Hemelrijk, J.

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Hemelrijk, J. (Jan)

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KONINKL. NEDERL. AKADEMIE VAN WETENSCHAPPEN - AMSTERDAM Reprinted from Proceedings, Series A, 55, No. 3 and Indag. Math., 14, No. 3, 1952 MATHEMATICS 12 A THEOREM ON THE SIGN TEST WHEN TIES ARE PRESENT BY J. HEMELRIJK (Communicated by Prof. D. VAN DANTzIG at the meeting of April 19, 1952) I. Introduction When x and y are two independently distributed random variables 1), tests for the hypothesis H0, that x and y have the same probability distribution, may be based on their difference x- y. Let a number (n) of pairs of independent observations (Xi., y1), •.. , (x,., y,.) be given, then the number of positive differences x,.- y,. (i = I, ... , n) may be used as a test criterion. The test based on this criterion is called the sign test 2). There seems to be some uncertainty concerning the best treatment of zero differences (often called "ties" in this context) when applying this test. W. J. DIXON and A. M. Moon (1946) propose to count half of the ties as positive and half as negative and W. J. DIXON and F. J. MASSEY Jr., (1951) advise to omit the ties from the observations. The object of this note is to prove, that omitting the ties results in a more powerful test than dividing them equally among the positive and negative observations. 2. Preliminary remarks Let ni, n2 and n0 = n - n1 - n2 denote the numbers of positive, negative and zero differences among n independent observations of x - y, and let p1, p2 and Po = 1 - Pi - p2 be the probabilities for each observation of being positive, negative and zero respectively. Then, under the hypo-thesis H0 to be tested, we have (1) The sign test with omission of the n0 ties is based on the conditional probability distribution of n1, under the condition n0 = n0 and under hypothesis H0 , which is (2) The critical region Z1 of this test, with level of significance a1, consists of those results of the n observations, for which one of the relations (3) 1 ) Random variables are denoted by bold type symbols, values assumed by random variables by the same symbols in normal type. 2) The sign test is probably the oldest test in existence. It has been applied already in 1710byJoHN ARBUTHNOT (cf. H. FREUDENTHAL (1951) andJ. TODHUNTER (1865) ). CENl'RUM -· SA 323 holds, where d(n0) (n0 = I, 2, ... , n) is the largest integer satisfying (4) When the ties are divided equally among the positive and the negative observations, the statistic (5) is used and the critical region Z2, with level of significance /¼, is defined as the set of those results, for which one of the relations (6} n~ < d' or n - d' < n~ holds, d' being the largest integer satisfying (7) In other words Z2 is defined as though the probability distribution of n;, under H0, were a binomial distribution with parameters n and p = ½-Notice that d' is independent of n0• The power functions of the two tests are (8) and it may easily be proved that (9) /J1(P, p) < lli for every p < ½- The analogous relation for /32 follows from the theorem which we are going to prove. Both tests thus satisfy the obvious require-ment that the probability of rejecting H0, when it is true, is at most equal to the level of significance. 3. A lemma For the proof of our theorem we shall use the following Lemma: If d, k and n are positive integers satisfying (10) then (11} d' = [d + ½k + ½] < ½n - ½, 3) 2k Jo (n~k) < .t (:). Proof: The proof is given by induction with respect to k. First consider even values of k, starting with k = 2. Then d' = d + 1. Using the relation 1) It should be kept in mind, that n = n0 + n1 + n 2 is a given positive integer. 2) The symbols Pi and p2 behind the vertical line denote that p1 and p 2 are the true probabilities of a positive and a negative difference respectively. 3 ) [u] denoting the largest integer not surpassing u. 324 we find d (n-2) tl+l (n) d (n-2) d (n-1) <1 (n-1) d+l (n) 42 -2 =42 -22 +22 -2 = •=0 ll •=0 V •=0 V •=0 V v=0 V •=0 V = 2 'f (n-2) _ 2 'f (n-2) + I (n-1) _ ,f (n-1) _ ( n ) = •=0 v •=0 v-1 •=0 v •=0 v-1 d+I - 2 i (n-2) - 2df (n-2) + i (n-1) - df (n-1) - ( n ) -- •=0 11 •=Q ll •=0 V •=0 V d+I -= 2 (n-2) + (n-1) _ ( n ) = (n-2) 2d+3-n d d d+I d d+I ' and this is < 0 according to (10) with d' = d + 1. Now if ( 11) is true fork = 2h we shall prove this inequality for k = 2h + 2, given (10), which for k = 2h + 2 assumes the form (12) d' == d + h + l < ½n - ½-The left hand member of the inequality to be proved is 22h+2 i (n-2h-2) = 22 { 2 2h i (n-2-2h) } . •=0 V •=0 11 The formula between curved brackets is equal to the left hand member of (11) when n- 2 is substituted for n and 2h fork. Therefore (11) may be applied if (10) is satisfied. However, for k = 2h and with n- 2 instead of n, (10) reduces to (12), which is a given relation. Thus 22h+2 ,t (n-:h-2) < 22 :r (n~2). The right hand member is again equal to the left hand member of (11), this time with k = 2 and n - 2 substituted for n. Again (10) reduces to (12) when these values are substituted and (11) may be applied once more. This gives d (n-2h-2) d+h+I (n) 22h+2 2 < 2 ' •=0 11 •=0 11 which was to be proved. For odd valueb of k (k = 2h- 1) the proof of the inequality, given (10), follows from (11) with k = 2h and n + I instead of n: 22h-I i (n-;--2h+l) = 2_1. 22h i (n+I-2h) < •=0 11 •=0 11 d+h (n) ( n ) d+h (n) -:z: _J_ <2 - •=0 j/ 2 d+h •=0 j/ ' (10) being satisfied for k = 2h and with n + l instead of n. This proves the lemma. 325 4. A theorem about the power of the two tests Theorem: If a:i < ~ < 1, then {32(p1, p2) < /31(p1, p2) for every p1 and p2 with Pi + p2 < 1. Proof: Consider the subset of possible results where n0 = le and denote the intersection of this set andZ;; by Z;;,1< (i = 1, 2; le= 1, ... , n). According to the definition of Z1 and Z2 the regions Z1,0 and Z 2,0 coincide if ~ = a:i, and Z1,0 :::> Z 2,0 when a1 > a2• We shall prove the latter relation. for le > 0. It then follows that Z1 :::> Z2 when a1 > a2 and this proves the theorem according to the definition of {31 and {32 (cf. (8)). According to (3) and (4) Z1,k consists of two parts, situated symmetrically with respect to the point n1 = ½(n- le), which we denote, for simplicity, by (13) n1 < d and n - le - d < ni, where d is the largest integer satisfying (14) From (5) and (6) we find that Z 2,k consists of the two parts (15) n1 < d' - ½le and n- ½fc- d' < n1, which are also situated symmetrically with respect to n1 = ½(n - le), with d' defined by (7). The proof of the relation Zu :::> Z 2,k will be given indirectly by proving that a1 would be smaller than a2 if this relation was untrue. Supposing Z 2.k to contain more values of n1 than Zu, we consider the region zi,1<, defined by the relation zi,k = Zu, which contains Zu. Denoting the value of d pertaining to Zu by d*, we remark that d* > d and we find from (13) and (15) (16) d' = [d* +½le+ ½J. Now d' < ½n - ½, for d' > ½n - ½ would imply a2 = 1 according to (7), and this has been excluded in the theorem. Consequently the lemma proved above may be applied and multiplying both sides of (11) by 2-n we get (17) d* k d' ( 21<-n L (n- ) < 2-n L n) • •=0 'V •=0 'V From d* > d, from the fact that dis the largest integer satisfying (14) and from (7) and ( 17) it follows that a2 > a1, which contradicts the premises of the theorem. 5. Remarks Numerical computation of special cases shows that in general, when ut = £½, Z1 contains more points than Z2, so that /31 > /32 for every p1 and p2 (p1 + p2 < 1). It may be proved, using the normal appoximation of the 326 binomial distribution, that the difference {J1 - {J2, given p1/p2 and the level of significance ni = C½, increases with increasing p0• We may remark further, that the application of the sign test is not limited to the. situation described in the introduction but may be used whenever the equality of the probabilities of two of the possible results of an experiment has to be tested. The test is consistent if and only if these probabilities are different. REFERENCES 1. DIXON, W. J. and A. M. Moon, The statistical sign test, Journ. Amer. Stat. Ass. 41, 557 -566 (1946). 2. --- and F. J. MAssEY, Jr., An introduction to statistical analysis, p. 248 (N.Y., 1951). 3. FREUDENTHAL, H., De opkomst der statistische denkwijze in de moderne weten-schap, Natuurkundige Voordrachten, nieuwe reeks 29, 57-65 (1951). 4. TODHUNTER, I., A history of the mathematical theory of probability, Camb. and Lond., 1865, p. 131, (Chelsea Puhl. Co., N.Y., 1949). Mathematical Centre, Amsterdarp, 

A theorem on the sign test when ties are present

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