11th bullet, multiple choice question, and last thoughts on the JSM

Posted on August 9, 2013 by Mayo

photo-on-8-4-13-at-3-40-pm1I. Apparently I left out the last bullet in my scribbled notes from Silver’s talk. There was an 11th. Someone sent it to me from a blog: revolution analytics:

11. Like scientists, journalists ought to be more concerned with the truth rather than just appearances. He suggested that maybe they should abandon the legal paradigm of seeking an adversarial approach and behave more like scientists looking for the truth.

OK. But, given some of the issues swirling around the last few posts, I think it’s worth noting that scientists are not disinterested agents looking for the truth—it’s only thanks to its (adversarial!) methods that they advance upon truth. Question: What’s the secret of scientific progress (in those areas that advance learning)?  Answer: Even if each individual scientist were to strive mightily to ensure that his/her theory wins out, the stringent methods of the enterprise force that theory to show its mettle or die (or at best remain in limbo). You might say, “But there are plenty of stubborn hard cores in science”. Sure, and they fail to advance. In those sciences that lack sufficiently stringent controls, the rate of uncorrected spin is as bad as Silver suggests it is in journalism. Think of social psychologist Diederik Stapel setting out to show what is already presumed to be believable. (See here and here and search this blog.).

There’s a strange irony when the same people who proclaim, “We must confront those all too human flaws and foibles that obstruct the aims of truth and correctness”, turn out to be enablers, by championing methods that enable flaws and foibles to seep through. It may be a slip of logic. Here’s a multiple choice question:

Multiple choice: Circle all phrases that correctly complete the “conclusion“:

Let’s say that factor F is known to obstruct the correctness/validity of solutions to problems, or that factor F is known to adversely impinge on inferences.

(Examples of such factors include: biases, limited information, incentives—of various sorts).

Factor F is known to adversely influence inferences.

Conclusion: Therefore any adequate systematic account of inference should _______

(a) allow F to influence inferences.
(b) provide a formal niche by which F can influence inferences.
(c) take precautions to block (or at least be aware of) the ability of F to adversely influence inferences.
(d) none of the above.

(For an example, see discussion of #7 in previous post.)

II. I may be overlooking sessions (inform me if you know of any), but I would have expected more on the statistics in the Higgs boson discoveries at the JSM 2013. Especially given the desire to emphasize the widespread contributions of statistics to the latest sexy science[i].  (At one point, I was asked about being part of a session on the five sigma effect in the Higgs boson discovery–not that I’m any kind of expert– by David Banks, because of my related blog posts (e.g., here), but people were already in other sessions. But I’m thinking about something splashy by statisticians in particle physics.) Did I miss? [ii]

III. I think it’s easy to see why lots of people showed up to hear Nate Silver: It’s fun to see someone “in the news”, be it from politics, finance, high tech, acting, TV, or, even academics–I, for one, was curious. I’m sure as many would have come out to hear Esther Duflo, Cheryl Sandberg, Fabiola Gionatti, or even Huma Abedin–to list some that happen to come to mind– or any number of others who have achieved recent recognition (and whose work intersects in some way with statistics). It’s interesting that I don’t see pop philosophers invited to give key addresses in yearly philosophy meetings; maybe because philosophers eschew popularity. I may be unaware of some; I don’t attend so many meetings.

IV. Other thoughts: I’ve only been to a handful of “official” statistics meetings. Obviously the # of simultaneous sessions makes the JSM a kind of factory experience, but that’s to be expected. But do people really need to purchase those JSM backpacks? I don’t know how much of the $400 registration fee goes to that, but it seems wasteful…. I saw people tossing theirs out, which I didn’t have the heart to do. Perhaps I’m just showing my outsider status.

V. Montreal: I intended to practice my French, but kept bursting into English too soon. Everyone I met (who lives there) complained about the new money and doing away with pennies in the near future. I wonder if we’re next.

[i]On Silver’s remark (in response to a “tweeted” question) that “data science” is a “sexed-up” term for statistics, I don’t know. I can see reflecting deeply over the foundations of statistical inference, but over the foundations of data analytics?

[ii] You don’t suppose the controversy about particle physics being “bad science” had anything to do with downplaying the Higgs statistics?

Categories: Higgs, Statistics, StatSci meets PhilSci | 5 Comments

What did Nate Silver just say? Blogging the JSM

Posted on August 6, 2013 by Mayo

imagesNate Silver gave his ASA Presidential talk to a packed audience (with questions tweeted[i]). Here are some quick thoughts—based on scribbled notes (from last night). Silver gave a list of 10 points that went something like this (turns out there were 11):

1. statistics are not just numbers

2. context is needed to interpret data

3. correlation is not causation

4. averages are the most useful tool

5. human intuitions about numbers tend to be flawed and biased

6. people misunderstand probability

7. we should be explicit about our biases and (in this sense) should be Bayesian?

8. complexity is not the same as not understanding

9. being in the in crowd gets in the way of objectivity

10. making predictions improves accountability

Just to comment on #7, I don’t know if this is a brand new philosophy of Bayesianism, but his position went like this: Journalists and others are incredibly biased, they view data through their prior conceptions, wishes, goals, and interests, and you cannot expect them to be self-critical enough to be aware of, let alone be willing to expose, their propensity toward spin, prejudice, etc. Silver said the reason he favors the Bayesian philosophy (yes he used the words “philosophy” and “epistemology”) is that people should be explicit about disclosing their biases. I have three queries: (1) If we concur that people are so inclined to see the world through their tunnel vision, what evidence is there that they are able/willing to be explicit about their biases? (2) If priors are to be understood as the way to be explicit about one’s biases, shouldn’t they be kept separate from the data rather than combined with them? (3) I don’t think this is how Bayesians view Bayesianism or priors—is it? Subjective Bayesians, I thought, view priors as representing prior or background information about the statistical question of interest; but Silver sees them as admissions of prejudice, bias or what have you. As a confession of bias, I’d be all for it—though I think people may be better at exposing other’s biases than their own. Only thing: I’d need an entirely distinct account of warranted inference from data.

This does possibly explain some inexplicable remarks in Silver’s book to the effect that R.A. Fisher denied, excluded, or overlooked human biases since he disapproved of adding subjective prior beliefs to data in scientific contexts. Is Silver just about to recognize/appreciate the genius of Fisher (and others) in developing techniques consciously designed to find things out despite knowledge gaps, variability, and human biases? Or not?

Share your comments and/or links to other blogs discussing his talk (which will surely be posted if it isn’t already). Fill in gaps if you were there—I was far away… (See also my previous post blogging the JSM). Photo on 8-4-13 at 3.40 PM

[i] What was the point of this, aside from permitting questions to be cherry picked? (It would have been fun to see ALL the queries tweeted.) The ones I heard were limited to: how can we make statistics more attractive, who is your favorite journalist, favorite baseball player, and so on. But I may have missed some, I left before the end.

For a follow-up post including an 11th bullet that I’d missed, see here. My first post on JSM13 (8/5/13) was here.

Categories: Error Statistics, Statistics | 42 Comments

At the JSM: 2013 International Year of Statistics

Posted on August 5, 2013 by Mayo

Photo on 8-4-13 at 3.40 PM“2013 is the International Year of Statistics” the JSM (Joint Statistical Meetings) brochures ring out! What does it mean?  Whatever it is, it’s exciting! Errorstatistics.com never took up this question, but it’s been on some of the blogs in my “Blog bagel”. So, Since I’m at the JSM here in Montreal, I may report on any clues. Please share your comments. I’m not a statistician, but a philosopher of science, and of inductive-statistical inference much more generally. So I have no dog in this fight, as they say. (Or do I? ) On the other hand, I have often rued “the decline of late in the lively and long-standing exchange between philosophers of science and statisticians” (see this post). [i] (We did have that one parody on “big data or pig data”.)

I know from Larry Wasserman (normaldeviate) that the “year of” label grows, at least in part, to help prevent Statistical Science being eclipsed by the fashionable “Big Data” crowd. In one blog he even spoke of “the end of statistics”. “Aren’t We Data Science?” Marie Davidian, president of the ASA, asks in a recent AmStatNews article.[ii] Davidian worries, correctly I’ve no doubt, that Big Dadaists may be collecting data with “little appreciation for the power of design principle. Statisticians could propel major advances through developments of ‘experimental design for the 21st century’!”.  This recalls Stan Young’s recent post:

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…..So often the problem is not with p-value technology, but with the design and conduct of the study.

So without statistical design principles, they may have wasted a decade!

Back to the JSM, I see they’ve even invited pollster Nate Silver to give the AMA presidential address. I thought he was more baseball stat expert/pundit/pollster than statistician, but some are calling him an “analytics rock star”. Never mind that there’s at least one extremely strange chapter (8) in his popular book (The Signal and the Noise). Here’s an excerpt from Wasserman’s review, which he titles:  “Nate Silver is a Frequentist: Review of The signal and the noise”:



I have one complaint. Silver is a big fan of Bayesian inference, which is fine. Unfortunately, he falls into that category I referred to a few posts ago. He confuses ‘Bayesian inference’ with ‘using Bayes’ theorem.’ His description of frequentist inference is terrible. He seems to equate frequentist inference with Fisherian significance testing, most using Normal distributions. Either he learned statistics from a bad book or he hangs out with statisticians with a significant anti-frequentist bias. Have no doubt about it: Nate Silver is a frequentist.[iii] (Wasserman)

I didn’t discuss Silver’s book on this blog, but looking up a few comments I made on other blogs, (e.g.,on a Gelman blog reviewing Silver), I see I am a bit less generous than Wasserman: “Frequentists, Silver alleges, go around reporting hypotheses like toads predict earthquakes and other “manifestly ridiculous” findings that are licensed by significance testing and data dredged correlations. (Silver, 253). But it is the frequentist who prevents such spurious correlations…. “  (Mayo) So Silver’s criticisms of frequents are way off base.  I was also slightly aghast at his Fisher ridicule and I poke fun at his “All-You-Need is Bayesian cheerleading. The simple use of Bayes Theorem solves all problems (he seems not to realize they too require statistical models)” I wrote.  It’s hard to tell if he’s just reporting or chiming in with those who advocate that schools stop teaching frequentist methods. Some statistical self-inflicted wounds perhaps? The other chapters look interesting, though I didn’t get too much further…(The Bayesian examples are all ordinary frequentist updating, it appears.)   If I can, I’ll go to Silver’s talk.

[i] In that post I wrote: “Philosophy of statistical science not only deals with the philosophical foundations of statistics but also questions about the nature of and justification for inductive-statistical learning more generally. So it is ironic that just as philosophy of science is striving to immerse itself in and be relevant to scientific practice, that statistical science and philosophy of science—so ahead of their time in combining the work of philosophers and practicing scientists—should see such dialogues become rather rare.  (See special topic here.)” (Mayo)

[ii] Some of the turf battles I hear about appear to reflect less substance than style (i.e., people being galvanized to use the latest meme in funding opportunities). Even in philosophy, the dept. head asked us to try and work it in.   In my view, rather than suggesting “Plato and Big Data”, they should be asking to highlight interconnections between statistical evidence, critical thinking, logic, ethics,  philosophy of science, and epistemology. That would advance our courses.

[iii] For example, Wasserman says, in his review of Silver:

One of the most important tests of a forecast — I would argue that it is the single most important one — is called calibration. Out of all the times you said there was a 40 percent chance of rain, how often did rain actually occur? If over the long run, it really did rain about 40 percent of the time, that means your forecasts were well calibrated.  (Wasserman)

Categories: Error Statistics | 5 Comments

Blogging (flogging?) the SLP: Response to Reply- Xi’an Robert

Posted on August 1, 2013 by Mayo
peeking through cover EGEK

Breaking through “the breakthrough”

Christian Robert’s reply grows out of my last blogpost. On Xi’an’s Og :

A quick reply from my own Elba, in the Dolomiti: your arguments (about the sad consequences of the SLP) are not convincing wrt the derivation of SLP=WCP+SP. If I built a procedure that reports (E1,x*) whenever I observe (E1,x*) or (E2,y*), this obeys the sufficiency principle; doesn’t it? (Sorry to miss your talk!)

Mayo’s response to Xi’an on the “sad consequences of the SLP.”[i]

This is a useful reply (so to me it’s actually not ‘flogging’ the SLP[ii]), and, in fact, I think Xi’an will now see why my arguments are convincing! Let’s use Xi’an’s procedure to make a parametric inference about q. Getting the report x* from Xi’an’s procedure, we know it could have come from E1 or E2. In that case, the WCP forbids us from using either individual experiment to compute the inference implication. We use the sampling distribution of TB.

Birnbaum’s statistic TB is a technically sufficient statistic for Birnbaum’s experiment EB  (the conditional distribution of Z given TB is independent of q). The question of whether this is the relevant or legitimate way to compute the inference when it is given that y* came from E2 is the big question. The WCP says it is not. Now you are free to use Xi’an’s procedure (free to Birnbaumize) but that does not yield the SLP. Nor did Birnbaum think it did. That’s why he goes on to say: “Never mind. Don’t use Xi’an’s procedure.  Compute the inference using E2  just as the WCP tells you to. You know it came from E2 . Isn’t that what David Cox taught us in 1958?”

Fine. But still no SLP!  Note it’s not that SP and WCP conflict, it’s WCP and Birnbaumization that conflict. The application of a principle will always be relative to the associated model used to frame the question.[iii]

These points are all spelled out clearly in my paper: [I can’t get double subscripts here. EB is the same as E-B][iv]

Given y*, the WCP says do not Birnbaumize. One is free to do so, but not to simultaneously claim to hold the WCP in relation to the given y*, on pain of logical contradiction. If one does choose to Birnbaumize, and to construct TB, admittedly, the known outcome y* yields the same value of TB as would x*. Using the sample space of EB yields: (B): InfrE-B[x*] = InfrE-B[y*]. This is based on the convex combination of the two experiments, and differs from both InfrE1[x*] and InfrE2[y*]. So again, any SLP violation remains. Granted, if only the value of TB is given, using InfrE-B may be appropriate. For then we are given only the disjunction: Either (E1, x*) or (E2, y*). In that case one is barred from using the implication from either individual Ei. A holder of WCP might put it this way: once (E,z) is given, whether E arose from a q-irrelevant mixture, or was fixed all along, should not matter to the inference; but whether a result was Birnbaumized or not should, and does, matter.

There is no logical contradiction in holding that if data are analyzed one way (using the convex combination in EB), a given answer results, and if analyzed another way (via WCP) one gets quite a different result. One may consistently apply both the EB and the WCP directives to the same result, in the same experimental model, only in cases where WCP makes no difference. To claim the WCP never makes a difference, however, would entail that there can be no SLP violations, which would make the argument circular. Another possibility, would be to hold, as Birnbaum ultimately did, that the SLP is “clearly plausible” (Birnbaum 1968, 301) only in “the severely restricted case of a parameter space of just two points” where these are predesignated (Birnbaum 1969, 128). But SLP violations remain.

Note: The final draft of my paper uses equations that do not transfer directly to this blog. Hence, these sections are from a draft of my paper.

[i] Although I didn’t call them “sad,” I think it would be too bad to accept the SLP’s consequences. Listen to Birnbaum:

The likelihood principle is incompatible with the main body of modern statistical theory and practice, notably the Neyman-Pearson theory of hypothesis testing and of confidence intervals, and incompatible in general even with such well-known concepts as standard error of an estimate and significance level. (Birnbaum 1968, 300)

That is why Savage called it “a breakthrough” result. In the end, however, Birnbaum could not give up on control of error probabilities. He held the SLP only for the trivial case of predesignated simple hypotheses. (Or, perhaps he spied the gap in his argument? I suspect, from his writings, that he realized his argument went through only for such cases that do not violate the SLP.)

[ii] Readers may feel differently.

[iii] Excerpt from a draft of my paper:
Model checking. An essential part of the statements of the principles SP, WCP, and SLP is that the validity of the model is granted as adequately representing the experimental conditions at hand (Birnbaum 1962, 491). Thus, accounts that adhere to the SLP are not thereby prevented from analyzing features of the data such as residuals, which are relevant to questions of checking the statistical model itself. There is some ambiguity on this point in Casella and R. Berger (2002):

Most model checking is, necessarily, based on statistics other than a sufficient statistic. For example, it is common practice to examine residuals from a model.  . . Such a practice immediately violates the Sufficiency Principle, since the residuals are not based on sufficient statistics. (Of course such a practice directly violates the [strong] LP also.) (Casella and R. Berger 2002, 295-6)

They warn that before considering the SLP and WCP, “we must be comfortable with the model” (296). It seems to us more accurate to regard the principles as inapplicable, rather than violated, when the adequacy of the relevant model is lacking.

Birnbaum, A.1968. “Likelihood.” In International Encyclopedia of the Social Sciences, 9:299–301. New York: Macmillan and the Free Press.

———. 1969. “Concepts of Statistical Evidence.” In Philosophy, Science, and Method: Essays in Honor of Ernest Nagel, edited by S. Morgenbesser, P. Suppes, and M. G. White, 112–143. New York: St. Martin’s Press.

Casella, G., and R. L. Berger. 2002. Statistical Inference. 2nd ed. Belmont, CA: Duxbury Press.

Mayo 2013, (http://arxiv-web3.library.cornell.edu/pdf/1302.7021v2.pdf)

Categories: Birnbaum Brakes, Statistics, strong likelihood principle | 9 Comments

New Version: On the Birnbaum argument for the SLP: Slides for my JSM talk

Posted on July 26, 2013 by Mayo

Picture 216 1mayo In my latest formulation of the controversial Birnbaum argument for the strong likelihood principle (SLP), I introduce a new symbol \Rightarrow  to represent a function from a given experiment-outcome pair, (E,z) to a generic inference implication.  This should clarify my argument (see my new paper).

(E,z) \Rightarrow InfrE(z) is to be read “the inference implication from outcome z in experiment E” (according to whatever inference type/school is being discussed).

A draft of my slides for the Joint Statistical Meetings JSM in Montreal next week are right after the abstract. Comments are very welcome.

Interested readers may search this blog for quite a lot of discussion of the SLP (e.g., here and here) including links to the central papers, “U-Phils” by others (e.g., here, here, and here), and amusing notes (e.g., Don’t Birnbaumize that experiment my friend, and Midnight with Birnbaum).

On the Birnbaum Argument for the Strong Likelihood Principle

Abstract

An essential component of inference based on familiar frequentist notions p-values, significance and confidence levels, is the relevant sampling distribution (hence the term sampling theory). This feature results in violations of a principle known as the strong likelihood principle (SLP), the focus of this paper. In particular, if outcomes x* and y* from experiments E1 and E2 (both with unknown parameter θ), have different probability models f1, f2, then even though f1(x*; θ) = cf2(y*; θ) for all θ, outcomes x* and y* may have different implications for an inference about θ. Although such violations stem from considering outcomes other than the one observed, we argue, this does not require us to consider experiments other than the one performed to produce the data. David Cox (1958) proposes the Weak Conditionality Principle (WCP) to justify restricting the space of relevant repetitions. The WCP says that once it is known which Ei produced the measurement, the assessment should be in terms of the properties of the particular Ei.      

The surprising upshot of Allan Birnbaum’s (1962) argument is that the SLP appears to follow from applying the WCP in the case of mixtures, and so uncontroversial a principle as sufficiency (SP). But this would preclude the use of sampling distributions. The goal of this article is to provide a new clarification and critique of Birnbaum’s argument. Although his argument purports that [(WCP and SP) entails SLP], we show how data may violate the SLP while holding both the WCP and SP. Such cases directly refute [WCP entails SLP].

Comments, questions, errors are welcome.

Full paper can be found here: http://arxiv-web3.library.cornell.edu/abs/1302.7021

Categories: Error Statistics, Statistics, strong likelihood principle | 20 Comments

Background Knowledge: Not to Quantify, But To Avoid Being Misled By, Subjective Beliefs

Posted on July 23, 2013 by Mayo

drapery6A low-powered statistical analysis of this blog—nearing its 2-year anniversary!—reveals that the topic to crop up most often—either front and center, or lurking in the bushes–is that of “background information”. The following was one of my early posts, back in Oct.30, 2011:

October 30, 2011 (London). Increasingly, I am discovering that one of the biggest sources of confusion about the foundations of statistics has to do with what it means or should mean to use “background knowledge” and “judgment” in making statistical and scientific inferences. David Cox and I address this in our “Conversation” in RMM (2011); it is one of the three or four topics in that special volume that I am keen to take up.

Insofar as humans conduct science and draw inferences, and insofar as learning about the world is not reducible to a priori deductions, it is obvious that “human judgments” are involved. True enough, but too trivial an observation to help us distinguish among the very different ways judgments should enter according to contrasting inferential accounts. When Bayesians claim that frequentists do not use or are barred from using background information, what they really mean is that frequentists do not use prior probabilities of hypotheses, at least when those hypotheses are regarded as correct or incorrect, if only approximately. So, for example, we would not assign relative frequencies to the truth of hypotheses such as (1) prion transmission is via protein folding without nucleic acid, or (2) the deflection of light is approximately 1.75” (as if, as Pierce puts it, “universes were as plenty as blackberries”). How odd it would be to try to model these hypotheses as themselves having distributions: to us, statistical hypotheses assign probabilities to outcomes or values of a random variable.

However, quite a lot of background information goes into designing, carrying out, and analyzing inquiries into hypotheses regarded as correct or incorrect. For a frequentist, that is where background knowledge enters. There is no reason to suppose that the background required in order sensibly to generate, interpret, and draw inferences about H should—or even can—enter through prior probabilities for H itself! Of course, presumably, Bayesians also require background information in order to determine that “data x” have been observed, to determine how to model and conduct the inquiry, and to check the adequacy of statistical models for the purposes of the inquiry. So the Bayesian prior only purports to add some other kind of judgment, about the degree of belief in H. It does not get away from the other background judgments that frequentists employ.

This relates to a second point that came up in our conversation when Cox asked, “Do we want to put in a lot of information external to the data, or as little as possible?” Continue reading →

Categories: Background knowledge, Error Statistics | Tags: , | Leave a comment

Guest Post: Larry Laudan. Why Presuming Innocence is Not a Bayesian Prior

Posted on July 20, 2013 by Mayo

DSCF3726“Why presuming innocence has nothing to do with assigning low prior probabilities to the proposition that defendant didn’t commit the crime”

by Professor Larry Laudan
Philosopher of Science*

Several of the comments to the July 17 post about the presumption of innocence suppose that jurors are asked to believe, at the outset of a trial, that the defendant did not commit the crime and that they can legitimately convict him if and only if they are eventually persuaded that it is highly likely (pursuant to the prevailing standard of proof) that he did in fact commit it. Failing that, they must find him not guilty. Many contributors here are conjecturing how confident jurors should be at the outset about defendant’s material innocence.

That is a natural enough Bayesian way of formulating the issue but I think it drastically misstates what the presumption of innocence amounts to.  In my view, the presumption is not (or at least should not be)  an instruction about whether jurors believe defendant did or did not commit the crime.  It is, rather, an instruction about their probative attitudes.wavy capital

There are three reasons for thinking this:

a). asking a juror to begin a trial believing that defendant did not commit a crime requires a doxastic act that is probably outside the jurors’ control.  It would involve asking jurors  to strongly believe an empirical assertion for which they have no evidence whatsoever.  It is wholly unclear that any of us has the ability to talk ourselves into resolutely believing x if we have no empirical grounds for asserting x. By contrast, asking juries to believe that they have seen as yet no proof of defendant’s guilt is an easy belief to acquiesce in since it is obviously true. Continue reading →

Categories: frequentist/Bayesian, PhilStatLaw, Statistics | 28 Comments

Msc Kvetch: A question on the Martin-Zimmerman case we do not hear

Posted on July 19, 2013 by Mayo

questionmark pinkThis is off topic, but a question I don’t hear people ask in regard to the Zimmerman case is: why didn’t any of the several people hearing screams intervene to stop the brawl? Never mind who was screaming, no one felt an obligation to intervene. Eye-witness John Good came out and said something like “stop” but then immediately ran back inside. Others could be heard saying, “don’t go out there”. I don’t say they should have joined the fight, but if a few people had gone outside and screamed or blew a whistle it probably would have been effective.

(I’ve been in such situations twice.)

Categories: msc kvetch | 5 Comments

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

Professor of Philosophy Resigns over Sexual Misconduct (rejected post)

Posted on July 13, 2013 by Mayo

Unknown-1My field (philosophy) is not known for the kinds of data frauds and retractions we’ve discussed on this blog, but scandals revolving around sexual harassment by male faculty are not rare, though I can’t think of another with a senior faculty resigning, at least not in recent times. This article is from 

A Few Words on the McGinn Imbroglio from the philosophy smoker blog (June 4, 2013)

As I guess we [in philosophy] all know, Colin McGinn has chosen to resign from the University of Miami rather than allow the University to proceed with an investigation into allegations of sexual misconduct involving a research assistant. The article at the Chronicle of Higher Ed is here (paywalled); Sally Haslanger has posted a PDF of the whole thing here. Discussion at NewApps here, here, here, and here; discussion at Feminist Philosophers here; discussion at Leiter here and here.

Briefly, what seems to have happened is this: McGinn had a Research Assistant who was a female graduate student. Last spring, the RA started feeling uncomfortable with McGinn. Then, last April, McGinn allegedly started sending her sexually explicit email messages, including one in which, according to the RA’s boyfriend and two unnamed faculty members, “McGinn wrote that he had been thinking about the student while masturbating.”* Wowza.

The RA then contacted the Office of Equality Administration. According to CHE, “after the university’s Office of Equality Administration and the vice provost for faculty affairs conducted an investigation, Mr. McGinn was given the option of agreeing to resign or having an investigation into the allegations against him continue in a public setting, several of the philosopher’s colleagues said.”

It’s hard to know exactly what to make of this. On one obvious interpretation, there’s a clearly implied threat: if you don’t resign, we’re going to publicly drag your name through the mud. And I’m not sure how normal the prospect of a “public” investigation is in this kind of circumstance. For example, if I recall correctly, the Oregon case from a couple of years ago involved an investigation that was supposed to have been kept private, and was made public only in violation of the University’s procedures. But procedures vary from institution to institution, and I don’t have any expertise here. I don’t really have any idea whether this is unusual or not, although my suspicion is that it is at least a little unusual.

It therefore seems reasonable to worry about whether the procedures Miami followed here were respectful of McGinn’s right to due process. But it’s worth emphasizing that the CHE article is not very clear about precisely what happened—for example, Leiter says that McGinn had legal representation and was acting on his lawyer’s advice, but the CHE doesn’t mention it. It is also worth emphasizing that the account in the CHE comes from unnamed “colleagues,” not McGinn or his representatives or any official source at the University. And this comment at Feminist Philosophers, the veracity of which I am not in a position to verify, makes the meeting seem at least a little less troubling. On that account, it was more like, we’ve got some pretty compelling, well-documented evidence of misconduct, which we are duty-bound to pursue; but we’d like to give you the opportunity to resign now and save us both a big headache.

Harrassment occurs between professors, and not just between professors and students, but without the obvious professor-student taboo, it is not taken especially seriously, in my experience. Naturally philosophers, being philosophers, some of them, will engage in deep philosophical discussion of the philosophical nature and justification of the infractions and even how it might have grown out of a legitimate philosophical research on the topic of the evolutionary development of the hand, in relation to its physical functions. Continue reading →

Categories: Rejected Posts, Uncategorized | 2 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

PhilStatLaw: Reference Manual on Scientific Evidence (3d ed) on Statistical Significance (Schachtman)

Posted on July 10, 2013 by Mayo

Memory Lane: One Year Ago on error statistics.com

A quick perusal of the “Manual” on Nathan Schachtman’s legal blog shows it to be chock full of revealing points of contemporary legal statistical philosophy.  The following are some excerpts, read the full blog here.   I make two comments at the end.

July 8th, 2012

Nathan Schachtman

How does the new Reference Manual on Scientific Evidence (RMSE3d 2011) treat statistical significance?  Inconsistently and at times incoherently.

Professor Berger’s Introduction

In her introductory chapter, the late Professor Margaret A. Berger raises the question of the role statistical significance should play in evaluating a study’s support for causal conclusions:

“What role should statistical significance play in assessing the value of a study? Epidemiological studies that are not conclusive but show some increased risk do not prove a lack of causation. Some courts find that they therefore have some probative value, 62 at least in proving general causation. 63”

Margaret A. Berger, “The Admissibility of Expert Testimony,” in RMSE3d 11, 24 (2011).

This seems rather backwards.  Berger’s suggestion that inconclusive studies do not prove lack of causation seems nothing more than a tautology.  And how can that tautology support the claim that inconclusive studies “therefore ” have some probative value? This is a fairly obvious logical invalid argument, or perhaps a passage badly in need of an editor.

…………

Chapter on Statistics

The RMSE’s chapter on statistics is relatively free of value judgments about significance probability, and, therefore, a great improvement upon Berger’s introduction.  The authors carefully describe significance probability and p-values, and explain:

“Small p-values argue against the null hypothesis. Statistical significance is determined by reference to the p-value; significance testing (also called hypothesis testing) is the technique for computing p-values and determining statistical significance.”

David H. Kaye and David A. Freedman, “Reference Guide on Statistics,” in RMSE3d 211, 241 (3ed 2011).  Although the chapter confuses and conflates Fisher’s interpretation of p-values with Neyman’s conceptualization of hypothesis testing as a dichotomous decision procedure, this treatment is unfortunately fairly standard in introductory textbooks.

Kaye and Freedman, however, do offer some important qualifications to the untoward consequences of using significance testing as a dichotomous outcome:

“Artifacts from multiple testing are commonplace. Because research that fails to uncover significance often is not published, reviews of the literature may produce an unduly large number of studies finding statistical significance.111 Even a single researcher may examine so many different relationships that a few will achieve statistical significance by mere happenstance. Almost any large data set—even pages from a table of random digits—will contain some unusual pattern that can be uncovered by diligent search. Having detected the pattern, the analyst can perform a statistical test for it, blandly ignoring the search effort. Statistical significance is bound to follow.

There are statistical methods for dealing with multiple looks at the data, which permit the calculation of meaningful p-values in certain cases.112 However, no general solution is available, and the existing methods would be of little help in the typical case where analysts have tested and rejected a variety of models before arriving at the one considered the most satisfactory (see infra Section V on regression models). In these situations, courts should not be overly impressed with claims that estimates are significant. Instead, they should be asking how analysts developed their models.113 ”

Id. at 256 -57.  This qualification is omitted from the overlapping discussion in the chapter on epidemiology, where it is very much needed. Continue reading →

Categories: P-values, PhilStatLaw, significance tests | Tags: , , , , | 6 Comments

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

Palindrome “contest” contest

Posted on June 29, 2013 by Mayo

 metablog old fashion typewriterWant to win one of these books? You may not have noticed that since May, the palindrome rules have gotten trivially easy. So since it’s Saturday night, and I’m giving a time extension to 14 July – Le Quatorze juillet—have some fun coming up with a palindrome. It only needs to include “Elba” and the word “contest”. For full bibiographies and complete rules, see palindrome page:

 .EGEK CoverSend your candidates to me at error@vt.edu. One of the winners under the older, much harder, rules is here.

Previous palindrome contests included:

runs test, omnibus, cycle, dominate, editor, data, Model, sample, random, probable, Bayes, confident, likely, error, decision, variable, integrate, maximal, median (comedian), interpret, action, code, predict, luck, assess, model, simple, null, bootstrap,minimum, wrong, prefer, dogma, (s)exist, email

with variations.

Categories: Announcement, Palindrome | 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

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