:ANTHROPOLOGIC AL REC-ORDS ;a ~~; U7.7 -8.5 CULTURE" ELE M T IS xxv ! .. s # re S . YO CULTUMELE ISTRIVTIOSULTS05 STATI CAL R URERESANTSRE C, DOUGIAS CHRE+,EN '"UNIVE OF CALIFORNIA PRESS ISKEEYAND LOS ANGELES e.~ u194 A-,; CULTURE ELEMENT DISTRIBUTIONS: XXV RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS BY C. DOUGLAS CHRETIEN ANTHROPOLOGICAL RECORDS Vol. 8, No. 5 ANTHROPOLOGICAL RECORDS EDITORS: A. L. KROEBER, E. W GIFFORD, R. H. LoWIE, R. L. OLSON Volume 8, No. 5, pp. 469-490 Submitted by the Editors October 23, I942 Issued January I3, I945 Price, 35 cents UNIVERSITY OF CALIFORNIA PRESS BERKELEY, CALIFORNIA CAMBRIDGE UNIVERSITY PRESS LONDON, ENGLAND MANUFACTURED IN THE UNITED STATES OF AMERICA CONTENTS Page roduction .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 ervation and hypothesis: the Chi-Square Test; probability ... . . . . . . . . . . . . . . 470 Driver's distribution of intradifferences .470 [ Re4;ed eleent lists 473 Fourfold tables .473 Introductory .474 Northwest California .475 Gulf of Georgia Salish .479 Western Indo-China .481 Discussion .482 Empirical confirmation .487 elusion ..... . . . . . . .......... ........ . 488 liography .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 .~~~~~~~~~~~~~~~~~~~~ [iii] CULTURE ELEMENT DISTRIBUTIONS: XXV RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS BY C. DOUGLAS CHRETIEN INTRODUCTION This study arose out of an investigation in statistical linguistics: I was seeking to understand why the statistical results achieved by Professor A. L. Kroeber and me for the relationships of Hittite to the Indo-European languages were meaningless.1 I was led by this problem to the Chi-Square Test, and thence to a combing of ethnological literature for criticisms of the statistical method. As a result of this reading, it seems to me that one of the most important con- siderations against the method is that of Professor Kluckhohn.2 His thesis is that "the use of formulae based upon probability theory must be regarded with scepticism."3 "This approach [the statistical] in its present form would seem to have certain very definite liabilities, of which the most serious is probably the fact that its present formulas are based mainly upon probability theory-- which may well be completely inapplicable to such a statistical universe."4 "I question, however, as at least premature the tendency to apply to our present data formulas based on 'chance' and with highly complex theoretical antecedents."5 It seemed to me that consideration had to be given to this objection. If it were valid for ethnology, it was likewise valid for linguistics. Data for inves- tigation, however, were already-at hand in the CED (or Culture Element Distribu- tion) papers, and in quantity far exceeding anything I had available in linguis- tics. For this reason I have used ethnological material. My interest, however, is not in specific ethnological results, but in the theory underlying the opera- tions which are employed when we bring in statistics, and its validity and appli- cability. Are we justified in applying to ethnology and linguistics the type of statistical analysis hitherto used, or is Kluckhohn right in holding these analy- ses invalid? If in seeking to answer these questions I seem to be taking a roundabout path and often to lose sight of my main object, I am doing so deliberately. The first requirement of an investigation into the validity and appropriateness of statistical methods is that the type of problem be clearly defined. We cannot disqualify the method on theoretical grounds alone. I have therefore developed various types of problems, giving the results achieved on the assumption that the method is valid, and reserve the main discussion of this latter point until near the end. 1Chretien, 1943. 2Kluckhohn, 1939. 3Ibid., 366. 4Ibid., 369. 5Ibid., 373. [469] OBSERVATION AND HYPOTHESIS: THE CHI-SQUARE TEST; PROBABILITY In handling the data of the culture element pen that the Hupa informants would "intradiffer distributions lists, it is common practice to as Driver called it,10 about the same element. classify the elements and determine the number of Likewise three pairs of informants, or all four. elements which fall into each class. The four- pairs, might intradiffer about the same element.. fold table used for determining association- Driver counted the occurrence of these intradi- coefficients is one example. Subdivision of the ferences, and I give his result in column II of,' element list into various groups of cultural ob- table 1.11 These are the actual frequencies of jects or activities, such as fishing, hunting, basketry, musical instruments, puberty rites, TABLE 1 etc., is another. It is frequently desirable to (i, intradifferences per element; i, ac- compare the number of elements observed to fall tual frequencies; m, theoretical fre into each class, or "actual frequencies, as they quencies; m/i, ratio of actual frequen are termed, with the "theoretical" or "hypo- cies to theoretical frequencies) thetical" frequencies, that is to say, the num- bers of elements which would fall into each class I II III IV if some hypothesis were in operation. I shall i m /m illustrate this procedure with three different O.... 477 455 1.05 kinds of material. 1.. 175 211 0.83 DRIVER'S DISTRIBUTION OF INTRADIFFERENCES 3.... 9 3 3.00 Let us first give our attention to a compari- son which Driver made in his The Reliability of Culture Element Data.7 The material upon which the present problem. Driver next calculated t this essay was based was gathered from four theoretical distribution of these intradiffer- tribes of northwest California and the adjacent ences on the assumption that their occurrence part of Oregon: the Hupa, Yurok, Tolowa, and without significance and purely at random. I Galice. For each of these tribes he had two in- give these figures in column III of table 1.13 formants, and hence two lists of data.8 These We now have a set of observed data and a -s lists were not all of the same length, and, as I of hypothetical data: what conclusions are to understand it, he made therefore two reductions. drawn from their comparison? Driver remarks t First the pair of lists for each tribe was re- the frequencies of elements showing two, three duced by omitting elements for which there was or four differences are very near to the chanc information (of either presence or absence) from frequencies.13 This may be a legitimate inter only one informant. Driver does not specifically pretation of the figures, but if it is, a cont state that he did this, but it seems implicit in dictory interpretation is just as legitimate. his treatment of the data. Next he made a fur- example, suppose we determine the ratios of a& ther reduction by eliminating all except "the 706 tual to theoretical values for each of the el elements which were positively or negatively re- given; we get the result given in column IV of ported for all four tribes."9 Although he does table 1. Since exact correspondence would gi not actually say so, I take it that each of his a ratio of 1.00, we see that this interpretati eight informants gave a response to each of these denies Driver's, and asserts that the actual elements. It is this list of 706 elements which ues which he stated to be nearest the hypothe concerns us for the present. cal values are really furthest from them. Th Taking the eight lists of these 706 elements, contradiction which my ratio results give to with the occurrence or absence of each element statement points, I believe, to the unreliabX noted, and comparing the two members of each and perhaps lack of meaning of both his resul tribal pair, Driver found that in the main they and mine, and indicates the need for a more agreed. But there was a certain amount of dis- cise method of drawing inferences from a corn agreement. One Yurok informant, for example, son of actual and theoretical frequencies., would report the presence of an element; the Such a more precise method is given us by other would deny it. It might or might not hap- device known as the Chi-Square Test (referred 6Yule, 1937, 416. hereafter by its Greek form X2). The various Y7Dle 1937 steps necessary to working it out are indica t Driver, 1938. in table 2. After setting down the actual-f - This is not literally true: for Galice, Bar- quencies (m) and the theoretical frequencies nett collected the data, not Driver, using two informants; for Tolowa, Driver had a single in- 10Ibid, 212. formant, and Drucker supplied the place of a' second informant by filling in an element list 1Ibid. from his data obtained by the usual field methods 12 Ibid. of ethnography. Driver, l938, 205. *3bd 9Ibid., 207. 3bd [470] CULTURE ELEM. DISTRIB.: XXV--CHRETIEN: RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS 471 or each class, we determine the difference be- value of P as indicating a level of significance. ween them (m-m), square it, and divide the P = 0.05 is widely used, also P = 0.01 and P = quare by the theoretical frequency.:14 The re- 0.001.1O These are known as the five per cent ult for each class, or "cell," as it is termed,15 level, the one per cent level, and the one-tenth s given in column VI of the table. These re- of one per cent level respectively. If P is ults are summed up, and their total is the value greater than the value of the level of signifi- f X2 for this particular table. (The reason for cance chosen, we say that the actual data do rot ouping the last three differences is explained depart sign'ificantly from the theoretical; if P n a footnote below.)'6 is less than this critical value, we sasy that the data do so depart. For example, if we threw a TABLE 2 pair of dice many times and recorded the results of each throw, we would get a certain distribu- ,itdfeep ele;,au fe tion table of results. If this table were com- i , intradifferences per element; m, actual fre- pared with the theoretical table for the same num- encies; m, theoretical frequencies) ________ ber of dice and throws, we could determine X2 and I i I P. If P indicated a significant departure from I II III IV V VI the theoretical, we would be justified in doubt- i i m m m-m (M-mf (M-m) ing the honesty of our dice, or of our throwing, m or of both. One of the dice might not be truly cubical, or its density not uniform, or it might 0 .... 477 455 22 484 1.064 have some sticky substance on one face, or some 1 .... 175 211 -36 1296 6.142 other disturbing factor might be present. 2.... 44 36 8 In the present instance X2 = 13.003. Consult- 3.... 9 3 6 225 5.797 ing appropriate tables,20 we find that P = 0.015. 4.... 1 0 1 'Whether we adopt the five per cent or the one per tal.. 706 705 13.003 cent level of significance, P shows that the ac- tual distribution of intradifferences departs significantly from the theoretical. If, however, Now X2 of itself tells us nothing additional we adopt the one-tenth of one per cent level, our what we know by inspection of the data, both actual data do not depart significantly from the tual and theoretical, themselves.'7 It is theoretical. The choice of level is arbitrary, ful, hovwever, because from it we determine and no absolute rule can be given for any par- thp,r value, known as P, which tells us'how ticular case. As I shall suggest below,21 the sely the actual data fit the theoretical.18 five per cent rule is adequate, the one per cent aries in value from zero to one; X2 varies in- more than adequate. sely to P from infinity to zero. If X2 is We may conclude, therefore, that, if our theo- o, P is one; the greater x becomes, the retical values are valid, and represent the dis- ser P approaches zero. Now we choose some tribution of intradifferences on the random or chance hypothesis, the actual data indicate that 4The process is cescribed in most textbooks on the intradifferences of informants do not occur tistics. See, e.g., Yule, 1937, 413 ff. at random but have some significance other than Ibid., that of mere chance. Some disturbing factor or Ibid., 413. combination of factors is at work. Driver sug- Yule advises (1937, 423) that the theoretical gests that "such factors migt be error in re- ue for any cell should be at least as large as corin on the part of the tborer; ver when less it should be grouped with one or n e other cells. In the present instance, if we error in response on the part of the informant; not group cells we get in column VI, last line, or true misunderstanding on the part of either value 1/0 = o, which makes it impossible to party. The fact that the questions are never ute X2. Grouping the last two lines gives given in exactly the same words by the ethnogra- =25.317, and hence P = .000014. The effect pe ntoocsosmasta h tmlsi grouping is to reduce X2, so if we err we do so pher on two occasions means that the stimulus is conservatively. not fully controlled."22 Assuming the validity "Clearly the quantities x [i.e., m-m] embody- of our theoretical data, we conclude from the test the information in the data about the dis- that some influence has been at work to affect the pancies between theory and fact" (Yule, 1937, 19Yule, 1937, 425. 20Pearson, 1924, table 12 8P is determined from tables: the most conveni- 21Since N- 500; see below, P. 487. for determ-aining it directly is Pearson, 1924, 22 le 12. We must first determine the number of 22Driver, 1938, 212. Cf. Kroeber in Barnett, ,rees of freedom, however. For a table of the 1937, 202: "The possible causes of the differ- we are using in this and the next section, ences in the twso theoretically identical lists numnber of degrees of freedom is one less than will be obvious: questions so worded as to be am- number of cells (counting grouped cells as a biguously construable; indifference, fatigue, gle cell). The nS of Pearson's table is the partial ignorance, or loss of memory by one or ,as the number of cells, that is, it is one the other informant; lack*of concreteness or than the number of degrees of~ freedom. cross-checking by the recorder, etc." 472 ANTHROPOLOGICAL RECORDS TABLE 3 TABLE 4 Galice Toliowa I 11111I IV V VI VIII II III IV V VI VI o rn i i-rn (ir) (rn rn a rn-rn (M-rn)__ 1. 162 78 19315 225 2.885 l..17I7 89 93 4 16 0.1 2.. 177 85 18 -67 4489 52.812 2.. 192 100 18 -82 6724 '67. 3.. 126 60 82 22 484 8.067 3.. 131 68 82 14 196 2. 4.. 93 45 56 11 121 2.689 4.. 94 49 56 7 49 1 5.. 57 27 35 8j 64 2.370 5.. 46 2435 11 121504 6.. 73 35 49 14 196 5.600 6.. 64 33 49 16 256 7. 7.. 89 43 64 21 I 441 10.256 7. 85 44 64 420 400 9. 8.. 19 9 6I -3 1 9 1.000 8.. 15 8 6 -2 4 0.115 9.. 91 44 69 25 625 14.205 9. 4 49 69 20 4008T 10.. 134 64 71 7 49 0.766 10.. 121 63 71 8 64 1.0 11. 938 40 2 4 0.105 11.. 85 44 40 -4 16 0.3 12.. 104 50 31 -19 361 7.220 12.. 56 29 31 2 4 0.1 13.. 132 63 50 -13 169 2.683 13.. 94 49 50 1 1 0. 14.. 124 60 42 -18 324 5.400 14.. 117 61 42 -19 361 5.9 Total 1460 701 706 116.058 Total 13661 710 706 10.l TABLE S TABLE 6 Yurok . __Hupa I II III IV V VI VII I II1111 IV V VI VII~ 2 o rn rnh -rnM (_-r)2 (irn rn i i-rn (-rn) 1.. 322 97 93 -4 16 0.165 1L. 315 113 93 -20 400 3.5 2.. 226 68 18 -50 2500 36.765 2.. 88 32 18 -14 196 64 3.. 240 72 82 10 100 1.389 3.. 222 80 82 2 4 0.01 4.. 208 62 56 -6 36 - 0.581 4.. 196 71 56 -15 225 3.11 5.. 105 32 35 3 9 0.281 5.. 105 .38 35 -3 9 0. 6.. 84 25 49 24 576 23.040 6.. 84 30 49 19 '361 12. 7. 155 47 64 17 289 6.149 7. 120 43 64 21 441 101. 8.. 76 23 6 -17 289 12.565 8.. 56 20 6 -14 196 9. 9.. 142 43 69 26 676 15.721 9.. 138 50 69 19 361 7. 10.. 203 61 71 10 100 1.639 10. 166 60 71 11 1212 11.. 142 43 40 -3 9 0.209 11. 123 44 40 -4 -16 0.0 12.. 126 38 31 -7 49 1.289 12.. 100 36 31 -5 25 0'. 13.. 171 51 50 -1 1 0.020 13.. 146 53 50 -3 9 0.1 14.. 137 41 42 1 1 0.024 14* _ ___ Totall 2337 1703 7061 ___ 99.837 Total 11859f 670 1664 _____ *Ornitted. Key to syrnbols used in above tables.-- 11 Death Stub: ~~~~~~~~~~~12 Social stratification, war 1 Subsistence 13 Sharnanisrn 2 Houses 14 Cerernonies 3 Navigation, technology, weapons Clm edns 4 Body and dress 51 I WevngD o Origi-nal urnreducedpt freque-nici of-P CULTURE ELEM. DISTRIB.: XLV--CBRETIEN: RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS 473 responses of the informant or the notes of the exactly comparable to the old. The frequencies ethnographer. Statistical methods do not go be- thus obtained are given in column III of the nd this: they do not tell us what the disturb- tables 3-6. For this investigation they are the ng factor is, but they do establish a disturbing "theoretical" values. They are the values which actor. I am not prepared to say how the reli- Driver's 706 element list would show if it cor- ility of the individual elements (which was responded exactly to the original lists, so far he object of Driver's investigation at this as distribution of types of elements is con- int) is affected by the results of the X2 test; cerned.2" Using the actual and "theoretical" fre- merely point out what it shows. One fact ap- quencies, I have computed the X2 values for each ars established, however. Driver's statement23 of these tables 3-6. Suimmarized they appear in hat "'the accumulation of differences is there- table 7. re mainly due to unknown factors whose cumula- ive effect produces distributions similar to TABLE 7 ohse of coins or dice" is clearly wrong. X2 p REDUCED ELEMENT LISTS Galice o .. 116 258 .000 000 For my second illustration I also etaploy the Yurok . 99.837 .000 000 ist of 706 elements. My previous problem was to Hupa.55.675 .000 000 termine whether intradifferences were distribu- d by chance. Now I seek to determine whether A word of comment is necessary on the zero values e 706 element list is a good sample of the data of P. They do not mean that P is zero; they ex- om which it was drawn. The problem is of im- press the limits of Pearson's table from which rtance because reduced element lists are not they were extracted. This table does not go be- frequent in the statistical treatments found yond the sixth decimal place: hence the values of the CED series. For example, in the North- P are less than 0.0000005. In other words, P st California study, "only those elements were does not reach even the one-tenth of one per cent ed which were represented by entries from at level of significance for any of these tables. ast five tribes";24 in the Gulf of Georgia Je can give a more precise meaning to our results. ish study, "all universal pluses and minuses The chances of Driver's 706 element list's aris- .were disregarded, as were pluses and blanks, ing by random sampling from the statistical popu- nuses and blanks, and those showing less than lations from which it was drawn are less than ve plus and minus symbols."25 The question five in ten million. As random samples of the ether reductions of this sort are legitimate Galice, Tolowa, Yurok, and Hupa data, Driver's uld seem to be important, and here I hope to 706 element list is extremely bad. ed some light upon it. Of course, Driver did not use his list to in- As I have already stated above, the list of tercorrelate these four tribes. I chose to ex- elements was compiled by omitting all ele- amine it because the data were conveniently avail- ts for which there were not responses from able. But I have illustrated, I hope, the danger four tribes. In table 1 of his "reliability" of arbitrary reductions of element lists, pro- Say26 Driver has grouped the elements of the vided, of course, that the original unreduced educed lists into various categories, with lists are actually good samples of the cultures frequencies of each tribe for each category. with vwhich they are concerned. This point will his table 227 he has done the same for the be touched upon again.29 t of 706 elements; here the frequencies are same for all four tribes: e.g., each tribe 93 elements classed as "subsistence," etc. FOURFOLD TABLES tables 3-6 on facinglpage, I give the compari- of the original data with the reduced. Be- For my third and final illustration and appli- e any comparison could be made, however, it cation of the X2 test, I shall use the fourfold necessary to reduce the data of Driver's tables upon which coefficients of association are le 1 proportionately to the size of the 706 based. I shall employ three bodies of data, ment list. For example, if Galice has 134 chosen to illustrate the problems which arise ments dealing with birth and mienstruation out when we deal with these tables. First, however, a total of 1460, in reduced form it would I must discuss in more general terms X2 as applied e 64 in a table of 706, if the new table were to fourfold tables. 3 28 23Driver, 1938, 212. 25To make a strictly comparable reduction we 24Driver 1939, 426. should see to it not only that the categories be 25 ' ' * ~~~~~~~~reduced proportionately, but likewise the pluses 25Barnett, 1939, 225. and minuses f~or each category. I have not done i6Driver, 1938, 214. so because I can make my point without it. l Ibid. 29See pp. 486fT. below. 474 ANTHROPOLOGICAL RECORDS Introductory It will be noted that the marginal frequencies of this table are the same as those in the table. In this type of problem the fourfold table of actual data. In fact, we have derived the supplies *the actual data; for the theoretical we theoretical frequencies from the marginal data. employ the hypothesis of statistical independence. Having now a set of actual and a set of theoreti I shall illustrate by an example. In Driver's cal frequencies, we can determine -2. Northwest California study we have for the pair For the fourfold table, however, we have a of tribes Chilula-Wiyot a table which I give as sor thi elimiates henece of d tbe830 short cut which eliminates the necessity of de-~. table 8. termining the theoretical frequencies. It can TABLE 8 be shown algebraically32 that for these tables Chilula-Wiyot X= N(ad-bc) (a+b)(c+d) (a+c) (b+d) a b a+b It is useful, nevertheless, to determine m for 463 285 748 this reason: the x2 function is a continuous f c d c+d tion; the fourfold table is discontinuous: heno c d c+d the value of x2 obtained from the formula above 352 1 846 ' 1198 l is only an approximation to the real value. Yat has introduced a correction which overcomes thi approximate character.33 He subtracts 1/2 frox a+c b+d 14 those cell values which exceed their independen values, and adds 1/2 to those which fall short. 815 1131 1946 In the Chilula-Wiyot example above, we should subtract 1/2 from the a and d values and add it These are the actual frequencies. What would to the b and c values, with this result: they be if Chilula and Wiyot were statistically independent, that is, if there were no signifi- 1946[(462.5 x 845.5) - (285.5 x 352.5)] cant association between them? To determine 748 x 1198 x 815 x 1131 this we calculate a value known as m, which is This procedure is complicated, but fortunately the smallest cell value, of the four values have again a short cut. For the corrected fo a b c d, which the table would show if statisti- or x2 as it is called, we may use this formul cal independence obtained.3i To get this we c take the smaller of the two marginal frequencies N(ad-bc?N/2) (a+b) and (c+d), multiply it by the smaller of (a+b)(c+d)(a+c)(b+d) the two marginal frequencies (a+c) and (b+d), and divide the product by N. In the present instance The sign of N/2 to be used is determined as fo3 it is lows: (1) determine m; (2) choose the plus sigu (a+b)(a+c) 748 x 815 (a) if m represents b or c and is less than th N 1946 313. actual frequency of the cell which it represen (b) if m represents a or d and is greater than; This is the "independence" value of the cell at actual frequency of the cell which it represen the intersection of the (a+b) row and the (a+c) (3) choose the minus sign: (a) if m represents column. The other cells are readily filled up or d and is less than the actual frequency ofl by subtracting 313 from the marginal values. 'e cell which it represents; (b) if m represents then have a series of hypothetical values which or c and is greater than the actual frequency. I give in table 9. the cell which it represents. These rules are simpler to employ than to describe. I do not TABLE 9 find the formula or the rules in any textbook, Chll-i ot- yohtcl.needne but they are readily derived algebraically frX Chilula-"diyot, Hypothetical Independence- bu thyaeraiydrvdagbaclyf Values the original X2 formula. I shall show below34, 30Driver, 1939, 430. a |b |a+b 31See Fisher and Yates, 1938, 3. 313 435 748 32See Pearson,' 1924, xxxiv; Fisher, 1938, 9 It should be noted that Pearson is wrong in r c d c+d spect to the number of degrees of freedom of fourfold tables. Hence using his tables is 502 696 1198 possible. His n* = 4 means three degrees of freedom, but' actually there is only one. Yule aL+c . b+d N|F 1937, 534-535, gives the appropriate table. i 815 | 1131 | 1946 l ~~33Fisher aLnd Yates, 1938, 3; Fisher, 1938,9 CULTURE ELEM. DISTRIB.: XXV--CBRETIE': RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS 475 hat the simplest procedure is to calculate X2 and If the value of X2 obtained from the data is hen to calculate Xc only when X2 falls below 15. less than the value given in table 10 for the If m is less than 10, it is desirable to em- level which we choose, we can state as our con- ioy a special device which will be mentioned be- clusion that the hypothesis of independence is ow in connection with the FUrer-Haimendorf not disproved, that is, the assumption that the ata.35 Since, however, I shall try to show two entities compared are unassociated is not at ethnological data of the type where this de- disproved. If, however, the value of X2 is *ce becomes appropriate are not suitable for greater than that of the level which we choose, e statistical method, I shall omit detailed we can assert that a significant association escription. exists. The me.aning of this interpretation will In the examples given under the two heads become clearer in the examples which now follow. river's Distribution of Intradifferences" . 470) and "Reduced Element Lists" (p. 473), I ye determined P in every instance. This is Northwest California ot necessary, however. It will be sufficient f we compare the value of X2 which we obtain For my first example I have drawn on the ma- ith the values which it takes at the various terial of Driver's study of Northwest Califor- vels of significance. These are given for the nia. 37 Driver himself did not treat this ma- urfold table in table 10.36 terial statistically; the quantitative results TABLE 10 given in appendices 1 and 2 of his study are by Kroeber. I have taken the a, b, c, d values X2 Table given in appendix 2, and have computed Q6, N, X2 and X_; my results appear in the following table Level of sign_ificance P_* X2 11. 5 per cent level . 0.05 3.841 1 per cent level .0.01 6.635 0.1 per cent level 0.001 10.827 TABLE 11 Northwest California .~~~~~~~~~~~~~~ . Tribes Q2 6 N X2 X lowa-Yurok 2 . . . . . . . . . . . . . . . . . 0.77 0.67 2103 458 456 Yurok 1 . . . . . . . . . . . . . . . . . .68 .57 2156 1 327 325 Karok2 ............. . | .64 .54 2058 263 262 Karok 1 . . . . . . . . . . . . . . . . . .65 .54 2008 257 261 Hupa 2 . .7 . . . . . . . . . . ...........3 .64 1891 355 353 Hupa 1 . . . . . . . . . . . . . . . . . .70 .60 1890 317 316 Chilula ............. .. 7. .72 .61 1975 351 349 Wiyot . . . . . . . . . . . . . . . . . . .66 .55 2142 291 289 Van Duzen ..... . . . . . . . . . . . .59 .48 2123 204 202 Chimariko . . . . . . . . . . . . . . . . .49 .37 1848 103 1 102 Sinkyone 1 . . . . . . . . . . . . . . . .59 .48 2146 222 1 221 Mattole . . . . . . . . . . . . . . . . . .45 .34 2168 110 109 Sinkyone 2 . . . . . . . . . . . . . . . .29 .22 1579 32 31 Coast Yuki . . . . . . . . . . . . . . . .00 .00 1987 0.001 0.000 Kato . . . . . . . . . . . . .01 .01 2106 0.038 Q.057 ok 2-Yurok 1 . . . . . . . . . . . . . . . . .93 .88 2127 995 992 Karok 2 . . . . . . . . . . . . . . . . . .89 .82 2008 749 747 Karok 1 . . . . . . . . . . . . . . . . . .79 .68 1947 446 444 Hupa 2 ..... . . . . . . . . . . . . .90 .83 1864 724 722 Hupa 1 . . . . . . . . . . . . . . . . . .88 .80 1954 691 689 Chilula ............. .. 7. .73 I .61. 1923 337 339 Wiyot . . . . . . . . . . . . . . . . . . .70 .59 2079 I 332 330 Van Duzen ............... . .44 .34 2049 92 92 Chimariko..32 .22 1786 38 37 Chimriko. . . . . . . . . . . . . . . . .2 .2 18 83 Sinkyone 1 . . . . . . . . . . . . . . . .35 .26 2097 65 65 Mattole . . . . . . . . . . . . . . . . . .19 .14 2125 16 16 Sinkyone 2 . . . . . . . . . . . . . . . -0..12 -0.08 1586 4.52 4.29 35See fn. 44. 36Fisher and Yates, 1938, 27. 37Driver, 1939. 476 ANTHROPOLOGICAL RECORDS Table 11 (Continued) Tribes Q 2 Q N Yurok 2-Coast Yuki . . . . . . . . . . . . . . . -0.31 -0.22 2026 34 34 Kato . . . . . . . . . . . . . . . . . . - .31 - .22 1994 48 47 Yurok 1-Karok 2 . . . . . . . . . . . . . . . . .90 .83 2098 827 824 Karok 1 ..81 .68 1947 446 44 Hupa 2 . . . . . . . . . . . . . . . . . .90 .83 1929 734 732 Hupa 1..89 .83 1931 733 731 Chilula . . . . . . . . . . . . . . . . . .71 .59 2005 330 328 Wiyot . . . . . . . . . . . . . . . . . . .62 .50 2125 241 240 Van Duzen . . . . . . . . . . . . . . . . .72 .59 2123 351 349 Chimariko . . . . . . . . . . . . . . . . .26 .16 1837 27 26 Sinkyone 1 . . . . . . . . . . . . . . . .32 .26 2147 57 57 Mattole . . . . . . . . . . . . . . . . . .09 .07 2171 4.08 3. Sinkyone 2 . . . . . . . . . . . . . . . - .14 - .10 1646 7.04 6.7 Coast Yuki . . . . . . . . . . . . . . . - .39 - .26 2027 61 61 Kato . . . . . . . . . . . . . . . . . . - .30 - .22 2102 46 45 Karok 2-Karok 1 . . . . . . . . . . . . . . . . .92 .84 1995 821 818 Hupa 2 ..89 .81 1887 687 685 Hupa 1 . . . . . . . . . . . . . . . . . .89 .81 1887 681 679 Chilula . . . . . . . . . . . . . . . . . .69 .59 1956 314 312 Wiyot . . . . . . . . . . . . . . . . . . .58 .48 2017 203 202 Van Duzen . . . . . . . . . . . . . . . . .57 .43 2035 167 166 Chimariko . . . . . . . . . . . . . . . . .54 .40 1750 122 121 Sinkyone 1 . . . . . . . . . . . . . . . .39 .31 2031 81 80 Mattole . . . . . . . . . . . . . . . . . .22 .16 1956 22 21 Sinkyone . . . . . . . . . . . . . . . . - .04 - .03 1574 0.54 0.4 Coast Yuki . . . . . . . . . . . . . . . -.29 - .22 1876 30 29 Kato. - .24 - .16 2008 28 28 Karok 1-Hupa 2..78 .67 1938 428 426 Hupa 1 . . . . . . . . . . . . . . . . . .84 .72 1846 483 481 Chilula . . . . . . . . . . . . . . . . . .77 .66 1944 415 413 'Wiyot . . . . . . . . . . . . . . . . . . .64 .52 1950 241 239 Karok 1-Van Duzen . . . . . . . . . . . . . . . .56 .43 1967 163 162 Chimariko . . . . . . . . . . . . . . . . .61 .45 1712 161 160 Sinkyone 1 . . . . . . . . . . . . . . . .45 .31 1965 87 86 Mattole . . . . . . . . . . . . . . . . . .24 .16 1984 25 24 Sinkyone 2 . . . . . . . . . . . . . . . - .03 - .02 1550 0.31 0. Coast Yuki . . . . . . . . . . . . . . . - .29 - .16 1838 27 26 Kato . . . . . . . . . . . . . . . . . . - .18 - .14 1853 14 14 Hupa 2-Hupa 1 . . . . . . . . . . . . . . . . . .99 .97 1876 1323 1320 Chilula . . . . . . . . . . . . . . . . . .86 .76 1791 544 542 Wiyot . . . . . . . . . . . . . . . . . . .58 .48 1862 182 181- Van Duzen . . . . . . . . . . . . . . . . .61 .48 1786 174 173 Chimariko . . . . . . .48 .34 1641 90 89 Sinkyone 1 . . . . . .46 .37 1879 109 108 Mattole . . . . . . . . . . . . . . . . . .13 .09 1803 6.50 6 Sinkyone 2 . . . . . . . . . . . . . . . .17 .12 1383 8.31 7. Coast Yuki . . . . . . . . . . . . . . . - .43 - .31 1645 64 63 Kato . . . . . . . . . . . . . . . . . . - .24 - .16 1867 27 27 Hupa 1-Chilula . . . . . . . . . . . . . . . . . .92 .83 1879 729 726, Wiyot . . . . . . . . . . . . . . . . . . .66 .55 1856 258 257 Van Duzen . . . . . . . . . . . . . . . . .66 .50 1874 214 212 Chimariko . . . . . . . . . . . . . . . . .47 .34 1633 84 83 Sinkyone 1 . . . . . . . . . . . . . . . .49 .40 1878 123 124 Mattole . . . . . . . . . . . . . . . . . .22 .16 1892 19 1 Sinkyone 2 . . . . . . . . . . . . . . . .12 .08 1448 4.28 4. Coast Yuki . . . . . . . . . . . . . . . - .29 - .22 1719 29 28 Kato . . . . . . . . . . . . . . . . . . - .18 - .14 1848 14 14 Chilula-Wiyot . . . . . . . . . . . . . . . . . .59 .48 1946 200 199 Van Duzen ..76 .64 1871 354 352- Chimariko . . . . . . . . . . . . . . . . .54 .40 1711 125 124 Sinkyone 1. .............. .55 .45 1855 165 164. Mattole. ................ .30 .16 1862 18 1 Sinkyone 2. .............. .42 .31 1529 63 2 Coast Yuki. .............. -.09 - .06 1721 2.47 2.i Kato . ................. 0.23 0.17 | 1834 22 12 CULTURE ELEM. DISTRIB.: XXV--CHRETIEN: RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS 477 Table 11 (Concluded) Tribes I ]2 Q 6 N c Wiyot-Van Duzen . . . . . . . . . . . . . . . . .38 .26 2189 74 74 Chimariko . . . . . . . . . . . . . . . . .55 .40 1837 136 135 Sinkyone 1 . . . . . . . . . . . . . . . .53 .43 2146 173 172 Mattole . . . . . . . . . . . . . . . . . .59 .45 2215 210 209 Sinkyone 2 . . . . . . . . . . . . . . . .21 .15 1639 15 14 Coast Yuki . . . . . . . . . . . .. . . . .07 - .05 1980 1.87 1.72 Kato . . . . . . . . . . . . . . . . . . .01 .01 2100 0.08 0.05 [Van Duzen-Chimariko . . . . . . . . . . . . . . .67 .52 1827 218 216 Sinkyone 1 . . . . . . . . . . . . . . . .79 .66 2151 463 461 Mattole . . . . . . . . . . . . . . . . . .61 .48 2136 206 204 Sinkyone 2 . . . . . . . . . . . . . . . .65 .50 1605 181 179 Coast Yuki . . . . . . . . . . . . . . . .21 .14 1941 14 14 Kato . . . .41 .31 2071 82 81 himariko-Sinkyone 1..74 .59 1839 298 296 lMattole . . . . . . . . . . . . . .60 .45 1865 166 164 Sinkyone 2 . . . . . . . . . . . . . . . .63 .48 1456 146 144 Coast Yuki . . . . . . . . . . . . . . . .38 .26 1735 49 48 Kato . . . . . . . . . . . . . . .37 .26 1819 54 53 inkyone 1-Mattole . . . . . . . . . . . . . . .64 .52 2257 261 260 Sinkyone 2 . . . . . . . . . . . . . . . .70 .57 1658 242 240 Coast Yuki . . . . . . . . . . . . . . . .32 .22 2001 32 32 Kato . . . . . . . . . . . . . . . . . . .45 .34 2140 112 111 ttole-Sinkyone 2 . . . . . . . . . . . . . . . .44 .31 1684 72 71 Coast Yuki . . . . . . . . . . . . . . . .42 .26 2089 73 72 Kato..3 .1 2185 976 Sio.... . . .'. . . . . . .43 .31 215 97 96 inkyone 2-Coast Yuki . . . . . . . . . . . . .83 .68 1704 385 382 Kato . . . . . . . . . . . . 72 .59 1716 269 267 oast Yuki-Kato . . . . . . 0.86 0.72 2042 540 538 From these results we may draw certain obser- reach the five per cent level but not the one tions. First of all we may notice that when per cent, and only seven fail to reach any level eir values exceed 12 there is no important of significance whatever. Since we are testing fference between x2 and Xc . Although I give the hypothesis of statistical independence, we e figures only to the nearest digit, I have can say that this hypothesis is definitely dis- mputed them more closely, and find that the proved for 107 of the pairs of tribes; and that imum difference between X2 and Xc is that for these 107 pairs show significant association when ok 2-Hupa 1, which is 3.67. Most of them are tested by the most rigorous level of significance. ch less, as will be readily seen by comparing It is interesting to examine more closely the e two coluimns in the table. These differences 13 pairs which do not reach the one-tenth of one e not important because the values of both X2 per cent level. Since they-all have X2 values d Xc are well beyond the critical value for below 15, I use the Xc values and carry them to level of significance in which we might be two or three decimal places. The pairs fall terested. It would seem that we can draw a naturally into three groups represented by tables rking rule from these results: let us choose 12 to 14 below. to be on the safe side, and say that when X2 ceeds it we do not need to apply Yates's cor- TABLE 12 tion. The procedure, then, will be to com- Pairs of Tribes Which Fail to Reach te X2 by the ordinary formula, and replace Level of Significance value thus obtained by Xc only when it falls --- - low 15. This will save a great deal of labor Tribes Q2 | 6 N c zch brings no gain commensurate with the added Tolowa-Coast Yuki.... 0.00 0.00 1987 0.000 fort. Tolowa-Kato .......... .01 .01 2106 .057 In the second place, we should notice that Karok 2-Sirnkyone 21.r. 9A... - .04 - .03 1574 .46 y 13 out of 120 results fail to reach the one- Karok 1-Sinkyone 2 .......... - .03 - .02 1550 .. 24 gth of one per cent level of significance. Of Chilula-Coast Yuki .......... - .09 - .06 1721 2.30 pse 13, two reach the one per cent level, four Wiyot-Coast Yuki . ........- .07 - .05 1980 1.72 I ~~~~~~~Wiyot-Kato .....0.011 0.011 2100 0.05 478 ANTHROPOLOGICAL RECORDS TABLE 13 to +0.13, Q6 from -0.08 to +0.09; the values still cluster about zero. The. one-tenth of Pairs of Tribes Wdhich Reach the S Per Centstl cute autzr. Teo-enhfo Pairs_____ of_Tribes ____ Reach the Per Cent per cent level extends them a little more: QA . Level but not the 1________t from -0. 14 to +0.17, Q6 from -0.10 to +0.12. Tribes Q_2 Q6 N X2 Now it is interesting and significant that no, Yurok 2-Sinkyone 2. I-0.12 | -0.08 1 586 4.i 29 other coefficients of the entire 120 fall wit Yurok 1-Mattole . ............. 09 .07 2171 3.90 these ranges; in other words, all the rest Hupa 2-Mattole . .... 13 ..09 | 1803 61i .925 with X2 in indicating significant association. Hupa 1-Sinkyone 2...1 0.12 0.081 1448 4.05 The results of both coefficients and X are Eup lSinyoe 2.......... 10.2 .. 008 148 i.0paarallel for the Northwest California data. i we should remind ourselves that in these data. N, that is, a+b+c+d, is large: it ranges from 1383 (Hupa 2-Sinkyone 2) to 2257 (Sinkyone 1- Pairs of Tribes Which Reach thel Per Cent Mattole). As we shall see presently,38 the Level but not the 0.1 Per Cent of N is important; hence we cannot say what Tribes Q, 6 N X2 otherwise we should be tempted to say, viz., Tribes______ _ _ _ Q2 |__ Q6 c X2 is a less convenient method of measuring Yurok 1-Sinkyone 2.. -0.14 -0.10 1646 6.76 sociation than q2 or Q6. Hupa 2-Sinkyone 2... 0.17 0.12 1383 7.98 In table 15 I give the Q6 values for these data. I star those values whose X2 value fai to reach the five per cent level of signific. If we adopt the five per cent level, we are It will be seen that they in no way affect interested to observe that the coefficients, both inferences which could be drawn from this ta q2 and q6, show the same results as X2 , namely, The tribes group themselves here precisely as the absence of significant association. q2 they do in Kroeber's grouping.39 ranges from -0.09 to +0.01, Q6 from -0.06 to +0.01. These values are close enough to zero to indicate lack of association. The one per cent See P. 482. level extends the range slightly: q2 from -0.12 Kroeber in Driver, 1939, 425-426. TABLE 15 Q6 Values for Northwest California Tribes To Y2 Yl K2 Kl H2 Hi Cl Wy- VD Cm S1 1Mt S2 CY KaD Tolowa ......... 67 57 54 54 64 60 61 554837 48 34 22 00* 0 Yurok 2 .67 88 82 68 83 80 61 59 34 22 26 14 08 22 2 Yurok 1 .57 88 83 68 83 83 59 50 59 16 26- 07 10 262 Karok 2 .54 82 83 84 81 81 59 48 43 40 31 16 0 3* 22 1, Karok 1 .54 68 68 84 67 72 66 52 43 45 31 16 02* 16 1 Hupa 2 .64 83 83 81 67' 97 76 48 48 34 37 09 12 31 L Hupa 1 .60 80 83 81 72 97 83 55 50 34 40 16 08 22 I Chilula .61 61 59 59 66 76 83 48 64 40 45 16 31 06* 1 Wiyot. 55 59 50 48 52 48 55 48 26 40 43 45 15 05* 0 Van Duzen .48 34 59 43 43 48 50 64 26 52 66 48 50 14 3 Chimariko .37 22 16 40 45 34 34 40 40 52 59 45 48 26 Sinkyone 1 . 48 26 26 31 31 37 40 45 43 66 59 52 57 22 3 Mattole .34 14 07 16 16 09 16 16 45 48 45 52 31 26 Sinkyone 2. 22 08 10 03* 02* 12 08 31 15 50 48 57 31 68 Coast Yuki. 00* 22 26 22 16 31 22 06* 05* 14 26 22 26 68 7 Kato .01* 22 22 16 14 16 14 17 01* 31 26 34 31 59 72 Decimal points are omitted. Negative coefficients are underlined. Starred values indicate pairs whose X2 values fail to reach the five per cent level of significance. CULTURE ELEM. DISTRIB.: XXV--CBRETIEN: RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS 479 Gulf of Georgia Salish level, and only 31 all levels. This is a strik- ing contrast to the Northwest California results. For my second example I use the data of Bar- 'de note, however, that whereas in the latter the nett's Gulf of Georgia Salish study.0 The re- value of N was always large (ranging from 1383 sults here are not so neat as they were for the to 2257), here it is often much smaller, ranging Northwest California material, as will be readily from 31 (Slaiamun 1-Slaiamun 2) to 1306 (Sechelt- seen if we inspect table 16. Squamish). .As we shall see presently,41 if we TABLE 16 Q6 Values for Gulf of Georgia Salish Tribes Pe Cx Kw Ho Ki Si S2 Se Sq WS ES Cw Na Pentlatch 83 61 09* 17 28* 13* 06* 00* 09* 22* 02* 06* Comox .83 35 28 31 66 35 22 02* 08* 09* J-* '13 Kwakiutl .61 35 11* 08* 60 05* 19* 16* 11* 22* 14* 23* Homalco .09* 28 11* 78 66 75 71 36 26 03* 05* 05* Klahuse .17 31 08* 78 62 77 35 08* 11 16* 11 01* Slaiamun l28* 66 60 66 62 94 70 34 29* 05* 05* 23 Slaiamun 2. 13* 35 05* 75 77 94 61 38 23 26* 05* 02* Sechelt .06* 22 19* 71 35 70 61 40 17 02* 03* 16 Squamish .00* 02* 16* 36 08* 34 38 40 54 05* 20 32 West Sanetch 09* *08* 11* 26 11 29* 23 17 54 60 72 52 East Sanetch ... 22* 09* 22* 03* 16* 05* 26* 02* 05* 60 65 55 Cowichan ..... 02* 05* 14* 05* 11 05* 05* 03* 20 72 65 88 Nanaimo . ....... .06* 13 23* 05* 01* 23* 02* 16* 32 52 55 88 Decimal points are omitted. Negative coefficients are underlined. Starred values indicate pairs whose X2 values fail to reach the five per cent level of significance. e starred values of the preceding table repre- classify the Salish figures by the size of N, a nt those pairs of tribes for which the X2 values pattern of meaning begins to emerge. Accordingly il to reach the five per cent level of signifi- I have prepared tables 17-20 on that basis. ce. It will be noted that these starred Q's equently are larger than other Q's not starred. TABLE 17 situation as presented in this table is very Gulf of Georgia Salish: N is less than 100 satisfactory, and at first glance interpreta- on is confused. As it will be necessary later (a) x2 fails to reach the five per cent level of break down the Salish results into categories, significance: do not give a table comparable to table 11 for N rthwest California. The information all ap- Tribes 6 X ars, however, sooner or later in tables 17-20 Pentlatch: Slaiamun 1 0.36 0.28 80 1.97 low. I have computed both X2 and X' , but in Kwakiutl: Slaiamun 2 .. .07 .05 91 .012 ew of what table 11 showed us, it seemed un- East Sanetch... - .29 - .22 78 1.08 cessary to give both values. It happens that Slaiamun 1: West Sanetch. .37 .29 97 2.87 give only Xc, but it would have been adequate East Sanetch .06 .05 60 .003 I had computed only x2 for those which reached Cowichan . .;. .07 .05 72 .001' or more. In the course of this section the Nanaimo . 0.31 0.23 69 1.02 rm X2 will refer to X2 The first thing that strikes us upon examin- (b) x2 reaches the five per cent level of sig- the Salish results is the large number of nificance: _____ _ s of tribes for wmhich X2 fails to attain one Tribes Q2 Q 6 N | 2 more levels of significance. Of a total of Tie 2 Q6 N X pairs, 38 fail to reach any level, 7 reach Comox: Slaiamun 1. . 0.76 0.66 77 14 five per cent level but not the one per cent, Kwakiutl: Slaiamun 1. .72 .60 58 7.98 1two the one per cent but not the one-tenth Homalco: Slaiamun 1 . .76 .66 96 18 one per cent. Thus only 40 pairs reach a Klahuse: Slaiamun 1. .74 .62 88 15 two the one per centbutnottheone-tenth HoSlaiamun 1: Slaiamun 2 .96 .94 31 15 oBarnett, 1939. Sechel ..80 .70 94 21 1 Table 25 below, p. 4,82. Squamish ...|0.441 0.34 99| 4.11 480 ANTHROPOLOGICAL RECORDS TABLE 18 (b) X2' reaches the f ive per cent level of sig- Gulf of Georgia Salish: N lies between 100 and 500 nificance:___ (a) X2 fails to reach the f ive per cent level of Tribes Q significance: C__ __ _ ___ Como x: Homal c o.. ....0.36 0.28 668 20 Tribes_______ _ -2 6 N - 2 Homalco:Klahuse...... .86 .78 765 248 Tribes Q2 Q6 N X2 ~~~~~~~~ ~ ~~Sechelt... ..... .81 .71 737 180 Pentlatch:Slaiamun 2 ... 0.16 0.13 280 1.51 Squamish .. ..... .45 .36 739 40 East Sanetch .. .. .28 .22 166 2.88 West Sanetch ..... .33 .26 712 19 Comox: East Sanetch. .12 .09 173 .379 Klahuse:West Sanetch.. .15 .11 956 5.26 Kwakiutl:Homalco.. -.. .14 .11 136 .384 Sechelt:West Sanetch.. .22 .17 966 11 Klahuse....... .12 .08 136 .268 Squamish:West Sanetch. .64 .54 915 57 Sechelt...... - .23 - 19 140 1.48 West Sanetch:Cowichan. .83 .72 761 102 Squamish .. .... .21 .16 137 1.06 Nanaimo.......0.62 0.52 860 51 West Sanetch .. .. .15 .11 136 .471 Cowichan.. .... - .18 -.14 110 .551 TABLE 20 Nanaimo .. ...31 .23 110 1.86 o Homalco:East Sanetch.. - .05 - .03 185 .025 Gulf of Georgia Salish: N is greater than 100 Klahiuse:East Sanetch... - .20 -.16 187 1.50 (a) X2 fails to reach the f ive per cent level Slaiamun 2:East Sanetch - .34 -.26 148 3.78 significance: _____ Cowichan ......- .06 -.05 278 .138 Nanaimo....... .02 .02 288 .000 TrbSQs Q.6 N X Sechelt:East Sanetch... - .03 -.02 193 .063 Pentlatch:Sechelt... 0.08 0.06 1-092 lP Squamish:East Sanetch.. I0.07 10.05 11871 0.053 Squamish .. ..... .00 .00 1087 Cowichan. . - ..03 - .02 1001 (b) X2 reaches the five per cent level of sig- C omo x: Squami sh .. .. .. .02 .02 1193 nificance: __ ____Cowichan... ..... .05 .05 1085 N V nK1ahuse:Squamish.. .... .10 .08 1289 2. Tribes Q.2 Q' N6X Nanaimo.......- .01 - .01 1149 Pentlatch:Kwakiutl .... 0.72 0.61 128 2 1 Sechelt:Cowichan.....-0.03 --0.03 1123 0. Comox:Kwakiutl .. .. .. .44 .35 123 5.56 Slaiamun 2 .. ... .48 .35 293 1 5 (b) X2 reaches the f ive per cent level of Si Homalco:Slaiamun 2 . .. .83 .75 322 90 nificance: __ ____ Klahuse:Slaiamun 2 . .. .85 .77 344 105 Slaiamuri 2:Sechelt . .. .71 .61 338 57 Tribes Q2 Q Squamish .. .... .48 .38 331 19 Pentlatch:Comox .. ...0.91 0.83 1076 41& West Sanetch .. .. .29 .23 332 6.41 Klahuse. ....... .22 .17 1066 13,< West Sanetch:East Sanetch .70 .60 180 16 Comox:Klahuse .. ..... .39 .31 119345 East Sanetch:Cowichan.. .74 .65 184 19 Sechelt........ .28 .22 112Z 2 Nanaimo....... 0.68 0.55 178 13 Nanaimo........ .16 .1.3 1117 6. Klahuse:Sechelt.... .. .46 .35 1286 6~ Cowichan ...... .14 -.11 1118 TABLE 19 Sechelt:Squamish...... .51 .40 1306 90 Gul o Gerga alih:N iesbewen 50 ndNanaimo..-.... .19 -.1.6 1157 5 Gul ofGeogiaSalsh N iesbeteen500an Squamish:Cowichan .. .. .26 .20 1130 81 1000 ~~~~~~~~~~~~Nanaimo....... .40 .32 1158 25 (a) X2 fails to reach theB five per cent level of Cowichan:Nanaimo .....0.94 10.88 11125 257~ significance: _____ Tribes N ~~~~~~~~~~~~~ ~We may extract from tables 17-20 some-int esting compariaons. Consider the following: Pentlatch:Homalco .. .. 0.13 0.09 573 2.05SlimnlWtSaec. 02 N97 Wiest Sanetch .. .. .13 .09 774 2.87 Nanaimo....... .07 .06 823 1.13 Kwakiutl:Nanaimo.......Q6 0. 23 N 110 Comox:West Sanetch ... .10 .08 882 2.05 Both of these pairs fail to reach the 5 per Homalco:Cowichan..... .06 .05 623 .464 level of significance. But these do: Nanaimo . 0.07 0.05 664 .586 ~~~~Klahuse:WJest Sanetch .....q6 0.11 N 9 Comox:Nanaimo .......*q6 0.13 N 1117 CULTURE ELEM. DISTRIB.: XXV--CHRETIEN: RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS 481 If we apply the 1/10 of 1 per cent level we get TABLE 22 .these startling comparisons: Western Indo-China: Intercorrelation of Tribes Kwakiutl:Slaiamun 1 . ...... Q6 0.60 N 58 Tribes Q 6 N __X_ This fails to reach the designated level. But 14:15 . 0.75 22 4.40 these do: 12:8 ..96 20 10.3 Sechelt:West Sanetch ...... Q6 0.17 N 966 12:13 ..65 24 3.15 Pentlatch:Klahuse ...... Qe 0.17 N 1066 8:13........... .76 21 3.99 Many others can be cited where Q6 is less than 4:6............ .60 27 3.06 0.60. Obviously we have an absurd state of af- 4:5............ .67 27 4.17 fairs whenQ6and X2contradict each other. 6:5............ .71 27 4.74 Values of Q6 like 0.23 and 0.29 indicate a de- 17:16 .1 0.94 18 8.00 gree of association better than independence. But according to X2, the hypothesis of inde- TABLE 23 -pendence is not contradicted. How are we to in- Western Indo-China: Intercorrelation of Element terpret such results? Before we attempt to answer this question, let us examine one more Groups and_Tribe set of data. Group: Tribe - 6 N X2 Western Indo-ChinA:14 ......... 0.88 22 7.54 Western Indo-China 15 .......... .91 20 7.91 For my third and final illustration I have 12 .......... .71 20 3.11 selected certain interrelationships from Furer- 8 .......... .43 19 0.49 Haimendorf's article on the tribes of western 13 .......... .40 24 0.65 Indo-China.42 This is one of the earlier sta- 17 .......... .26 18 0.03 tistical studies in ethnology, antedating, at C:12..26 20 0.04 least in respect to publication, all those of 8 ..40 19 0.11 the CED series, as well as a number of others. 13..52 24 1.28 It is in the Czekanowski tradition, and explicit 4 .......... .75 24 3.36 eference to him is made.43 The table of dis- 6 ..61 25 2.33 ributions comprises 27 elements and 17 tribes. . .45 23 0.69 early all possible intercorrelations are made: 17 .......... .59 18 1.12 lement with element, tribe with tribe, group of 16 .......... .26 23 0.06 lements with tribe, language family with tribal 9 ..48 21 0.63 oup. I have selected 49 of these intercorre- 10 ..26 22 0.03 ations for examination, and give the data in 11 .......... .43 22 0.57 ables 21-24. 2 .......... 0.55 24 1.68 TABLE 21 stern Indo-China: Intercorrelation of Elements TABLE 24 Western Indo-China: Intercorrelation of Language Elements Q6 N x2 Families and Tribal Groups 12:3 ............ 0.50 15 0.49 12:25 ........... .73 11 .73 Language family: Q6 N X 3:25. 73 11 .73 Tribal group Austro-Asiatic: I .... 0.88 15 1.03 2:9 ..60 10 .41 Middle and South 2:17 ..71 10 .62 Assam: I .......... - .13 15 .267 2:23. .71 10 .62 II.... .... .13 15 .267 9:17 ..78 15 1.66 ......... . . .60 15 .77 9:23 ........... .50 14 .34 IV ..-.--...- .13 15 .267 17:23 ........... .94 14 3.97 V .- .82 15 2.78 I . 1 5 I2 ...... .34 15 .079 8:13 .......... 0.61 15 0.20 1 . . 34 15 .079 North Assam: I . - .11 15 .71 42 Fiurer-Haimendorf 1934. IV ..I V .88 15 1.03 A43 . 'Arakan Hills and 4Ibid., 4s22. Burma: I ...... ..- .37 15 .000 44Fisher and Yates , 1938, 3-4 , give a special . ..II .......12 15 .237 bthod to be used for determining X2 when m<(10. V.0.... . .096 15 5.91 used this method for the Furer-Haimendorf _____________ .______ _____ ata, but got no results different from those ibtained from Xc In the tables 21-24 I use Xc exclusively, since c ~~~~~~~~all values are less than 154 I do not specifi- 482 ANTHROPOLOGICAL RECORDS cally name the tribes or elements, but use or what we may call for convenience nonsignifi Furer-Haimendorf's numbers by which he refers to cant Q6's. In table 25 I subdivide them accord them.45 Since our interest is in numerical re- ing to N and give their range. The broken 1n sults and not in the tribes or elements, it seems on either side of the center are boundary lines unnecessary to give their names. of levels of significance; but they are exclu- Looking over the X2 column, we find that only sive, not inclusive: thus the area lying bet ten pairs attain the five per cent level of sigz the five per cent lines represents values of , nificance, and of these only four attain the one fcr fourfold tables which do not reach the five per cent level. No pairs reach the one-tenth of per cent level: it is the area outside these cne per cent level. The range of coefficients lines which represents significance. of these significant pairs is from 0.67 to 0.96. Befcre interpreting these figures we should` The rest of the material, all of which fails to remember that a zero value of Q6 means the sa reach even the five per cent level, exhibits the thing as a value of X2 which fails to reach a widest kind of range of Q's, from -0.82 to -0.88. level of significance, that is, lack of sig- Such a striking contradiction between what X2 nificant association. We should take this and Q6 tell us demands an explanation. Such an statement with a grain of common sense. We explanation will be attempted in the remainder would not assert that a Q6 = 0.00 means lack o. of this paper. With it the criticisms of Kluck- association and a Q6 = 0.01 means significant- hohn will be considered and, I hope, answered. association, any more than we would say that Q6 = 0.87 means a greater degree of associatio Discussion than q6 = 0.86. We do not and cannot measure Let us begin by resurveying theproblem. We culture relationships with the accuracy that an. Let us begin by resurveying the problem. We eniermaue natmbl itn46 W have just examined 247 relationships based on engineer measures an automobile piston. We fourfold tables. For each of these we have de- may say, as a working rule, that values which termined X2 and Both of these functions are cluster about zero indicate lack of associatio termined X2 and 46 Both of these functions are No unn otbe2,w bev htfr > . > . 1l ~~~~~~~Now turning to table 25, we observe that fo.,; measures of association; furthermore they are low values of N the range of nonsignificant Q6 related: is very large, even if we use only the five per ad-bc cent level. It is not until N passes the 500 let rhk = V(a-b)(c-d)(a-c)(b-d) mark that these values of Q6 can be said to then X2 = Nr7 cluster about zero. If therefore we regard hk these small values of 16 as indicating no sig-, and Q6 = sin( 2 rhk) nificant association, we can say that X2 at t a five per cent level and Q6 agree in respect to Nevertheless they do not always give parallel re- showing lack of association when N is greater sults. Thus, as we found above, a high Q6 indi- than 500. Furthermore, consultation of tables! cating close association may go with a low X2 11, 19, and 20 will show that no significant TABLE 25 Range of Nonsignificant Values of Q6 -~~~~~~ Indo-China: N1 < 30 I I -0.82 0.88 1 0.96 1 0.96 | Salish: 30 < NK< 100 I - .22 .29 I .34 1 .60 100 < N< 500 I I I - .26 .23 .35 500< Na+b ATe naay use the sane illustration of ;Ne,,roes in. which reduces readily to -;riculture. If there is no significant associa- tion between Negroes arid agriculture in the ad> bc South, we should expect the proportion of Negroes or among the farmers to be the same as the propor- ad-bc> O. tion of NJegroes among those who are not farmers. The relationship ad-bc enters into several 'v.e call this positive association, or simply measures used in ethnology, either as the nuLler- association.55 It is the condition which must ator (r Q2) or as part of the nu-merator hold if rhk and Q,6 are to have positive values" (X2, <6?* Kluckhohn has pointed this out in a On the other hand, if the proportion of statement wvhich is literally accurate. He says Negroes among the farmers was less than their that Q2, Q6, and rhk are based on the product u law o algeraic robablity.2 Thi is tue 5Yule., 1937, 35, derives the product law ff law of algebraic probability. . This is true, our second definition of independence. since we derived ad-bc = 0 from the product law 54 Curiously enough Kluckhohn, while believii' 4,. that the ideas of independence and associatio 4'"When we apply these formulas to ethnological rest on the product law, seems to overlook th# data we are introducing into our comparisons fact that X2 rests on the same foundations as factors which do not rest on the simple quanti- independence and association, for he suggests< fication of the data themselves, but which pro- that "it seems possible that the best statist ceed from complex considerations of statistical cal procedure in historical ethn logy would be. theory" (Kluckhohn, 1939, 359). apply this test of significance lX2T and then, 50hi definition is by Yule, 1937, 35. proceed to make inferences directly from the : ' ' * ~~~~~~~~without using coefficients of association" K 5~This definition is by Yule, 1912, 58O. hohn, 1939, 35S). 52Kluckhohn, 1939, 349.. ~Yule, 1912, 580. CULTURE ELEM. DISTRIB.: XXV--CHRETIEN: RELIABILITY OF STATISTICAL PROCEDURES AID RESULTS 485 proportion in the population, we should assert covert assumptions":60 (1) "That all correla- that not only was there no association between tions (in default of specific evidence to the Negroes and agriculture, but some factor or contrary) are expressions of historico-geographi- group of factors-was at work to keep or dis- cal connections and continuities."811 (2) "That, courage Negroes from being farmers. Algebrai- were it not for these 'historical accidents,' cally, the relation is this and environmental situations, the distributions would be quite at random and without coherence."62 a < +b Or, in other words, "apart from such historical or environmental interrelationship, culture 'which reduces to traits can be independent of each other."63 It N, seems clear that Kluckhohn has transferred bodily acf< bc or the.abstract idea of independence of S and R to ad-bc <0 ethnology without inquiring what ethnological meaning such abstract independence possesses. In that is, ad-bc is negative. We call this nega- other words, the obvious next step is to inter- !tive association or dissociation.56 It is the pret the abstract ideas of independence and as- condition which must hold if rhk and Q6 are to sociation in terms of ethnology. Take our ex- have negative values. X2, being a square, is ample of the ilegroes and farming: if the two are ,always positive. "independent," we do not say that lNegroes are It is interesting to examine the limits of independent of farmers or farmers independent association and dissociation. Suppose S and R of Negroes. V-e seek to interpret the situation. are identical: then b = 0 and c = 0, since t-here We conclude that there are no special factors ,are no elements of S which are not R (that is, either to encourage Negroes to be farmers or to Mno +-), and no elements of R which are not S discourage them. Yet, if I understand his state- (that is, no -+). Furthermore, every element ment of it, Kluckhohn is interpreting ihdepend- of R belongs to S and vice versa. Then rhk = 1, ence to mean that Negroes can be independent of Q6= 1, and X2 = N. This is complete associa- farmers. tion, or complete positive association.57 Thus What ethnological meanings are to be given to as rhk moves from zero to one, association the terms independence, association, and disso- moves from independence to complete identity. ciation? Not being an ethnologist, I am not Suppose, however, that no element of S is an competent to answer this question except in the lement of R, and no element of R is an element crude way that would occur to anyone. Two tribes f S; suppose further that between them they with a very high Q6 would naturally show a large xhaust the element list, that is, not only are number of a and d values as-compared with b and here no a's, there are also no d's. Then rhk c; the inference of historical connection is -1, Q6 = -1, and X2 = N. This is complete hardly extravagant. Likewise if two tribes dissociation or complete negative association.58 showed a very large negative q6, the number of hus as rhk moves from zero to minus one, asso- b and c values would be large as compared with iation moves from independence to complete a and d. The inference of little or no his- issociation. torical connection is not overbold. It is the So far I have been dealing with the terms in- middle values of the scale, the region of "in- ependence and association quite without regard dependence," which need careful examination. In o ethnology. To return to Kluckhohn's criti- my linguistic study referred to above,64 I have isms, it seems to me that at this point his sought to interpret these values for linguistics osition is weak. Speaking of certain "quite as indicating a lack of influence of one language undamental premises which seem to have been ac- upon the other. The positive end of the scale epted almost unquestioningly,"59 he gives, is held to mean a high degree of converging in- ong others, the following "covert or partly fluence, the negative end a high degree of di- verging influence. The middle of the scale is 56Ibid. thus a region of indifference. This interpreta- .5 s tion may perhaps hold for ethnology: I do not onditions, but also when b or c alone is zero. know, but I suggest that a careful analytic uch a condition could arise when one entity was study of the contents of element lists for dif- ompletely associated with a second, but not vice ferent values of 46 would throw light upon the ersa; for example, we may associate an individual meaning of independence and association for ompletely with his nation, but the culture of he individual is never as large as the culture ethnology. f the whole nation. 60Ibid., 350. R8Q2 shows complete dissociation under these 61Ibid., 358 onditions, but also when a or d alone is zero. Vch a result might be v'ery misleading. This is 6Ibid. he weakness of Q2. 631bid., 359. 59Kluckhohn, 1939, 358. 6 4Chretien, 1943. 486 ANTHROPOLOGICAL RECORDS But leaving the question of precise ethno- It is at precisely this point that Kluckho logical interpretation aside, we must notice that criticism of the application of probability all our definitions of independence given above theory to ethnology would appear to be most co made one very important assumption, namely, that gent. As he pointed out,66 we are employing the marginal frequencies of the fourfold table here not mathematical probability as we were were actual representations of S and R. This above in connection with the product law, but may be illustrated by the Chilula-Wiyot table frequency probability, which may be defined in already given. Let us repeat it here (table 28), these terms: "If on taking any very large num- with the values changed to percentages. ber N out of a series- of cases in which an eve A is in question, A happens on,pN occasions, TABLE 28 probability of the event is - id to be p."67 Chilula TWiyot This is clearly an induction from experience, and hence has an empirical basis. It underlies many types of statistical judgment, of which Not life insurance, traffic accidents, crime, etc., Wiyot Wiyot are good examples. Kluckhohn's criticism at this point is that ethnologists have assumed l lla b lla+b t that they were dealing with an N which was suf ficiently large, or they have ignored the rel Chilula || 23 15 ll 38 l vancy of the size of N to the validity of re- sults.68 In the fourfold tables which I have examined above, N has varied from 10 to 2257. ||c d l l c+d l Obviously 10 is not a "very large number"; is c d c+d ~~~~~~~~2257?' Not G~h lula 11 19 43 l 1 62 | In answering this question we must be caref not to fall into the mistake of applying theo in any slap-dash fashion. N is not a very 1 a+c b+d N number in the abstract if it is only 2257. Itl may however be a very large number for ethnol 42 58 100 cal problems. Let us take an example. Yule gives a fourfold table dealing with the imoot _____________ ___tion of tubercular cattle.69 Here N is 30. 1 is quite probable that the high correlation b tween inoculation and recovery which this tabi When we use this table to compute x2 or Q6, we shows is valid. There are fairly simply analy are assuming that in the complete element uni- able connections between the two. W4ith life i verse of Northwest California, Chilula would surance, on the other hand, an actuarial table show 38 per cent plus and 62 per cent minus en- based on 30 cases would be ridiculous. The qug tries, and Wiyot 42 per cent plus and 58 per tion is to be answered, then, by pointing out-' cent minus entries. We make this assumption in that over and over again in the CED studies .tU this way: both x2 and Q6 are measures of devia- tistical results have agreed with nonstatistic tion of the table from independence values; judgments when N was 500 or more--even at tine these independence values are determined from when it was less. Now Driver has criticized the marginal frequencies, which are thus taken this kind of confirmation: "It is easy to see as representing the actual proportions of plus that if statistical methods must be tested by. and minus for each tribe. But do they? If we more subjective methods they are less valid t compare these Chilula and Wiyot values with the latter."70 It seems to me that Driver is. those obtained before the element lists were re- wrong on two counts. First the test of any 65 fidta duced for comparative purposes, we find that method is and must be empirical. The test of' Chilula had 43 per cent plus, an increase of 5 the actuarial tables is not theoretical; it is1 per cent, and Wiyot 46 per cent plus, an in- that the insurance companies- which use them do crease of 4 per cent. Nor is there good or com- not go bankrupt. The test of the statistical, pelling reason to suppose a priori that these method in ethnology is that it works, that unreduced lists reflect the facts of the com- its use is properly safeguarded ethnologists plete element universe. The element lists of not make absurd statements on the basis of it., the CED series are certainly not random samples _ of the complete universe. Yet the reliability 66Kluckhohn 1939, 360, footnote 74. of both X2 and Q6 depends on the element lists' 67 being good samples. Plummer, 1940, 1. 68Kluckhohn, 1939, 35'6. 69Yule, 1937, 48. 65Driver, 1939, 427. 70Driver, 1939, 304.| CULTURE ELEM. DISTRIB.: XXV--CHRETIEN: RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS 487 Secondly it is not to be supposed that every time TABLE 29 the method is used an empirical test must be Gulf of Georgia Salish ifound for it. The statistical method is rela- tively new, its limitations and applications are Pe Cx Ho Kl Se Sq WS Cw Na not fully understood, the conditions of its va- Pentlatch.83 17 lidity are still somewhat uncertain, obvious Comox .......... 83 28 31 32 13 faults are not all corrected. Under such cir- Homalo ........8 28 78 71 36 2 6 cumstances empirical confirmation is indispen- Klahuse ........ 17 31 78 35 11 11 sable. But the time will come when this will no Sechelt ........ 22 71 35 40 17 16 longer be generally necessary. Then ethnolo- Squamish 36 40 54 20 32 gists will know precisely how to use the method, West Sanetch ....... 6 11 17 54 72 52 or they will have given it up entirely. Cowichan 11 20 72 88 The question of the size of N is also of mo- Nanaimo w n....... 13 16 32 52 88 ment in deciding upon the choice of a level of significance. The five per cent level is widely We note that Kwakiutl, Slaiamun 1, Slaiamun 2, used, and if we keep N above 500 we shall prob- and East Sanetch are entirely omitted. If we ably make no mistake in using it. But we may compare this diagram with Barnett's diagram 1,72 advance this further consideration in its favor. . > . * ~~we find that they are identical except that he Let us suppose that the marginal frequencies rep- includes Slaiamun 2 between Klahuse and Sechelt. resent the universe accurately. Let us suppose It is interesting to quote Kroeber's note on also that X2 is 3.841, and P therefore 0.05. thiS: Then there are five chances in a hundred that tS: the actual table could have arisen by random The shortest lists are Slaiamun 1, 129 elements; sampling from a universe in which S and R were Kwakiutl, 217; East Sanetch, 295. The coeffi- independent.1 This is one chance in twenty, a cients for the first two of these fall quite igh degree of improbability when N is 500 or randomly, as compared with geography and known ighrdegree of improba bilit ethnography. For the East Sanetch, the fit of lore. If however N iS 10 or 15 or 27, values the coefficients to [ethnographic and geographic] hich it had in the Furer-Haimendorf data, one expectability is roughly right but only fair. Ln twenty is an extremely low degree of improba- These tribes have therefore been omitted from ility. For this reason it seems safe to use the diagram. Slaiamun 2 is the next smallest he five per cent level when N is greater than list, based on 572 elements. The fit of this is 00. But for tables where N is less than that, conformable to all other known facts: its coeffi- oth the likelihood that the actual data arise cients range themselves in size to accord with y chance, and the larger unreliability of the geography about as well as the coefficients for any other tribe. The Slaiamun 2 list is there- ginal frequencies would advise us to eschew fore reliable.73 uch tables altogether. Why then have I omitted it, when Kroeber states Empirical Confirmation that its N is over 500? The answer is that it The procedure which I sought to enforce above is never over 500 in the fourfold tables. Here The rocdur whch souht o efore aove it ranges from 31 (with Slaiamun 1) to 338 (witl ay be confirmed empirically by reference once S echel. Tf ac .thataits cefficiets agre re to the Salish material. Let us take all Sechelto. The fact that lts coeffns ients agree hQ values where X2 reaches the five per cent with nonstatistacal conclusgons means probably vl an N is grae _hn50 n rag that the fit iS accidentally g ood. kt least I l an N disgreaerm than5 a rn should not care, as a general rule, to rely on hem in the diagram form of table 29. tables this small. It is interesting that Barnett and Kroeber, operating on grounds quite different from mine, basing their inclusions and exclusions on em- 71p = 0.05 is the same as saying that the pirical fit, arrived at much the same pipture robability is five in a hundred; P = 0.01, one of the Salish data as I did, operating on grounds n a hundred; P = 0.001, one in a thousand. largely theoretical. It is especially interest- 72sBarnett, 1939, 224. ing that Kroeber hit upon the N = 500 rule quite 73Kroeber in Barnett, 1939, 226. independently and for utterly different reasons. CONCLUSION In the course of this study the X2 test has struct a proto-element list for, let us say, been applied to three different kinds of material. Northwest California, or not. If it can be do The first of these, intradifferences of inform- and if such a list covers the culture of the. ants, gives rise to a specific problem which can area representatively, many problems of compar come about only when duplicate lists of data are ability would disappear. It should be remar available. For this reason I do not comment on furthermore, that the culture represented by it further. The other two types of material, a list need never have existed historically. however, lead to problems of more general in- Linguistic scholars in their severer critical., terest. -moments grant the artificiality of their "Ur- My second illustration was of reduced element sprachen"; this does not diminish their utilit lists. Here I chose a rather extreme and per- however.7 The proto-element list is a framei haps unfair example. Driver's reduced list of not necessarily, or in every detail, a reality, 706 elements was not intended to be used for I make the suggestion of an ethnological proto- intercorrelations of tribes.74 Nevertheless, element list, but only a competent ethnologist reducing element lists for whatever purpose in- can handle it. troduces problems which have not been adequately My third illustration dealt with fourfold considered. The whole question needs investiga- tables and coefficients. Here I trust I have;. tion from the ethnological, not merely the sta- removed doubts occasioned by Kluckhohn's crit tistical, point of view. The element lists are cisms. I trust also that I have pointed outv most certainly samples of the total element where further investigation is needed. Prim population of the area in question;75 they are in importance, it seems to me, is the need to. most certainly not random samples;78 but it does determine more precisely the meaning of the not follow that they are not gopd or representa- scale of association. All association studie; tive samples. The successful fit of coefficients to date have confined their attention to the8 to nonstatistical judgments which has occurred so high positive values. Such a procedure clear many times in the CED series shows that the lists does not exhaust the possibilities of infere are in the main fairly good. Nevertheless, if from a table of coefficients. Here again, t the statistical method is to be of service, it competent ethnologist must handle the proble" must eventually free itself from the necessity Meanwhile, if the coefficients are based on N' of empirical confirmation; hence it would seem of 500 or more, it seems safe to use them, at that the question of representative element least as they have been used thus far.' lists needs attention. The situation in linguis- tics may help here. In linguistics we start with a proto-element list, a list of the charac- 741t was used, however for intracorrelatio teristics of the reconstructed parent language. of eac tie, horesult not marelytto We~~~~~~~~~ the copr th,agtrlngae ihec of each tribe, with results not markedly diffez We then compare thze daughter languages with a each ent from those given by the unreduced lists. other, element by element, on the basis of their 75i t adherence to, or departure from, the parent. 5In general, I believe it legitimate to adherence to , or.departure from, the parent. sume that any number of elements is a sample- For example, Ur-Germanic has *i: this is pre- some very much larger totality" (Driver, 193*, served in West Saxon and Old High German; we put 206). down plus for both. Ur-GJermanic also has *e 76The various descriptions in the CED serts (long open Q): West Saxon keeps this; Old High of the way element lists grow in the field German changes it to a; we put down plus for this. West Saxon and minus for Old High German. I do 77A good discussion of this, with refereno not know whether it would be possible to con- will be found in Buck, 1926. [488]~~~~~~~~~~~~~~~~~~~~~~fb CULTURE ELEM. DISTRIB.: XKV--CHRETIEN: RELIABILITY OF STATISTICAL PROCEDURES AND RESULTS 489 TABLE 30 Values of rhk and Q6 at Various Levels of Significance and for Different Values of N 5% 1% 0.1% N _ rhk Q 6 hk Q6 rhk Q6 10 0.62 0.83 0.81 0.96 20 .44 .64 .58 .79 0.74 0.92 30 .36 .54 .46 .66 .60 .81 40 .31 .47 .41 .60 .51 .72 50 .28 .43 .36 .54 .47 .67 60 .25 .38 .33 .50 .42 .61 70 .23 .35 .31 .47 .38 .56 80 .22 .34 .29 .44 .37 .55 90 .21 .32 .27 .41 .35 .52 100 .19 .29 .26 .40 .33 .50 200 .14 .22 .18 .28 .23 .35 300 .11 .17 .15 .23 .19 .29 400 .10 .16 .13 .20 .16 .25 500 .09 .14 .12 .19 .15 .23 600 .08 .13 .11 .17 .13 .20 700 .074 .12 .10 .16 .12 .19 800 .Q69 .11 .090 .14 .12 .19 900 .065 .10 .086 .14 .11 .17 1000 .062 .09 .081 .13 .10 .16 1100 .059 .09 .078 .12 .098 .15 1200 .056 .09 .074 .12 .095 .15 1300 .055 .09 .071 .11 ,091 .14 1400 .052 .08 .069 .11 .088 .14 1500 .051 .08 .066 .10 .085 .13 1600 .049 .08 .064 .10 .082 .13 1700 .048 .08 .062 .1C .080 .13 1800 .046 .07 .061 .10 .078 .12 1900 .045 .07 .058 .09 .075 .12 2000 .044 .07 .058 .09 .074 .12 2100 .042 .07 .056 .09 .072 .11 2200 .041 .06 .055 .09 .070 .11 2300 .041 .06 .054 .09 .069 .11 2400 .040 .06 .053 .08 .067 .11 2500 .038 .06 .051 .08 .066 .10 3000 .038 .06 .046 .07 .060 .09 4000 .031 .05 .041 .06 .051 .08 5000 .028 .04 .036 .06 .047 .07 6000 .025 .04 .033 .05 .042 .07 7000 .023 .04 .031 .05 .038 .06 8000 .022 .04 .029 .05 .037 .06 9000 .021 .03 .027 .04 .035 .06 10000 0.019 0.03 0.026 0.04 0.033 0.05 BIBLIOGRAPHY Abbreviations: Fisher, R. 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