On Significance (2000-2009)

"When significance tests are used and a null hypothesis is not rejected, a major problem often arises - namely, the result may be interpreted, without a logical basis, as providing evidence for the null hypothesis." (David F Parkhurst, "Statistical Significance Tests: Equivalence and Reverse Tests Should Reduce Misinterpretation", BioScience Vol. 51 (12), 2001)

"If you flip a coin three times and it lands on heads each time, it's probably chance. If you flip it a hundred times and it lands on heads each time, you can be pretty sure the coin has heads on both sides. That's the concept behind statistical significance - it's the odds that the correlation (or other finding) is real, that it isn't just random chance." (T Colin Campbell, "The China Study", 2004)

"Many statistics texts do not mention this and students often ask, ‘What if you get a probability of exactly 0.05?’ Here the result would be considered not significant, since significance has been defined as a probability of less than 0.05 (<0.05). Some texts define a significant result as one where the probability is less than or equal to 0.05 ( 0.05). In practice this will make very little difference, but since Fisher proposed the ‘less than 0.05’ definition, which is also used by most scientific publications, it will be used here." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"The dual meaning of the word significant brings into focus the distinction between drawing a mathematical inference and practical inference from statistical results." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"A common statistical error is to summarize comparisons by statistical significance and then draw a sharp distinction between significant and nonsignificant results. The approach of summarizing by statistical significance has a number of pitfalls, most of which are covered in standard statistics courses but one that we believe is less well known. We refer to the fact that changes in statistical significance are not themselves significant. A small change in a group mean, a regression coefficient, or any other statistical quantity can be neither statistically significant nor practically important, but such a change can lead to a large change in the significance level of that quantity relative to a null hypothesis." (Andrew Gelman & Hal Stern, "The Difference between 'Significant' and 'Not Significant' Is Not Itself Statistically Significant", The American Statistician Vol. 60 (4), 2006

"A type of error used in hypothesis testing that arises when incorrectly rejecting the null hypothesis, although it is actually true. Thus, based on the test statistic, the final conclusion rejects the Null hypothesis, but in truth it should be accepted. Type I error equates to the alpha (α) or significance level, whereby the generally accepted default is 5%." (Lynne Hambleton, "Treasure Chest of Six Sigma Growth Methods, Tools, and Best Practices", 2007)

"For the study of the topology of the interactions of a complex system it is of central importance to have proper random null models of networks, i.e., models of how a graph arises from a random process. Such models are needed for comparison with real world data. When analyzing the structure of real world networks, the null hypothesis shall always be that the link structure is due to chance alone. This null hypothesis may only be rejected if the link structure found differs significantly from an expectation value obtained from a random model. Any deviation from the random null model must be explained by non-random processes." (Jörg Reichardt, "Structure in Complex Networks", 2009)

On Significance (1950-1974)

"In the examples we have given [...] our judgment whether P was small enough to justify us in suspecting a significant difference [...] has been more or less intuitive. Most people would agree [...] that a probability of .0001 is so small that the evidence is very much in favour. . . . Suppose we had obtained P = 0.1. [...] Where, if anywhere, can we draw the line? The odds against the observed event which influence a decision one way or the other depend to some extent on the caution of the investigator. Some people (not necessarily statisticians) would regard odds of ten to one as sufficient. Others would be more conservative and reserve judgment until the odds were much greater. It is a matter of personal taste." (G U Yule & M G Kendall, "An introduction to the theoryof statistics" 14th ed., 1950)

"It will, of course, happen but rarely that the proportions will be identical, even if no real association exists. Evidently, therefore, we need a significance test to reassure ourselves that the observed difference of proportion is greater than could reasonably be attributed to chance. The significance test will test the reality of the association, without telling us anything about the intensity of association. It will be apparent that we need two distinct things: (a) a test of significance, to be used on the data first of all, and (b) some measure of the intensity of the association, which we shall only be justified in using if the significance test confirms that the association is real." (Michael J Moroney, "Facts from Figures", 1951)

"The main purpose of a significance test is to inhibit the natural enthusiasm of the investigator." (Frederick Mosteller, "Selected Quantitative Techniques", 1954)

"Null hypotheses of no difference are usually known to be false before the data are collected [...] when they are, their rejection or acceptance simply reflects the size of the sample and the power of the test, and is not a contribution to science." (I Richard Savage, "Nonparametric Statistics", Journal of the American Statistical Association 52, 1957)

"[...] to make measurements and then ignore their magnitude would ordinarily be pointless. Exclusive reliance on tests of significance obscures the fact that statistical significance does not imply substantive significance." (I Richard Savage, "Nonparametric Statistics", Journal of the American Statistical Association 52, 1957)

"[...] the tests of null hypotheses of zero differences, of no relationships, are frequently weak, perhaps trivial statements of the researcher’s aims [...] in many cases, instead of the tests of significance it would be more to the point to measure the magnitudes of the relationships, attaching proper statements of their sampling variation. The magnitudes of relationships cannot be measured in terms of levels of significance." (Leslie Kish, "Some statistical problems in research design", American Sociological Review 24, 1959)

"There are instances of research results presented in terms of probability values of ‘statistical significance’ alone, without noting the magnitude and importance of the relationships found. These attempts to use the probability levels of significance tests as measures of the strengths of relationships are very common and very mistaken." (Leslie Kish, "Some statistical problems in research design", American Sociological Review 24, 1959)

"The null-hypothesis significance test treats ‘acceptance’ or ‘rejection’ of a hypothesis as though these were decisions one makes. But a hypothesis is not something, like a piece of pie offered for dessert, which can be accepted or rejected by a voluntary physical action. Acceptance or rejection of a hypothesis is a cognitive process, a degree of believing or disbelieving which, if rational, is not a matter of choice but determined solely by how likely it is, given the evidence, that the hypothesis is true." (William W Rozeboom, "The fallacy of the null–hypothesis significance test", Psychological Bulletin 57, 1960)

"The null hypothesis of no difference has been judged to be no longer a sound or fruitful basis for statistical investigation. […] Significance tests do not provide the information that scientists need, and, furthermore, they are not the most effective method for analyzing and summarizing data." (Cherry A Clark, "Hypothesis Testing in Relation to Statistical Methodology", Review of Educational Research Vol. 33, 1963)

"[...] the test of significance has been carrying too much of the burden of scientific inference. It may well be the case that wise and ingenious investigators can find their way to reasonable conclusions from data because and in spite of their procedures. Too often, however, even wise and ingenious investigators [...] tend to credit the test of significance with properties it does not have." (David Bakan, "The test of significance in psychological research", Psychological Bulletin 66, 1966)

"[...] we need to get on with the business of generating [...] hypotheses and proceed to do investigations and make inferences which bear on them, instead of [...] testing the statistical null hypothesis in any number of contexts in which we have every reason to suppose that it is false in the first place." (David Bakan, "The test of significance in psychological research", Psychological Bulletin 66, 1966)

"Significance levels are usually computed and reported, but power and confidence limits are not. Perhaps they should be." (Amos Tversky & Daniel Kahneman, "Belief in the law of small numbers", Psychological Bulletin 76(2), 1971)

"The emphasis on significance levels tends to obscure a fundamental distinction between the size of an effect and its statistical significance." (Amos Tversky & Daniel Kahneman, "Belief in the law of small numbers", Psychological Bulletin 76(2), 1971)

"[...] too many users of the analysis of variance seem to regard the reaching of a mediocre level of significance as more important than any descriptive specification of the underlying averages. Our thesis is that people have strong intuitions about random sampling; that these intuitions are wrong in fundamental respects; that these intuitions are shared by naive subjects and by trained scientists; and that they are applied with unfortunate consequences in the course of scientific inquiry. We submit that people view a sample randomly drawn from a population as highly representative, that is, similar to the population in all essential characteristics. Consequently, they expect any two samples drawn from a particular population to be more similar to one another and to the population than sampling theory predicts, at least for small samples." (Amos Tversky & Daniel Kahneman, "Belief in the law of small numbers", Psychological Bulletin 76(2), 1971)

On Significance (-1949)

"It is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not. Deviations exceeding twice the standard deviation are thus formally regarded as significant." (Ronald A Fisher, "Statistical Methods for Research Workers", 1925)

"If one in twenty does not seem high enough odds, we may, if we prefer it, draw the line at one in fifty (the 2 per cent point), or one in a hundred (the 1 per centp oint). Personally, the writer prefers to set a low standard of significance at the 5 per cent point, and ignore entirely all results which fail to reach this level. A scientific fact should be regarded as experimentally established only if a properly designed experiment rarely fails to give this level of significance," (Ronald A Fisher, 1926)

"An observation is judged significant, if it would rarely have been produced, in the absence of a real cause of the kind we are seeking. It is a common practice to judge a result significant, if it is of such a magnitude that it would have been produced by chance not more frequently than once in twenty trials. This is an arbitrary, but convenient, level of significance for the practical investigator, but it does not mean that he allows himself to be deceived once in every twenty experiments. The test of significance only tells him what to ignore, namely all experiments in which significant results are not obtained. He should only claim that a phenomenon is experimentally demonstrable when he knows how to design an experiment so that it will rarely fail to give a significant result. Consequently, isolated significant results which he does not know how to reproduce are left in suspense pending further investigation." (Ronald A Fisher, "The Statistical Method in Psychical Research", Proceedings of the Society for Psychical Research 39, 1929)

"What the use of P [the significance level] implies, therefore, is that a hypothesis that may be true may be rejected because it has not predicted observable results that have not occurred." (Harold Jeffreys, "Theory of Probability", 1939)

"As usual we may make the errors of I) rejecting the null hypothesis when it is true, II) accepting the null hypothesis when it is false. But there is a third kind of error which is of interest because the present test of significance is tied up closely with the idea of making a correct decision about which distribution function has slipped furthest to the right. We may make the error of III) correctly rejecting the null hypothesis for the wrong reason." (Frederick Mosteller, "A k-Sample Slippage Test for an Extreme Population", The Annals of Mathematical Statistics 19, 1948)

"Errors of the third kind happen in conventional tests of differences of means, but they are usually not considered, although their existence is probably recognized. It seems to the author that there may be several reasons for this among which are 1) a preoccupation on the part of mathematical statisticians with the formal questions of acceptance and rejection of null hypotheses without adequate consideration of the implications of the error of the third kind for the practical experimenter, 2) the rarity with which an error of the third kind arises in the usual tests of significance." (Frederick Mosteller, "A k-Sample Slippage Test for an Extreme Population", The Annals of Mathematical Statistics 19, 1948)

"If significance tests are required for still larger samples, graphical accuracy is insufficient, and arithmetical methods are advised. A word to the wise is in order here, however. Almost never does it make sense to use exact binomial significance tests on such data - for the inevitable small deviations from the mathematical model of independence and constant split have piled up to such an extent that the binomial variability is deeply buried and unnoticeable. Graphical treatment of such large samples may still be worthwhile because it brings the results more vividly to the eye." (Frederick Mosteller & John W Tukey, "The Uses and Usefulness of Binomial Probability Paper?", Journal of the American Statistical Association 44, 1949)