Statistical Concepts in Their Relation to Reality–E.S. Pearson

11 August 1895 – 12 June 1980

This is my third and final post marking Egon Pearson’s birthday (Aug. 11). The focus is his little-known paper: “Statistical Concepts in Their Relation to Reality” (Pearson 1955). I’ve linked to it several times over the years, but always find a new gem or two, despite its being so short. E. Pearson rejected some of the familiar tenets that have come to be associated with Neyman and Pearson (N-P) statistical tests, notably the idea that the essential justification for tests resides in a repeated applications or long-run control of rates of erroneous interpretations–what he termed the “behavioral” rationale of tests. In an unpublished letter E. Pearson wrote to Birnbaum (1974), he talks about N-P theory admitting of two interpretations: behavioral and evidential:

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

(Nowadays, it might be said that some people concentrate to an absurd extent on “science-wise error rates” in their view of statistical tests as dichotomous screening devices.) Continue reading

Categories: Egon Pearson, phil/history of stat, Philosophy of Statistics | Tags: , , | Leave a comment

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

Continuing with posts on E.S. Pearson in marking his birthday, I reblog this guest post by Aris Spanos. 

Egon Pearson’s Neglected Contributions to Statistics

by Aris Spanos

    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:

Xk ∽ NIID(μ,σ²), k=1,2,…,n,…             (1)

where ‘NIID(μ,σ²)’ stands for ‘Normal, Independent and Identically Distributed with mean μ and variance σ²’. These procedures include the ‘optimal’ estimators of μ and σ², Xbar and s², and the pivotal quantities:

(a) τ(X) =[√n(Xbar- μ)/s] ∽ St(n-1),  (2)

(b) v(X) =[(n-1)s²/σ²] ∽ χ²(n-1),        (3)

where St(n-1) and χ²(n-1) denote the Student’s t and chi-square distributions with (n-1) degrees of freedom.

The question of ‘how these inferential results might be affected when the Normality assumption is false’ was originally raised by Gosset in a letter to Fisher in 1923:

“What I should like you to do is to find a solution for some other population than a normal one.”  (Lehmann, 1999)

He went on to say that he tried the rectangular (uniform) distribution but made no progress, and he was seeking Fisher’s help in tackling this ‘robustness/sensitivity’ problem. In his reply that was unfortunately lost, Fisher must have derived the sampling distribution of τ(X), assuming some skewed distribution (possibly log-Normal). We know this from Gosset’s reply:

“I like the result for z [τ(X)] in the case of that horrible curve you are so fond of. I take it that in skew curves the distribution of z is skew in the opposite direction.”  (Lehmann, 1999)

After this exchange Fisher was not particularly receptive to Gosset’s requests to address the problem of working out the implications of non-Normality for the Normal-based inference procedures; t, chi-square and F tests.

In contrast, Egon Pearson shared Gosset’s concerns about the robustness of Normal-based inference results (a)-(b) to non-Normality, and made an attempt to address the problem in a series of papers in the late 1920s and early 1930s.

This line of research for Pearson began with a review of Fisher’s 2nd edition of the 1925 book, published in Nature, and dated June 8th, 1929.  Pearson, after praising the book for its path breaking contributions, dared raise a mild criticism relating to (i)-(ii) above:

“There is one criticism, however, which must be made from the statistical point of view. A large number of tests are developed upon the assumption that the population sampled is of ‘normal’ form. That this is the case may be gathered from a very careful reading of the text, but the point is not sufficiently emphasised. It does not appear reasonable to lay stress on the ‘exactness’ of tests, when no means whatever are given of appreciating how rapidly they become inexact as the population samples diverge from normality.” (Pearson, 1929a)

Fisher reacted badly to this criticism and was preparing an acerbic reply to the ‘young pretender’ when Gosset jumped into the fray with his own letter in Nature, dated July 20th, in an obvious attempt to moderate the ensuing fight. Gosset succeeded in tempering Fisher’s reply, dated August 17th, forcing him to provide a less acerbic reply, but instead of addressing the ‘robustness/sensitivity’ issue, he focused primarily on Gosset’s call to address ‘the problem of what sort of modification of my tables for the analysis of variance would be required to adapt that process to non-normal distributions’. He described that as a hopeless task. This is an example of Fisher’s genious when cornered by an insightful argument. He sidestepped the issue of ‘robustness’ to departures from Normality, by broadening it – alluding to other possible departures from the ID assumption – and rendering it a hopeless task, by focusing on the call to ‘modify’ the statistical tables for all possible non-Normal distributions; there is an infinity of potential modifications!

Egon Pearson recognized the importance of stating explicitly the inductive premises upon which the inference results are based, and pressed ahead with exploring the robustness issue using several non-Normal distributions within the Pearson family. His probing was based primarily on simulation, relying on tables of pseudo-random numbers; see Pearson and Adyanthaya (1928, 1929), Pearson (1929b, 1931). His broad conclusions were that the t-test:

τ0(X)=|[√n(X-bar- μ0)/s]|, C1:={x: τ0(x) > cα},    (4)

for testing the hypotheses:

H0: μ = μ0 vs. H1: μ ≠ μ0,                                             (5)

is relatively robust to certain departures from Normality, especially when the underlying distribution is symmetric, but the ANOVA test is rather sensitive to such departures! He continued this line of research into his 80s; see Pearson and Please (1975).

Perhaps more importantly, Pearson (1930) proposed a test for the Normality assumption based on the skewness and kurtosis coefficients: a Mis-Specification (M-S) test. Ironically, Fisher (1929) provided the sampling distributions of the sample skewness and kurtosis statistics upon which Pearson’s test was based. Pearson continued sharpening his original M-S test for Normality, and his efforts culminated with the D’Agostino and Pearson (1973) test that is widely used today; see also Pearson et al. (1977). The crucial importance of testing Normality stems from the fact that it renders the ‘robustness/sensitivity’ problem manageable. The test results can be used to narrow down the possible departures one needs to worry about. They can also be used to suggest ways to respecify the original model.

After Pearson’s early publications on the ‘robustness/sensitivity’ problem Gosset realized that simulation alone was not effective enough to address the question of robustness, and called upon Fisher, who initially rejected Gosset’s call by saying ‘it was none of his business’, to derive analytically the implications of non-Normality using different distributions:

“How much does it [non-Normality] matter? And in fact that is your business: none of the rest of us have the slightest chance of solving the problem: we can play about with samples [i.e. perform simulation studies], I am not belittling E. S. Pearson’s work, but it is up to you to get us a proper solution.” (Lehmann, 1999).

In this passage one can discern the high esteem with which Gosset held Fisher for his technical ability. Fisher’s reply was rather blunt:

“I do not think what you are doing with nonnormal distributions is at all my business, and I doubt if it is the right approach. … Where I differ from you, I suppose, is in regarding normality as only a part of the difficulty of getting data; viewed in this collection of difficulties I think you will see that it is one of the least important.”

It’s clear from this that Fisher understood the problem of how to handle departures from Normality more broadly than his contemporaries. His answer alludes to two issues that were not well understood at the time:

(a) departures from the other two probabilistic assumptions (IID) have much more serious consequences for Normal-based inference than Normality, and

(b) deriving the consequences of particular forms of non-Normality on the reliability of Normal-based inference, and proclaiming a procedure enjoys a certain level of ‘generic’ robustness, does not provide a complete answer to the problem of dealing with departures from the inductive premises.

In relation to (a) it is important to note that the role of ‘randomness’, as it relates to the IID assumptions, was not well understood until the 1940s, when the notion of non-IID was framed in terms of explicit forms of heterogeneity and dependence pertaining to stochastic processes. Hence, the problem of assessing departures from IID was largely ignored at the time, focusing almost exclusively on departures from Normality. Indeed, the early literature on nonparametric inference retained the IID assumptions and focused on inference procedures that replace the Normality assumption with indirect distributional assumptions pertaining to the ‘true’ but unknown f(x), like the existence of certain moments, its symmetry, smoothness, continuity and/or differentiability, unimodality, etc. ; see Lehmann (1975). It is interesting to note that Egon Pearson did not consider the question of testing the IID assumptions until his 1963 paper.

In relation to (b), when one poses the question ‘how robust to non-Normality is the reliability of inference based on a t-test?’ one ignores the fact that the t-test might no longer be the ‘optimal’ test under a non-Normal distribution. This is because the sampling distribution of the test statistic and the associated type I and II error probabilities depend crucially on the validity of the statistical model assumptions. When any of these assumptions are invalid, the relevant error probabilities are no longer the ones derived under the original model assumptions, and the optimality of the original test is called into question. For instance, assuming that the ‘true’ distribution is uniform (Gosset’s rectangular):

Xk ∽ U(a-μ,a+μ),   k=1,2,…,n,…        (6)

where f(x;a,μ)=(1/(2μ)), (a-μ) ≤ x ≤ (a+μ), μ > 0,

how does one assess the robustness of the t-test? One might invoke its generic robustness to symmetric non-Normal distributions and proceed as if the t-test is ‘fine’ for testing the hypotheses (5). A more well-grounded answer will be to assess the discrepancy between the nominal (assumed) error probabilities of the t-test based on (1) and the actual ones based on (6). If the latter approximate the former ‘closely enough’, one can justify the generic robustness. These answers, however, raise the broader question of what are the relevant error probabilities? After all, the optimal test for the hypotheses (5) in the context of (6), is no longer the t-test, but the test defined by:

w(X)=|{(n-1)([X[1] +X[n]]-μ0)}/{[X[1]-X[n]]}|∽F(2,2(n-1)),   (7)

with a rejection region C1:={x: w(x) > cα},  where (X[1], X[n]) denote the smallest and the largest element in the ordered sample (X[1], X[2],…, X[n]), and F(2,2(n-1)) the F distribution with 2 and 2(n-1) degrees of freedom; see Neyman and Pearson (1928). One can argue that the relevant comparison error probabilities are no longer the ones associated with the t-test ‘corrected’ to account for the assumed departure, but those associated with the test in (7). For instance, let the t-test have nominal and actual significance level, .05 and .045, and power at μ10+1, of .4 and .37, respectively. The conventional wisdom will call the t-test robust, but is it reliable (effective) when compared with the test in (7) whose significance level and power (at μ1) are say, .03 and .9, respectively?

A strong case can be made that a more complete approach to the statistical misspecification problem is:

(i) to probe thoroughly for any departures from all the model assumptions using trenchant M-S tests, and if any departures are detected,

(ii) proceed to respecify the statistical model by choosing a more appropriate model with a view to account for the statistical information that the original model did not.

Admittedly, this is a more demanding way to deal with departures from the underlying assumptions, but it addresses the concerns of Gosset, Egon Pearson, Neyman and Fisher much more effectively than the invocation of vague robustness claims; see Spanos (2010).


Bartlett, M. S. (1981) “Egon Sharpe Pearson, 11 August 1895-12 June 1980,” Biographical Memoirs of Fellows of the Royal Society, 27: 425-443.

D’Agostino, R. and E. S. Pearson (1973) “Tests for Departure from Normality. Empirical Results for the Distributions of b₂ and √(b₁),” Biometrika, 60: 613-622.

Fisher, R. A. (1915) “Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population,” Biometrika, 10: 507-521.

Fisher, R. A. (1921) “On the “probable error” of a coefficient of correlation deduced from a small sample,” Metron, 1: 3-32.

Fisher, R. A. (1922a) “On the mathematical foundations of theoretical statistics,” Philosophical Transactions of the Royal Society A, 222, 309-368.

Fisher, R. A. (1922b) “The goodness of fit of regression formulae, and the distribution of regression coefficients,” Journal of the Royal Statistical Society, 85: 597-612.

Fisher, R. A. (1925) Statistical Methods for Research Workers, Oliver and Boyd, Edinburgh.

Fisher, R. A. (1929), “Moments and Product Moments of Sampling Distributions,” Proceedings of the London Mathematical Society, Series 2, 30: 199-238.

Neyman, J. and E. S. Pearson (1928) “On the use and interpretation of certain test criteria for purposes of statistical inference: Part I,” Biometrika, 20A: 175-240.

Neyman, J. and E. S. Pearson (1933) “On the problem of the most efficient tests of statistical hypotheses”, Philosophical Transanctions of the Royal Society, A, 231: 289-337.

Lehmann, E. L. (1975) Nonparametrics: statistical methods based on ranks, Holden-Day, San Francisco.

Lehmann, E. L. (1999) “‘Student’ and Small-Sample Theory,” Statistical Science, 14: 418-426.

Pearson, E. S. (1929a) “Review of ‘Statistical Methods for Research Workers,’ 1928, by Dr. R. A. Fisher”, Nature, June 8th, pp. 866-7.

Pearson, E. S. (1929b) “Some notes on sampling tests with two variables,” Biometrika, 21: 337-60.

Pearson, E. S. (1930) “A further development of tests for normality,” Biometrika, 22: 239-49.

Pearson, E. S. (1931) “The analysis of variance in cases of non-normal variation,” Biometrika, 23: 114-33.

Pearson, E. S. (1963) “Comparison of tests for randomness of points on a line,” Biometrika, 50: 315-25.

Pearson, E. S. and N. K. Adyanthaya (1928) “The distribution of frequency constants in small samples from symmetrical populations,” Biometrika, 20: 356-60.

Pearson, E. S. and N. K. Adyanthaya (1929) “The distribution of frequency constants in small samples from non-normal symmetrical and skew populations,” Biometrika, 21: 259-86.

Pearson, E. S. and N. W. Please (1975) “Relations between the shape of the population distribution and the robustness of four simple test statistics,” Biometrika, 62: 223-241.

Pearson, E. S., R. B. D’Agostino and K. O. Bowman (1977) “Tests for departure from normality: comparisons of powers,” Biometrika, 64: 231-246.

Spanos, A. (2010) “Akaike-type Criteria and the Reliability of Inference: Model Selection vs. Statistical Model Specification,” Journal of Econometrics, 158: 204-220.

Student (1908), “The Probable Error of the Mean,” Biometrika, 6: 1-25.

Categories: Egon Pearson, Statistics | Leave a comment

Behavioral vs Evidential Interpretations of N-P tests: E.S. Pearson’s Statistical Philosophy: Belated Birthday Wish

E.S. Pearson

This is a belated birthday post for E.S. Pearson (11 August 1895-12 June, 1980)–one of my statistical heroes. It’s basically a post from 2012 which concerns an issue of interpretation (long-run performance vs probativeness) that’s badly confused these days. Yes, I know I’ve been neglecting this blog as of late, because I’m busy planning our workshop: The Statistics Wars and Their Casualties (22-23 September, online). See I will reblog some favorite Pearson posts in the next few days.


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.) PearsonCases 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.

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

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

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

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

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

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

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

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

Three Steps in the Original Construction of Tests

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

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

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

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

Pearson warns that:

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

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

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

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

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

Neyman Was the More Behavioristic of the Two

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

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

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

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

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

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

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

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

These points on Pearson are discussed in more depth in my book Statistical Inference as Severe Testing (SIST): How to Get Beyond the Statistics Wars (CUP 2018). You can read and download the entire book for free during the month of August 2022 at the following link:



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

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

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

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

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

Categories: E.S. Pearson, Error Statistics | Leave a comment

Read It Free: “Stat Inference as Severe Testing: How to Get Beyond the Stat Wars” during August

CUP will make the electronic version of my book, Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (2018), available to access for free from August 1-31 at this link: However, they will confirm the link closer to August, so check this blog on Aug 1 for any update, if you’re interested. (July 31, the link works!) (August 5, the link is working. Let me know if you have problems getting in.) Blurbs of the 16 tours in the book may be found here: blurbs of the 16 tours.

Here’s a CUP interview from when the book first came out.

Categories: Announcement, SIST | Leave a comment

The Statistics Wars and Their Casualties Workshop-Now Online

The Statistics Wars
and Their Casualties 

22-23 September 2022
15:00-18:00 pm London Time*


To register/receive notification of updates for the workshop, please fill out the registration/notification form here.

*These will be sessions 1 & 2, there will be two more
The future online sessions (3 & 4) to be announced.

Yoav Benjamini (Tel Aviv University), Alexander Bird (University of Cambridge), Mark Burgman (Imperial College London),  Daniele Fanelli (London School of Economics and Political Science), Roman Frigg (London School of Economics and Political Science),
Stephan Guettinger
 (London School of Economics and Political Science), David Hand (Imperial College London), Margherita Harris (London School of Economics and Political Science), Christian Hennig (University of Bologna), Daniël Lakens (Eindhoven University of Technology), Deborah Mayo (Virginia Tech), Richard Morey (Cardiff University), Stephen Senn (Edinburgh, Scotland), Jon Williamson (University of Kent) Continue reading

Categories: Announcement, Error Statistics | Leave a comment

Happy Birthday Sir David R. Cox

15 July 1924-18 January 2022

HAPPY BIRTHDAY SIR DAVID COX!  Today is David Cox’s birthday, he would have been 98 years old today. Below is a remembrance I contributed to Significance when he died, with a link to others in that same issue.

“In celebrating Cox’s immense contributions, we should recognise how much there is yet to learn from him” Continue reading

Categories: Sir David Cox | 1 Comment

10 years after the July 4 statistical discovery of the the Higgs & the value of negative results


Today marks a decade since the discovery on July 4, 2012 of evidence for a Higgs particle based on a “5 sigma observed effect”. CERN celebrated with a scientific symposium (webcast here). The observed effect refers to the number of excess events of a given type that are “observed” in comparison to the number that would be expected from background alone—which they can simulate in particle detectors. Because the 5-sigma standard refers to a benchmark from frequentist significance testing, the discovery was immediately imbued with controversies that, at bottom, concerned statistical philosophy. Continue reading

Categories: Error Statistics | 2 Comments

D. Lakens responds to confidence interval crusading journal editors


In what began as a guest commentary on my 2021 editorial in Conservation Biology, Daniël Lakens recently published a response to a recommendation against using null hypothesis significance tests by journal editors from the International Society of Physiotherapy Journal. Here are some excerpts from his full article, replies (‘response to Lakens‘), links and a few comments of my own. Continue reading

Categories: stat wars and their casualties, statistical significance tests | 12 Comments



Continue reading

Categories: Error Statistics | 1 Comment

Too little, too late? The “Don’t say significance…” editorial gets a disclaimer (ii)


Someone sent me an email the other day telling me that a disclaimer had been added to the editorial written by the ASA Executive Director and 2 co-authors (Wasserstein et al., 2019) (“Moving to a world beyond ‘p < 0.05′”). It reads:


The editorial was written by the three editors acting as individuals and reflects their scientific views not an an endorsed position of the American Statistical Association.

Continue reading

Categories: ASA Guide to P-values, ASA Task Force on Significance and Replicability, editorial COIs, WSL 2019 | 19 Comments

Philosophy of socially aware data science conference

I’ll be speaking at this conference in Philly tomorrow. My slides are also below.


PDF of my slides: Statistical “Reforms”: Fixing Science or Threats to Replication and Falsification. Continue reading

Categories: Announcement, Philosophy of Statistics, socially aware data science | Leave a comment

D. Mayo & D. Hand: “Statistical significance and its critics: practicing damaging science, or damaging scientific practice?”



Prof. Deborah Mayo, Emerita
Department of Philosophy
Virginia Tech


Prof. David Hand
Department of Mathematics Statistics
Imperial College London

Statistical significance and its critics: practicing damaging science, or damaging scientific practice?  (Synthese)

[pdf of full paper.] Continue reading

Categories: Error Statistics | 3 Comments

Paul Daniell & Yu-li Ko commentaries on Mayo’s ConBio Editorial

I had been posting commentaries daily from January 6, 2022 (on my editorial “The Statistics Wars and Intellectual conflicts of Interest”, Conservation Biology) until Sir David Cox died on January 18, at which point I switched to some memorial items. These two commentaries from what Daniell calls my ‘birthday festschrift’ were left out, and I put them up now. (Links to others are below.)

Continue reading

Categories: Mayo editorial, stat wars and their casualties | 1 Comment

3 Commentaries on my Editorial are being published in Conservation Biology



There are 3 commentaries soon to be published in Conservation Biology on my editorial, “The statistics wars and intellectual conflicts of interest” also published in Conservation Biology. Continue reading

Categories: Mayo editorial, significance tests | Tags: , , , , | Leave a comment

A statistically significant result indicates H’ (μ > μ’) when POW(μ’) is low (not the other way round)–but don’t ignore the standard error


1. New monsters. One of the bizarre facts of life in the statistics wars is that a method from one school may be criticized on grounds that it conflicts with a conception that is the reverse of what that school intends. How is that even to be deciphered? That was the difficult task I set for myself in writing Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (CUP, 2008) [SIST 2018]. I thought I was done, but new monsters keep appearing. In some cases, rather than see how the notion of severity gets us beyond fallacies, misconstruals are taken to criticize severity! So, for example, in the last couple of posts, here and here, I deciphered some of the better known power howlers (discussed in SIST Ex 5 Tour II) I’m linking to all of this tour (in proofs). Continue reading

Categories: power, reforming the reformers, SIST, Statistical Inference as Severe Testing | 16 Comments

Do “underpowered” tests “exaggerate” population effects? (iv)


You will often hear that if you reach a just statistically significant result “and the discovery study is underpowered, the observed effects are expected to be inflated” (Ioannidis 2008, p. 64), or “exaggerated” (Gelman and Carlin 2014). This connects to what I’m referring to as the second set of concerns about statistical significance tests, power and magnitude errors. Here, the problem does not revolve around erroneously interpreting power as a posterior probability, as we saw in the fallacy in this post. But there are other points of conflict with the error statistical tester, and much that cries out for clarification — else you will misunderstand the consequences of some of today’s reforms.. Continue reading

Categories: power, reforming the reformers, SIST, Statistical Inference as Severe Testing | 16 Comments

Join me in reforming the “reformers” of statistical significance tests


The most surprising discovery about today’s statistics wars is that some who set out shingles as “statistical reformers” themselves are guilty of misdefining some of the basic concepts of error statistical tests—notably power. (See my recent post on power howlers.) A major purpose of my Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (2018, CUP) is to clarify basic notions to get beyond what I call “chestnuts” and “howlers” of tests. The only way that disputing tribes can get beyond the statistics wars is by (at least) understanding correctly the central concepts. But these misunderstandings are more common than ever, so I’m asking readers to help. Why are they more common (than before the “new reformers” of the last decade)? I suspect that at least one reason is the popularity of Bayesian variants on tests: if one is looking to find posterior probabilities of hypotheses, then error statistical ingredients may tend to look as if that’s what they supply.  Continue reading

Categories: power, SIST, statistical significance tests | Tags: , , | 2 Comments

Happy Birthday Neyman: What was Neyman opposing when he opposed the ‘Inferential’ Probabilists? Your weekend Phil Stat reading


Today is Jerzy Neyman’s birthday (April 16, 1894 – August 5, 1981). I’m reposting a link to a quirky, but fascinating, paper of his that explains one of the most misunderstood of his positions–what he was opposed to in opposing the “inferential theory”. The paper, fro 60 years ago,Neyman, J. (1962), ‘Two Breakthroughs in the Theory of Statistical Decision Making‘ [i] It’s chock full of ideas and arguments. “In the present paper” he tells us, “the term ‘inferential theory’…will be used to describe the attempts to solve the Bayes’ problem with a reference to confidence, beliefs, etc., through some supplementation …either a substitute a priori distribution [exemplified by the so called principle of insufficient reason] or a new measure of uncertainty” such as Fisher’s fiducial probability. It arises on p. 391 of Excursion 5 Tour III of Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (2018, CUP). Here’s a link to the proofs of that entire tour. If you hear Neyman rejecting “inferential accounts,” you have to understand it in this very specific way: he’s rejecting “new measures of confidence or diffidence”. Here he alludes to them as “easy ways out”. He is not rejecting statistical inference in favor of behavioral performance as is typically thought. It’s amazing how an idiosyncratic use of a word 60 years ago can cause major rumblings decades later. Neyman always distinguished his error statistical performance conception from Bayesian and Fiducial probabilisms [ii]. The surprising twist here is semantical and the culprit is none other than…Allan Birnbaum. Yet Birnbaum gets short shrift, and no mention is made of our favorite “breakthrough” (or did I miss it?). You can find quite a lot on this blog searching Birnbaum. Continue reading

Categories: Bayesian/frequentist, Neyman | Leave a comment

Power howlers return as criticisms of severity

Mayo bangs head

Suppose you are reading about a statistically significant result x that just reaches a threshold p-value α from a test T+ of the mean of a Normal distribution

 H0: µ ≤  0 against H1: µ >  0

with n iid samples, and (for simplicity) known σ.  The test “rejects” H0 at this level & infers evidence of a discrepancy in the direction of H1.

I have heard some people say:

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

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

I have heard other people say:

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

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

Which is correct, from the perspective of the frequentist error statistical philosophy? Continue reading

Categories: Statistical power, statistical tests | Tags: , , , , | 7 Comments

Insevere Tests of Severe Testing (iv)


One does not have evidence for a claim if little if anything has been done to rule out ways the claim may be false. The claim may be said to “pass” the test, but it’s one that utterly lacks stringency or severity. On the basis of this very simple principle, I build a notion of evidence that applies to any error prone inference. In this account, data x are evidence for a claim C only if (and only to the extent that) C has passed a severe test with x.[1] How to apply this simple idea, however, and how to use it to solve central problems of induction and statistical inference requires careful consideration of how it is to be fleshed out. (See this post on strong vs weak severity.) Continue reading

Categories: Error Statistics | 2 Comments

Blog at