Monthly Archives: August 2015

The Paradox of Replication, and the vindication of the P-value (but she can go deeper) 9/2/15 update (ii)

Posted on August 31, 2015 by Mayo
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The unpopular P-value is invited to dance.

  1. The Paradox of Replication

Critic 1: It’s much too easy to get small P-values.

Critic 2: We find it very difficult to get small P-values; only 36 of 100 psychology experiments were found to yield small P-values in the recent Open Science collaboration on replication (in psychology).

Is it easy or is it hard?

You might say, there’s no paradox, the problem is that the significance levels in the original studies are often due to cherry-picking, multiple testing, optional stopping and other biasing selection effects. The mechanism by which biasing selection effects blow up P-values is very well understood, and we can demonstrate exactly how it occurs. In short, many of the initially significant results merely report “nominal” P-values not “actual” ones, and there’s nothing inconsistent between the complaints of critic 1 and critic 2.

The resolution of the paradox attests to what many have long been saying: the problem is not with the statistical methods but with their abuse. Even the P-value, the most unpopular girl in the class, gets to show a little bit of what she’s capable of. She will give you a hard time when it comes to replicating nominally significant results, if they were largely due to biasing selection effects. That is just what is wanted; it is an asset that she feels the strain, and lets you know. It is statistical accounts that can’t pick up on biasing selection effects that should worry us (especially those that deny they are relevant). That is one of the most positive things to emerge from the recent, impressive, replication project in psychology. From an article in the Smithsonian magazine “Scientists Replicated 100 Psychology Studies, and Fewer Than Half Got the Same Results”:

The findings also offered some support for the oft-criticized statistical tool known as the P value, which measures whether a result is significant or due to chance. …

The project analysis showed that a low P value was fairly predictive of which psychology studies could be replicated. Twenty of the 32 original studies with a P value of less than 0.001 could be replicated, for example, while just 2 of the 11 papers with a value greater than 0.04 were successfully replicated. (Link is here.)

Continue reading →

Categories: replication research, reproducibility, spurious p values, Statistics

3 YEARS AGO (AUGUST 2012): MEMORY LANE

Posted on August 24, 2015 by Mayo
3 years ago...
3 years ago…

MONTHLY MEMORY LANE: 3 years ago: August 2012. I mark in red three posts that seem most apt for general background on key issues in this blog.[1] Posts that are part of a “unit” or a group of “U-Phils” count as one (there are 4 U-Phils on Wasserman this time). Monthly memory lanes began at the blog’s 3-year anniversary in Sept, 2014. We’re about to turn four.

August 2012

[1] excluding those reblogged fairly recently.

[2] Larry Wasserman’s paper was “Low Assumptions, High dimensions” in our special RIMM volume.

Categories: 3-year memory lane, Statistics

How to avoid making mountains out of molehills, using power/severity

Posted on August 20, 2015 by Mayo
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A classic fallacy of rejection is taking a statistically significant result as evidence of a discrepancy from a test (or null) hypothesis larger than is warranted. Standard tests do have resources to combat this fallacy, but you won’t see them in textbook formulations. It’s not new statistical method, but new (and correct) interpretations of existing methods, that are needed. One can begin with a companion to the rule in this recent post:

(1) If POW(T+,µ’) is low, then the statistically significant x is a good indication that µ > µ’.

To have the companion rule also in terms of power, let’s suppose that our result is just statistically significant at a level α. (As soon as the observed difference exceeds the cut-off the rule has to be modified). 

Rule (1) was stated in relation to a statistically significant result x (at level α) from a one-sided test T+ of the mean of a Normal distribution with n iid samples, and (for simplicity) known σ:   H0: µ ≤  0 against H1: µ >  0. Here’s the companion:

(2) If POW(T+,µ’) is high, then an α statistically significant x is a good indication that µ < µ’.
(The higher the POW(T+,µ’) is, the better the indication  that µ < µ’.)

That is, if the test’s power to detect alternative µ’ is high, then the statistically significant x is a good indication (or good evidence) that the discrepancy from null is not as large as µ’ (i.e., there’s good evidence that  µ < µ’).

An account of severe testing based on error statistics is always keen to indicate inferences that are not warranted by the data, as well as those that are. Not only might we wish to indicate which discrepancies are poorly warranted, we can give upper bounds to warranted discrepancies by using (2).

POWER: POW(T+,µ’) = POW(Test T+ rejects H0;µ’) = Pr(M > M*; µ’), where M is the sample mean and M* is the cut-off for rejection. (Since it’s continuous, it doesn’t matter if we write > or ≥.)[i]

EXAMPLE. Let σ = 10, n = 100, so (σ/√n) = 1.  Test T+ rejects H0 at the .025 level if  M  > 1.96(1).

Find the power against µ = 2.3. To find Pr(M > 1.96; 2.3), get the standard Normal z = (1.96 – 2.3)/1 = -.84. Find the area to the right of -.84 on the standard Normal curve. It is .8. So POW(T+,2.8) = .8.

For simplicity in what follows, let the cut-off, M*, be 2. Let the observed mean M0 just reach the cut-off  2.

The power against alternatives between the null and the cut-off M* will range from α to .5. Power exceeds .5 only once we consider alternatives greater than M*, for these yield negative z values.  Power fact, POW(M* + 1(σ/√n)) = .84.

That is, adding one (σ/ √n) unit to the cut-off M* takes us to an alternative against which the test has power = .84. So, POW(T+, µ = 3) = .84. See this post.

 By (2), the (just) significant result x is decent evidence that µ< 3, because if µ ≥ 3, we’d have observed a more statistically significant result, with probability .84.  The upper .84 confidence limit is 3. The significant result is much better evidence that µ< 4,  the upper .975 confidence limit is 4 (approx.), etc. 

Reporting (2) is typically of importance in cases of highly sensitive tests, but I think it should always accompany a rejection to avoid making mountains out of molehills. (However, in my view, (2) should be custom-tailored to the outcome not the cut-off.) In the case of statistical insignificance, (2) is essentially ordinary power analysis. (In that case, the interest may be to avoid making molehills out of mountains.) Power analysis, applied to insignificant results, is especially of interest with low-powered tests. For example, failing to find a statistically significant increase in some risk may at most rule out (substantively) large risk increases. It might not allow ruling out risks of concern. Naturally, what counts as a risk of concern is a context-dependent consideration, often stipulated in regulatory statutes.

NOTES ON HOWLERS: When researchers set a high power to detect µ’, it is not an indication they regard µ’ as plausible, likely, expected, probable or the like. Yet we often hear people say “if statistical testers set .8 power to detect µ = 2.3 (in test T+), they must regard µ = 2.3 as probable in some sense”. No, in no sense. Another thing you might hear is, “when H0: µ ≤  0 is rejected (at the .025 level), it’s reasonable to infer µ > 2.3″, or “testers are comfortable inferring µ ≥ 2.3”.  No, they are not comfortable, nor should you be. Such an inference would be wrong with probability ~.8. Given M = 2 (or 1.96), you need to subtract to get a lower confidence bound, if the confidence level is not to exceed .5 . For example, µ > .5 is a lower confidence bound at confidence level .93.

Rule (2) also provides a way to distinguish values within a 1-α confidence interval (instead of choosing a given confidence level and then reporting CIs in the dichotomous manner that is now typical).

At present, power analysis is only used to interpret negative results–and there it is often called “retrospective power”, which is a fine term, but it’s often defined as what I call shpower). Again, confidence bounds could be, but they are not now, used to this end [iii].

Severity replaces M* in (2) with the actual result, be it significant or insignificant. 

Looking at power means looking at the best case (just reaching a significance level) or the worst case (just missing it). This is way too coarse; we need to custom tailor results using the observed data. That’s what severity does, but for this post, I wanted to just illuminate the logic.[ii]

One more thing:  

Applying (1) and (2) requires the error probabilities to be actual (approximately correct): Strictly speaking, rules (1) and (2) have a conjunct in their antecedents [iv]: “given the test assumptions are sufficiently well met”. If background knowledge leads you to deny (1) or (2), it indicates you’re denying the reported error probabilities are the actual ones. There’s evidence the test fails an “audit”. That, at any rate, is what I would argue.

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[i] To state power in terms of P-values: POW(µ’) = Pr(P < p*; µ’) where P < p* corresponds to rejecting the null hypothesis at the given level.

[ii] It must be kept in mind that statistical testing inferences are going to be in the form of µ > µ’ =µ0 + δ,  or µ ≤ µ’ =µ0 + δ  or the like. They are not to point values! (Not even to the point µ =M0.) Take a look at the alternative H1: µ >  0. It is not a point value. Although we are going beyond inferring the existence of some discrepancy, we still retain inferences in the form of inequalities. 

[iii] That is, upper confidence bounds are too readily viewed as “plausible” bounds, and as values for which the data provide positive evidence. In fact, as soon as you get to an upper bound at confidence levels of around .6, .7, .8, etc. you actually have evidence µ’ < CI-upper. See this post.

[iv] The “antecedent” of a conditional refers to the statement between the “if” and the “then”.

OTHER RELEVANT POSTS ON POWER

Categories: fallacy of rejection, power, Statistics

Statistics, the Spooky Science

Posted on August 19, 2015 by Mayo

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I was reading this interview Of Erich Lehmann yesterday: “A Conversation with Erich L. Lehmann”

Lehmann: …I read over and over again that hypothesis testing is dead as a door nail, that nobody does hypothesis testing. I talk to Julie and she says that in the behaviorial sciences, hypothesis testing is what they do the most. All my statistical life, I have been interested in three different types of things: testing, point estimation, and confidence-interval estimation. There is not a year that somebody doesn’t tell me that two of them are total nonsense and only the third one makes sense. But which one they pick changes from year to year. [Laughs] (p.151)…..

DeGroot: …It has always amazed me about statistics that we argue among ourselves about which of our basic techniques are of practical value. It seems to me that in other areas one can argue about whether a methodology is going to prove to be useful, but people would agree whether a technique is useful in practice. But in statistics, as you say, some people believe that confidence intervals are the only procedures that make any sense on practical grounds, and others think they have no practical value whatsoever. I find it kind of spooky to be in such a field.

Lehmann: After a while you get used to it. If somebody attacks one of these, I just know that next year I’m going to get one who will be on the other side. (pp.151-2)

Emphasis is mine.

I’m reminded of this post.

Morris H. DeGroot, Statistical Science, 1986, Vol. 1, No.2, 243-258

 

 

Categories: phil/history of stat, Statistics

Severity in a Likelihood Text by Charles Rohde

Posted on August 15, 2015 by Mayo
Mayo elbow

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I received a copy of a statistical text recently that included a discussion of severity, and this is my first chance to look through it. It’s Introductory Statistical Inference with the Likelihood Function by Charles Rohde from Johns Hopkins. Here’s the blurb:

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This textbook covers the fundamentals of statistical inference and statistical theory including Bayesian and frequentist approaches and methodology possible without excessive emphasis on the underlying mathematics. This book is about some of the basic principles of statistics that are necessary to understand and evaluate methods for analyzing complex data sets. The likelihood function is used for pure likelihood inference throughout the book. There is also coverage of severity and finite population sampling. The material was developed from an introductory statistical theory course taught by the author at the Johns Hopkins University’s Department of Biostatistics. Students and instructors in public health programs will benefit from the likelihood modeling approach that is used throughout the text. This will also appeal to epidemiologists and psychometricians. After a brief introduction, there are chapters on estimation, hypothesis testing, and maximum likelihood modeling. The book concludes with sections on Bayesian computation and inference. An appendix contains unique coverage of the interpretation of probability, and coverage of probability and mathematical concepts.

It’s welcome to see severity in a statistics text; an example from Mayo and Spanos (2006) is given in detail. The author even says some nice things about it: “Severe testing does a nice job of clarifying the issues which occur when a hypothesis is accepted (not rejected) by finding those values of the parameter (here mu) which are plausible (have high severity[i]) given acceptance. Similarly severe testing addresses the issue of a hypothesis which is rejected…. . ”  I don’t know Rohde, and the book isn’t error-statistical in spirit at all.[ii] In fact, inferences based on error probabilities are often called “illogical” because they take into account cherry-picking, multiple testing, optional stopping and other biasing selection effects that the likelihoodist considers irrelevant. I wish he had used severity to address some of the classic howlers he delineates regarding N-P statistics. To his credit, they are laid out with unusual clarity. For example a rejection of a point null µ= µ0 based on a result that just reaches the 1.96 cut-off for a one-sided test is claimed to license the inference to a point alternative µ= µ’ that is over 6 standard deviations greater than the null. (pp. 49-50). But it is not licensed. The probability of a larger difference than observed, were the data generated under such an alternative is ~1, so the severity associated with such an inference is ~ 0. SEV(µ <µ’) ~1. 

[i]Not to quibble, but I wouldn’t say parameter values are assigned severity, but rather that various hypotheses about mu pass with severity. The hypotheses are generally in the form of discrepancies, e..g,µ >µ’

[ii] He’s a likelihoodist from Johns Hopkins. Royall has had a strong influence there (Goodman comes to mind), and elsewhere, especially among philosophers. Bayesians also come back to likelihood ratio arguments, often. For discussions on likelihoodism and the law of likelihood see:

How likelihoodists exaggerate evidence from statistical tests

Breaking the Law of Likelihood ©

Breaking the Law of Likelihood, to keep their fit measures in line A, B

Why the Law of Likelihood is Bankrupt as an Account of Evidence

Royall, R. (2004), “The Likelihood Paradigm for Statistical Evidence” 119-138; Rejoinder 145-151, in M. Taper, and S. Lele (eds.) The Nature of Scientific Evidence: Statistical, Philosophical and Empirical Considerations. Chicago: University of Chicago Press.

Click to access 2006Mayo_Spanos_severe_testing.pdf

Categories: Error Statistics, Severity

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

Posted on August 14, 2015 by Mayo
egon pearson

E.S. Pearson

Are methods based on error probabilities of use mainly to supply procedures which will not err too frequently in some long run? (performance). Or is it the other way round: that the control of long run error properties are of crucial importance for probing the causes of the data at hand? (probativeness). I say no to the former and yes to the latter. This, I think, was also the view of Egon Sharpe (E.S.) Pearson (11 Aug, 1895-12 June, 1980). I reblog a relevant post from 2012.

Cases of Type A and Type B

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

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

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

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

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

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

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

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

Categories: 3-year memory lane, phil/history of stat | Tags:

A. Spanos: Egon Pearson’s Neglected Contributions to Statistics

Posted on August 11, 2015 by Mayo
egon pearson swim

11 August 1895 – 12 June 1980

Today is Egon Pearson’s birthday. I reblog a post by my colleague Aris Spanos from (8/18/12): “Egon Pearson’s Neglected Contributions to Statistics.”  Happy Birthday Egon Pearson!

    Egon Pearson (11 August 1895 – 12 June 1980), is widely known today for his contribution in recasting of Fisher’s significance testing into the Neyman-Pearson (1933) theory of hypothesis testing. Occasionally, he is also credited with contributions in promoting statistical methods in industry and in the history of modern statistics; see Bartlett (1981). What is rarely mentioned is Egon’s early pioneering work on:

(i) specification: the need to state explicitly the inductive premises of one’s inferences,

(ii) robustness: evaluating the ‘sensitivity’ of inferential procedures to departures from the Normality assumption, as well as

(iii) Mis-Specification (M-S) testing: probing for potential departures from the Normality  assumption.

Arguably, modern frequentist inference began with the development of various finite sample inference procedures, initially by William Gosset (1908) [of the Student’s t fame] and then Fisher (1915, 1921, 1922a-b). These inference procedures revolved around a particular statistical model, known today as the simple Normal model: Continue reading →

Categories: phil/history of stat, Statistics, Testing Assumptions | Tags: , , ,

Statistical Theater of the Absurd: “Stat on a Hot Tin Roof”

Posted on August 8, 2015 by Mayo
metablog old fashion typewriter

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Memory lane: Did you ever consider how some of the colorful exchanges among better-known names in statistical foundations could be the basis for high literary drama in the form of one-act plays (even if appreciated by only 3-7 people in the world)? (Think of the expressionist exchange between Bohr and Heisenberg in Michael Frayn’s play Copenhagen, except here there would be no attempt at all to popularize—only published quotes and closely remembered conversations would be included, with no attempt to create a “story line”.)  Somehow I didn’t think so. But rereading some of Savage’s high-flown praise of Birnbaum’s “breakthrough” argument (for the Likelihood Principle) today, I was swept into a “(statistical) theater of the absurd” mindset.(Update Aug, 2015 [ii])

The first one came to me in autumn 2008 while I was giving a series of seminars on philosophy of statistics at the LSE. Modeled on a disappointing (to me) performance of The Woman in Black, “A Funny Thing Happened at the [1959] Savage Forum” relates Savage’s horror at George Barnard’s announcement of having rejected the Likelihood Principle!

Barnard-1979-picture

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The current piece also features George Barnard. It recalls our first meeting in London in 1986. I’d sent him a draft of my paper on E.S. Pearson’s statistical philosophy, “Why Pearson Rejected the Neyman-Pearson Theory of Statistics” (later adapted as chapter 11 of EGEK) to see whether I’d gotten Pearson right. Since Tuesday (Aug 11) is Pearson’s birthday, I’m reblogging this. Barnard had traveled quite a ways, from Colchester, I think. It was June and hot, and we were up on some kind of a semi-enclosed rooftop. Barnard was sitting across from me looking rather bemused.

The curtain opens with Barnard and Mayo on the roof, lit by a spot mid-stage. He’s drinking (hot) tea; she, a Diet Coke. The dialogue (is what I recall from the time[i]):

 Barnard: I read your paper. I think it is quite good.  Did you know that it was I who told Fisher that Neyman-Pearson statistics had turned his significance tests into little more than acceptance procedures? Continue reading →

Categories: Barnard, phil/history of stat, Statistics | Tags: , , , ,

Neyman: Distinguishing tests of statistical hypotheses and tests of significance might have been a lapse of someone’s pen

Posted on August 5, 2015 by Mayo
Neyman April 16, 1894 – August 5, 1981

April 16, 1894 – August 5, 1981

“Tests of Statistical Hypotheses and Their Use in Studies of Natural Phenomena” by Jerzy Neyman

ABSTRACT. Contrary to ideas suggested by the title of the conference at which the present paper was presented, the author is not aware of a conceptual difference between a “test of a statistical hypothesis” and a “test of significance” and uses these terms interchangeably. A study of any serious substantive problem involves a sequence of incidents at which one is forced to pause and consider what to do next. In an effort to reduce the frequency of misdirected activities one uses statistical tests. The procedure is illustrated on two examples: (i) Le Cam’s (and associates’) study of immunotherapy of cancer and (ii) a socio-economic experiment relating to low-income homeownership problems.

Neyman died on August 5, 1981. Here’s an unusual paper of his, “Tests of Statistical Hypotheses and Their Use in Studies of Natural Phenomena.” I have been reading a fair amount by Neyman this summer in writing about the origins of his philosophy, and have found further corroboration of the position that the behavioristic view attributed to him, while not entirely without substance*, is largely a fable that has been steadily built up and accepted as gospel. This has justified ignoring Neyman-Pearson statistics (as resting solely on long-run performance and irrelevant to scientific inference) and turning to crude variations of significance tests, that Fisher wouldn’t have countenanced for a moment (so-called NHSTs), lacking alternatives, incapable of learning from negative results, and permitting all sorts of P-value abuses–notably going from a small p-value to claiming evidence for a substantive research hypothesis. The upshot is to reject all of frequentist statistics, even though P-values are a teeny tiny part. *This represents a change in my perception of Neyman’s philosophy since EGEK (Mayo 1996).  I still say that that for our uses of method, it doesn’t matter what anybody thought, that “it’s the methods, stupid!” Anyway, I recommend, in this very short paper, the general comments and the example on home ownership. Here are two snippets: Continue reading →

Categories: Error Statistics, Neyman, Statistics | Tags:

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