1377 lines
22 KiB
Plaintext
1377 lines
22 KiB
Plaintext
Journal of the American Statistical Association
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ISSN: 0162-1459 (Print) 1537-274X (Online) Journal homepage: https://www.tandfonline.com/loi/uasa20
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Extended Tables of the Wilcoxon Matched Pair Signed Rank Statistic
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Robert L. McCornack
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To cite this article: Robert L. McCornack (1965) Extended Tables of the Wilcoxon Matched Pair Signed Rank Statistic, Journal of the American Statistical Association, 60:311, 864-871, DOI: 10.1080/01621459.1965.10480835 To link to this article: https://doi.org/10.1080/01621459.1965.10480835
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Published online: 10 Apr 2012. Submit your article to this journal Article views: 63 View related articles Citing articles: 3 View citing articles
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Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=uasa20
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EXTENDED TABLES OF THE WILCOXON MATCHED PAIR SIGNED RANK STATISTIC
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ROBERTL. MCCORNACK System Development Corporation
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A table of critical values of the Wilcoxon Signed Rank Statistic is presented for N =4(1)100 pairs of observations a t one-tail probability levels of .00005, .0005, .0025, .005(.005).025, .050(.025).150, and .20(.05).45. Probabilities were computed accurately t o a t least 6 digits, regardless of the location of the decimal point. Therefore, all critical values are correct as tabled. Normal approximation probabilities were found to be biased, too small a t the .05 level but too large at the .005 and ,0005 probability levels. Approximation errors were less than 10% for all N >35 for the .05, ,025, and ,005 one-tail probability levels; but this degree of accuracy was not achieved a t the ,0005 level for N=100.
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1. INTRODUCTION
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IN THE comparison of two treatments, a sample of N pairs of observations may be available. Wilcoxon [la, 13,141has suggested a procedure whereby
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the differences ( X i - Y i ) are ranked in order of size, disregarding sign. Each
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rank is then given the sign of the corresponding original difference. The sum of
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the positive ranks, T , is then the test static. The test seems to be in widespread
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use. Its use is discussed in many general reference works, for example
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[ I , 7, 8, 9, l o ] ; and bibliographies are available [6, 111.
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The sampling distribution of T is discrete and symmetric. Let F t , N =the
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number of ways to choose positive distinct integral summands, none greater
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than N , whose sum equals t ; where t = O , 1, . . . , N ( N + 1 ) / 2 . The number of
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sums, not all different, is 2N. For trivially small values of N these frequencies
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are immediately apparent. For N = 3 , there are two ways (3 and l f 2 ) t o
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obtain a sum of 3 and only one way to obtain each of the other possible sums;
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so that F3,3=2 and F 0 , 3 = F 1 , 3 = F 2 , 3 = F 4 , 3 = Fb,g=F6,3= 1. Given this distri-
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bution for N = 3 , the distribution for all succeeding sample sizes may be
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determined from the recursion formula:
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+ F ~ , N= F t , ~ - i F~-N.N-I
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(1)
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where if (t- N ) <0, Ft-N,N-I =0. And the cumulative one-tail probability (Y is:
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The sampling distribution of T may be simply tabulated, but the publication problem is formidable. At first, the tables [e.g., 12, 131 were restricted to small values of N and to a few probability levels. Computational accuracy was also a problem; for example, slight errors appear in the majority of the probabilities in one table [12]. At the present time, much more extensive and accurate tables are available to the user. Dixon and Massey [1, pp. 443-4441 give the
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864
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TABLES O F WILCOXON SIGNED RANK STATISTICS 100-
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SG5
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-40-
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X
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N -60-
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0,
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cr' w
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-80-
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-100-
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E
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-220-' I
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1
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1
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I
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I
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20
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30
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46
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50
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60
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I
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I
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I
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I
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70
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80
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90
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100
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SAMPLE SIZE
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FIGUR1E. Error as the normal approximation less the exact probahility a t four probahility levels.
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one-tail cumulative probability to three decimals for each value of T for which the probability is between .005 and .125, for every sample size up to N =20. Owen [5,pp. 326-3301 has used equations (I) and ( 2 ) to prepare a table which gives the one-tail cumulative probability to three decimals for every sample size up to N=20 for each value of T , t=0(1)180. I n effect, this tabulates every possible value of T up through N = 18, and virtually the entire distribution for sample sizes of 19 and 20. The most recent table by Wilcoxon, Katti and
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Wilcox [15, pp. 62-64] extends the sample size coverage up to N =50, concen-
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trating upon four points in the distribution. One-tail cumulative probabilities arc given to four decimals for the two values of 7' which bracket the significance levels of .05, .025, .01, and .005. With the exception of six errors' in the Dixon and Massey table, these three tables check perfectly against each other and with the probability distributions computed by the author.
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The purpose of this paper is to present a table of critical values of T that extends coverage to more probability levels and to larger values of N . A sec-
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1 These appear t o be ronnding errors, and the correct probahilit1p8rnaybe obtained by adding ,001 for T =45, for N =16. 17, and 18; for T=F4, N=19; for T=63,N =20; and by subtracting .001 for T = 2 6 , N =17.
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866
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AMERICAN STATISTICAL ASSOCIATION JOURNAL, SEPTEMBER 1985
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TABLE a. CRITICAL VALUES OF T , T H E WILCOXON SIGNED RANK
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STATISTIC, WHERE T IS T H E LARGEST INTEGER SUCH THAT
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P r [ T l t l N ] +, THE CUMULATIVE ONE-TAIL PROBABILITY
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-
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2 a .15
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.10
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.05
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.04
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.03
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.02
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.Ol
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a005
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-001 -0001
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a .075
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.050
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.C25
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.020 .015
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.010 - 0 0 5
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.0025
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.0005 .00005
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4
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0
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5
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1
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0
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6
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2
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2
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0
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0
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7
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4
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3
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2
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1
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0
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0
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8
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7
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5
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3
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3
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2
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1
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0
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9
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9
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8
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5
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5
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4
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3
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1
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10
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12
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10
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8
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7
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6
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5
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3
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11
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16
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13
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10
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9
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8
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7
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5
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12
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19
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17
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13
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12
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11
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9
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7
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13
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24
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21
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17
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16
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14
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12
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9
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14
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28
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25
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21
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19
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18
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15
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12
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15
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33
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30
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25
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23
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21
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19
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15
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16
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39
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35
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29
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28
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26
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23
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19
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17
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45
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41
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34
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33
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30
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27
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23
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18
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51
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47
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40
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38
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35
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32
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27
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19
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58
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53
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46
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43
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41
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37
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32
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20
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65
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60
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52
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50
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47
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43
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37
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2L
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73
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67
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58
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56
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53
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49
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42
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22
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81
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75
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65
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63
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59
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55
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48
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23
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89
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83
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73
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70
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66
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62
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54
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24
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98
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91
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81
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78
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74
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69
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61
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25
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10.8
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100
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89
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86
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a2
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76
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68
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26
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118
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110
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98
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94
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9C
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84
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75
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27
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128
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119
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1@7
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103
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99
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92
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83
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28
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138
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130
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116
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112
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108
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101
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91
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29
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150
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140
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126
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122
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117
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110
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100
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30
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161
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151
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137
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132
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127
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120
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109
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31
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173
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163
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147
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143
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137
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130
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118
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32
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186
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175
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159
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154
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148
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140
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128
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33
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199
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187
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170
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165
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159
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151
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138
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3 4.
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212
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200
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182
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177
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17 1
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162
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148
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35
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226
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213
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195
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189
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182
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173
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159
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36
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240
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227
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208
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202
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195
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185
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171
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37
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255
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241
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221
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215
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208
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336
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44
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45
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389
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371
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343
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407
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389
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361
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352
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342
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328
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307
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427
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407
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370
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359
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345
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48
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446
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426
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396
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388
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377
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3 62
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339
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49
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466
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446
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415
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406
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394
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379
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355
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50
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487
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466
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434
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425
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413
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397
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373
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0 1 3 5 7 9 12 15 19 23 27 32 37 42 48 54 60 67 74 82 90 98 107 116 126 136 146 157 168 180 192 204 2 17 230 244 258 272 287 302 318 334 3 50
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0 1 2 4 6 8 11 14 18 21 25 30
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35
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40 45
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51 57 64
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|
71 7a 86 94 102 111 120 130 140 150 161 172 183 195 207
|
|
220 233 246
|
|
260 274 289 304
|
|
|
|
0 2 3 5 8 10 13 17 20 24 28 33 38 43 49 55 61 68 74 82 90 98 106 115 124 133 143 153 164
|
|
17.5 186 198
|
|
210 223 235 249
|
|
|
|
ondary purpose is to investigate the adequacy of the normal approximation at a few selected significancelevels.
|
|
2. COMPUTATIONAL METHODS AND ACCURACY
|
|
- - - The complete frequency distribution forevery value of T for N =4, , 100
|
|
was computed using formula (1).Each one-tail cumulative probability a was computed by means of formula (2). All calculations were performed on an IBM 7090 digital computer. For selected values of a,Table 1gives the largest value of T for which the cumulative probability is less than or equal to 01. The value of a appears as the column heading; and 2a is the corresponding two-tail probability. Since the distribution is symmetric, and to save space, only half of the distribution has been tabled. However, an additional column is given in Table 1 to facilitate work with the other tail of the distribution. The maximum value of T is N ( N + 1 ) / 2 , and this quantity is given in the first column under
|
|
|
|
TABLKS O F WILCOXON SIGNED RANK STATISTICS
|
|
|
|
867
|
|
|
|
TABLE 1 (Continued)
|
|
|
|
2 a -15
|
|
|
|
.075
|
|
|
|
51
|
|
|
|
508
|
|
|
|
52
|
|
|
|
530
|
|
|
|
53
|
|
|
|
551
|
|
|
|
54
|
|
|
|
574
|
|
|
|
55
|
|
|
|
597
|
|
|
|
56
|
|
|
|
620
|
|
|
|
57
|
|
|
|
644
|
|
|
|
58
|
|
|
|
668
|
|
|
|
59
|
|
|
|
693
|
|
|
|
60
|
|
|
|
718
|
|
|
|
61
|
|
|
|
744
|
|
|
|
62
|
|
|
|
770
|
|
|
|
63
|
|
|
|
796
|
|
|
|
64
|
|
|
|
023
|
|
|
|
65
|
|
|
|
851
|
|
|
|
66
|
|
|
|
879
|
|
|
|
67
|
|
|
|
907
|
|
|
|
68
|
|
|
|
936
|
|
|
|
69
|
|
|
|
965
|
|
|
|
70
|
|
|
|
995
|
|
|
|
71
|
|
|
|
1025
|
|
|
|
72
|
|
|
|
1056
|
|
|
|
73
|
|
|
|
1087
|
|
|
|
74
|
|
|
|
1119
|
|
|
|
75
|
|
|
|
1151
|
|
|
|
76
|
|
|
|
1183
|
|
|
|
77
|
|
|
|
1216
|
|
|
|
78
|
|
|
|
1250
|
|
|
|
79
|
|
|
|
1284
|
|
|
|
80
|
|
|
|
1318
|
|
|
|
81
|
|
|
|
1353
|
|
|
|
82
|
|
|
|
1389
|
|
|
|
83
|
|
|
|
1424
|
|
|
|
84
|
|
|
|
1461
|
|
|
|
85
|
|
|
|
1497
|
|
|
|
86
|
|
|
|
1535
|
|
|
|
87
|
|
|
|
1572
|
|
|
|
88
|
|
|
|
1610
|
|
|
|
89
|
|
|
|
1649
|
|
|
|
90
|
|
|
|
1688
|
|
|
|
91
|
|
|
|
1728
|
|
|
|
92
|
|
|
|
1768
|
|
|
|
93
|
|
|
|
180R
|
|
|
|
94
|
|
|
|
1849
|
|
|
|
95
|
|
|
|
1891
|
|
|
|
96
|
|
|
|
1932
|
|
|
|
97
|
|
|
|
1975
|
|
|
|
98
|
|
|
|
2018
|
|
|
|
99
|
|
|
|
2061
|
|
|
|
100
|
|
|
|
2105
|
|
|
|
.10
|
|
-050
|
|
486 50 7 529 550 573 595 618 642 666 690 715 74 1 767 793 820 847 875 903 93 1 960 990 1020 1050 1081 1112 1144 1176 1209 1242 1276 1310 1345 1380 1415 1451 1487 1524 1561 1599 1638 1676 171 5 1755 1795 1836 1877 1918 1960 2003 2045
|
|
|
|
-05
|
|
.025
|
|
453 473 494 514 536 557 579 602 625 648 672 697 721 74 7 772 798 825 852 879 907 936 964 994 1023 1053 1084 1115 1147 1179 1211 1244 1277 1311 1345 1380 1415 1451 1487 1523 156@ 1597 1635 1674 1712 1752 1791 1832 1R72 1913 1955
|
|
|
|
.04
|
|
.020
|
|
444 463 483 504 525 546 568 590 613 636 659 683 708 733 758 784 810 837 864 891 919 948 977 1006 1036 1066 1097 1128 1160 1192 1224 1257 1291 1325 1359 1394 1429 1464 1501 1537 1574 1612 1650 1688 1727 1766 1806 1846 1887 1928
|
|
|
|
-03
|
|
.015
|
|
432 451 471 491 511 532 554 575 598 620 644 667 691 716 741 766 792 818 845 872 900 928 956 985 1014 1044 1075 I105 1136 1168 1200 1233 1266 1299 1333 1367 1402 1437 1473 1509 1545 1582 1620 1658 1696 1735 1774 1814 1854 1894
|
|
|
|
.02
|
|
-010
|
|
416 434 454 473 493 514 535 556 578 600 623 646 669 693 718 742 768 793 819 646 873 901 928 957 986 1015 1044 1075 1105 1136 1168 1200 1232 1265 1298 1332 1366 1400 1435 1'471 1507 1543 1580 1617 1655 1693 1731 1770 1810 1850
|
|
|
|
-01
|
|
-005
|
|
390 .408 427 445 465 484 504 525 546 567 589 611 634 657 681 70 5 729 754 779 805 831 858 884 912 940 968 997 1026 1056 1086 1116 1147 1178 1210 1242 1275 1308 1342 1376 1410 1445 1480 1516 1552 1589 1626 1664 1702 1740 1779
|
|
|
|
-005
|
|
.0025
|
|
367 384 402 420 4 39 457 477 497 517 537 558 580 602 624 647 6 70 694 718 142 767 192 818 844 871 898 925 953 981 1010 1039 1069 1099 1129 1160 1191 1223 1255 1288 1321 1355 1389 1423 1458 1493 1529 1565 1601 1638 1676 1714
|
|
|
|
.001
|
|
.0005
|
|
319 335 35 1 368 385 402 420 438 457 476 495 515 535 556 577 599 621 643 666 689 712 736 76 1 786 811 836 862 889 916 943 97 1 999 1028 1057 1086 1116 1146 1177 1208 1240 1271 1304 1337 1370 1404 1438 1472 1507 1543 1578
|
|
|
|
-0001
|
|
-00005
|
|
262 216 291 305 321 336 352 360 385 402 419 437 456 474 493 513 533 553 573 594 616 638 660 683 706 729 753 777 802 827 R52 8 78 904 931 958 985 1013 1042 1070 1100 1129 1159 1189 1220
|
|
1347 1380 1413
|
|
|
|
the heading M A X . For a given N , if the probability is a that T l t , then the probability is a that T 2 M A X - t.
|
|
The problem of computational accuracy merited special attention for two reasons. First, many quite large numbers such as 2lo0were represented by a much smaller number of digits. Second, because of the recursive technique employed, the roundoff errors present at low values of N affect accuracy at all higher values of AT. Of course, ordinary internal checks such as the cumulative probability reaching unity a t the appropriate value of T werc used. Howevcr, principal reliance was placed upon the commonly used technique [3, p. 251 of performing all calculations with 8 significant digits and again with 16 significant digits. Since the use of more significant digits should result in greater accuracy, it may be argued that the number of places which agree is a minimum to the number of correct digits. Cumulative probabilities computed with 8 and 1 G digits were found to agree to at least 6 digits in every case, regardless of the
|
|
|
|
868
|
|
|
|
- -
|
|
|
|
- N
|
|
4 5 6 7 8 9 10 11 12 13
|
|
14 15 16 17
|
|
18 19 20 21 22
|
|
23 24 25 26 27 28
|
|
29 30 31
|
|
32 33 34 35 36 37 38 39
|
|
40 41 42 43
|
|
44 45 46 47 48 49
|
|
-50
|
|
|
|
10 15 21 28 36 45 55 66
|
|
78 91
|
|
105 120 136 153 171 190 2 10 231 253 276 300 325 351 378 406 435 465 496 528 561 595 6 30 666 703 741 780 820 861 903 946 990 1035 1081 1128 1176 1225 1275
|
|
|
|
AMERICAN STATISTICAL ASSOCIATION JOUENAL, SEPTEMBEIt 1965
|
|
TABLE 1 (Continued)
|
|
|
|
I0
|
|
x -45
|
|
4 6 9 12 16 20 25 31 36 43 49 57 65 73 82 91 101 111 122 133 145 157 169 183 196 211 22 5 240 256 272 289 306 324 342 36 1 380 400 420 440 4h 1 403 505 528 55 1 575 599 623
|
|
|
|
.t 10
|
|
.40
|
|
3 5 8 11 15 19 24 29 35 41 47 54 62 70 79 88 97 107 118 129 140 152 164 177 191 205 219 234 249 265 281 298 316 334 352 371 390 410 430 451 472 494 516 539 562 586 610
|
|
|
|
: I0
|
|
.35
|
|
3 5 8 I1 14
|
|
la
|
|
23 27 33 39 45 52 59 67 75 84 93 103 113 124 135 147 159 172 185 198 212 227 242 258 274 290 307 325 343 36 1 380 399 419 440 46 1 482 504 526 549 572 596
|
|
|
|
.t
|
|
.3C
|
|
2 4 7 10 13 17 21 26 31 37 43 50 57 64 12 a1 90 99 1C9 119 130
|
|
tE
|
|
166 179 192 ZC6 220 235 250 265 282 298 315 333 351 370 389 408 428 449 470 491 513 535 558 582
|
|
|
|
50
|
|
.25
|
|
2 4 6 9
|
|
12 16 20 24 29 35 40 47 54 61 69 77 86 95 104 114 125 136 148 160 172 185 198 212 226 241 257 272 289 305 323 340 358 377 396 416 436 456 477 499 521 543 566
|
|
|
|
.4 10
|
|
.20
|
|
2 3 5 8 11 14 18 22 27 32 38 44 50 57 65 73 81 90 99
|
|
109 119 130 141 153 165 177 i90 204 218 232 2 47 262 278 294 311 328 346 364 383 402 42 1 44 1 462 483 504 526 549
|
|
|
|
.3 0
|
|
.15
|
|
1 2 4 7 9 12 16 20 24 29 35 40 47 53 60 60 76 84 93 102 112 123 133 144 156 168 181 194 207 221 2 35 2 50 265 281 291 3 14 331 349 367 386 405 424 444 464 485 507 529
|
|
|
|
.2 5
|
|
,125
|
|
1 2 4 6 9 12 15 19 23 2a 33 38 44 51 58 65 73 81 90
|
|
99 108 118 129 140 151 163 175 188 201 215 229 243 258 274 290 306 323 340 358 376 395 414 434 454 474 495 517
|
|
|
|
.2 0
|
|
.lo0
|
|
0 2 3 5 8 10 14 17 21 26 31 36 42 48 55 62 69 77 86 94 104 113 124 134 145 151 169 181 194 207 221 235 250 265 26 1 287 313 330 348 365 384 402 422 441 462 482 503
|
|
|
|
location of the decimal point. The cumulative probabilities were never enough different to modify the value of T shown in Table 1.It is concluded that Table 1 values are correct as tabled and that the cumulative probabilities upon which Table 1 depends were correct to a t least the first 6 non-zero digits.
|
|
|
|
3. THE NORMliL APPROXIMATION
|
|
It has been suggested [12] that z = ( T - T ) / s , where T=N(N+1)/4 and sz=N(2N+ 1)( N + 1)/24, rapidly approaches Normal (0, 1) as N increases. It
|
|
is often assumed that for N >25 the normal approximation is quite satisfactory.
|
|
Fellingham and Stoker [2] present some numerical results concerning the adequacy of the normal approximation and another approximation based upon an Edgeworth series. However, they did not have exact probabilities available to them for N > 2 5 . Since the two-tail significance levels of .lo, .05, .01, and .001 are of particular interest to statisticians, it was decided to compare the exact and normal approximation probabilities at the corresponding onc-tail probabilities of .05, 5, .02.005, and .0005. The exact probabilitics werc ohtaiiicd
|
|
|
|
TABLES OF WILCOXON SIGNED B A S K STATISTICS
|
|
|
|
869
|
|
|
|
TABLE 1 (Continued)
|
|
|
|
51 52
|
|
53 54
|
|
55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90.
|
|
91 92 93 94 95 96 97 98 99
|
|
- 100
|
|
|
|
1326 1378
|
|
1431 1485
|
|
1540 1596 1653 1711 1770 1830 1891 1953 2016 2080 2145 2211 2278 2346 2415 2485 2556 2628 2701 2775 2850 2926 3003 308 1 3160 3240 3321 3403 3486 3570 3655 374 1 3828 3916 40C5 4095
|
|
4186 4278 4371 4465 4560 4656 4753 4851 4950 5050
|
|
|
|
-90
|
|
1 -45
|
|
648 67% 700 727
|
|
7 5c
|
|
781 81c: 838 867 897 927 957 989 102c 1052 1085 ilia 1151 1185 1222 1255 1290 1327 1363 140C 1438 1476 1514 1553 1593 1633 1673 17 14 1756 1798 184L 1883 1927 1971 2015 2060 2106 2152 2198 2245 2292 2340 2389 2438 2487
|
|
|
|
.80
|
|
-40
|
|
635 660 686 712 739 766 793 822 850 879 909 939 970 1001 1032 1065 1097 1130 1164 1198 1233 1268 1303 1339 1376 1413 1450 148R 1527 1566 1605 1645 1686 1727 1768 1810 1853 1896 1933 1983 2028 2073 2118 2164 2210 2257 2305 2353 2401 2450
|
|
|
|
-70
|
|
-35
|
|
621 645 671 696 123 749 7 77 804 833 861 8 9J 920 9 50 981 1012 1044 1076 1109 1142 1175 1209 1244 1279 1315 1351 1387 1424 1462 150C 1538 1577 1617 1657 1697 1738 1780 1822 1864 1907 1950 1994 2039 2083 2129 2175 2221 2268 2315 2363 2411
|
|
|
|
.60
|
|
.30
|
|
606 630 655 680 706 732 759 786 814 842 871 900 93c 963 99 1 1022 1C54 icn6 1118 1151 1185 1219 1254 1289 1324 1360 1397 1434 1471 1509 1548 1586 lb26 1666 1706 1747 1789 1830 1873 1916 1959 2C03 2047 2092 2137 2183 2229 2276 2323 2371
|
|
|
|
-50
|
|
-25
|
|
590 613 638 663 688 714 740 767 794 822 850 879 908 938 968 998 1029 1061 1091 1126 1159 1192 1226 1261 1296 1331 1367 1403 1440 1478 1516 1554 1593 1632 1672 1712 1753 1794 1836 1878 1921 1964 2008 2052 2097 2142 2187 2233 2280 2327
|
|
|
|
-40
|
|
.20
|
|
572 595 619 643 668 693 719 745 772 799 827 855 883 912 942 972 1003 1034 1065 1097 1129 1162 1196 1230 1264 1299 1334 1370 1406 1443 1480 1518 1556 1595 1634 1673 1713 1754 1795 1837 1879 1921 1964 2007 2051 2096 2141 2186 2232 2278
|
|
|
|
-30
|
|
-15
|
|
551 574 597 620 645 669 694 720 746 773 800 827 855 883 912 942 971
|
|
ioaz
|
|
1032 1064 1095 1128 I160 1193 1227 1261 1296 1331 1366 1432 1439 1476 1513 1551 1589 1628 1667 1707 1747 1788 1829 1871 1913 1956 1999 2043 2087 2131 2176 2222
|
|
|
|
-25
|
|
.125
|
|
539 561 584 607 631 655 b80 705 731 757 784 811 838 866 895 924 953 983 1013 1044 1076 1107 1140 1172 1205 1239 1273 1308 1343 1378 1414 1451 1488 1525 1563 1602 1640 1680 1720 1760 1801 1842 1883 1926 1968 2011 2055 2099 2144 2189
|
|
|
|
.20
|
|
.lo0
|
|
525 547 569 592 615 639 664 688 714 739 76 5 792 819 847 875 903 932 962 992 1022 1053 1084 1116 1148 1181 1214 1247 1282 1316 1351 1387 1423 1459 1496 1533 1571 1609 1648 1688 1727 1767 1808 1849 1891 1933 1976 2019 2062 2106 2151
|
|
|
|
from equations (I) and ( 2 ) for each value of T shown in Table 1for the selected probability levels for each N from 20 to 100.
|
|
Probabilities for the normal approximation were computed from a formula given by Hastings [4, p. 1691which has a maximum absolute error of 15X
|
|
Figures 1and 2 summarize the results of these comparisons. The actual values of the normal approximation errors shown in Figure 1 were computed as the normal minus the exact probability. Positive errors thus represent overestimates by the normal approximation. It is clear that the normal approximation probabilities are uniformly too low a t the .06 level, and uniformly too high a t the .005 and .0005 levels. At the .025 level the approximation is too low up to an N of about 45 and thereafter too high. These results suggest that a single correction for continuity, such as that given by Tate and Clelland 19, p. 1021 or Walsh [ll, p. 1481, will sometimes improve and sometimes worsen the approximation depeiidirig 011 the probability level.
|
|
|
|
870
|
|
|
|
AMERICAN STATISTICAL ASSOCIATION JOURNAL, SEPTEMBER 1965
|
|
|
|
100-
|
|
95 -
|
|
-90
|
|
|
|
85-
|
|
-80 -75
|
|
-70 -65
|
|
60 -
|
|
|
|
soi 55-
|
|
e
|
|
50-
|
|
c
|
|
5 45-
|
|
w
|
|
' 40-
|
|
|
|
35-
|
|
30 -
|
|
|
|
25-
|
|
-20 -15
|
|
|
|
10-
|
|
|
|
5-
|
|
|
|
0-
|
|
|
|
P 015
|
|
|
|
-
|
|
|
|
~
|
|
|
|
-5
|
|
|
|
< P: 05 1
|
|
|
|
I
|
|
|
|
I
|
|
|
|
I
|
|
|
|
I
|
|
|
|
I
|
|
|
|
I
|
|
|
|
I
|
|
|
|
20
|
|
|
|
30
|
|
|
|
40
|
|
|
|
50
|
|
|
|
60
|
|
|
|
70
|
|
|
|
80
|
|
|
|
90
|
|
|
|
100
|
|
|
|
SAMPLE SIZE
|
|
|
|
FIGURE2. Normal approximation error as a percentage of the exact probability at four probability levels.
|
|
|
|
The actual error was divided by the exact probability and multiplied by 100 to yield the per cent error values shown in Figure 2. It is a matter of judgment as to when the approximation becomes close enough. If one insists upon the error being 10% or less in absolute value, then the normal approximation is satisfactory for the .05 and .025 levels for all N from 20 to 100. The results for the .025 level are quite striking, the per cent error being less than 1%for every N > 2 0 and less than .1% for every N from 39 to 53. This may account for some of the extremely optimistic conclusions concerning the normal approxi-
|
|
|
|
TABLES OF WILCOXON SIGNED RANK STATISTICS
|
|
|
|
871
|
|
|
|
mation [e.g., 7, p. 791, since the two-tail .05 level (one-tail .025 level) is the typical focus of attention. The normal approximation errors at the .005 level drop rapidly from 18% at N =20 to less than 10% for every N > 3 5 . The situation is quite different at the .0005 level where the per cent error does not drop below 50% until N =33, and for N = 100 it is still 14%.
|
|
|
|
REFERENCES
|
|
111 Dixon, W. J. and Massey, F. J., Jr., Introduction to Statistical Analysis. New York:
|
|
McGraw-Hill Book Co., 1957. 121 Fellingham, S. A. and Stoker, D. J., “An approximation for the exact distribution of
|
|
the Wilcoxon test for symmetry,” Journal of the American Statistical Association, 59 (1964), 899-905. [3] Hamming, R. W., Numerical Methods for Scientists and Engineers, New York: McGraw-Hill Book Co., 1962. [4] Hastings, C., Jr., Approximations for Digital Computers. Princeton, N. J. : Princeton University Press, 1955. [5] Owen, D. B., Handbook of Statistical Tables. Reading, Mass.: Addison-Wesley Publishing Co., 1962. [6] Savage, I. R., Bibliography of Nonparametric Statistics. Cambridge, Mass.: Harvard University Press, 1962. 171 Siegel, Sidney, Nonparametric Xtatistics. New York: McGraw-Hill Book Co., 1956. [S] Snedecor, G. W., Statistical Methods. Ames, Iowa: Iowa State College Press, 1956. [9] Tate, M. W. and Clelland, R. C., Nonparametric and Shortcut Statistics. Danville, Ill. :Interstate Printers and Publishers, Inc., 1957. [lo] Walker, H. M. and Lev, Joseph, Statistical Inference. New York: Henry Holt and Co., 1953. [ l l ] Walsh, J. E., Handbook of Nonparametric Statistics. Princeton, N. J.: D. Van Nostrand Co., 1962. [12] Wilcoxon, F., “Individual comparisons by ranking methods, Biometrics Bulletin, 1 (1945), 80-3. [13] Wilcoxon, F., “Probability tables for individual comparisons b y ranking methods, ” Biometrics, 3 (1947), 119-22. [14] Wilcoxon, F., Some Rapid Approximate Statistical Procedures. Stamford, Conn.: American Cyanamid Co., 1949. [15] Wilcoxon, F., Katti, S. K. and Wilcox, R. A., Critical Values and Probability Levels for the Wilcoxon Rank Sum Test and the Wilcoxon Signed Rank Test. Pearl River, N. Y.: American Cyanamid Co. and Florida State University, 1963.
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