key: cord-022879-j6cecioe
authors: Fager, Edward W.
title: Determination and Analysis of Recurrent Groups
date: 1957-10-01
journal: Ecology
DOI: 10.2307/1943124
sha: 
doc_id: 22879
cord_uid: j6cecioe

nan

The primary information available from an ecological survey usually consists of a mass of sampling results involving many localities and many species. In order to investigate the interspecific processes which may be involved in controlling the abundance and distribution of the species found, it is useful to be able to group together those species which frequently occurred together in the samples, which were in this sense a nearly constant part of each other's biological environment. The grouping will be most valuable as a classificatory device if it consists of defined units obtained by a procedure which can be repeated. Whatever the method of grouping employed, the sampling procedure is certain to introduce subjective elements; the size of sample, the number and selection of sampling sites and the sampling methods used will always be influenced by the judgment of the person doing the study and they will in turn influence the composition of the groups. If, however, the sampling conditions are explicitly given, the probable extent and character of these influences can be taken into account and a meaning given to the groupings in this context. Some of the methods of grouping which have been suggested will be briefly reviewed (see also Macfadyen, 1954 ) and a new method will be outlined which overcomes some of the difficulties REVIEW oF SoME METHODS OF GROUPING Animals have often been grouped on the basis of vegetation or of various physical or chemical factors in the environment, but many difficulties are involved in vegetational classification or in deciding what is a critical limit in respect to a certain physical or chemical factor. Furthermore, animal groupings do not always conform to vegetation or other factors ; a group may extend over several recognizable vegetation or soil divisions or there may be several distinct assemblages of animal species within an environmental complex which is uniform in many other respects. The characterization of the habitat in which an animal group is often found must include the associated plants and other factors, but it seems better to use the animals as the primary basis for their grouping. Gisin ( 1947 Gisin ( , 1949 Gisin ( , 1951 has suggested a procedure using what he has called "differential" species. In the hands of an experienced worker this method can lead to useful groupings, but the choice of the species which will be designated as "differential" is so subjective that comparison between groupings by different workers, even if based on the same material, may be impossible. Moreover, using this method groups may be suggested the constituent species of which so seldom occur together that it seems unlikely that they could have any relation to each other. For example, Gisin ( 1947) in his rearrangement of Agrell's ( 1941) Swedish Lapland Collembola suggested four groups. Members of three of these were found together moderately often. The fourth group contained four species, representatives of which were recorded in a total of thirty-one samples ; no sample contained all four, one sample contained three, five samples contained different pairs and twenty-five samples contained only one of the species.

The subjective element can be reduced if the grouping is based on a defined index of affinity. Perhaps the simplest expressions which might be used for this purpose are those proposed by Jaccard ( 1908) and Sprenson ( 1948) as measures of the similarity between floras. As given, however, neither differentiates between a coefficient based on ten samples and one based on one thousand samples, though the latter would surely be a more reliable estimate. The expressions also fail to take account of changes in the relative number of total occurrences of the two species; as long as the sum of the total occurrences remains constant, the proportions can change without affecting the value of the coefficient.

A method based on correlation between pairs of species is precisely definable but, as Cole ( 1949) has pointed out, product moment correlation cannot be used directly because sampling results for most species are not normally distributed in regard to numbers of individuals per sample; there are qsually a few outsize samples which will have a disproportionate effect on the calculations. Transformation of the data or the use of rank correlation could overcome this difficulty, but any method which involves a measure of abundance may, in certain cases, not lead to the desired results : two species may always occur together and never separately and yet, unless there is a nearly constant relation between the relative numbers of individuals of the two species, a correlation coefficient will indicate no relationship even though they are a constant part of each other's biological environment. For this reason it seems best to base species groupings upon presence and absence alone and to consider abundance relations within such groupings after they have been determined. In his paper, Cole ( 1949) discussed the coefficients of association which had been proposed up to that time and indicated the deficiencies of each. He then proposed a new coefficient using only presence and absence and based on a 2 x 2 con-ti~Ig-ency table for which a Chi-square test of sigmficance can be calculated. A basic assumption of his expression is that all samples taken should be included; i.e., the probabilities of occurrence of the two species being tested for association are assumed to be uniform throughout the area sampled. Under this assumption, the Chi-square indicates the significance of the departure of the observed number of joint occurrences from the number expected on the hypothesis of independent distribution of the two species over all the samples. Although Cole did not use his coefficient for grouping species, it has been so used several times since. The following example shows that it is not a satisfactory criterion for grouping if the groups are to be composed of species which form a nearly constant part of each other's biological environment: Species 1 and 2 show no evidence of association when examined by Cole's method and yet they nearly always occur together-over 80% of the occurrences of each are in company with the other -and should be considered together in any grouping based on this set of samples. On the other hand, species 3 and 4 are significantly associated by Cole's criterion even though their affinity is rather low-less than 40% of the occurrences of each are in company with the other. In general, two species which occur in most of the samples taken will show little or no evidence of association and two species with a low percentage of joint occurrences can show significant evidence of association if they are rare enough or if a large enough number of samples in which they could not occur are included. Bray ( 19S6) has recently noted these difficulties connected with the use of Cole's index and has attempted to overcome them by considering only quadrats within plots in which both species occurred. Macan ( 19S4) using the method without modification found a very large number of positive associations in his analysis of collections of Corixidae, a finding which, as he indicated, was unsatisfactory because the objective analysis produced a less well-defined picture than that which could be obtained from a subjective technique based on a knowledge of habitat features.

An expression of the sort suggested by Cole has certain desirable properties; in particular, it is sensitive to changes in the relative frequency of occurrence of the two species and a "significance A t-test is preferred to Chi square because it can be used as a one-tailed test [ 1.64S ,....., p (.OS) ] for positive affinity; negative affinity, being based on the failure to find a species, seems potentially subject to too many unavoidable errors.

There are both mathematical and biological reasons for setting an upper limit on the value of nn which will be considered with a given nA: the closer the ratio nnjnA is to 1, the closer the normal approximation will be to the true value; keeping the ratio close to 1 ensures that whenever evidence of affinity is obtained the number of joint occurrences will be a considerable proportion of the total number of occurrences of both of the species. This is a desirable property for an index to be used as a basis for grouping if the purpose is to put together organisms which are a nearly const·mt part of each other's biological environment. An upper limit of 2 for the ratio nn/nA is therefore sug-level" can be defined for use as an objective criterion of the presence of affinity. It can be adapted for use as an index of affinity if it is assumed that the probability of finding species A is nAjnA + nB and of finding species B is nBjnA + nB, where nA is the total number of occurrences of species A and nB is the total number of occurrences of species B;

i.e., the probalilities of occurrence of the two species are related to the sum of their occurrences (nA + nB) rather than to the total number of samples taken. The use of this assumption removes the premium on rarity and also makes it possible to find evidence of affinity between two species which occur in most of the samples taken. It seems neither more nor less arbitrary than the use of the total number of samples for the latter is usually determined arbitrarily by the person doing the work, often mainly on grounds of practicality or subjective judgment that the number taken is sufficient.

If the preceding assumption is used and the species are assigned to the letters so that nA is less than or equal to ne, the number of joint occurrences (J), on the hypothesis of independent distribution, will have a hypergeometric distribution with expected J equal to nAnejnA + ne and variance equal to (nAnB) 2 j(nA + ne) 2 (nA + ne-1).

For values of nA greater than 10 and p and q not too different from 1j2, a normal approximation can be used to test the significance of the deviation of the observed number of joint occurrences from the expected number. For this purpose, the following expression can be taken as a normal deviate with mean 0 and unit variance (Pearson 1947 ):

gested. The use of such a limit has the disadvantage of preventing the finding of affinity between such a pair of species as a rare parasite and its relatively abundant host, but the existence of a relationship of this kind can, in any case, probably be satisfactorily established only by field observation and breeding studies.

Minimum values of J which are significant at the .OS level have been calculated for values of nA from S to 10 using exact probabilities obtained by Leslie's method (19SS) . Similar values have been calculated for each tenth value of nA from 20 to 100 using the normal approximation. Both sets are given in Table 1 in the Appendix. These values and intermediate ones which can be estimated by interpolation will make it possible to quickly sort pairs of species into those which certainly show evidence of affinity, those which cer-tainly do not and those which are doubtful and require calculation. When applied to the examples used in the discussion of Cole's coefficient, the index proposed above indicated significant evidence of affinity between species 1 and 2 but none between species 3 and 4. It is, therefore, a better index of the pro-portion of occurrences which are joint occurrences and should provide a more satisfactory basis for grouping. It may be repeated that Cole did not develop his index as a basis for grouping and that the two indices measure different things: Cole's index measures the departure of the two species from independence of distribution on the assumption that the probability of occurrence of each is constant over all the samples taken ; the index proposed in this paper provides an objective measure for the word "frequently" in the statement "these species frequently occurred together."

The procedure to be outlined for the determination of recurrent groups may appear complicated but with a little practice it can be gone through at least as rapidly as the "fitting by eye" of a trellis diagram. It has the virtue of repeatability ; given the same primary information, two workers will always arrive at the same groups. This means that if several workers using similar sampling methods make studies in different localities and find different recurrent groups, there is some assurance that the differences are real and not the result of differences in judgment or emphasis. Any dichotomous index of relationship between species can be used as a basis for grouping by this procedure; the meaning of the groups will be determined by the nature of the attributes measured by the index.

In this paper, a recurrent group is defined as one which satisfies the following requirements :

( 1) The evidence for affinity is significant at the .05 level for all pairs of species within the group.

(2) The group includes the greatest possible number of species. ( 3) If several groups with the same number of members are possible, those are selected which will give the greatest number of groups without members in common. ( 4) If two or more groups with the same number of species and with members in common are possible, the one which occurs as a unit in the greater number of samples is chosen.

These requirements are to be taken in order; i.e., the group or groups which fulfill requirements 1 and 2 are determined and then, if there are several possible combinations, a choice is made on the basis of 3, followed, if necessary, by 4. Such groups will represent the largest, most frequent, separate units within which all the species formed a nearly constant part of each other's biological environment. They would be a likely choice if one wished to make an intensive study of interspecific processes which are quantitatively important in the habitats sampled.

The following procedure will give a recurrent group or groups as defined above. The affinity information for the species concerned may be set out in a trellis diagram as shown for the example (Table I) . If there are many species, it has been found more convenient to record the information for each species on a separate card. . =non-affinity

The species are put in order in terms of the number of other species with which they have affinity. In the example this order is A, B, C, . . . P, Q. Starting with the species with the largest number of affinities, species are counted in the direction of decreasing number of affinities until the number of species counted (X) exceeds the number of affinities ( Y) of the last species counted. In the example this occurs at species H where X = 8 and Y = 6.

Two possible relationships between X and Y must now be considered:

If this is true, as it is in the example, the largest potential group will contain Y + 1 species ; 7 species in the example. It cannot contain more because species H has affinities with only 6 other species and can, therefore, only form a group of 7 species satisfying requirement ( 1). If species H is omitted, only 7 species are left. Because the species were put Ecology, Vol. 38, No. 4 in order in terms of the number of other species with which they had affinity, the species beyond H will either have the same number of affinities as H, in which case they are also potential members of a group of 7 species, or fewer affinities than H, in which case they cannot be included in a group of 7 species satisfying requirement

( 1). If species H had had affinities with 7 other species, the largest potential group would have contained the first 8 species, A to H.

[X > Y + 2] If this is true, the largest potential group will contain the X -1 species preceding the last one counted. For example, if species H had had affinities with 5 or fewer other species, the largest potential group would have been composed of the 7 species, A to G, for H and any species beyond it could only form a group containing 6 or fewer species.

The counting procedure thus selects those species which are potentially members of a group satisfying requirements ( 1) and (2). These must be further examined to determine whether such a group can be formed. Two preliminary tests of the possibility of its formation can be applied.

The first test can be stated as follows : In order to form a group of Z members from V potential members, there must be at least Z of these with more than Z-2 affinities with others of the potential group. For this test the affinities of each of the selected species with all others of the potential group are tabulated. This tabulation for the example is shown below :

A B C D E F G H V = 8 ;Z = 7. 7 6 6 6 4 4 4 5

There are only 4 species which have more than 5 affinities and, therefore, a group of 7 species cannot be formed.

Passing the preceding test is a necessary but not sufficient condition for the formation of a group as can be seen by consideration of the following : If the interrelationships among the 8 potential members had been such that there was affinity between all species except the four pairs, A-B, C-D, E-F, and G-H, the tabulated affinities would have all been 6 and yet no group containing 7 species and satisfying requirement ( 1) could have been formed. On the other hand, if the species A to G had had affinity with all others except H, the tabulated affinities would have been seven 6's and one 0 and a recurrent group of 7 species could have been formed. In general, the affinities lacked by the species needed to form the group must be accounted for by species which are not necessary for its formation.

The second test determines whether this condi-tion is met. It can be expressed algebraically as follows : In order to form a group of Z members from V potential members the following inequality must hold:

(V-1) (2Z-V) < 1 + ~ Z largest affinities -~ the rest of the affinities. Applied to the two cases discussed in the preceding paragraph, it gives the following results:

incorrect, a group of 7 cannot be formed.

(8-1) (2[7] -8) < 1 + 7(6)-1(0); correct, a group of 7 can be formed.

This is a necessary and sufficient condition for the formation of a recurrent group when V = Z or Z + 1 ; it is necessary but not sufficient when V = Z + 2 or more for in this case the two sums on the right hand side of the expression can vary independently to some extent, the more so as there are more possible pairs of species within the category "rest." Although passing these tests does not make the potential group a certainty in all cases, the two tests will prevent unnecessary work in many instances in which no group can be formed.

Returning now to the example, no group of 7 species could be formed so the possibility of the next smaller group, 6 species, is investigated. Two additional species, J and K, must now be considered for, having affinities with 5 other species, they could be members of a group of 6. The affinities within the group of 10 species are shown in the first line of the tabulation below. Both tests indicate that it may be possible to form a recurrent group of 6 species : there are over 6 species with more than 4 affinities; (10-1)(2[6] -10) is less than 1 + (9 + 8 + 8 + 7 + 5 + 5) -( 5 + 5 + 4 +4). The affinity interrelationships must now be examined in detail in order to make the final determination. Perhaps the easiest way to do this is to eliminate those species which do not have more than Z -2 affinities, G and K in the example ; repeat the tabulation as shown in the second line below, eliminate species which now do not have more than Z -2 affinities and repeat until no more species can be eliminated. If at the end of this process V is less than Z + 2, the second test can be applied as a both necessary and sufficient condition for the formation of a recurrent group. V=lO Z=6 7 6 7 6 5 3 5 5 V= 8 Z=6 6 6 6 6 5 4 5 V= 7 Z=6 5 15 5 5 5 5 V= 6 Z=6

In the last line (6-1) (2[6] -6) is less than 1 + 6(5) -0 and as V = Z this is a necessary and sufficient condition for the formation of a recurrent group. Therefore, the species A-E, J constitute the group. The presence of J in the group emphasizes the necessity of considering all species which, on the basis of the number of other species with which they have affinities, might be part of the group. The initial counting procedure takes care of this automatically for the largest potential group, but the appropriate species must be added if this group cannot be formed and the next smaller must be considered. As there were no alternative groups of 6 satisfying requirements (1) and (2), requirements (3) and (4) do not have to be considered. The species not included in the group are now examined to see if any have affinities only with members of the recurrent group. This is true of L and Q. Such species are considered associates of the recurrent group.

The procedure is repeated on the remaining species, omitting their affinities with members of the first recurrent group (Table II) . The largest potential group contains 4 species; X = 4 and Y = 3 at species N. When the affinities within this group of 4 are tabulated, however, P and N have only 2 and a group of 4 species cannot be formed. All 7 of the species must be included when the next smaller group, 3 species, is considered. None can be eliminated on the basis of the first test and as (7 -1) (2 [ 3] -7) is less than 1 + ( 5 + 4 + 4) -( 3 + 2 + 2 + 2) a group may be possible. When the interrelationships are examined in detail, the following groups are found to satisfy requirements ( 1) and (2) : FGN, FGP, FHP, FKP, GMN. Requirement (3) will be satisfied if GMN and either FHP or FKP are selected. In order to decide between the latter two, the original sampling results would have to be investigated on the basis of requirement ( 4) . It will be assumed here that FHP occurred as a unit more often than did FKP. The relation- ships between the species might be set out as shown in Figure 1 . This method has been applied to several studies reported in the literature and to an unpublished survey of the invertebrate fauna of decaying wood done by the author. In all cases the groups determined were consistent with what was known of the species' requirements, preferences, etc. The invertebrates from decaying wood were grouped using code numbers for the species in order to eliminate the possibility of bias ; the groups found were in general agreement with impressions gained during sorting and counting.

Recurrent groups based on the index of affinity proposed in this paper and determined by the procedure outlined above are composed of species which very frequently form a part of each other's biological environment. Such interspecific concepts as dominance, concordance and correlation should, therefore, be particularly meaningful when examined within these groups. The several methods of analysis suggested below use ranking procedures which, as Kendall ( 1955) has shown, are very useful for detecting general trends in material which includes some extreme values, as sampling results from natural populations nearly always do.

Before the analytical methods can be applied, samples representative of each recurrent group must be selected. For groups containing only a few species, selection can be based on the requirement that all species in the group are present in the sample. For groups containing many species, this may too greatly reduce the number of samples and some compromise must be accepted. The requirement should be set at a high percentage present but the exact value to be used will depend on the sampling results which are available. It should be explicitly stated in any case. Table III presents some sampling results, related to actual observations but somewhat modified in order to show the different results which DoMINANcE Numerical dominance is often expressed as a mean percentage of total individuals, obtained from the summed percentages of the samples considered separately. Such an expression, using percentages calculated for samples with different total numbers of individuals, is hard to interpret and gives no information about the constancy of the relation. The difficulties of interpretation are increased if the species are dissimilar in size, activity, etc. It is, therefore, suggested that determinations of dominance be restricted to species within recurrent groups which are, or appear to be, similar in regard to size, activity, food requirements, etc. and that the determination be based on ranks within samples, summed over the set of samples. This is equivalent to n things (species) ranked m times (number of samples in the set). The concordance among the rankings can be tested by the statistic W (Kendall 1955, pp. 94-102) , which can range from 0 to 1 ; from no agreement to perfect agreement among the rankings. Details of the calculation of W and of an F-test of its significance are given in the Appendix. The use of ranks within samples gives every sample an equal voice in the decision and eliminates the bias which one or two samples with exceptionally large or small percentages may impose on the per cent method.

Rankings within samples for the species A, B and C and D and E are shown in the second part of Table III. For The value of W is significant at the .01 level showing that there is concordance among the samples; i.e., the dominance relations between the species tend to be constant over the set of samples. The sum of ranks is for each species the best estimate, in the sense of least squares, of the over-all rank of that species and if there is significant evidence of concordance the species may be ordered by these sums. The results may be stated as C >A, B; i.e., C is constantly dominant to A and B, and A is probably dominant to B but their sums of ranks are so nearly equal that the relative positions are less certain.

The value of W[W = 0.04; F(n1 = 0.8; n2 = 7.2) = 0.375 (p > .20)] is not significant for D and E indicating that dominance relation between these two species is not constant.

Relative abundance over all the samples is related to but not the same as numerical dominance. For example, assume that the numbers of individuals of two species found in 10 samples were as follows: Species x would be found to be constantly dominant to species y and yet their relative abundances would not be significantly different. The difference between the relative abundances could be tested by the usual t-test of the difference between the means, but if there were any outsize samples they would exert a disproportionate influence. As these are often present among samples taken from natural populations, it is probably better to use the distribution-free statistic U and a t-test based on it (Hoel1954, pp. 291-3). This tests the hypothesis that the distributions of the two populations of values are the same against the alternative that one distribution is situated to the left (lower valued) of the other. It is not affected by outsize samples. Details of the calculation will be found in the Appendix.

In the case of species A and B the observed value of U is not significantly different from the expected value. There is, therefore, no evidence that the distribution of abundances of A was different from that of B. As is apparent by inspection, C was significantly more abundant than either A or B. The observed value of U is significant at the .OS level, indicating that E was significantly more abundant than D even though determination of dominance had shown that it was not constantly dominant to D.

The concordance among the species of a recurrent group on what constitutes a good habitat can be tested by the same methods ( W) as were used in examining numerical dominance except that in this case ranking is done within species and the meaning of n (number of samples in the set) and 1n (number of species) are the reverse of those used in the previous test. If the observed number of individuals of a species is taken as an estimate of the goodness of a sample for that species, the samples can be ranked in regard to each species ; this eliminates difficulties due to differences between species in average abundance, efficiency of collection, etc. The sum of these ranks over all species of the group is for each sample the best estimate, in the sense of least squares, of the rank of that sample among the samples selected as representative of the group. If there is significant evidence of concordance, the samples can be put in order.

Rankings within species are shown in the third part of Table III. For the recurrent group considered as a unit; showing that there is agreement among the species on what constitutes a good or bad habitat. The samples can be put in order of decreasing goodness as follows: 6 > 9, 5 > 2, 3, 1, 4, 7, 10 > 8. An examination of the other characteristics of these samples, especially the extremes, might reveal a good deal concerning the preferences and requirements of the group as a whole.

Rank correlations between species should also be calculated because the preceding generalization may miss significant relations: if, in a group consisting of four species two rankings are alike and the other two are exactly the opposite, the concordance will be zero although the six possible pairs of species will give four perfect negative correlations and two perfect positive correlations. Because of the possibility of parent correlations, nonsense correlations, etc., the interpretation of correlation analysis must always be viewed with reservations. If, however, correlation coefficients are calculated only for those pairs of species for which there are other reasons for expecting interrelationships, they may serve as useful guides to the mature and constancy of the relations. Kendall's (1955, pp. 34-8, 55) The nearly significant positive correlation between the two species of poduromorph Collembola, AC, might be due to chance, might indicate that the species have a favourable effect on each other, or might result from their liking the same conditions and not interfering. The significant positive correlation shown by the presumed prey-predator pair, CJ, indicates that the predator did tend to congregate in those places where its prey was abundant. The absence of significant negative correlation between similar species means that in none of the pairs of species was an increase in the number of individuals of one always accompanied by a proportionate decrease in the number of individuals of the other. More evidence than the presence or absence of correlation is necessary before biological relationships can be claimed, but such an analysis will suggest those which might best repay further investigation.

This paper presents a defined, repeatable method for grouping together species which are frequent components of each other's biological environment. Such a method makes it possible to compare groups found in different habitats or at different times or localities. As the composition of these recurrent groups is influenced by the method of sampling used and by the "level of significance" required of the index of affinity, the groups are abstractions. It has, however, been found that if sampling is related to the size, activity, etc., of the organisms and the significance requirement is made stringent, the groups formed have ecological unity in the sense of significant intragroup agreement on what constitutes a good or bad habitat. It seems reasonable, therefore, to consider them as natural assemblages, somewhat artificially delimited but nonetheless real.

Because the recurrent groups are composed of species which very often occur together, they represent particularly appropriate units within which to examine interspecific relationships. The analytical procedures suggested for this examination have two purposes; the provision of a summary description of relationships and the suggestion of working hypotheses upon which to base further field and laboratory work. The use of methods employing ranking gives each species an equal voice in the analyses, removes many of the difficulties due to non-normality of distributions and makes it possible to determine the constancy of general trends.

A new index of affinity between species, based on presence and absence, is proposed and a table is provided from which the significance of an observed number of joint occurrences can be estimated.

Using this index and a four-part definition of a recurrent group, a procedure is outlined which leads to the largest, most frequent, separate groups within which all species formed a nearly constant part of each other's biological environment.

Ranking methods are suggested for the examination of interspecific concepts such as numerical dominance, relative abundance, concordance and correlation within these groups.

The essentials of the methods of testing for con-cordance ( W) and correlation ( -rb) and of the U test are given below for the convenience of those to whom the texts may not be readily available. The expressions given for the first two tests include the corrections required when tied ranks are present. In both cases, t or u represent the extent of ties present; e.g., if a ranking has 2 members with one rank and 3 others with another rank, the summation is 2 ( 4 -1) + 3 ( 9 -1) in the case of Wand 2(2 -1) + 3(3-1) in the case of "b· For the calculation of W all ties are considered together ; for the calculation of 'rb the extent of ties in one ranking is represented by t and that in the other by u. When tied ranks are absent, the summations are 0 and the expressions are considerably simplified.

Concordance (Kendall 1955) : If n things are ranked m times, the ranks for each of the n things are summecl and the sum of the squares of the deviations of these sums from their mean is represented by S, the concordance between the rankings ( W) is given by the following expression :

w-12s m 2 n(n 2 -1)ml'Jt(t 2 -1). This does not have to be modified for tied ranks unless the number and extent of ties is large.

To determine significance the usual table of the variance ratio (F) is entered with degrees of freedom n 1 and n2• Correlation (Kendall 1955) : The rankings are arranged so that one is in the natural order-1, 2, 3, ... n. Following this arrangement, scoring is based entirely on the other ranking. Starting with the lefthand member, +1 is scored for every larger rank and -1 for every smaller rank to the right of it. After this has been done for each member of this ranking, -rb is calculated from the sum of the scores ( S) as follmvs : 2S 'rb = y'[n(n-1)l'Jt(t-1) ][n(n-1)l'Ju(u-1)] . T is the sum of ranks of the y's when the x's andy's are ranked together. These expressions can be used in a one-tailed t-test of the hypothesis that the distributions of the two populations are the same against the alternative that the x population is situated to the left (lower valued) of the y population. For m and n equal to 8 or more, the following expression may be taken as a normal deviate with 0 mean and unit variance :

This test is nearly as efficient as the usual t-test when applied to normally distributed populations ; as it is distribution-free, it is applicable to nonnormal distributions. 

Example: If nA =20 and nB=30, a J of at least 16 is necessary for significant evidence of affinity; if nA=25 and nB=35 (nB/nA=1.4), minimum significant J can be

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