The Statistical Sign Test Author(s): W. J. Dixon and A. M. Mood Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 41, No. 236 (Dec., 1946), pp. 557- 566 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/2280577 . Accessed: 19/02/2013 17:39 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association. http://www.jstor.org This content downloaded on Tue, 19 Feb 2013 17:39:22 PM All use subject to JSTOR Terms and Conditions THE STATISTICAL SIGN TEST* W. J. DIXON Universityof Oregon A. M. MOOD Iowa State College This paper presentsand illustratesa simplestatistical test forjudging whether one of two materialsor treatmentsis bet- terthan the other.The data to whichthe test is applied consist of paired observationson the two materials or treatments. The test is based on the signs of the differencesbetween the pairs of observations. It is immaterialwhether all the pairs of observationsare comparableor not. However,when all the pairs are compar- able, there are more efficienttests (the t test, for example) whichtake account of the magnitudesas well the signsof the differences.Even in this case, the simplicityof the sign test makes it a usefultool fora quick preliminaryappraisal of the data. In thispaper the resultsof previously published work on the sign test have been included,together with a table of signifi- cance levels and illustrativeexamples. INTRODUCTION IN EXPERIMENTAL investigations,it is oftendesired to compare two materials or treatmentsunder various sets of conditions.Pairs of observations(one observationfor each of the two materials or treat- ments) are obtained for each of the separate sets of conditions.For example,in comparingthe yieldof two hybridlines of corn,A and B, one might have a few results fromeach of several experimentscar- ried out under widely varyingconditions. The experimentsmay have been performedon differentsoil types, with differentfertilizers, and in differentyears withconsequent variations in seasonal effectssuch as rainfall,temperature, amount of sunshine,and so forth.It is supposed that both lines appeared equally oftenin each block ofeach experiment so that the observedyields occur in pairs (one yield foreach line) pro- duced under quite similar conditions. The above example illustratesthe circumstancesunder which the signtest is mostuseful: (a) There are pairs of observationson two thingsbeing compared. * This paper is an adaptation of a memorandumsubmitted to the Applied MathematicsParel by the StatisticalResearch Group,Princeton University. The StatisticalResearch Group operatedunder a contractwith the Officeof ScientificResearch and Development,and was directedby the Applied Mathematic Panel ofthe National DefenseResearch Committee. 557 This content downloaded on Tue, 19 Feb 2013 17:39:22 PM All use subject to JSTOR Terms and Conditions 558 AMERICAN STATISTICAL ASSOCIATION (b) Each of the two observationsof a given pair arose under similar conditions. (c) The differentpairs were observedunder differentconditions. This last conditiongenerally makes the t test invalid. If this were not the case (that is, if all the pairs of observationswere comparable), the t test would ordinarilybe employedunless therewere otherreasons, for example,obvious non-normality,for not using it. Even whenthe t test is the appropriatetechnique many statisticians like to use the sign test because of its extremesimplicity. One merely counts the numberof positiveand negative differencesand refersto a table of significancevalues. Frequently the question of significance may be settled at once by the sign test withoutany need for calcula- tions. It should be pointed out that, strictlyspeaking, the methodsof this paper are applicable only to the case in which no ties in paired com- parisons occur. In practice,however, even when ties would not occur if measurementswere sufficientlyprecise, ties do occur because meas- urementsare often made only to the nearest unit or tenth of a unit forexample. Such ties should be includedamong the observationswith half of thembeing counted as positive and halfnegative. Finally, it is assumed that the differencesbetween paired observa- tions are independent,that is, that the outcome of one pair of obser- vations is in no way influencedby the outcome of any otherpair. PROCEDURE Let A and B representtwo materialsor treatmentsto be compared. Let x and y representmeasurements made on A and B. Let the num- ber of pairs of observationsbe n. The n pairs of observationsand their differencesmay be denoted by: (XI, Yl1), (X2, y2), ***,(X., Yn) and ... - X1 - Yl, X2 - Y2 * Xn Yn. The sign test is based on the signs of these differences.The letter r willbe used to denote the numberof times the less frequentsign occurs. If some ofthe differencesare zero,half of them will be givena plus sign and half a minussign. As an example ofthe type ofdata forwhich the sign test is appropri- ate, we may considerthe followingyields of two hybridlines of corn obtained fromseveral differentexperiments. In this example n =28 and r=7. This content downloaded on Tue, 19 Feb 2013 17:39:22 PM All use subject to JSTOR Terms and Conditions THE STATISTICAL SIGN TEST 559 If thereis no differencein the yieldingability of the two lines,the positiveand negativesigns should be distributedby thebinomial dis- tributionwith p= . The null hypothesishere is that each difference has a probabilitydistribution (which need not be thesame for all dif- ferences)with median equal to zero.This nullhypothesis will obtain, forinstance, if each differenceis symmetricallydistributed about a meanof zero, although such symmetry is not necessary.The nullhy- pothesiswill be rejectedwhen the numbersof positiveand negative signsdiffer significantly from equality. YIELDS OF TWO HYBRID LINES OF CORN Experiment Yield of Sign of Experiment Yield of Sign of Number A B x-y Number A B x-v 1 47.8 46.1 + 4 40.8 41.3 _ 48.6 50.1 - 39.8 40.8 _ 47.6 48.2 - 42.2 42.0 + 43.0 48.6 - 41.4 42.5 - 42.1 43.4 _ 41.0 42.9 - 5 38.9 39.14 39.0 39.4 - 2 28.9 38.8 - 37.5 37.3 + 29.0 31.1 - 27.4 28.0 - 6 36.8 37.5 - 28.1 27.5 + 35.9 37.3 - 28.0 28.7 _ 33.6 34.0 - 28.3 28.8 26.4 26.3 + 7 39.2 40.1 - 26.8 26.1 + 39.1 42.6 - 8 33.3 32.4 + 30.6 31.7 _ Table 1 givesthe critical values of r forthe 1, 5, 10,and 25 percent levelsof significance.A discussionof how thesevalues are computed maybe foundin the appendix.A value ofr less thanor equal to that in thetable is significantat thegiven per cent level. Thus in the example above where n = 28 and r = 7, there is sig- nificanceat the 5% level,as shownby Table 1. That is, the chances are only1 in 20 ofobtaining a value ofr equal to or less than8 when thereis no real differencein the yieldsof the two linesof corn.It is concluded,therefore, at the 5% levelof significance,that the two lines havedifferent yields. In general,there are no valuesof r whichcorrespond exactly to the levelsof significance 1, 5, 10, 25 per cent.The values givenare such that theyresult in a level of significanceas close as possibleto, but notexceeding 1, 5, 10,25 percent. Thus? the test is a littlemore strict, This content downloaded on Tue, 19 Feb 2013 17:39:22 PM All use subject to JSTOR Terms and Conditions TABLE 1 TABLE OF CRITICAL VALUES OF r FOR THE SIGN TEST Per Cent Level of Per Cent Level of Significance Significance n 1 5 10 25 n 1 5 10 25 1 _ - - - 51 15 18 19 20 2 - _ - 52 16 18 19 21 3 - _ - 0 53 16 18 20 21 4 - _ - 0 54 17 19 20 22 5 - - 0 0 55 17 19 20 22 6 - 0 0 1 56 17 20 21 23 7 - 0 0 1 57 18 20 21 23 8 0 0 1 1 58 18 21 22 24 9 0 1 1 2 59 19 21 22 24 10 0 1 1 2 60 19 21 23 25 11 0 1 2 3 61 20 22 23 25 12 1 2 2 3 62 20 22 24 25 13 1 2 3 3 63 20 23 24 26 14 1 2 3 4 64 21 23 24 26 15 2 3 3 4 65 21 24 25 27 16 2 3 4 5 66 22 24 25 27 17 2 4 4 5 67 22 25 26 28 18 3 4 5 6 68 22 25 26 28 19 3 4 5 6 69 23 25 27 29 20 3 5 5 6 70 23 26 27 29 21 4 5 6 7 71 24 26 28 30 22 4 5 6 7 72 24 27 28 30 23 4 6 7 8 73 25 27 28 31 24 5 6 7 8 74 25 28 29 31 25 5 7 7 9 75 25 28 29 32 26 6 7 8 9 76 26 28 30 32 27 6 7 8 10 77 26 29 30 32 28 6 8 9 10 78 27 29 31 33 29 7 8 9 10 79 27 30 31 33 30 7 9 10 11 80 28 30 32 34 31 7 9 10 11 81 28 31 32 34 32 8 9 10 12 82 28 31 33 35 33 8 10 11 12 83 29 32 33 35 34 9 10 11 13 84 29 32 33 36 35 9 11 12 13 85 30 32 34 36 36 9 11 12 14 86 30 33 34 37 37 10 12 13 14 87 31 33 35 37 38 10 12 13 14 88 31 34 35 38 39 11 12 13 15 89 31 34 36 38 40 11 13 14 15 90 32 35 36 39 41 11 13 14 16 91 32 35 37 39 42 12 14 15 16 92 33 36 37 39 43 12 14 15 17 93 33 36 38 40 44 13 15 16 17 94 34 37 38 40 45 13 15 16 18 95 34 37 38 41 46 13 15 16 18 96 34 37 39 41 47 14 16 17 19 97 35 38 39 42 48 14 16 17 19 98 35 38 40 42 49 15 17 18 19 99 36 39 40 43 50 15 17 18 20 100 36 39 41 43 For n > 100,approximate values of r may be foundby takingthe nearest integer less than in -k v n, wherek =1.3, 1, .82, .58 forthe 1, 5,10, 25 per centvalues respectively.A closerapproximation to the values of r is obtained fromi(n -1)-k -4 In+1 and the more exact values of k, 1.2879, .9800, .8224, .5752.