{"id":47531,"date":"2020-05-15T15:19:28","date_gmt":"2020-05-15T14:19:28","guid":{"rendered":"https:\/\/blogs.bmj.com\/bmj\/?p=47531"},"modified":"2020-05-25T18:02:04","modified_gmt":"2020-05-25T17:02:04","slug":"jeffrey-aronson-when-i-use-a-word-testing-hypotheses-in-testing-times","status":"publish","type":"post","link":"https:\/\/blogs.bmj.com\/bmj\/2020\/05\/15\/jeffrey-aronson-when-i-use-a-word-testing-hypotheses-in-testing-times\/","title":{"rendered":"Jeffrey Aronson: When I Use a Word . . . testing hypotheses in testing times"},"content":{"rendered":"<p><span style=\"font-weight: 400\">All hypotheses are unequal; some are more unequal than others.<\/span><\/p>\n<p><span style=\"font-weight: 400\">I have <\/span><a href=\"http:\/\/blogs.bmj.com\/bmj\/2016\/01\/29\/jeffrey-aronson-when-i-use-a-word-apothecaries\" target=\"_blank\" rel=\"noopener noreferrer\"><span style=\"font-weight: 400\">previously<\/span><\/a><span style=\"font-weight: 400\"> discussed the Indo-European root <\/span><span style=\"font-weight: 400\">DHE<\/span><span style=\"font-weight: 400\">. The basic root, the so-called e-grade form, <\/span><span style=\"font-weight: 400\">DHE, <\/span><span style=\"font-weight: 400\">means to set or put down, to make or shape. But vowels change readily, by the mechanism that <\/span><a href=\"http:\/\/blogs.bmj.com\/bmj\/2015\/05\/01\/jeffrey-aronson-when-i-use-a-word-grimms-law\" target=\"_blank\" rel=\"noopener noreferrer\"><span style=\"font-weight: 400\">Jakob Grimm<\/span><\/a><span style=\"font-weight: 400\"> called ablaut (literally an off-sound). So there is also an o-grade form, <\/span><span style=\"font-weight: 400\">DHO, and a <\/span><span style=\"font-weight: 400\">zero-grade form, <\/span><span style=\"font-weight: 400\">DH\u018f<\/span><span style=\"font-weight: 400\">, in which the final vowel is a neutral vowel sound called a schwa, after the Hebrew vowel of that name. The schwa, represented by an inverted e (lower-case \u0259 or upper-case <\/span><span style=\"font-weight: 400\">\u018f<\/span><span style=\"font-weight: 400\">) typically occurs in weakly stressed syllables, like the <\/span><i><span style=\"font-weight: 400\">e<\/span><\/i><span style=\"font-weight: 400\"> in \u201chypothesis\u201d (\/h\u028c\u026a\u02c8p\u0252\u03b8\u0259s\u1d7bs\/). These various forms can also have prefixes and suffixes and may be reduplicated. This gives rise to a myriad of words, examples of which follow.<\/span><\/p>\n<p><span style=\"font-weight: 400\">DHE <\/span><span style=\"font-weight: 400\">gives deed, misdeed, and the obsolete alms-deed; as Queen Margaret says of Richard Duke of York in Henry VI, \u201cMurder is thy alms-deed\u2014Petitioners for blood thou ne\u2019er putt\u2019st back\u201d. <\/span><span style=\"font-weight: 400\">DHO<\/span><span style=\"font-weight: 400\"> gives us do, doing, and done. <\/span><span style=\"font-weight: 400\">DHEM <\/span><span style=\"font-weight: 400\">gives deem and theme, and <\/span><span style=\"font-weight: 400\">DHOM <\/span><span style=\"font-weight: 400\">gives doom and words ending -dom, like leechdom. Abdomen is conjecturally from the Latin abdere (ab + dare), to put away or conceal. <\/span><span style=\"font-weight: 400\">DHEK <\/span><span style=\"font-weight: 400\">gives theca (Greek<\/span> <span style=\"font-weight: 400\">\u03b8\u03ae\u03ba\u03b7), a sheath that encloses an organ, and apothecary, from the Greek \u1f00\u03c0\u03bf\u03c4\u03b9\u03b8\u03ad\u03bd\u03b1\u03b9 to put away and hence \u1f00\u03c0\u03bf\u03b8\u03ae\u03ba\u03b7 a storehouse. And reduplication,<\/span><span style=\"font-weight: 400\"> DHE-DH\u018f, <\/span><span style=\"font-weight: 400\">gives the Greek verb \u03c4\u03b9\u03b8\u03ad\u03bd\u03b1\u03b9, to put or place, which yields the trilogy of dialectical materialism, thesis, antithesis, and synthesis.<\/span><\/p>\n<p><span style=\"font-weight: 400\">\u201cTheory\u201d is thought to come from two Greek words meaning to see, \u03b8\u03b5\u1fb6\u03c3\u03b8\u03b1\u03b9 and \u03cc\u03c1\u03ac\u03b5\u03b9\u03bd. However, I venture to suggest that it may have come from \u03c4\u03b9\u03b8\u03ad\u03bd\u03b1\u03b9 + \u03cc\u03c1\u03ac\u03b5\u03b9\u03bd, to place in view.<\/span><\/p>\n<p><span style=\"font-weight: 400\">So a hypothesis, literally \u201cplaced underneath\u201d (\u1f51\u03c0\u03cc), is a foundation, and hence the basis of an argument or a supposition. When it first entered English in the late 16<\/span><span style=\"font-weight: 400\">th<\/span><span style=\"font-weight: 400\"> century it meant a particular case of a general proposition, and later the proposition itself. In logic it meant a supposition or condition forming the antecedent of a proposition. When Sir Thomas Browne used the word in his <\/span><i><span style=\"font-weight: 400\">Pseudodoxia Epidemica<\/span><\/i><span style=\"font-weight: 400\"> in 1646, he meant, invoking the definition in the <\/span><i><span style=\"font-weight: 400\">Oxford English Dictionary<\/span><\/i><span style=\"font-weight: 400\">, \u201cA provisional supposition from which to draw conclusions that shall be in accordance with known facts, and which serves as a starting-point for further investigation by which it may be proved or disproved and the true theory arrived at.\u201d<\/span><\/p>\n<p><span style=\"font-weight: 400\">This definition, unrevised since 1899, needs some attention. Hypotheses are indeed typically based on known facts and serve as starting points for further investigation, but only as such. It is easy to formulate explanations of phenomena once they have been observed, but wrong to accept them as gospel without further testing. However, to assert, as the <\/span><i><span style=\"font-weight: 400\">OED<\/span><\/i><span style=\"font-weight: 400\">\u2019s definition does, that by such testing a hypothesis can thereby be proved is misleading. As Karl Popper taught us in <\/span><i><span style=\"font-weight: 400\">The Logic of Scientific Discovery<\/span><\/i><span style=\"font-weight: 400\"> (1959), originally published in German in 1934 as <\/span><i><span style=\"font-weight: 400\">Logik der Forschung<\/span><\/i><span style=\"font-weight: 400\">, the proper way to test a scientific hypothesis is to try, not to prove it, but to disprove it, to falsify it, as Popper put it. Here is what he wrote:\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400\">\u201cA scientist constructs hypotheses, or systems of theories, and tests them against experience by observation and experiment.\u201d Denying the validity of induction, he suggests that \u201cinference to theories, from singular statements which are \u2018verified by experience\u2019 \u2026 is logically inadmissible.\u201d He then goes on:<\/span><\/p>\n<p><a href=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/Karl_Popper.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-47532 \" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/Karl_Popper.png\" alt=\"\" width=\"799\" height=\"358\" srcset=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/Karl_Popper.png 1169w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/Karl_Popper-300x134.png 300w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/Karl_Popper-1024x459.png 1024w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/Karl_Popper-768x344.png 768w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/Karl_Popper-640x287.png 640w\" sizes=\"auto, (max-width: 799px) 100vw, 799px\" \/><\/a><\/p>\n<p><span style=\"font-weight: 400\">But today Popper\u2019s principle is being abandoned. During the pandemic, researchers, presumably supported by the ethics committees that scrutinize their proposed studies, have decided that masked, randomized, placebo-controlled trials are not necessary in testing treatments for a serious infection, with few grounds for assuming that they will be efficacious, or, even if efficacious, that the balance of benefit and harm will be favourable.<\/span><\/p>\n<p><span style=\"font-weight: 400\">In Greek<\/span> <span style=\"font-weight: 400\">\u1f51\u03c0\u03cc\u03b8\u03b5\u03c3\u03b9\u03c2 meant, among several things, a proposal or a proposed action, a suggestion or advice, an excuse or pretext, a practical problem or a supposition. Aristotle used it to mean the assumption of the existence of some scientific object. Today this meaning seems to be being espoused once more. Instead of testing their hypotheses, instead of trying to falsify them, researchers are trying to prove them. Next week I shall discuss how we should be grading therapeutic hypotheses and testing them in different ways.<\/span><\/p>\n<p><em><strong>Jeffrey Aronson<\/strong>\u00a0is a clinical pharmacologist, working in the Centre for Evidence Based Medicine in Oxford\u2019s Nuffield Department of Primary Care Health Sciences. He is also president emeritus of the British Pharmacological Society.<\/em><\/p>\n<p><em><strong>Competing interests:<\/strong> none declared.<\/em><\/p>\n<p>&nbsp;<\/p>\n<table>\n<tbody>\n<tr style=\"border: 1px solid black\">\n<td style=\"border: 1px solid black;text-align: center\">\n<p style=\"text-align: center\"><b>This week&#8217;s interesting integer: 274<\/b><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"text-align: left\"><span style=\"font-weight: 400\">274 is a Tribonacci number. The series of Fibonacci numbers, first described by Leonardo of Pisa (Fibonacci) begins with 0, 1 and continues by adding the two previous numbers: 0, 1, 1, 2, 3, 5, 8, &#8230; The series of Tribonacci numbers starts with 0, 0, 1 and continues by adding the three previous numbers: 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, &#8230;<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"text-align: left\"><span style=\"font-weight: 400\">274 is a semiprime, i.e. the product of two primes, 2 \u00d7 137. It is also an interprime, the average of two consecutive primes, 271 and 277.The sum of its prime factors (2 + 137 = 139) is also prime.<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"text-align: left\"><span style=\"font-weight: 400\">274 is a Smith number, so called when Albert Wilansky noticed that the phone number (493-7775) of his brother-in-law Harold Smith had an interesting property: the sum of its digits is the same as the sum of the digits of its prime factors (3, 5, 5, and 65837)<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400\">274 = 2 \u00d7 137 and 2 + 7 + 4 = 2 + 1 + 3 + 7 = 13<\/span><\/p>\n<p><span style=\"font-weight: 400\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Smith numbers are also called joke numbers. Numbers of this type that do not have repeated prime factors are also hoax numbers; 274 is the eleventh Smith number and the thirteenth hoax number.<\/span><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"text-align: left\"><span style=\"font-weight: 400\">274 is the sum of three non-zero squares in three ways:\u00a0<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400\">274 = 3<sup>2<\/sup> + 3<sup>2<\/sup> + 16<sup>2<\/sup> = 3<sup>2<\/sup> + 11<sup>2<\/sup> + 12<sup>2<\/sup> = 7<sup>2<\/sup> + 9<sup>2<\/sup> + 12<sup>2<\/sup><\/span><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li><span style=\"font-weight: 400\">274 is the sum of four squares, a<sup>2<\/sup>, b<sup>2<\/sup>, c<sup>2<\/sup>, and d<sup>2<\/sup>, such that a<sup>2<\/sup>+ b<sup>2<\/sup>, c<sup>2<\/sup> + d<sup>2<\/sup>, a<sup>2<\/sup> + d<sup>2<\/sup>, and b<sup>2<\/sup> + c<sup>2<\/sup> are all prime: 274 = 13<sup>2<\/sup> + 10<sup>2<\/sup> + 2<sup>2<\/sup> + 1<sup>2<\/sup><\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400\">13<sup>2<\/sup> + 10<sup>2<\/sup> = 269; 2<sup>2<\/sup> + 1<sup>2<\/sup> = 5; 13<sup>2<\/sup> + 2<sup>2<\/sup> =173; 10<sup>2<\/sup> + 1<sup>2<\/sup><span style=\"font-weight: 400\"> = 101<\/span><\/span><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"text-align: left\"><span style=\"font-weight: 400\">The square of 274 is the sum of 24 consecutive squares:<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400\">274<sup>2<\/sup> = 44<sup>2<\/sup> + 45<sup>2<\/span><span style=\"font-weight: 400\"> + 46<sup>2<\/sup> + \u2026\u202665<sup>2<\/sup> + 66<sup>2<\/sup> + 67<sup>2<\/sup><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"text-align: left\"><span style=\"font-weight: 400\">The proper divisors of a number are all the numbers that divide it apart from the number itself. 274 is the sum of the squares of 1, 2, 3, 4, 6, 8, and 12, which are the proper divisors of 24.<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"text-align: left\"><span style=\"font-weight: 400\">274 is a member of <\/span><a href=\"https:\/\/thatsmaths.com\/2017\/09\/14\/moessners-magical-method\/\" target=\"_blank\" rel=\"noopener noreferrer\"><span style=\"font-weight: 400\">Moessner\u2019s Magic Factorial Triangle<\/span><\/a><span style=\"font-weight: 400\">. In 1951 the mathematician Alfred Moessner discovered a method for generating series of consecutive squares, cubes, fourth powers, and so on. The variant of his method shown here generates the series of factorial numbers n! ( i.e. 1, 1 \u00d7 2, 1 \u00d7 2 \u00d7 3, \u2026.. 1 \u00d7 2 \u00d7 3 \u00d7 \u2026. \u00d7 n). Here\u2019s the method:<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left\"><span style=\"font-weight: 400\">\u00a0 \u00a0<strong>A)\u00a0<\/strong> Write the counting numbers from 1 onwards in a row. Cover the triangular numbers (1, 1 + 2, 1 + 2 + 3, etc); these are coloured yellow in the diagram below.<\/span><\/p>\n<p style=\"text-align: left\"><span style=\"font-weight: 400\">\u00a0 \u00a0<strong>B)\u00a0<\/strong> Add the numbers that are left uncovered in row A to make the partial sums shown in row B. Cover the numbers to the right of each section of numbers (coloured blue).<\/span><\/p>\n<p style=\"text-align: left\"><span style=\"font-family: inherit;font-size: inherit;background-color: transparent\">\u00a0 \u00a0<strong>C)<\/strong>\u00a0 Add the uncovered numbers as before and write the partial sums in row D. Cover the numbers to the right (green).<\/span><\/p>\n<p style=\"text-align: left\"><span style=\"font-family: inherit;font-size: inherit;background-color: transparent\">\u00a0 <strong>\u00a0D)\u00a0<\/strong> \u00a0Repeat (orange).<\/span><\/p>\n<p style=\"text-align: left\"><span style=\"font-family: inherit;font-size: inherit;background-color: transparent\">\u00a0 \u00a0<strong>E)\u00a0<\/strong> \u00a0Repeat (mauve).<\/span><\/p>\n<p><span style=\"font-weight: 400\">The numbers on the extreme left of each row are the factorials 1!, 2!, 3!, 4!, and 5!<\/span><\/p>\n<p><span style=\"font-weight: 400\">The process can be continued ad infinitum by writing more numbers to the right of row A and continuing the downward progression.<\/span><\/p>\n<p><a href=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-47533 \" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_1.png\" alt=\"\" width=\"896\" height=\"318\" srcset=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_1.png 1026w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_1-300x106.png 300w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_1-1024x363.png 1024w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_1-768x272.png 768w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_1-640x227.png 640w\" sizes=\"auto, (max-width: 896px) 100vw, 896px\" \/><\/a><\/p>\n<p><span style=\"font-weight: 400\">The coloured numbers above can be rearranged in a triangle, as shown below; the red numbers are the factorials and the orange numbers are the sums of the numbers in the corresponding lines of the triangle, in each case one less than the factorial that starts the next line.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-47535\" src=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_2.png\" alt=\"\" width=\"804\" height=\"350\" srcset=\"https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_2.png 804w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_2-300x131.png 300w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_2-768x334.png 768w, https:\/\/blogs.bmj.com\/bmj\/files\/2020\/05\/274_2-640x279.png 640w\" sizes=\"auto, (max-width: 804px) 100vw, 804px\" \/><\/a><\/p>\n<p><span style=\"font-weight: 400\">Moessner\u2019s magical process is akin to that of logarithms\u2014it turns powers into multiplications and multiplications into additions.<\/span><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>All hypotheses are unequal; some are more unequal than others. I have previously discussed the Indo-European root DHE. The basic root, the so-called e-grade form, DHE, means to set or [&#8230;]<\/p>\n<p><a class=\"btn btn-secondary understrap-read-more-link\" href=\"https:\/\/blogs.bmj.com\/bmj\/2020\/05\/15\/jeffrey-aronson-when-i-use-a-word-testing-hypotheses-in-testing-times\/\">More&#8230;<\/a><\/p>\n","protected":false},"author":1,"featured_media":38359,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5762],"tags":[],"class_list":["post-47531","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-jeff-aronsons-words"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Jeffrey Aronson: When I Use a Word . . . testing hypotheses in testing times - The BMJ<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.bmj.com\/bmj\/2020\/05\/15\/jeffrey-aronson-when-i-use-a-word-testing-hypotheses-in-testing-times\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Jeffrey Aronson: When I Use a Word . . . testing hypotheses in testing times - The BMJ\" \/>\n<meta property=\"og:description\" content=\"All hypotheses are unequal; some are more unequal than others. 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