Monthly Archives: August 2017

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

11 August 1895 – 12 June 1980

Continuing with my Egon Pearson posts in honor of his birthday, I reblog a post by Aris Spanos:  Egon Pearson’s Neglected Contributions to Statistics“. 

    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 genius 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).

References

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: E.S. Pearson, phil/history of stat, Spanos, Testing Assumptions | 2 Comments

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

egon pearson

E.S. Pearson (11 Aug, 1895-12 June, 1980)

This is a belated birthday post for E.S. Pearson (11 August 1895-12 June, 1980). It’s basically a post from 2012 which concerns an issue of interpretation (long-run performance vs probativeness) that’s badly confused these days. I’ll blog some E. Pearson items this week, including, my latest reflection on a historical anecdote regarding Egon and the woman he wanted marry, and surely would have, were it not for his father Karl!

HAPPY BELATED BIRTHDAY EGON!

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. 

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.

“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)

__________________________

 

References:

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: highly probable vs highly probed, phil/history of stat, Statistics | Tags: | Leave a comment

Thieme on the theme of lowering p-value thresholds (for Slate)

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Here’s an article by Nick Thieme on the same theme as my last blogpost. Thieme, who is Slate’s 2017 AAAS Mass Media Fellow, is the first person to interview me on p-values who (a) was prepared to think through the issue for himself (or herself), and (b) included more than a tiny fragment of my side of the exchange.[i]. Please share your comments.

Will Lowering P-Value Thresholds Help Fix Science? P-values are already all over the map, and they’re also not exactly the problem.

 

 

Illustration by Slate

                 Illustration by Slate

Last week a team of 72 scientists released the preprint of an article attempting to address one aspect of the reproducibility crisis, the crisis of conscience in which scientists are increasingly skeptical about the rigor of our current methods of conducting scientific research.

Their suggestion? Change the threshold for what is considered statistically significant. The team, led by Daniel Benjamin, a behavioral economist from the University of Southern California, is advocating that the “probability value” (p-value) threshold for statistical significance be lowered from the current standard of 0.05 to a much stricter threshold of 0.005.

P-values are tricky business, but here’s the basics on how they work: Let’s say I’m conducting a drug trial, and I want to know if people who take drug A are more likely to go deaf than if they take drug B. I’ll state that my hypothesis is “drugs A and B are equally likely to make someone go deaf,” administer the drugs, and collect the data. The data will show me the number of people who went deaf on drugs A and B, and the p-value will give me an indication of how likely it is that the difference in deafness was due to random chance rather than the drugs. If the p-value is lower than 0.05, it means that the chance this happened randomly is very small—it’s a 5 percent chance of happening, meaning it would only occur 1 out of 20 times if there wasn’t a difference between the drugs. If the threshold is lowered to 0.005 for something to be considered significant, it would mean that the chances of it happening without a meaningful difference between the treatments would be just 1 in 200.

On its face, this doesn’t seem like a bad idea. If this change requires scientists to have more robust evidence before they can come to conclusions, it’s easy to think it’s a step in the right direction. But one of the issues at the heart of making this change is that it seems to assume there’s currently a consensus around how p-value ought to be used and this consensus could just be tweaked to be stronger.

P-value use already varies by scientific field and by journal policies within those fields. Several journals in epidemiology, where the stakes of bad science are perhaps higher than in, say, psychology (if they mess up, people die), have discouraged the use of p-values for years. And even psychology journals are following suit: In 2015, Basic and Applied Social Psychology, a journal that has been accused of bad statistical (and experimental) practice, banned the use of p-values. Many other journals, including PLOS Medicine and Journal of Allergy and Clinical Immunology, actively discourage the use of p-values and significance testing already.

On the other hand, the New England Journal of Medicine, one of the most respected journals in that field, codes the 0.05 threshold for significance into its author guidelines, saying “significant differences between or among groups (i.e P<.05) should be identified in a table.” That may not be an explicit instruction to treat p-values less than 0.05 as significant, but an author could be forgiven for reading it that way. Other journals, like the Journal of Neuroscience and the Journal of Urology, do the same.

Another group of journals—including Science, Nature, and Cell—avoid giving specific advice on exactly how to use p-values; rather, they caution against common mistakes and emphasize the importance of scientific assumptions, trusting the authors to respect the nuance of any statistics tools. Deborah Mayo, award-wining philosopher of statistics and professor at Virginia Tech, thinks this approach to statistical significance, where various fields have different standards, is the most appropriate. Strict cutoffs, regardless of where they fall, are generally bad science.

Mayo was skeptical that it would have the kind of widespread benefit the authors assumed. Their assessment suggested tightening the threshold would reduce the rate of false positives—results that look true but aren’t—by a factor of two. But she questioned the assumption they had used to assess the reduction of false positives—that only 1 in 10 hypotheses a scientist tests is true. (Mayo said that if that were true, perhaps researchers should spend more time on their hypotheses.)

But more broadly, she was skeptical of the idea that lowering the informal p-value threshold will help fix the problem, because she’s doubtful such a move will address “what almost everyone knows is the real cause of nonreproducibility”: the cherry-picking of subjects, testing hypothesis after hypothesis until one of them is proven correct, and selective reporting of results and methodology.

There are plenty of other ways that scientists are testing to help address the replication crisis. There’s the move toward pre-registration of studies before analyzing data, in order to avoid fishing for significance. Researchers are also now encouraged to make data and code public so a third party can rerun analyses efficiently and check for discrepancies. More negative results are being published. And, perhaps most importantly, researchers are actually conducting studies to replicate research that has already been published. Tightening standards around p-values might help, but the debate about reproducibility is more than just a referendum on the p-value. The solution will need to be more than that as well.

 ___

 [i] We did not discuss that recent test ban(“Don’t ask don’t tell”).  If we had, I might have pointed him to my post on “P-value madness”. 

Link to Nick Thieme’s Slate article:Will Lowering P-Value Thresholds Help Fix Science? P-values are already all over the map, and they’re also not exactly the problem.”

Categories: P-values, reforming the reformers, spurious p values | 14 Comments

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