J. Berger

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

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

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

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

Continue reading

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

High error rates in discussions of error rates: no end in sight

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waiting for the other shoe to drop…

“Guides for the Perplexed” in statistics become “Guides to Become Perplexed” when “error probabilities” (in relation to statistical hypotheses tests) are confused with posterior probabilities of hypotheses. Moreover, these posteriors are neither frequentist, subjectivist, nor default. Since this doublespeak is becoming more common in some circles, it seems apt to reblog a post from one year ago (you may wish to check the comments).

Do you ever find yourself holding your breath when reading an exposition of significance tests that’s going swimmingly so far? If you’re a frequentist in exile, you know what I mean. I’m sure others feel this way too. When I came across Jim Frost’s posts on The Minitab Blog, I thought I might actually have located a success story. He does a good job explaining P-values (with charts), the duality between P-values and confidence levels, and even rebuts the latest “test ban” (the “Don’t Ask, Don’t Tell” policy). Mere descriptive reports of observed differences that the editors recommend, Frost shows, are uninterpretable without a corresponding P-value or the equivalent. So far, so good. I have only small quibbles, such as the use of “likelihood” when meaning probability, and various and sundry nitpicky things. But watch how in some places significance levels are defined as the usual error probabilities —indeed in the glossary for the site—while in others it is denied they provide error probabilities. In those other places, error probabilities and error rates shift their meaning to posterior probabilities, based on priors representing the “prevalence” of true null hypotheses.

Begin with one of his kosher posts “Understanding Hypothesis Tests: Significance Levels (Alpha) and P values in Statistics” (blue is Frost): Continue reading

Categories: highly probable vs highly probed, J. Berger, reforming the reformers, Statistics | 1 Comment

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

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

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

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

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

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

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

Categories: J. Berger, power, reforming the reformers, S. Senn, Statistical power, Statistics | 36 Comments

High error rates in discussions of error rates (1/21/16 update)

27D0BB5300000578-3168627-image-a-27_1437433320306

waiting for the other shoe to drop…

Do you ever find yourself holding your breath when reading an exposition of significance tests that’s going swimmingly so far? If you’re a frequentist in exile, you know what I mean. I’m sure others feel this way too. When I came across Jim Frost’s posts on The Minitab Blog, I thought I might actually have located a success story. He does a good job explaining P-values (with charts), the duality between P-values and confidence levels, and even rebuts the latest “test ban” (the “Don’t Ask, Don’t Tell” policy). Mere descriptive reports of observed differences that the editors recommend, Frost shows, are uninterpretable without a corresponding P-value or the equivalent. So far, so good. I have only small quibbles, such as the use of “likelihood” when meaning probability, and various and sundry nitpicky things. But watch how in some places significance levels are defined as the usual error probabilities and error rates—indeed in the glossary for the site—while in others it is denied they provide error rates. In those other places, error probabilities and error rates shift their meaning to posterior probabilities, based on priors representing the “prevalence” of true null hypotheses. Continue reading

Categories: highly probable vs highly probed, J. Berger, reforming the reformers, Statistics | 11 Comments

Are P Values Error Probabilities? or, “It’s the methods, stupid!” (2nd install)

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Despite the fact that Fisherians and Neyman-Pearsonians alike regard observed significance levels, or P values, as error probabilities, we occasionally hear allegations (typically from those who are neither Fisherian nor N-P theorists) that P values are actually not error probabilities. The denials tend to go hand in hand with allegations that P values exaggerate evidence against a null hypothesis—a problem whose cure invariably invokes measures that are at odds with both Fisherian and N-P tests. The Berger and Sellke (1987) article from a recent post is a good example of this. When leading figures put forward a statement that looks to be straightforwardly statistical, others tend to simply repeat it without inquiring whether the allegation actually mixes in issues of interpretation and statistical philosophy. So I wanted to go back and look at their arguments. I will post this in installments.

1. Some assertions from Fisher, N-P, and Bayesian camps

Here are some assertions from Fisherian, Neyman-Pearsonian and Bayesian camps: (I make no attempt at uniformity in writing the “P-value”, but retain the quotes as written.)

a) From the Fisherian camp (Cox and Hinkley):

For given observations y we calculate t = tobs = t(y), say, and the level of significance pobs by

pobs = Pr(T > tobs; H0).

….Hence pobs is the probability that we would mistakenly declare there to be evidence against H0, were we to regard the data under analysis as being just decisive against H0.” (Cox and Hinkley 1974, 66).

Thus pobs would be the Type I error probability associated with the test.

b) From the Neyman-Pearson N-P camp (Lehmann and Romano):

“[I]t is good practice to determine not only whether the hypothesis is accepted or rejected at the given significance level, but also to determine the smallest significance level…at which the hypothesis would be rejected for the given observation. This number, the so-called p-value gives an idea of how strongly the data contradict the hypothesis. It also enables others to reach a verdict based on the significance level of their choice.” (Lehmann and Romano 2005, 63-4) 

Very similar quotations are easily found, and are regarded as uncontroversial—even by Bayesians whose contributions stood at the foot of Berger and Sellke’s argument that P values exaggerate the evidence against the null. Continue reading

Categories: frequentist/Bayesian, J. Berger, P-values, Statistics | 32 Comments

U-Phil: Deconstructions [of J. Berger]: Irony & Bad Faith 3

Memory Lane: 2 years ago:
My efficient Errorstat Blogpeople1 have put forward the following 3 reader-contributed interpretive efforts2 as a result of the “deconstruction” exercise from December 11, (mine, from the earlier blog, is at the end) of what I consider:

“….an especially intriguing remark by Jim Berger that I think bears upon the current mindset (Jim is aware of my efforts):

Too often I see people pretending to be subjectivists, and then using “weakly informative” priors that the objective Bayesian community knows are terrible and will give ridiculous answers; subjectivism is then being used as a shield to hide ignorance. . . . In my own more provocative moments, I claim that the only true subjectivists are the objective Bayesians, because they refuse to use subjectivism as a shield against criticism of sloppy pseudo-Bayesian practice. (Berger 2006, 463)” (From blogpost, Dec. 11, 2011)
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Andrew Gelman:

The statistics literature is big enough that I assume there really is some bad stuff out there that Berger is reacting to, but I think that when he’s talking about weakly informative priors, Berger is not referring to the work in this area that I like, as I think of weakly informative priors as specifically being designed to give answers that are _not_ “ridiculous.”

Keeping things unridiculous is what regularization’s all about, and one challenge of regularization (as compared to pure subjective priors) is that the answer to the question, What is a good regularizing prior?, will depend on the likelihood.  There’s a lot of interesting theory and practice relating to weakly informative priors for regularization, a lot out there that goes beyond the idea of noninformativity.

To put it another way:  We all know that there’s no such thing as a purely noninformative prior:  any model conveys some information.  But, more and more, I’m coming across applied problems where I wouldn’t want to be noninformative even if I could, problems where some weak prior information regularizes my inferences and keeps them sane and under control. Continue reading

Categories: Gelman, Irony and Bad Faith, J. Berger, Statistics, U-Phil | Tags: , , , | 3 Comments

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