Brief Communications 1537

Transcription

1 Brief Communications 1537 Lauvi Vainu u, the chief who, in due course, made available to me his own genealogy book (tusi gafa) was, in fact, the pulenu u, or mayor, of Sa anapu. In so doing, he voluntarily went against the ruling of the jono (of which he was the leader), and merely to help the enquiries of a young ethnographer who had come to live with his aiga and studyhis people. This action I then regarded, as I still do, as one of especial friendship. Lauvi Vainu u and I (because of this and many other shared experiences) were bound by the deepest of feelings, and I remember him, for he is now dead, as one of the closest and best friends I have ever had. DEREK FREEMAN Instilde of Adttanced Studies, Australian Sational University KEFERENCES CITED EMBER, MELVIN 1964 Reply to Freeman: a rejoiner. American Anthropologist 66: FREEMAN, DEREK 1943 The Seuao cave. Journal of the Polynesian Society 52: The social structure of a Samoan village community. Ms. University of London Some observations on kinship and political authority in Samoa. American Anthropologist 66: GRATTAN, 1;. J. H An introduction to Samoan custom. Apia, Samoa Printing & Publishing Co. MURDOCK, GEORGE PETER 1957 World ethnographic sample. American Anthropologist 59: SAHLINS, MARSHALL D Social stratification in Polynesia. Seattle, University of Washington Press That s not what I said: reply to Derek Freeman. American Anthropologist 66: A NOTE OK FISHER S EXACT TEST Fisher s exact test is a common means of determining the probability that it given arrangement of values in a 2 X 2 table could have occurred by chance if the relationship between the variables defining the table is random. Unfortunately, many statistics texts give improper or incomplete instructions for performing the test and it is not uncommon for instructors of courses in statistics to perpetuate errors found in the texts. In interviewing two instructors of statistics in departments of sociology and one in a department of psychology, I found that all three taught the incorrect methods described in the textbooks (Siege1 1956: , Zelditch 1959: ) which they emplo>-ed. The usual instructions for performing the Fisher test are as follows: First compute the probability of obtaining the arrangement of values in a given 2x2 table. Sest compute the probability of obtaining every more extreme set of arrangements with the marginal frequencies of the table held constant. The Fisher coefficient is the sum of the probabilities for the original arrangement plus the sum of the probabilities for every set constituting a more extreme set of arrangements. The probability of any one arrangement is deter-

2 1538 A mericaia Anthropologist mined by the formula (a+ b)!( c + d)!(a+ c)! (b + d)! N!a!b!c!d! where a, b, c, and d are arranged as follows:! :1 I b I [67, The major problem that one encounters in using these instructions is in determining what represents a lmore extreme arrangement than the original set of figures. From the idea of using the Fisher test to test the null hypothesis, it logically follows that a more extreme arrangement means every arrangement that has less of a probability of occurring than the original arrangement. It is equally clear, however, that the instructions contained in many basic texts do not utilize this definition. Ordinarily the more extreme arrangements are defined as those which are obtained by starting with the original arrange- TABLE 1. ALL POSSIBLE ARRANGEMENTS OF CELL FREQUENCIES IN A 2x2 TABLE WITH FIXED MARGINAL FREQUENCIES (From Walker and Lev, 1953) A Arrangement _-I_- 8 0 Probability.OOO ! 18! 8! 81 O! 26! 18! 16! 8! ! 17! 9! 7! l! I 16 34! 16! lo! 6! 2! I 15 34! 15! ll! 5! 3! ! 18! 16! = I 14 34! 14! 12! 41 4! 3 5 = i 13 34! 13! 13! 5! 3! c -- D E _- I? G I 12 H I I - 16 I = ! 14! 12! 6! 2! 26! 18! --_ 16! 8! = ! 15! ll! 7! l! _- 26! 18! 16! 81 -=, ! 16! lo! 8! O! -- Total =

3 Brief Communications 1539 ment, finding the cell containing the smallest number, reducing this cell and the one diagonally opposite to it by 1, and increasing the remaining 2 cells by 1 each. This process is repeated until the smallest cell is reduced to zero. Thus, if the original set of arrangements is that given by F in Table 1, the more extreme sets of arrangements are represented by G, H, and I. It is clear that the above instructions cannot be followed for tables in which the smallest frequency is found in each of two cells which are not diagonally opposed as, e.g., in the following arrangement: I I I It is also clear that in following the usual procedure there will be cases in which the arrangements constructed in the above manner will actually have a greater probability of occurring than the original arrangement. This will happen in many cases where the product of the cells of the diagonal defined by the smallest cell is greater than the product of the cells of the other diagonal. Anyone may easily determine this for himself, The proper method for performing the Fisher s exact test is as follows: Given the arrangement if X is directly proportional to Y, we will say that a positive association exists. If X is inversely proportional to Y, we will say that a negative association exists. If ad>bc, then X is directly proportional to Y. If ad<bc, then X is inversely proportional to Y. If we predict that X and Y are positively associated, then if ad<bc we are wrong and no test of significance is necessary. If, however, we obtain a positive association (ad> bc), it is necessary to determine the probability of having obtained the given arrangement and every more extreme set in which ad> bc with the marginal totals of our tables remaining fixed. To do this we successively subtract 1 from cells b and c and add 1 to cells a and d until either b or c equals 0. An examination of the following arrangement will show that this procedure is not necessarily equivalent to reduction of the cells in the diagonal defined by the smallest cell, since in this case the diagonal which will be reduced does not include the smallest cell: 1 I I

4 1540 American Anthropologist [67, If we predict that an association will be negative, we are correct only if ad<bc. If we are correct, then we procede to compute the Fisher s coefficient as above, except, in this instance, the a and d cells are decreased and the b and c cells are increased. The above tests are both one-tailed, i.e., since we predict the direction of the association, it is not necessary to compute probabilities with respect to the tail opposite to that predicted. In the event that we do not predict the direction of association it is necessary to perform a two-tailed test. A twotailed test is merely a means of determining the sum of the probabilities of every set of arrangements in a given 2x2 table that has a probability of occurring equal to or less than the original set. Some of these will be in tables in which X and Y are positively associated and some will be in tables in which X and Y are negatively associated. To do a two-tailed test we first determine the probability far the original arrangement. We then obtain all sets of arrangements that are possible by performing the operations for both negative and positive associations as described above. The probabilities for every set of tables equal to or greater than the probability for the given table are added to the probability of the given table to determine the Fisher coefficient. The above is the exact method for performing a two-tailed test. An approximation to the exact two-tailed test may be made by performing a onetailed test and doubling the results. This method will tend to err conservatively, i.e., it will ordinarily yield a higher probability than the exact method. The above methods are illustrated here in respect to Table 1: If we predict that a positive association will exist between X and Y, and in our observations of the data we observe that distribution C exists between our variables, then since ad>bc the association is in the predicted direction. We then add the probabilities for obtaining the observed arrangement to the probabilities for obtaining every more extreme positive set of arrangements to obtain the Fisher coefficient. We therefore add the probabilities for A and B to the probability for C and the result is If we predict that a negative association exists between X and Y and we observe arrangement F, then, since ad < bc, the association is in the predicted direction. The Fisher s coefficient is obtained by adding the probability of the observed arrangement to the probabilities for each of the more extreme negative arrangements. \h e therefore add the probabilities for G, H, and I to the probability for F and the result is If we do not predict the direction of association between X and Y, and the observed distribution is G, then we add every arrangement that has a probability equal to or less than the probability of the observed arrangement to the probability of the observed arrangement. We therefore add the probabilities for A, B, C, H, and I to the probability for G, and the result is The use of digital computers makes it practical to use the Fisher s test in most instances where such approximations as Chi2 were formerly used. The author has available a program for the IBAl 7010 that will compute both one

5 Brief Communications 1541 and two-tailed Fisher coefficients for 2x2 tables with N s equal to or less than 999. A listing of this program will be sent on request. ALLAX D. COULT University of Texas REFERENCES CITED SIEGEL, SlDNEY 1956 Non parametric statistics for the behavioral sciences. New York, McGraw-Hill. WALKER, H. M. and J. LEV 1953 Statistical inference. New York, Henry Holt and Co. ZELDITCII, MORRIS, JR X basic course in Sociological statistics. New York, Henry Holt and Co. THE COKSANGUISEAL HOUSEHOLD AND RIATRIFOCALITY The matrifocal family is a subject which has attracted a great deal of attention in recent years, and as a classificatory device has been utilized so indiscriminately that many might wonder whether the concept has any utility at all. Peter Kunstadter s A Survey of the Consanguine or Matrifocal Family (1963), in which he uses a definition of the matrifocal family supposedly first proposed by me (Solien 1959), has recently called forth critical comment by two authors in the same issue of the American Anthropologist (Boyer 1964; Randolph 1964). Since their criticisms of Kunstadter largely center on his definition of matrifocality, and since all three acknowledge me as the originator of this definition, I feel called upon to enter the discussion, not only in my own defense, but in the hope of furthering our understanding of matrifocality. I must first point out that Kunstadter improperly quoted me. The definition which he took from my dissertation refers specifically, not to the matrifocal family, but to the consanguineal household, which is quite a different thing. I said, The consanguineal household is a co-residential kinship group which includes no regularly present male in the role of husband-father. Rather, the effective and enduring relationships within the group are those existing between consanguineal kin (Solien 1959 : Abstract). A consanguineal household may or may not be matrifocal as well-probably in most cases it is-and conversely, a matrifocal family may quite well include a regularly present person in the role of husband-father. In another publication I tried to clarify the conceptual differences between a household on the one hand, and a family on the other (Solien 1960). I believe that confusion between these two phe- L nomena lies at the heart of Royer s disagreement with Kunstadter. Since apparently none of the three authors referred to above saw any more than an abstract of my thesis, I would here like to present a fuller description of what I was talking about before going on to further comments and criticisms. The central problem under discussion in my dissertation is the unusual type of household organization that characterizes Black Carib culture. This form merits close study because it appears to exist in various societies. In some cases its existence as a type has not been formulated by persons examining the cul-