must do for physical products. I regard statistics as having been woefully negligent of cognitive psychology in this regard. In particular, widespread adoption and vigorous defense of a statistical method or philosophy is no more evidence of its scientific value than widespread adoption and vigorous defense of a religion is evidence of its scientific value. 
That should bring us to alternatives. I am aware of no compelling data showing that other approaches would have done better, but I do find compelling the arguments that at least some of the problems would have been mitigated by teaching a dualist approach to statistics, in which every procedure must be supplied with both an accurate frequentist and an accurate Bayesian interpretation, if only to reduce prevalent idiocies like interpreting a two-sided P-value as “the” posterior probability of a point null hypothesis.

 Nicole Jinn
 (to Sander Greenland)

 What exactly is this ‘dualist’ approach to teaching statistics and why does it mitigate the problems, as you claim? (I am increasingly interested in finding more effective ways to teach/instruct others in various age groups about statistics.)
I have a difficult time seeing how effective this ‘dualist’ way of teaching could be for the following reason: the Bayesian and frequentist approaches are vastly different in their aims and the way they see statistics being used in (natural or social) science, especially when one looks more carefully at the foundations of each methodology (e.g., disagreements about where exactly probability enters into inference, or about what counts as relevant information). Hence, it does not make sense (to me) to supply both types of interpretation to the same data and the same research question! Instead, it makes more sense (from a teaching perspective) to demonstrate a Bayesian interpretation for one experiment, and a frequentist interpretation for another experiment, in the hopes of getting at the (major) differences between the two methodologies.

Mayo

Sander. Thanks for your comment. 
Interestingly, I think the conglomeration of error statistical tools are the ones most apt at dealing with human limitations and foibles: they give piecemeal methods to ask one question at a time (e.g., would we be mistaken to suppose there is evidence of any effect at all? mistaken about how large? about iid assumptions? about possible causes? about implications for distinguishing any theories?). The standard Bayesian apparatus requires setting out a complete set of hypotheses that might arise, plus prior probabilities in each of them (or in “catchall” hypotheses), as well as priors in the model…and after this herculean task is complete, there is a purely deductive update: being deductive it never goes beyond the givens. Perhaps the data will require a change in your prior—this is what you must have believed before, since otherwise you find your posterior unacceptable—thereby encouraging the very self-sealing inferences we all claim to deplore. Continue reading →

Categories: Bayesian/frequentist, Error Statistics, P-values, Statistics | 21 Comments

What do these share in common: m&ms, limbo stick, ovulation, Dale Carnegie? Sat night potpourri

Posted on June 22, 2013 by Mayo
images-2

For entertainment only

I had said I would label as pseudoscience or questionable science any enterprise that regularly permits the kind of ‘verification biases’ in the laundry list of my June 1 post.  How regularly? (I’ve been asked)

Well, surely if it’s as regular as, say, much of social psychology, it goes over the line. But it’s not mere regularity, it’s the nature of the data, the type of inferences being drawn, and the extent of self-scrutiny and recognition of errors shown (or not shown). The regularity is just a consequence of the methodological holes. My standards may be considerably more stringent than most, but quite aside from statistical issues, I simply do not find hypotheses well-tested if they are based on “experiments” that consist of giving questionnaires. At least not without a lot more self-scrutiny and discussion of flaws than I ever see. (There may be counterexamples.)

Attempts to recreate phenomena of interest in typical social science “labs” leave me with the same doubts. Huge gaps often exist between elicited and inferred results. One might locate the problem under “external validity” but to me it is just the general problem of relating statistical data to substantive claims.

Experimental economists (expereconomists) take lab results plus statistics to warrant sometimes ingenious inferences about substantive hypotheses.  Vernon Smith (of the Nobel Prize in Econ) is rare in subjecting his own results to “stress tests”.  I’m not withdrawing the optimistic assertions he cites from EGEK (Mayo 1996) on Duhem-Quine (e.g., from “Rhetoric and Reality” 2001, p. 29). I’d still maintain, “Literal control is not needed to attribute experimental results correctly (whether to affirm or deny a hypothesis). Enough experimental knowledge will do”.  But that requires piece-meal strategies that accumulate, and at least a little bit of “theory” and/or a decent amount of causal understanding.[1]

I think the generalizations extracted from questionnaires allow for an enormous amount of “reading into” the data. Suddenly one finds the “best” explanation. Questionnaires should be deconstructed for how they may be misinterpreted, not to mention how responders tend to guess what the experimenter is looking for. (I’m reminded of the current hoopla over questionnaires on breadwinners, housework and divorce rates!) I respond with the same eye-rolling to just-so story telling along the lines of evolutionary psychology.

I apply the “Stapel test”: Even if Stapel had bothered to actually carry out the data-collection plans that he so carefully crafted, I would not find the inferences especially telling in the least. Take for example the planned-but-not-implemented study discussed in the recent New York Times article on Stapel:

 Stapel designed one such study to test whether individuals are inclined to consume more when primed with the idea of capitalism. He and his research partner developed a questionnaire that subjects would have to fill out under two subtly different conditions. In one, an M&M-filled mug with the word “kapitalisme” printed on it would sit on the table in front of the subject; in the other, the mug’s word would be different, a jumble of the letters in “kapitalisme.” Although the questionnaire included questions relating to capitalism and consumption, like whether big cars are preferable to small ones, the study’s key measure was the amount of M&Ms eaten by the subject while answering these questions….Stapel and his colleague hypothesized that subjects facing a mug printed with “kapitalisme” would end up eating more M&Ms.

Stapel had a student arrange to get the mugs and M&Ms and later load them into his car along with a box of questionnaires. He then drove off, saying he was going to run the study at a high school in Rotterdam where a friend worked as a teacher.

Stapel dumped most of the questionnaires into a trash bin outside campus. At home, using his own scale, he weighed a mug filled with M&Ms and sat down to simulate the experiment. While filling out the questionnaire, he ate the M&Ms at what he believed was a reasonable rate and then weighed the mug again to estimate the amount a subject could be expected to eat. He built the rest of the data set around that number. He told me he gave away some of the M&M stash and ate a lot of it himself. “I was the only subject in these studies,” he said.

He didn’t even know what a plausible number of M&Ms consumed would be! But never mind that, observing a genuine “effect” in this silly study would not have probed the hypothesis. Would it? Continue reading →

Categories: junk science, Statistics | 5 Comments

Stanley Young: better p-values through randomization in microarrays

Posted on June 19, 2013 by Mayo

I wanted to locate some uncluttered lounge space for one of the threads to emerge in comments from 6/14/13. Thanks to Stanley Young for permission to post this. 

YoungPhoto2008 S. Stanley Young, PhD
Assistant Director for Bioinformatics
National Institute of Statistical Sciences
Research Triangle Park, NC

There is a relatively unknown problem with microarray experiments, in addition to the multiple testing problems. Samples should be randomized over important sources of variation; otherwise p-values may be flawed. Until relatively recently, the microarray samples were not sent through assay equipment in random order. Clinical trial statisticians at GSK insisted that the samples go through assay in random order. Rather amazingly the data became less messy and p-values became more orderly. The story is given here:
http://blog.goldenhelix.com/?p=322
Essentially all the microarray data pre-2010 is unreliable. For another example, Mass spec data was analyzed Petrocoin. The samples were not randomized that claims with very small p-values failed to replicate. See K.A. Baggerly et al., “Reproducibility of SELDI-TOF protein patterns in serum: comparing datasets from different experiments,” Bioinformatics, 20:777-85, 2004. So often the problem is not with p-value technology, but with the design and conduct of the study.

experim_design6

Please check other comments on microarrays from 6/14/13.

Categories: P-values, Statistics | Tags: , , | 9 Comments

P-values can’t be trusted except when used to argue that P-values can’t be trusted!

Posted on June 14, 2013 by Mayo

images-1Have you noticed that some of the harshest criticisms of frequentist error-statistical methods these days rest on methods and grounds that the critics themselves purport to reject? Is there a whiff of inconsistency in proclaiming an “anti-hypothesis-testing stance” while in the same breath extolling the uses of statistical significance tests and p-values in mounting criticisms of significance tests and p-values? I was reminded of this in the last two posts (comments) on this blog (here and here) and one from Gelman from a few weeks ago (“Interrogating p-values”).

Gelman quotes from a note he is publishing:

“..there has been a growing sense that psychology, biomedicine, and other fields are being overwhelmed with errors … . In two recent series of papers, Gregory Francis and Uri Simonsohn and collaborators have demonstrated too-good-to-be-true patterns of p-values in published papers, indicating that these results should not be taken at face value.”

But this fraudbusting is based on finding statistically significant differences from null hypotheses (e.g., nulls asserting random assignments of treatments)! If we are to hold small p-values untrustworthy, we would be hard pressed to take them as legitimating these criticisms, especially those of a career-ending sort.

…in addition to the well-known difficulties of interpretation of p-values…,…and to the problem that, even when all comparisons have been openly reported and thus p-values are mathematically correct, the ‘statistical significance filter’ ensures that estimated effects will be in general larger than true effects, with this discrepancy being well over an order of magnitude in settings where the true effects are small… (Gelman 2013)

But surely anyone who believed this would be up in arms about using small p-values as evidence of statistical impropriety. Am I the only one wondering about this?*

CLARIFICATION (6/15/13): Corey’s comment today leads me to a clarification, lest anyone misunderstand my point. I am sure that Francis, Simonsohn and others would never be using p-values and associated methods in the service of criticism if they did not regard the tests as legitimate scientific tools. I wasn’t talking about them. I was alluding to critics of tests who point to their work as evidence the statistical tools are not legitimate. Now maybe Gelman only intends to say, what we know and agree with, that tests can be misused and misinterpreted. But in these comments, our exchanges, and elsewhere, it is clear he is saying something much stronger. In my view, the use of significance tests by debunkers should have been taken as strong support for the value of the tools, correctly used. In short, I thought it was a success story! and I was rather perplexed to see somewhat the reverse.

______________________

*This just in: If one wants to see a genuine quack extremist** who was outed long ago***, see Ziliac’s article declaring the Higgs physicists are pseudoscientists for relying on significance levels!( in the Financial Post 6/12/13).

**I am not placing the critics referred to above under this umbrella in the least.

***For some reviews of Ziliac and McCloskey, see widgets on left. For their flawed testimony on the Matrixx case, please search this blog.

Categories: reforming the reformers, Statistical fraudbusting, Statistics | 43 Comments

Mayo: comment on the repressed memory research

Posted on June 11, 2013 by Mayo

freud mirror espHere are some reflections on the repressed memory articles from Richard Gill’s post, focusing on Geraerts, et.al.,(2008).

1. Richard Gill reported that “Everyone does it this way, in fact, if you don’t, you’d never get anything published: …People are not deliberately cheating: they honestly believe in their theories and believe the data is supporting them and are just doing their best to make this as clear as possible to everyone.”

This remark is very telling. I recommend we just regard those cases as illustrating a theory one believes, rather than providing evidence for that theory. If we could mark them as such, we can stop blaming significance tests for playing a role in what are actually only illustrative attempts, or to strengthen someone’s beliefs about a theory.

2. I was surprised the examples had to do with recovered memories. Wasn’t that entire area dubbed a pseudoscience way back (at least 15-25 years ago?) when “therapy induced” memories of childhood sexual abuse (CSA) were discovered to be just that—therapy induced and manufactured? After the witch hunts that ensued (the very accusation sufficing for evidence), I thought the field of “research” had been put out of its and our misery. So, aside from having used the example in a course on critical thinking, I’m not up on this current work at all. But, as these are just blog comments, let me venture some off-the-cuff skeptical thoughts. They will have almost nothing to do with the statistical data analysis, by the way…

3. Geraerts, et.al., (2008, 22) admit at the start of the article that therapy-recovered CSA memories are unreliable, and the idea of automatically repressing a traumatic event like CSA implausible. Then mightn’t it seem the entire research program should be dropped? Not to its adherents! As with all theories that enjoy the capacity of being sufficiently flexible to survive anomaly (Popper’s pseudosciences), there’s some life left here too. Maybe , its adherents reason, it’s not necessary for those who report “spontaneously recovered” CSA memories to be repressors, instead they merely be “suppressors” who are good at blocking out negative events. If so, they didn’t automatically repress but rather deliberately suppressed: “Our findings may partly explain why people with spontaneous CSA memories have the subjective impression that they have ‘repressed’ their CSA memories for many years.” (ibid., 22).

4. Shouldn’t we stop there? I would. We have a research program growing out of an exemplar of pseudoscience being kept alive by ever-new “monster-barring” strategies (as Lakatos called them). (I realize they’re not planning to go out to the McMartin school, but still…) If a theory T is flexible enough so that any observations can be interpreted through it, and thereby regarded as confirming T, then it is no surprise that this is still true when the instances are dressed up with statistics. It isn’t that theories of repressed memories are implausible or improbable (in whatever sense one takes those terms). It is the ever-flexibility of these theories that renders the research program pseudoscience (along with, in this case, a history of self-sealing data interpretations). Continue reading →

Categories: junk science, Statistical fraudbusting, Statistics | 7 Comments

Richard Gill: “Integrity or fraud… or just quesionable research practices?”

Posted on June 8, 2013 by Mayo
Professor Gill

Professor Gill

Professor Richard Gill
Statistics Group
Mathematical Institute
Leiden University
http://www.math.leidenuniv.nl/~gill/

I am very grateful to Richard Gill for permission to post an e-mail from him (after my “dirty laundry” post) along with slides from his talk, “Integrity or fraud… or just questionable research practices?” and associated papers. I record my own reflections on the pseudoscientific nature of the program in one of the Geraerts et.al., papers in a later post.

I certainly have been thinking about these issues a lot in recent months. I got entangled in intensive scientific and media discussions – mainly confined to the Netherlands  – concerning the cases of social psychologist Dirk Smeesters and of psychologist Elke Geraerts.  See: http://www.math.leidenuniv.nl/~gill/Integrity.pdf

And I recently got asked to look at the statistics in some papers of another … [researcher] ..but this one is still confidential ….

The verdict on Smeesters was that he like Stapel actually faked data (though he still denies this).

The Geraerts case is very much open, very much unclear. The senior co-authors Merckelbach, McNally of the attached paper, published in the journal “Memory”, have asked the journal editors for it to be withdrawn because they suspect the lead author, Elke Geraerts, of improper conduct. She denies any impropriety. It turns out that none of the co-authors have the data. Legally speaking it belongs to the University of Maastricht where the research was carried out and where Geraerts was a promising postdoc in Merckelbach’s group. She later got a chair at Erasmus University Rotterdam and presumably has the data herself but refuses to share it with her old co-authors or any other interested scientists. Just looking at the summary statistics in the paper one sees evidence of “too good to be true”. Average scores in groups supposed in theory to be similar are much closer to one another than one would expect on the basis of the within group variation (the paper reports averages and standard deviations for each group, so it is easy to compute the F statistic for equality of the three similar groups and use its left tail probability as test statistic. Continue reading →

Categories: junk science, Statistical fraudbusting, Statistics | 5 Comments

Anything Tests Can do, CIs do Better; CIs Do Anything Better than Tests?* (reforming the reformers cont.)

Posted on June 6, 2013 by Mayo

Having reblogged the 5/17/12 post on “reforming the reformers” yesterday, I thought I should reblog its follow-up: 6/2/12.

Consider again our one-sided Normal test T+, with null H0: μ < μ0 vs μ >μ0  and  μ0 = 0,  α=.025, and σ = 1, but let n = 25. So M is statistically significant only if it exceeds .392. Suppose M (the sample mean) just misses significance, say

Mo = .39.

The flip side of a fallacy of rejection (discussed before) is a fallacy of acceptance, or the fallacy of misinterpreting statistically insignificant results.  To avoid the age-old fallacy of taking a statistically insignificant result as evidence of zero (0) discrepancy from the null hypothesis μ =μ0, we wish to identify discrepancies that can and cannot be ruled out.  For our test T+, we reason from insignificant results to inferential claims of the form:

μ < μ0 + γ

Fisher continually emphasized that failure to reject was not evidence for the null.  Neyman, we saw, in chastising Carnap, argued for the following kind of power analysis:

Neymanian Power Analysis (Detectable Discrepancy Size DDS): If data x are not statistically significantly different from H0, and the power to detect discrepancy γ is high (low), then x constitutes good (poor) evidence that the actual effect is < γ. (See 11/9/11 post).

By taking into account the actual x0, a more nuanced post-data reasoning may be obtained.

“In the Neyman-Pearson theory, sensitivity is assessed by means of the power—the probability of reaching a preset level of significance under the assumption that various alternative hypotheses are true. In the approach described here, sensitivity is assessed by means of the distribution of the random variable P, considered under the assumption of various alternatives. “ (Cox and Mayo 2010, p. 291):

This may be captured in :

FEV(ii): A moderate p-value is evidence of the absence of a discrepancy d from Ho only if there is a high probability the test would have given a worse fit with H0 (i.e., a smaller p value) were a discrepancy d to exist. (Mayo and Cox 2005, 2010, 256).

This is equivalently captured in the Rule of Acceptance (Mayo (EGEK) 1996, and in the severity interpretation for acceptance, SIA, Mayo and Spanos (2006, p. 337):

SIA: (a): If there is a very high probability that [the observed difference] would have been larger than it is, were μ > μ1, then μ < μ1 passes the test with high severity,…

But even taking tests and CIs just as we find them, we see that CIs do not avoid the fallacy of acceptance: they do not block erroneous construals of negative results adequately. Continue reading →

Categories: CIs and tests, Error Statistics, reformers, Statistics | Tags: , , , , , , , | Leave a comment

Do CIs Avoid Fallacies of Tests? Reforming the Reformers (Reblog 5/17/12)

Posted on June 5, 2013 by Mayo

The one method that enjoys the approbation of the New Reformers is that of confidence intervals. The general recommended interpretation is essentially this:

For a reasonably high choice of confidence level, say .95 or .99, values of µ within the observed interval are plausible, those outside implausible.

Geoff Cumming, a leading statistical reformer in psychology, has long been pressing for ousting significance tests (or NHST[1]) in favor of CIs. The level of confidence “specifies how confident we can be that our CI includes the population parameter m (Cumming 2012, p.69). He recommends prespecified confidence levels .9, .95 or .99:

“We can say we’re 95% confident our one-sided interval includes the true value. We can say the lower limit (LL) of the one-sided CI…is a likely lower bound for the true value, meaning that for 5% of replications the LL will exceed the true value. “ (Cumming 2012, p. 112)[2]

For simplicity, I will use the 2-standard deviation cut-off corresponding to the one-sided confidence level of ~.98.

However, there is a duality between tests and intervals (the intervals containing the parameter values not rejected at the corresponding level with the given data).[3]

“One-sided CIs are analogous to one-tailed tests but, as usual, the estimation approach is better.”

Is it?   Consider a one-sided test of the mean of a Normal distribution with n iid samples, and known standard deviation σ, call it test T+.

H0: µ ≤  0 against H1: µ >  0 , and let σ= 1.

Test T+ at significance level .02 is analogous to forming the one-sided (lower) 98% confidence interval:

µ > M – 2(1/ √n ).

where M, following Cumming, is the sample mean (thereby avoiding those x-bars). M – 2(1/ √n ) is the lower limit (LL) of a 98% CI.

Central problems with significance tests (whether of the N-P or Fisherian variety) include:

(1) results are too dichotomous (e.g., significant at a pre-set level or not);

(2) two equally statistically significant results but from tests with different sample sizes are reported in the same way  (whereas the larger the sample size the smaller the discrepancy the test is able to detect);

(3) significance levels (even observed p-values) fail to indicate the extent of the effect or discrepancy (in the case of test T+ , in the positive direction).

We would like to know for what values of δ it is warranted to infer  µ > µ0 + δ. Continue reading →

Categories: confidence intervals and tests, reformers, Statistics | Tags: , , , | 7 Comments

Some statistical dirty laundry

Posted on June 1, 2013 by Mayo

Objectivity 1: Will the Real Junk Science Please Stand Up?I finally had a chance to fully read the 2012 Tilberg Report* on “Flawed Science” last night. The full report is now here. Here are some stray thoughts…

1. Slipping into pseudoscience.
The authors of the Report say they never anticipated giving a laundry list of “undesirable conduct” by which researchers can flout pretty obvious requirements for the responsible practice of science. It was an accidental byproduct of the investigation of one case (Diederik Stapel, social psychology) that they walked into a culture of “verification bias”[1]. Maybe that’s why I find it so telling. It’s as if they could scarcely believe their ears when people they interviewed “defended the serious and less serious violations of proper scientific method with the words: that is what I have learned in practice; everyone in my research environment does the same, and so does everyone we talk to at international conferences” (Report 48). So they trot out some obvious rules, and it seems to me that they do a rather good job.

One of the most fundamental rules of scientific research is that an investigation must be designed in such a way that facts that might refute the research hypotheses are given at least an equal chance of emerging as do facts that confirm the research hypotheses. Violations of this fundamental rule, such as continuing an experiment until it works as desired, or excluding unwelcome experimental subjects or results, inevitably tends to confirm the researcher’s research hypotheses, and essentially render the hypotheses immune to the facts…. [T]he use of research procedures in such a way as to ‘repress’ negative results by some means” may be called verification bias. [my emphasis] (Report, 48).

I would place techniques for ‘verification bias’ under the general umbrella of techniques for squelching stringent criticism and repressing severe tests. These gambits make it so easy to find apparent support for one’s pet theory or hypotheses, as to count as no evidence at all (see some from their list ). Any field that regularly proceeds this way I would call a pseudoscience, or non-science, following Popper. “Observations or experiments can be accepted as supporting a theory (or a hypothesis, or a scientific assertion) only if these observations or experiments are severe tests of the theory” (Popper 1994, p. 89). [2] It is unclear at what point a field slips into the pseudoscience realm.

2. A role for philosophy of science?
I am intrigued that one of the final recommendations in the Report is this:

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. Based on these insights, research Master’s students and PhD students must receive practical training from their supervisors in the application of the rules governing proper and honest scientific research, which should include examples of such undesirable conduct as data massage. The Graduate School must explicitly ensure that this is implemented.

A philosophy department could well create an entire core specialization that revolved around “the relevant basic principles of philosophy of science, methodology, ethics and statistics that enable the responsible practice of science” (ideally linked with one or more other departments).  That would be both innovative and fill an important gap, it seems to me. Is anyone doing this?

3. Hanging out some statistical dirty laundry.images
Items in their laundry list include:

  • An experiment fails to yield the expected statistically significant results. The experiment is repeated, often with minor changes in the manipulation or other conditions, and the only experiment subsequently reported is the one that did yield the expected results. The article makes no mention of this exploratory method… It should be clear, certainly with the usually modest numbers of experimental subjects, that using experiments in this way can easily lead to an accumulation of chance findings…. Continue reading →
Categories: junk science, spurious p values, Statistics | 6 Comments

K. Staley: review of Error & Inference

Posted on May 29, 2013 by Mayo

kent-staleyK. W. Staley
Associate Professor
Department of Philosophy,
Saint Louis University

(Almost) All about error


BOOK REVIEW Metascience (2012) 21:709–713 DOI 10.1007/s11016-011-9618-1E & I Cover 2
Deborah G. Mayo and Aris Spanos (eds): Error and inference: Recent exchanges on experimental reasoning, reliability, objectivity, and rationality. New York: Cambridge University Press, 2010, xvii+419 pp

The ERROR’06 (experimental reasoning, reliability, objectivity, and rationality) conference held at Virginia Tech aimed to advance the discussion of some central themes in philosophy of science debated by Deborah Mayo and her more-or-less friendly critics over the years. The volume here reviewed brings together the contributions of these critics and Mayo’s responses to them (with Mayo’s collaborator Aris Spanos). (I helped with the organization of the conference and, with Mayo and Jean Miller, edited a separate collection of workshop papers that were presented there, published as a special issue of Synthese.) My review will focus on a couple of themes I hope to be of interest to a broad philosophical audience, then turn more briefly to an overview of the entire collection. The discussions in Error and Inference (E&I) are indispensable for understanding several current issues regarding the methodology of science.

The remarkably useful introductory chapter lays out the broad themes of the volume and discusses ‘‘The Error-Statistical Philosophy’’. Here, Mayo and Spanos provide the most succinct and non-technical account of the error-statistical approach that has yet been published, a feature that alone should commend this text to anyone who has found it difficult to locate a reading on error statistics suitable for use in teaching.

Mayo holds that the central question for a theory of evidence is not the degree to which some observation E confirms some hypothesis H but how well-probed for error a hypothesis H is by a testing procedure T that results in data x0. This reorientation has far-reaching consequences for Mayo’s approach to philosophy of science. On this approach, addressing the question of when data ‘‘provide good evidence for or a good test of’’ a hypothesis requires attention to characteristics of the process by means of which the data are used to bear on the hypothesis. Mayo identifies the starting point from which her account is developed as the ‘‘Weak Severity Principle’’ (WSP):

Data x0 do not provide good evidence for hypothesis H if x0 results from a test procedure with a very low probability or capacity of having uncovered the falsity of H (even if H is incorrect). (21)

The weak severity principle is then developed into the full severity principle (SP), according to which ‘‘data x0 provide a good indication of or evidence for hypothesis H (just) to the extent that test T has severely passed H with x0’’ where H passes a severe test T with x0 if x0 ‘‘agrees with’’ H and ‘‘with very high probability, test T would have produced a result that accords less well with H than doesx0, if H were false or incorrect’’ (22). This principle constitutes the heart of the error-statistical account of evidence, and E&I, by including some of the most important critiques of the principle, provides a forum in which Mayo and Spanos attempt to correct misunderstandings of the principle and to clarify its meaning and application.

The appearance in the WSP of the disjunctive phrase ‘‘a very low probability or capacity’’ (my emphasis) indicates a point central to much of this clarificatory work. The error-statistical account is resolutely frequentist in its construal of probability. It is commonly held (including by some frequentists) that the rationale for frequentist statistical methods lies exclusively in the fact that they can sometimes be shown to have low error rates in the long run. Throughout E&I, Mayo insists that this ‘‘behaviorist rationale’’ is not applicable when it comes to evaluating a particular body of data in order to determine what inferences may be warranted. That evaluation rests upon thinking about the particular data and the inference at hand in light of the capacity of the test to reveal potential errors in the inference drawn. Frequentist probabilities are part of how one models the error-detecting capacities of the process. As Mayo explains in a later chapter co-authored with David Cox, tests of hypotheses function analogously to measuring instruments: ‘‘Just as with the use of measuring instruments, applied to a specific case, we employ the performance features to make inferences about aspects of the particular thing that is measured, aspects that the measuring tool is appropriately capable of revealing’’ (257).

One of the most fascinating exchanges in E&I concerns the role of severe testing in the appraisal of ‘‘large-scale’’ theories. According to Mayo, theory appraisal proceeds by a ‘‘piecemeal’’ process of severe probing for specific ways in which a theory might be in error. She illustrates this with the history of experimental tests of theories of gravity, emphasizing Clifford Will’s parametrized post-Newtonian (PPN) framework, by means of which all metric theories of gravity can be represented in their weak-field, slow-motion limits by means of ten parameters. Experimental work on gravity theories then severely tests hypotheses about the values of those parameters. Rather than attempting to confirm or probabilify the general theory of relativity (GTR), the aim is to learn about the ways in which GTR might be in error, more generally to ‘‘measure how far off what a given theory says about a phenomenon can be from what a ‘correct’ theory would need to say about it’’ (55).

Alan Chalmers and Alan Musgrave both challenge this view. According to Chalmers, no general theory, whether ‘‘low level’’ or ‘‘high level’’, can pass a severe test because the content of theories surpasses whatever empirical evidence supports them. As a consequence, Chalmers argues, Mayo’s severe-testing account of scientific inference must be incomplete because even low-level experimental testing sometimes demands relying on general theoretical claims. Similarly, Musgrave accuses Mayo of holding that (general) theories are not tested by ‘‘testing their consequences’’, but that ‘‘all that we really test are the consequences’’ (105), leaving her with ‘‘nothing to say’’ about the assessment, adoption, or rejection of general theories (106). Continue reading →

Categories: Error Statistics, Statistics | Tags: , | 1 Comment

A.Birnbaum: Statistical Methods in Scientific Inference

Posted on May 27, 2013 by Mayo

Birnbaum: born May 27, 1923

Today is (statistician) Allan Birnbaum’s birthday. He lived to be only 53 [i]. From the perspective of philosophy of statistics and philosophy of science, Birnbaum is best known for his work on likelihood, the Likelihood Principle [ii], and for his attempts to blend concepts of likelihood with error probability ideas to obtain what he called “concepts of statistical evidence”. Failing to find adequate concepts of statistical evidence, Birnbaum called for joining the work of “interested statisticians, scientific workers and philosophers and historians of science”–an idea I would heartily endorse!  While known for attempts to argue that the (strong) Likelihood Principle followed from sufficiency and conditionality principles, a few years after publishing this result, he seems to have turned away from it, perhaps discovering gaps in his argument.

NATURE VOL. 225 MARCH 14, 1970 (1033)

LETTERS TO THE EDITOR

Statistical methods in Scientific Inference

 It is regrettable that Edwards’s interesting article[1], supporting the likelihood and prior likelihood concepts, did not point out the specific criticisms of likelihood (and Bayesian) concepts that seem to dissuade most theoretical and applied statisticians from adopting them. As one whom Edwards particularly credits with having ‘analysed in depth…some attractive properties” of the likelihood concept, I must point out that I am not now among the ‘modern exponents” of the likelihood concept. Further, after suggesting that the notion of prior likelihood was plausible as an extension or analogue of the usual likelihood concept (ref.2, p. 200)[2], I have pursued the matter through further consideration and rejection of both the likelihood concept and various proposed formalizations of prior information and opinion (including prior likelihood).  I regret not having expressed my developing views in any formal publication between 1962 and late 1969 (just after ref. 1 appeared). My present views have now, however, been published in an expository but critical article (ref. 3, see also ref. 4)[3] [4], and so my comments here will be restricted to several specific points that Edwards raised.

 If there has been ‘one rock in a shifting scene’ or general statistical thinking and practice in recent decades, it has not been the likelihood concept, as Edwards suggests, but rather the concept by which confidence limits and hypothesis tests are usually interpreted, which we may call the confidence concept of statistical evidence. This concept is not part of the Neyman-Pearson theory of tests and confidence region estimation, which denies any role to concepts of statistical evidence, as Neyman consistently insists. The confidence concept takes from the Neyman-Pearson approach techniques for systematically appraising and bounding the probabilities (under respective hypotheses) of seriously misleading interpretations of data. (The absence of a comparable property in the likelihood and Bayesian approaches is widely regarded as a decisive inadequacy.) The confidence concept also incorporates important but limited aspects of the likelihood concept: the sufficiency concept, expressed in the general refusal to use randomized tests and confidence limits when they are recommended by the Neyman-Pearson approach; and some applications of the conditionality concept. It is remarkable that this concept, an incompletely formalized synthesis of ingredients borrowed from mutually incompatible theoretical approaches, is evidently useful continuously in much critically informed statistical thinking and practice [emphasis mine].

While inferences of many sorts are evident everywhere in scientific work, the existence of precise, general and accurate schemas of scientific inference remains a problem. Mendelian examples like those of Edwards and my 1969 paper seem particularly appropriate as case-study material for clarifying issues and facilitating effective communication among interested statisticians, scientific workers and philosophers and historians of science.

Allan Birnbaum
New York University
Courant Institute of Mathematical Sciences,
251 Mercer Street,
New York, NY 10012

Birnbaum’s confidence concept, sometimes written (Conf), was his attempt to find in error statistical ideas a concept of statistical evidence–a term that he invented and popularized. In Birnbaum 1977 (24), he states it as follows:

(Conf): A concept of statistical evidence is not plausible unless it finds ‘strong evidence for J as against H with small probability (α) when H is true, and with much larger probability (1 – β) when J is true.

Birnbaum questioned whether Neyman-Pearson methods had “concepts of evidence”  simply because Neyman talked of “inductive behavior” and Wald and others cauched statistical methods in decision-theoretic terms. I have been urging that we consider instead how the tools may actually be used, and not be restricted by the statistical philosophies of founders (not to mention that so many of their statements are tied up with personality disputes, and problems of “anger management”). Recall, as well, E. Pearson’s insistence on an evidential construal of N-P methods, and the fact that Neyman, in practice, spoke of drawing inferences and reaching conclusions (e.g., Neyman’s nursery posts, links in [iii] below). Continue reading →

Categories: Likelihood Principle, phil/history of stat, Statistics | Tags: | 3 Comments

Schachtman: High, Higher, Highest Quality Research Act

Posted on May 26, 2013 by Mayo

wavy capitalSince posting on the High Quality Research act a few weeks ago, I’ve been following it in the news, have received letters from professional committees (asking us to write letters), and now see that  Nathan A. Schachtman, Esq., PC posted the following on May 25, 2013 on his legal blog*:

NAS-3“The High Quality Research Act” (HQRA), which has not been formally introduced in Congress, continues to draw attention. See“Clowns to the left of me, Jokers to the right.”  Last week, Sarewitz suggests that “the problem” is the hype about the benefits of pure research and the let down that results from the realization that scientific progress is “often halting and incremental,” with much research not “particularly innovative or valuable.”  Fair enough, but why is this Congress such an unsophisticated consumer of scientific research in the 21st century?  How can it be a surprise that the scientific community engages in the same rent-seeking behaviors as do other segments of our society? Has it escaped Congress’s attention that scientists are subject to enthusiasms and group think, just like, … congressmen?

Nature published an editorial piece suggesting that the HQRA is not much of a threat. Daniel Sarewitz, “Pure hype of pure research helps no one, ” 497 Nature 411 (2013).

Still, Sarewitz believes that the HQRA bill is not particularly threatening to the funding of science:

“In other words, it’s not a very good bill, but neither is it much of a threat. In fact, it’s just the latest skirmish in a long-running battle for political control over publicly funded science — one fought since at least 1947, when President Truman vetoed the first bill to create the NSF because it didn’t include strong enough lines of political accountability.”

This sanguine evaluation misses the effect of the superlatives in the criteria for National Science Foundation funding:

“(1) is in the interests of the United States to advance the national health, prosperity, or welfare, and to secure the national defense by promoting the progress of science;

(2) is the finest quality, is ground breaking, and answers questions or solves problems that are of utmost importance to society at large; and

(3) is not duplicative of other research projects being funded by the Foundation or other Federal science agencies.” Continue reading →

Categories: evidence-based policy, PhilStatLaw, Statistics | Tags: | 12 Comments

Gelman sides w/ Neyman over Fisher in relation to a famous blow-up

Posted on May 24, 2013 by Mayo
3-d red yellow puzzle people (E&I)

blog-o-log

Andrew Gelman had said he would go back to explain why he sided with Neyman over Fisher in relation to a big, famous argument discussed on my Feb. 16, 2013 post: “Fisher and Neyman after anger management?”, and I just received an e-mail from Andrew saying that he has done so: “In which I side with Neyman over Fisher”. (I’m not sure what Senn’s reply might be.) Here it is:

“In which I side with Neyman over Fisher” Posted by  on 24 May 2013, 9:28 am

As a data analyst and a scientist, Fisher > Neyman, no question. But as a theorist, Fisher came up with ideas that worked just fine in his applications but can fall apart when people try to apply them too generally.gelman5

Here’s an example that recently came up.

Deborah Mayo pointed me to a comment by Stephen Senn on the so-called Fisher and Neyman null hypotheses. In an experiment with n participants (or, as we used to say, subjects or experimental units), the Fisher null hypothesis is that the treatment effect is exactly 0 for every one of the n units, while the Neyman null hypothesis is that the individual treatment effects can be negative or positive but have an average of zero.

Senn explains why Neyman’s hypothesis in general makes no sense—the short story is that Fisher’s hypothesis seems relevant in some problems (sometimes we really are studying effects that are zero or close enough for all practical purposes), whereas Neyman’s hypothesis just seems weird (it’s implausible that a bunch of nonzero effects would exactly cancel). And I remember a similar discussion as a student, many years ago, when Rubin talked about that silly Neyman null hypothesis. Continue reading →

Categories: Fisher, Statistics, Stephen Senn | Tags: , | 10 Comments

Mayo: Meanderings on the Onto-Methodology Conference

Posted on May 19, 2013 by Mayo

mayo blackboard b&w 2Writing a blog like this, a strange and often puzzling exercise[1], does offer a forum for sharing half-baked chicken-scratchings from the back of frayed pages on themes from our Onto-Meth[2] conference from two weeks ago[3]. (The previous post had notes from blogger and attendee, Gandenberger.)

Onto-Meth conference

Onto-Meth conference

Several of the talks reflect a push-back against the idea that the determination of “ontology” in science—e.g., the objects and processes of theories, models and hypotheses—is (or should strive to correspond to?)  “real” objects in the world and/or what is approximately the case about them. Instead, at least some of the speakers wish to liberate ontology to recognize how “merely” pragmatic goals, needs, and desires are not just second-class citizens, but can and do (and should?) determine the categories of reality. Well there are a dozen equivocations here, most of which we did not really discuss at the conference.

In my own half of the Spanos-Mayo (D & P presentation[4]) I granted and even promoted the idea of a methodology that was pragmatic while also objective, so I’m not objecting to that part. The measurement of my weight is a product of “discretionary” judgments (e.g., to weigh in pounds with a scale having a given precision), but it is also a product of how much I really weigh (no getting around it). By understanding the properties of methodological tools and measuring systems, it is possible to “subtract out” the influence of the judgments to get at what is actually the case. At least approximately. But that view is different, it seems to me, from someone like Larry Laudan (at least in his later metamorphosis). Even though he considers his “reticulated” view a fairly hard-nosed spin on the Kuhnian idea of scientific paradigms as invariably containing an ontology (e.g., theories), a methodology, and (what he called) an “axiology” or set of aims (OMA), Laudan seems to think standards are so variable that what counts as evidence is constantly fluctuating (aside from maybe retaining the goal of fitting diverse facts). So I wonder if these pragmatic leanings are more like Laudan or more like me (and my view here, I take it, is essentially that of Peirce). I am perfectly sympathetic to the piecemeal “locavoracity” idea in Ruesche, by the way.

My worry, one of them, is that all kinds of rival entities and processes arise to account for (accord with, predict, and purportedly explain) data and patterns in data, and don’t we need ways to discriminate them? During the open discussion, I mentioned several examples, some of which I can make out all scrunched up in the corners of my coffee-logged program, such as appeals to “cultural theories” of risk and risk perceptions. These theories say appeals to supposedly “real” hazards, e.g, chance of disease, death, catastrophe, and other “objective” risk assessments are wrong.  They say it is not only possible but preferable (truer?) to capture attitudes toward risks, e.g., GM foods, nuclear energy, climate change, breast implants, etc. by means of one or another favorite politico-cultural grid-group categories (e.g., marginal-individualists, passive-egalitarians, hierarchical-border people, fatalists,  etc.). (Your objections to these vague category schemes are often taken as further evidence that you belong in one of the pigeon-holes!) And the other day I heard a behavioral economist declare that he had found the “mechanism” to explain deciding between options in virtually all walks of life using a regression parameter, he called it beta, and guess what? beta = 1/3! He proved it worked statistically too. He might be right, he had a lot of data. Anyway, in my deliberate attempt to trigger discussion at the conference end, I was wondering if some of the speakers and/or attendees (Danks, Woodward, Glymour? Anyone?) had anything to say about cases that some of us might wish to call reification. Continue reading →

Categories: O & M conference, Statistics | 10 Comments

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