Chapter 9

Analysis of Amphibian Biodiversity Data

Lee-Ann C. Hayek

Introduction tion of a situation and its definition in opera- tional terms can lead to a correct choice for data In this chapter I discuss problems of use, misuse, analysis. When in doubt about correct statistical and problematical interpretation of data, spe- procedures, or when irregularities arise, investi- cifically for amphibian biologists. I focus on gators should always seek the advice of a mathe- statistical procedures that are appropriate for matical statistician with experience m biological amphibian sampling and monitoring and con- applications. Theory is always well behaved; sider limiting factors and assumptions of statisti- data never are. cal models in terms suitable for observational and other studies. The number of formulas I present is negligible, and they involve only sim- Species richness ple algebraic statements. This chapter is not all- encompassing. Presence-Absence Data No technique is universally appropriate, and standardization, as used here, does not imply Amphibian data may be recorded on a nominal, that only a single method of analysis exists. In- ordinal, interval, or ratio scale (see "Measure- deed, in some cases a good amount of work will ment Scales and Statistical Analysis," Chap- have to be done before any choice can be made. ter 4). Data measured against the first two scale Only a mathematical and probabilistic evalua- types are called nonmetric; those measured

207 208 CHAPTER 9

against the latter two are called interval, metric, Table 19. Generalized 2x2 Frequency Table or continuous. Single inventories can lead di- Using Counts rectly to nonmetric data by providing a total State, State, habitat, or species count with a designation of species as habitat, species A abundant, common, or rare; a list of species by or name; or a record of presence or absence of each species B Present Absent Total species within a selected habitat or sample. In some instances, the results of a monitoring effort Present a b a + b also may be presented as nonmetric data because Absent C d c + d of the nontrivial reduction in effort relative to a Total a + c b + d n study design calling for interval data. Results can be obtained quickly or inexpensively with a preliminary analysis of the research effort, and a given area. Fourfold (or 2 y. 2) contingency problems with sampling techniques can be tables are especially convenient for summariz- detected. ing data regarding the relationship between two Nonmetric data also may be the only possible species, populations, habitats, or localities (e.g., or reasonable choice when the precision with Table 19), because the number of categories which the data will be collected is in question• (four) is less than the sample size (number of for example, if collector reliability is low. In observations). Pairwise comparisons for all addition, with some sampling methods (e.g., combinations of species over an entire data set audio strip transects and night driving, discussed are a realistic means by which to summarize a in Chapters 6 and 7, respectively), measurement large set of observations and to examine inter- or observation error can be high, and the record- specific relationships. When associations among mg of precise continuous variables may be prob- species are not obvious and a priori assumptions lematic, unnecessary, or even deleterious to the are neither desirable nor possible, methods such study. Alternatively, for certain studies a contin- as Ä-mode and g-mode cluster analyses (e.g., uous variable may be categorized after it has Gower 1967; Hazel 1970; Anderberg 1973; been recorded, as when age data for juveniles or Sneath and Sokal 1973) can be used to identify adults are grouped. relationships. Presence-absence data can be coded using the The amount of information in data sets that numbers i and 0 for computer input. Whether a merely record presence and absence is consid- species is rare or abundant, its simple presence is ered to be minimal, and information is presumed indicated by a 7. Such data are termed two-state, to be lost when a continuous variable has its binary, categorical, or nonmetric, as well as scale categorized. Presumably, the smaller the nominal, number of categories, the greater the loss of A list of presence-absence data for species in information. Several authors have addressed this a particular area provides information on geo- question (Bonham-Carter 1965; Buzas 1967, graphic or habitat distribution of those species, 1972a; Erez and Gill 1977). Gill and Tipper but little else. These data can, however, provide (1978) tested the hypothesis that the results ob- information on species interactions that may be tained firom a study with metric variables can be particularly useful in explaining observed distri- obtained from binary nonmetric data. They di- butions. Such interactions are generally exam- chotomized and reanalyzed the data from six ined by means of pairwise comparisons of large multi varia te studies and showed agreement presence-absence among species or localities in between the pairs of analyses that ranged from Analysis (^ Biodiversity Data 209

75% to 97%. They concluded that for those 1. Similarity-dissimilarity coefficients. These studies, nonmetric binary data were entirely coefficients reflect the proportion of the adequate. sample that represents mutual occurrence (Hohn 1976). Formulas for their calculation do not include specific consideration of Measures of Association joint absences of variables (species). Measures, or coefficients, of association have 2. Matching coefficients. Formulas for these indi- been in existence since at least the late 1800s. ces include joint absences or non-occurrences They are appropriate for use with binary data and do not discriminate between positive and and thus are especially important for fau nal negative association (Hohn 1976). comparisons because the data from such stud- 3. Traditional association coefficients. Usu- ies are often in binary form. Ancillary data on ally, these coefficients are based upon the presence or absence of taxonomic characters, chi-square statistic (Table 20: section C) or environmental states, observer concordance, a statistic computationally related to chi- and chemical properties also can be recorded square, or they have an underlying assump- in binary form and analyzed with measures of tion of normality. These coefficients are association. calculated with formulas that include Joint Measures of association have several impor- absences, and they can be evaluated against tant applications in amphibian studies: expres- a probability distribution. sion of faunal resemblance as similarity or dissimilarity among species observed at two or Coefficients in the first two categories treat more sample sites; assessment of the degree of a sample as if it were devoid of sampling or coexistence in different localities or habitats of measurement error. They can be based loosely particular species of amphibians, based on their or strictly upon a probabilistic model. They frequency of co-occurrence in a series of sam- are not test statistics, however, and have no ples or observations; and measurement of the related probability or sampling distributions. similarity or dissimilarity of habitat or micro- Alternatively, the traditional coefficients as- habitat choices or conditions for selected spe- sess the probability that an observed number cies. Additionally, these measures are useful for of co-occtirrences between two species has condensing data on the resemblance of species, arisen by chance. habitats, or samples and for summarizing large Some argue that before a measure of associa- amounts of species data, tion is calculated, a significance test should be applied. The test would be used to indicate the CLASSES OF COEFFICIENTS existence (at a stated probability level) of an Resemblance, similarity, matching, and asso- underlying relationship between the species, ciation are the adjectives most commonly ap- habitats, or localities being considered. It would plied to the class of coefficients used to describe how likely the observed result is under describe two-state presence-absence data. Dis- the null hypothesis of no association and would tinct disciplines have spawned a vast array of help to determine if an association were "real." such coefficients, many of which are redun- Both the test statistic and its significance level dant. The 46 most popular measures are pre- are influenced by sample size. For any given sented in Table 20. I have grouped them into altemative hypothesis, chi-square (or an equiva- three categories, that are not mutually ex- lent) increases and the significance level de- clusive but that facilitate consideration. creases as sample size increases. The binary 210 CHAPTER 9

Table 20. Measures of Association

Relationship to # Coefficient Reference Alternative names Formula others A. Similarity coefTicients 1 Simpson Simpson Simpson coefficient a Equals Mountfotü's (1943) Degree of Faunal a + min(6, c) (#8) if population Resemblance follows a log distribution

2 Kulczynski Kulczynski Second Kulczynski \{ a 1 a'\ Can be written as the (1927) coefficient 2 a+b a+c arithmetic mean Driver and Kroeber \ J ofSinpson's(#l) coefficient andBraun- Blanquet's(#10)

3 Ochiai/ Ochiai Ochiai coefficieni a Same as the cosine Otsuka (1957) Otsuka coefficient coefficient for [(a+ *)(a+ £)]"" binary data Actually attributed to Otsuka in the Ochiai reference 4 Dice Dice (1945) Dice coefficient íSJ a Equivalent to the Coincidence index 2a + b + c a + ^Aib + c) additive inverse S0rensen(1948) of the Bray-Cur coelficient tis Coefficient on Quotient of binary data similarity Equivalent to Czekanowski 1-Nonmetric (1912) coefficient (#13) Burt coefficient Goodman & (Peters 1968) Kruskal's(19S4) Pirlot coefficient Xr = 2(Dice)-l (Peters 1968) Whittaker(1952) index Bray (1956) index of amplitudinal conespondence The asymmetric 4a Dice's Dice(1945) (a)IndexB|A = -^- asym- a + c versions of Dice metric (#4) indices (b) Index A|B=^^

Jaccard Jaccard Jaccard coefficient (1901) CoefÎKienl of floral a + b + c communities Coefficient of community Similarity index (Sneath 1957) Analysis of Biodiversity Data 211

Max/Min°' Known condition large Nonlinearity- Max E- sample Description Range linearity" Invariance" Min Max Coefficient'^ theory

The proportion of joint occun-ences or (0,1) Linear Yes Yes Yes No matches compared with the minimum number of occurrences for one species The proportion of all localities in which both of a par of species have been found relative to the smallest number of localities in which one species was found The arithmetic mean of the proportions (0,1) Linear Yes Yes Yes No of species jointly present over their separate marginals Devised to avoid the selection of a minimum mismatch in Simpson's (#1) formula The geometric mean of the proportions of (0,1 ) Linear Yes Yes Yes Yes Yes species present jointly relative to their separate marginals

The harmonic mean of the proportions of (0,1) Linear Yes Yes Yes Yes Yes species present jointly relative to the weighted total Weights mismatches less than matches so that weight of a frequency is twice the combined weight of b and c May be interpreted as a conditional probability

Index (a): the proportion of time or (0,1) Linear Yes Yes Yes No samples that A occurs when it was associated with B Index (b): the proportio of time or samples that B occurs when it was associated with A Devised to differ depending on the species used; to be used for a one- sided or nonreciprocal species dependency The proportion of localities in which both (0,1) Convex Yes Yes Yes Yes Yes species occur relative to the total number of localities in which any species was found

(Continued) 212 CHAPTER 9

Table 20. {Continued)

Relationship to # Coefficient Reference Alternative names Formula others 6 Sokal and Sokal and a a Sneath Sneath a + 2b + 2c a + 2(b-Y •c) UN; (1963)

7 Kulczynski Kulczynski First Kulczynski a (1927) coefficient b + c

8 Mountford Mountford I 2a An estimate of the (1962) 2bc + ab + ac~ multiplicative a inverse of ^A(ab + ac) + be Fisher's a When the popula- tion follows a log distribution, Simpson's Index (#1)= Mount- ford's 1= 1/Fisher's a 9 Correlation Sorgenfrei The square of ratio (1958) (a + c)(a + b) Ochiai/ Otsuka's (#3) 10 aun- Braun- Inconeclly called a Blanquet Blanquet Jaccard (#5) by a + max(i),c) (1932) Peters (1968)

11 Fager Pager and Pager and McGowan Ochiai/Otsuka's max(i),c) McGowan coefficient Vi 2 (#3) plus a [(fl + cXa + h)] (1963) correction factor

12 Savage Savage Coefficient of a The additive (1960) difference a •+ max(ii,c) inverse of Braun- Blanquet's(#10) 13 Nonmetric Sneath and b + c Equals 1-Dice Soka! 2a + b + c (#4) (1973)

14 McCon- McCon- a^-bc A scale- naughey naughey (a + b)(a + c) transformed (1965) Kulczynski (#2) coefficient Equals 2(Kulczynski)-l 15 Johnson Johnson Twice the (1967) a+b a+c Kulczynski (#2) coefficient Analysis of Biodiversity Data 213

Max/Min" Known condition large Noniinearity- Max E- sample Description Range linearity" Invariance" Min Max CoefTicient'^ theory Devised as an alternative to Jaccard's (0,1) Convex Yes Yes Yes Yes (#5) by differentially weighting the mismatches

The rate of joint occurrences per number (0, «>) Convex Yes Yes Yes No of mismatches

Derived from Fisher's a diversity index, C0.= Convex No Yes No No the parameter estimate from the log series distribution The actual similarity index was I = 1/a, for which the formula in this table was suggested as an approximation

Not defíned if a = c or if a = ii (ftï) Convex Yes Yes Yes Yes

The proportion of joint occurrences (0,1) Linear Yes Yes Yes No compared with the maximum number of oceuTTences for one species The proportion of all localities in which both of a pair of species have been found relative to the largest number of localities in which one species was found Derived from Ochiai/Otsuka (#3) (-", 1) Linear No No Yes The geometric mean of the proportions of co-occurrence for each of the two species with a correctino factor for the pairs of rare species (sample size) Authors said > ,5 meant association or affinity. ®1) Linear Yes Yes Yes

A scaled Euclidean distance for binary (0,1) Linear Yes Yes Yes data The proportion of mismatches over a weighted sum of all occurrences (-1,1) Linear Yes Yes Yes

The unweighted sum of die two basic (0,2) N/A Yes Yes Yes proportions involving joint occurrences

(Continued) 214 CHAPTER 9

Table 20. (Continued)

Relationship to # Coefficient Reference Alternative names Formula others 16 Forties Forbes For a' = the expected value of a, (1907) a na a' (a + b)(a + c)

17 Gilbert and Gilbert and log a - log 71 - log 1^^ - lea ^^ log of Forbes's Wells Wells (#16) (1966) 18 Forbes Forbes ng - (a + b)(a + c) When ad > be, (1925) Omat - ú n • min(b,c) ~(a + b)[a + c) equals Cole's (#45) 19 Tarwid Tarwid a-a' m-(a-¥ b)(a + c) (1960) a + a' na + {a + b}{a + c) 20 Resemblance Preston equation (1962a) coefficient Hagmeier and Stults where 1 - z is the coefficient (1964) B. Matching coefficients 21 Simple Sokal and Soka! and Michenei fl +

24 Sokal and Sokal and a + d lid = 0, equals the Sneath Sneath b + c first Kulczynski UNj (1963) coefficient (#7) Numerator is index of crude agree- ment (see #21) 25 Sokal and Sokal and 2(a + ii) If íí = 0, equals Sneath Sneath 2i,a + d) + (b + c) Dice's (#4) UN, (1963)

26 Russell and Russell and If íí = 0, equals Rao Rao a+b+c+d n Jaccard's (#5) (1948) Analysis of Biodiversity Data 215

Max/Min" Known condition large Nonlinearity- MaxE- sample Description Range linearity" Invariance" Min Max Coefficient' theory Compares the number of associated (0,~) Linear Yes Yes No occurrences of the two species in a series of random samples with the number of such co-occurrences expected by chance When this equals 1, shows that the two species occur together in exactly the number of samples expected by chance Is the ratio of actual to calculated joint occurrences Min not defined for a = 0 (_M,oi.) Concave Yes No No For max, log n-log a

(-JO,«) Convex Yes No No

(-1,1) Concave Yes No No

Equation is transcendental with no Yes general solution, only a sei of approximate solutions Assumes individuals are distributed according to log normal distribution

An alternative to Jaccard's (#5) by (0,1) Linear Yes Yes Yes Yes including negative matches or joint absences in both numerator and denominator The proportion of ( 1,1 ) and (0,0) agreements in the totals of n species comparisons Square of the geometric mean of the four (0,1) Convex Yes Yes Yes Yes basic proportions in the table

The arithmetic mean of the four basic (0,1) Linear Yes Yes Yes No proportions in the table Equals Vi when degree of agreement is exactly that predicted by chance A comparison of the joint events with the (0.«) Convex Yes Yes No No single species events or of matches with mismatches Max not defined for A = c

The weighted proportion of joint events (0,1) Convex Yes Yes Yes No compared with the (weighted) sum of all events Mismatches are given half the weight of matches. An alternative to Jaccard's (#5) by (0,1) Linear No Yes No No inclusion of ii, the joint absences, in denominator The proportion of joint occurrences in the total n comparisons

{Continued) 216 CHAPTER?

Table 20. {Continued)

Relationship to # Coefficient Reference Alternative names Formula others 27 Rogers and Rogers and a + d If d = 0, equals Taniinoto Tanimoto a + d + 2(b + c) UN, (#6) (1960) Reflects a com- pletely different situation from UN; (#25) Index of cm de agreement in numerator (see #21) 28 Hamann Hamann G-Index a + d-{b+c) Is a scale-trans- (1961) G-Index of agreement a + d+(b + c) formed SMC HoUey and Guilford (#21) to (1964) change the range 2(#21)-1 Mathematically related to Yule's Q(#43) Equals the (j) coef- ficient (#42) only when the mar- ginal proportions all equal .5 29 Baroni- Baroni- S* ••fad + a-h-c Is (#30) scale Urbani Urbani •4ad + a + b + c transformed to and Buser and Buser change the range (1976) 2(#30)-l 30 Baroni- Baroni- S** Vüd-HíJ Urbani Urbani V•i -híi-hÈ-h£• and Buser and Buser (1976)

31 Total Sneath Binary distance b-vc For binary data, the difference (1968) coefficient a+b+c+d square rcwt is Sokal binary distance equivalent to coefficient average Euclidean distance coefficient Related to Non- metric (#13) without weight- ing the joint occurrences and by including the joint absences 32 Pattern Sneath 2

Max/Min"' Known condition large Noniinearity- MaxE- sample Description Range linearit)''' Invariance" Min Max Coefficient theory The proportion of joint events compared (0,1) Convex Yes Yes Yes Yes with the weiglited sum of all events

Mismatches are given twice the weight of matches.

Requires no assumptions about the data (-1,1) Linear Yes Yes Yes Yes

Based on the probability of agreement of responses

Not a correlation coefficient even though it can equal 41 in one case

Max not defined iia + d = b + c.

Devised as an adjustment to Hamann's (-1,1) Concave No Yes Yes No (#28) to correct the problem that Hamann's does not approach 1 as a + d approaches n when a*0. Distribution varies with n (0,1) Concave Yes Yes Yes No Devised to express similarity as the dif- ference of positive weighted matches and equal but negative weighted mismatches The proportion of mismatches to the total (1,0) Linear No No No

Equals 1 if a = (i = 0 and* = c (0,1) Linear Yes No No

(0,.57) Concave (in Yes Yes Yes Absolute Value)

{Continued) 218 CHAPTER 9

Table 20. {Continued)

Relationship to # Coefficient Reference Alternative names Formula others 34 Michael Michael Called McEven and 4{ad - be) (1921) Michael by Cole (1949) with no reference 35 Faith Faith (1983) C-coefficient

2n n

36 Omega Yule (1912) Coefficient of 2 37 Eyraud Eyraud q-[(q + fa)(q+C)] (1936) (a + b){c + (i)(a + cYb + d)

C. Traditional association measures 38 x'statistic Binomial index of njad - bcf dispersion {a + b){c + d){a + c)(è + d)

39 Coefficient Pearson V {ad-bcf The X statistic of mean (1901) Pearson's mean {a + b){c-^d){a + c){b + d) scaled by the square Doolittle square contingency total= X/n contin- (1885) Discriminate For the 2 X 2 table, gency association ratio is Pearson's p' Doolittle derived it as asymmetric version of Pearce's (#44). Tschouproff's J.2 \ t= -r.V(r-l)(c-l)^ Sakbda's p* P/Pm., = min(r,c)i)>^ [min(r,c)- !](! + <

40 Cramer's D Cramer {ad-bcf \iflmm(r,c) - 1 (1946) {a + b){c + d)(a + c)(h + d)\mm (r,c) - 1]

41 Pearson's Pearson ad-be An estimate of contin- (1904) ^ Pearson's p [(a + b){c + d){a + c){b +d) + (ad- bc:f] gency coefficient Equals l-<^*

Equals 1+xX

Equals Anaiysis of Biodiversity Data 219

Max/Min" Known condition targe Nonlinearity- \iaxE- sample Description Range linearity Invariance Min Max CoefFicient thecH-y The factor {ad-hc) in numerator relates this (-1,1) Convex Yes Yes Yes 10 traditional statistical association measures.

Is author's "asymmetric" measure to count (0,«) Linear Yes Yes No shared presence as 1, a mismatch as -1, and a shared absence as 0 Author recommends il when one-state homogeneity is to be maximized. That is, groups with large number of matches and few mismatches are desirable (-1,1) Concave Yes Yes Yes

(-1,0) N/A No No Yes

Range depends cm table's dimensions (0, n[min(r, c) - 1]), so Can attain its maximum only when each for 2 X 2 table it is row and column contains all nonzero (0,«) entries Even though this is a x'dividing out the [0, min(r, c) - 1], so for factor of total size, its range still 2x liable it is(0,l) depends on the table's dimensions. When there are no missing data, it is of little consequence to scale x^-

Developed this nonparametric statistic as a (0,1 ) scaled version of an estimate of (¡i '^ (#39) Range is independent of table dimensions. Can be seen as the maximum likelihood (0,1) estimator of p (correlation coefficient) under multinominal sampling Proposed by Pearson to transform the scale on his first index (#39)

(Continued) 220 CHAPTER 9

Tïible 20. (Continued)

Relationship to # Coefficient Reference Alternative names Formula others

42 Phi Pearson ad-be Equals (xV•)** coefficient (1901) Yule's Q' [(a + £•){& + d)(a + b)<.c + d)]'^ Equals V4?" Yule (1911) Pearson's correlation Boas (1909) coefficient p Tschouproff Product sum (1919) correlation (Yule [(r-lXc-l)] 1912) Fourfold point (tetrachoric) coefficient (Pearson and Heron 1913) McEwen 's coefficient Kendall and Stuart's 1)

43 Yule's Q Yule (19(X)) ad-DC n ad + bc ad + bc

44 Pearce Pearce J ad- be a__ b A special case of (1884) (j) {a + c){b + d)~ a + c h + d the phi Youden G coefficient (1950) Youden coefficient (#42) if row totals and column totals, respectively, are equal

45 C, Cole (1949) Index of interspecific For ad S be: association ad-be

(a + b)(b + d) ^ /v TnaK For be >ad,d> a: with the sign of ad-be ad-bc {a + b)(a + c) For a > d: ad- be (h + d)(c + d) Analysis of Biodiversity Data 221

Max/Min"'* Known condition large Nonlinearity- Max£- sample Description Range linearity" Invariance" Min Max Coefficient'^ theorj' Range still depends on die table's dimensions. . Jmin(r,c) - • <1 niin(r,c) Not defined for ¡2 = d = 0 or for è = c = 0 Perason derived this as the correlation (-1,1) between errors in the position of means of two variables when each is measured in terms of its own standard deviation. Yule separately (apparently differing notation caused the problem) derived it as "a theoretical value" of the correlation coefficient. This use is correctly severely criticized. The probability interpietation of Yule's Q is the probability that two individuals selected have their A and B states in the same order, minus the probability that they have different A and B states in opposite orders. When ad = ÍIC then Q = 0, a value (-1,1) midway between the two extremes. Yule wroie: "Formoderate association, this coefficient gives much the larger values." Heron (1911) said it should be used only when a + b = c + d and a + i- = b + d. When equals 0, is equivalent to (-1,1) randomized prediction jsing just the row marginal frequencies, i.e., corresponds to independence. Equals a d •r+7•^-1a+b b+d

Is the difference between the conditional column-wise probabilities Basic assumption is that false positives (c cell) are as undesirable as false negatives (d cell). Not independent of the frequencies (-1,1)

{Continued} 222 CHAPTERS

Table 20. {Continued)

Relationship to Coeflkient Reference Alternative names Formula others 46 C, Hurlbert ad-be ' Xmm (1969) \aä-bc\ ls± with the sign of ad-be with the sign of ad-bc When table yields a statistically significant result, x'min = id (observed value) and C-, = Cjj.

"See the section on "Properties of Association Measures" for explanations, and see Appendix in this chapter for examples,

''These columns indicate whether the coefficient attains its maximum and minimum values at the respeaive end points of its range.

Jansen and Vegelius (1981). Dice (1945) showed that Forbes' coefficient measures the degree to which the association of two species conforms to expectation on the basis of chance, not the degree of association itself. Analysis of Biodiversity Data 223

Max/Min"^ Known condition large Nontinearity- MaxE- sample Description Range linearity" Invariance" Min Max Coefficient' thewy Measures the degree to which two (-1,1) species'joint occurrences are more or less frequent than based on chance alone 224 CHAPTER 9

measure, in contrast, describes the size of the Table 21. Generalized 2x2 Frequency Table association (or effect) and cannot be influenced Using Proportions by sample size. State or species A A measure of association should be selected State or in keeping with the questions being asked; no species B Present Absent Total single coefficient is sufficient for all amphibian Present A B (a + h)ln data. The most common approach to selection = A-vB appears to be familiarity with a similar study. I Absent C D (c + d)ln outline here a rationale for selection of a coeffi- = C + D cient in keeping with the inventory and monitor- ing objectives. Total (a + c)ln (h + d)/n 1.00 =A+C =B+D I have not considered the distance metric, a measure of spatial relationship in coordinate space, because it generally is not appropriate for the unordered nominal data of our discussion. A 2 X 2 table can be rewritten in terms of the Besides, similarity coefficients (under certain related proportions. In this case, A = a/n, B = bin, circumstances) can be transformed quite simply C = cin, D ^ din, zxia A + B + C + D = \ into distance metrics. These coefficients are (Table 21). Altematively, we could write these treated by Legrendre and Legrendre (1983). with symbols for the proportion in the /th row and yth column, p,j,j= 1,2. NOTATION AND EVALUATION Binary measures must be evaluated from I have restricted discussion specifically to pair- the standpoint of operationalization, that is, wise comparisons, that is, to contingency tables the meaning in the study of association. Is this with 2 rows (R) and 2 columns (C) (2x2 tables), word synonymous just with joint presence because they are the basis for most of the coeffi- events? Are two species "highly" associated cients. Nevertheless, much of the discussion is when they occur in a large number of localities applicable to measures based upon generalized together and never alone? Would specimens of tables (R X C, for any number of rows and col- Rana ingerí and R. blythi be associated if they umns). The most acceptable and easily general- co-occurred in 40% of the samples and also ized presentation of two-state data is in terms of were absent from 40% of the samples in which the four cells of a 2 x 2 frequency table, in which one might have expected them? Answers to a = frequency of positive matches, joint such questions form the basis for selection of presence, or co-occurrence (1, 1); d = frequency the appropriate measure of association for a of negative matches, joint absences, or non- specific sampling situation. All aspects of the co-occurrences (0, 0); and b and c = frequency study must be considered and the related of mismatches [(0, l)and(l,0)] (Table 19), with hypotheses and research questions operationa- n equaling the number of amphibians, individual lized in order to arrive at the proper choice. elements, samples, or characters in the study (i.e., n = a + b + c + d). Every possible associa- CLASSIFYING DATA tive relationship between states, habitats, or spe- Confusion exists in the ecological literature cies A and B (usually called factors or variables) about four aspects of the classification of data is expressible in terms of these four quantities. for use with binary measures. Each of these fac- Much of the redundancy among the coefficients tors influences the choice of an index for a par- is due to the lack of a common notational system. ticular amphibian study and its interpretation. Analysis of Biodiversity Data 225

UNDERLYING CONTINUA. For data arranged in Presence or absence of inherent ordering of a contingency table, the factors (species, habi- the data is an important consideration in the se- tats) describing the columns and rows could be lection of association measures. Some measures assumed to arise from an underlying continuum. assume an ordered structure in the data, and For example, species occurrences could be some traditional association measures can statis- based upon population distributions defined on a tically test for its existence. However, in a 2 x 2 continuum of depth or vegetation height, or, in table, the two possible orderings of the catego- other words, a specific nonrandom abundance ries can alter only the measure's sign; they do distribution of amphibian species could exist not provide important statistical information to within the breeding habitats being sampled. The the analysis. value of such an assumption is disputed. Pearson and Heron (1913) argued that it is most often ASYMMETRY. Asymmetry in binary data occurs justified, whereas Yule (1912) believed that it is when one of the two states is interpreted as most often misleading. When all factors from a "more informative" than the other. This would study originate from a multivariate normal dis- be the case with presence-absence data in which tribution on an underlying continuum (unlike the state of "presence" provides more informa- our fourfold tables), most investigators agree tion on species similarity than does the fre- that the measure of association should be based quency of absences. When an investigator is on a correlation coefficient (e.g., tetrachoric cor- reasonably assured, a priori, that a causal rela- relation coefficient). tionship exists and runs in one direction only, or if the coding of the character can be reversed ORDER. A fundamental ordering may exist be- with no sacrifice of information, then the data tween (or among) the classifications of a fac- must be viewed in an asymmetric manner. tor. For example, aquatic or montane habitats Many authors (e.g., Sneath and Sokal 1973; occupied by frog species can be ordered by Baroni-Urbani and Buser 1976) have argued not depth, elevation, ambient temperature, time, only that absences contribute less information or abundance. The latter three classifications than presences, but also that shared absences normally use interval or continuous variables. have no part in a measure of similarity. Some Depth and elevation can be categorized on or- authors (e.g., Green 1974; Ludwig and Reynolds dinal or interval scales. In ordinal classifica- 1988) have stated that any index of bioassocia- tions, categories are intrinsically ordered (e.g., tion should be independent of the number of shallow, deep; at sea level, above sea level), joint absences (= d in the frequency table). An but not quantitatively comparable. Shallow is inspection of the formulas given in Table 20 less than deep, but by an unknown amount. An shows that the inclusion or exclusion of shared interval scale results if shallow and deep are absences (t/) in a formula may not be relevant defined by number of meters from a precisely because

Table 22. Frequency Distribution ot Eleutherodactylus guentheri Frogs among Four Microhabitats, by Age and Sex

Microliabitat

In leaf Exposed on On vegetation On vegetation Age and sex litter ground <0.5m >0.5m

Juvenile males 0.25 0.00 0.00 0.00 Juvenile females 0.00 0.25 am 0.00 Adult males 0.00 0.00 0.00 0.25 Adult females 0.00 0.00 ^m 0.00

The d cell then represents the number of joint in the leaf litter with those collected from ex- absences of a pair of species that had been ex- posed ground or vegetation (Table 23). An alter- pected in similar samples, or the absence of a native would be to combine terrestrial and species from a pair of habitats in which it had arboreal microhabitats and compare adults and been expected. Use of the d cell frequency juveniles between them (Table 24). It could be speaks to the chance that an occurrence could argued, equally, that habitat preferences of the reasonably have occurred but did not. age groups were associated with higher and lower humidity rather than cover or height CATEGORY DEFINITIONS. Because the opera- above ground. The change from a more subtle to tional definitions of the categories used in a a coarser description of habitat can affect the study can affect the degree of association, the magnitude of the measure. Clearly, without pre- definitions should be carefully foimulated in cise definitions, inventory data cannot be sum- Iteeping with the expected uses of the final con- marized in a useful way with any binary clusions (Goodman and Kruskal 1954). This is coefficient. particularly important because the same dala set, combined in different ways, yields quite differ- Table 23. Alternate 1 of Collapsed ent tables and coefficients and can provide dis- Microhabitat Frequency of Occurrence Data tinctly different impressions of the degree of for Eleutherodactylus guentheri. Contrasting association between factors. Specimens from Leaf Litter with Those from For example, consider a study that calls for Exposed Sites, for Juveniles and Adults collecting Eleutherodactylus guentheri from Microhabitat four microhabitats, in which the number of frogs collected in each varies by sex and age Exposed on (Table 22). Because the tabled array of data in- ground or on vegetation cludes a number of empty cells, the researcher Age In leaf litter above ground may wish to collapse the table to reduce the number of categories with zero values. He or she Juveniles 0.25 0.2S could combine the data over categories, for ex- Adults 0.00 0.50 ample, contrasting juveniles and adults collected Analysis of Biodiversity Data 227

Table 24. Alternate 2 of Collapsed the species are associated, then a:>¡=(a + h){a + Microhabitat Frequency of Occurrence Data c)/n. and the difference could be written as £> = for Eleutherodactylus gueniheri, Contrasting a-(a + b)(a + c)/n = {ad-bc)¡n, which is the term Specimens from Terrestrial Microhabitats with found in the formula for calculating the usual Those from Arboreal Microliabitats, for chi-square statistic (x^) and all of its monotoni- Juveniles and Adults cally related statistics (see Table 20: section C). Microhabitat The x^ test is used to determine whether two Terrestrial Arboreal species (or attributes) are independent. How- Age microhabitats microhabitats ever, the term independent in this context does not imply the usual stochastic independence of Juveniles 0.50 0.00 the variables used to generate the 2 x 2 table Adults 0.00 0.50 (Kendall and Stuart 1973). Rather, in the context of association measures, independence of the variables indicates species independence (or In addition, it is important to remember that a lack of association), which may or may not coin- test of significance in tables (e.g., with a chi- cide with stochastic independence. Likewise, al- square approximation) assumes that the sample though results of tests of association correspond is random and that the category designations are to intuitive notions of association and indepen- chosen a priori. Pooling of tabled values affects dence in many situations, this is not always the case. randomness with undeterminable consequences and may result in meaningless calculations with- Consider an example of two species of Cros- out statistical interpretation. sodactylodes captured in an arboreal bromeliad sampling session (Table 25). If the species are independent (lack association), the frequency in CONCEPTUAL RELATIONSHIPS cell a, which indicates joint occurrences, should Another problem that interferes with the formula- be (a + b)(a + c)/n) or 50 • 49/100 = 24.5. In fact, tion of an operational definition of association (and the observed frequency of a is 48, greater than its measures) is the confusion of terms used in the expected under independence, so the two species literature to describe association. I briefly discuss may be said to be positively associated in this the relationships of some of these terms below. microhabitat. Alternatively, consider tadpoles of Pseudacris crucifer and Rana clamitans (Table ASSOCIATION AND INDEPENDENCE. In the fre- 26) dipnetted from a series of ponds. If these two quency notation for 2 X 2 tables, if no association species are independent, then 50 • 60/75, or 40, exists between species A and species B, an equal proportion of species B should be found among the samples with species A and the samples Table 25. The Occurrence of Two Species of without species A. This relationship can be writ- Frogs {Crossodaclylodes spp.) in Bromeliads ten as a/(a + c) = b/{h + d) = (a + b)/n. By f C. bokermanni rearranging this set of equations, we can alterna- izecksohni tively say that a = (a + b)(a + c)/n. This algebraic Present Absent Total statement shows that when two species are not Present 48 2 50 associated, the proportion of joint occurrences Absent 1 49 50 (a) is equivalent to the product of the proportions Total 49 51 100 of each separate species in the entire sample. If 228 CHAPTER 9

Table 26. Data from 75 Dipnet Samples of the relative numbers of individuals of the two Tadpoles of Two Species of Frogs; Arranged in species are constant or nearly so (Cole 1949). a 2 X 2 Frequency Table Using Counts The assumption that the abundances (frequen- cies) of two positively associated species in- Hyla crucifer Rana crease or decrease together is unrealistic if their clamitans Present Absent Total association is merely the result of similar habitat requirements (Hurlbert 1969). Present 40 10 50 The correlation coefficient also is not a rea- Absent 20 5 25 sonable indicator of association when the table Total 60 15 75 is asymmetric. Low correlation values do not have significance, but that does not imply, nec- joint occurrences are expected. In this instance, essarily, lack of species relationship (e.g., the observed frequency of joint occurrences {a) Legrendre and Legrendre 1983). With site-by- is 40, and we conclude that the tadpoles are not site classifications, many of the estimated corre- associated. lations are small, because a few species are If the data in Table 25 are considered as a abundant and others are rare (e.g., Clifford and sample from an infinite population and are sub- Stephenson 1975). jected to a chi-square test of independence, then highly significant association is indicated, in TRADITIONAL ASSOCIATION MEASURES For agreement with our previous statement. The continuous data, the strength of an association cross-classified dipnet tadpole data (Table 26) can be assessed by the reduction in variance in yield a nonsignificant test result, which corre- a dependent variable that results from know- spwnds to a decision in favor of independence. ing the value of the independent variable. For Note, however, that in the Crossodactylodes categorical variables, the variance quantity, example, for which we indicated association, strictly speaking, is not defined. Historically, 48% of the frogs sampled co-occurred, whereas three approaches have been used to find a sub- in the tadpole example, for which we indicated stitute quantity for the desired variance (Hays independence, co-occurrence was approxi- 1973): statistical independence between the 2 mately 53%. Investigators should always pro- two attributes (measures based upon the x vide operational definitions of terms before statistic), predictive association, and informa- selecting or applying a measure. Otherwise, log- tion theory. Each approach, in turn, leads to a ical inconsistencies may result. measure by which to assess association strength. ASSOCIATION AND CORRELATION. A relation- None of these measures depends upon the ship exists between the intuitive notion of asso- order of the categories. The first approach (x^) is ciation and the statistical measure of correlation the only method commonly used m ecological in a 2 X 2 table. However, association analyses studies of amphibians. Association is measured group species based upon mutual presence, by the difference between the probability of a whereas correlation and covariance coefficients joint occurrence and the product of the two mar- give a picture of the linear correlation between ginal probabilities. If two events are statistically fluctuations in abundances of two species. Thus, independent, the difference should be zero. even if two species co-occur 100% of the time, a Pearson and Yule used this fact in the develop- correlation coefficient (e.g., Pearson's or a vari- ment of what is commonly known as the

Meaningful interpretation of measures of as- tive gain in information about one species, given sociation based on the x^ statistic is difficult. information of the other. None of the measures has a straightforward Association then would be defined operation- probabilistic interpretation, and none is widely ally as an increase in information or a decrease accepted in the statistical literature. In addition, in uncertainty. Sampling theory exists for this the range of many of these measures (Table 20: type of measure, and tests of no association are section C) depends on the table dimensions, so possible. the measures are not really comparable across tables of different size. PROPERTIES OF ASSOCIATION MEASURES The two other approaches to the study of asso- Empirical and theoretical aspects of association ciation are not widely encountered in the am- measures related to the goals of the study should phibian literature and will be mentioned only be thoroughly investigated prior to selection of a briefly. The first deals with predictive associa- measure. Some important mathematical argu- tion and assesses the relative utility of one factor ments for selection of a particular measure of for improving the ability to predict the second association do exist, but most have been ad- factor's categories. Two factors (e.g., species) vanced to support informed opinion. Likewise, are associated if one factor is of value for reduc- certain measures are appropriate for a unique ing the error when predicting the other (Good- problem but not for general use. Below and in man and Kruskal 1954). Goodman and Kruskal Table 20,1 consider some aspects of association (1979) developed an asymmetric measure of indices that appear to contribute most often to predictive association, lambda {X). This measure the problem of selecting an appropriate measure. is intuitively appealing for amphibian studies. However, lambda depends on the marginal fre- RANGE OF THE MEASURE Many authors argue quencies and may not be suitable for compara- that a particular range of a measure (i.e., values the tive studies across species or habitats unless measure may assume, such as ail values between 1 sample sizes and, therefore, marginal fi-equen- and -1) is "best" when, in fact, it is merely accept- cies are equal. able convention. Most of the measures in Table 20 The final approach to studying association range between limits of 0 and 1 or of 1 and -1, uses an information theory strategy to derive a altiiough some measures can assume any real-val- variance analogue for binary data (see, e.g., ued positive number or are undefined (e.g., cannot Lambert and Williams 1966; Wallace and equal 0 in the range -1 to 1 ) about particular points. Boulton 1968; Field 1969). The derived mea- The importance of ranges per se is minimal, be- sure is related to diversity indices and other cause it is quite simple to transform either of the indices for numerical data. One diversity two most popular ranges to the other with a simple index that can be used is Shannon's index H, linear transformation. When a coefficient Cl is which is a measure of the information given defined on (0,1) and coefficient C2 is defined on by the sample. When used with qualitative (1,-1), we use: data, H can be written as an analogue of the variance of the distribution, becoming larger Cl = (1/2)(C2 + 1) as the distribution spreads out along the axis C2 = 2C1-1 and smaller as it nears the average. In general, the greater the variance, the less the informa- It seems reasonable to require that the limits of tion. Because of this tendency, an investigator the range be known and achievable and that the can define an asymmetric measure of the rela- coefficient assume the central or minimum value 230 CHAPTER 9

of its range when factors are not associated. Be- smallest values in its range must be defined. yond this, the scale should be easily interpret- Maximum values of association indices able to the user. The popularity of the (0,1) should occur when h = c = 0 (e.g., Jansen and interval coincides with the natural preference to Vegelius 1981). Minima can occur when a = d avoid negative numbers and may be preferable = 0 (Jansen and Vegelius 1981), but a stronger when values that are symmetric about zero are condition would be that only a - 0 (Kendall not a consideration. and Stuart 1973; Hubalek 1982). These conditions are inherent attributes of the INVARIANCE OR HOMOGENEITY. Homogeneity coefficients. means, simply, that the value of a descriptive mea- In the "Range Conditions" section of the Ap- sure is constant as long as the factors occur in the pendix at the end of this chapter, I show how same proportions (Jansen and Vegelius 1981). each of the measures varies as some conditions Seemingly, no cogent biological reason argues for for attaining the limits of the range (maximum the use of a measure that is not invariant in this and minimum) are varied. The values in the sense. Each coefficient is examined for this condi- cross-classified tables are purposefully extreme tion empirically in the Appendix at the end of this to highlight the dangers of uninformed selection chapter (see "Examples for Invariance"), or of selection not based upon the operational meaning of the study. These values should be LINEARITY. The value of a coefficient should examined carefully prior to selection of a coeffi- increase as the relationship from nonassociation cient and in light of the study to be done. Ques- to association strengthens. The form of this tions such as the following should be asked: change is important for interpretation of a mea- (1) Should the two species of the study be re- sure in a given study. As a minimum condition, it garded as similar or associated largely on the appears sensible to expect that a coefficient in- basis of their shared presence only in the se- crease or decrease in a systematic way relative to lected locality? (2) Should association be de- association. In a mathematical sense, however, fined as a combination of joint occurrences and systematic may be defined in a variety of ways. joint absences of the two species? (3) Which of Linearity in measures of association means the coefficients•Simpson, Dice, or Simple equal amounts of change in the value of the Matching (Table 20), for example•best adheres coefficient when values of joint occurrence to my preconceptions concerning the nature of change by a factor of one. When an amphibian the relationship being studied? worker selects a coefficient as a measure of To illustrate what Table 20 tells us, consider ecological resemblance, the change in values the latter question, focusing on the maximum. probably should be linear for ease of interpre- The Simpson coefficient attains its maximum tation, because such a relationship appears to under a wider set of conditions than do the other correspond best to intuition. Examples of lin- two coefficients. The Dice coefficient and the earity, and an evaluation of this attribute in each simple matching coefficient become 1 when b = coefficient are given in the Appendix at the end c = 0, whereas Simpson's demands only that b - of this chapter (see "Examples for Linearity 0 or c = 0. Conditions on a and d do not affect the Conditions"). ability of any of the three coefficients to attain its maximum. Thus, although the three indices vary CONDITIONS ON MAXIMA AND MINIMA. The between the same limits of 0 and 1, the range set of conditions under which the value of an does not depend solely on the proportion of spe- association measure will equal the largest and cies found together. Analysis of Biodiversity Data 231

MatJiematically, it is acceptable that condi- concept of independence between factors. Al- tions under which maximum and minimum val- though simple and intuitively appealing, they are ues are attained differ, but empirically, the not apphcable in all cases, as I show in the sec- researcher most definitely should be awa^e of tion on Species Abundance, below. If a test indi- the specific conditions under which the measure used fluctuates. cates a dependence, the investigator must ask what that dependence means for the study. The If only one of the possible h and c cell fre- selection of an operational definition of associa- quencies must equal zero to achieve maximum tion in keeping with the character of the study is values of association, then the association could most important at this stage. be called undistributed because all A's are B'& Dice (1945) warned that the abundance of a but not all ß's are A's. Put in more concrete species and the social behavior of its individuals terms, if all frogs are amphibians, undistributed can affect the frequency with which the species association exists between frogs and amphibi- appears in a sample and, therefore, can affect ans. The problem of what to do with the many measures of association. Interaction or correla- amphibians that are not frogs is unresolved. If tion among the specimens that form the cell alternatively, we require that two cell frequen- counts,'for reasons extraneous to the association cies equal zero (¿) = 0 and c = 0) before a coeffi- analysis, can also affect these measures. How- cient can assume its upper limit, then the ever, if the samples are drawn at random and are coefficient will equal 1 (the maximum) only if representative of the species, the cell frequencies all A's are B's and all 5's are A\ This kind of may be subjected to association analysis. It is, association is termed proper association and is therefore, important to emphasize that measures also the absolute statistical association of Ken- of association should be applied only across dall and Stuart (1973). comparable samples. For example, coefficients cannot be used to compare samples taken from MONOTONIC RELATIONSHIP AMONG MEASURES. breeding habitats with samples from nonbreed- A monotonie relationship exists between binary ing habitats, because the a priori expectation is coefficients when they are related by an inequal- one of marked difference (i.e., nonassociation). ity•that is, one coefficient is always less than or The fundamental feature of the approach is the equal to the other, regardless of the values in the assumption of an association of the two species, observed table. Almost all of the traditional asso- observable when their presence-absence is re- ciation measures are monotonically related to x^, corded in the same sample. as well as to each other (see, e.g., Hubalek Association analysis cannot usually be ap- 1982). When comparative analyses are run with plied to samples from a very large and heteroge- different coefficients, it is possible to confuse neous area unless the species and habitats predetermined monotonicities with differences sampled have been judiciously selected. Alterna- caused by the data, unless the relationships of tively, if a sample plot is relatively small, species coefficients are known. are likely to occur in only a small proportion of the space, and all will appear to be rare. Hohn WARNINGS ABOUT INTERPRETATION (1976) showed that when two taxa occur in a Significance tests of the chi-square type are majority of samples, a similarity coefficient will given wide attention in ecological and bioasso- be high even if the taxa can be described by an ciational studies, but they are not a panacea. effectively random distribution; such a coeffi- They produce approximate results dependent cient lacks meaning because the total number of upon sample size that indicate a vaguely defined samples is not considered. The most efficient 232 CHAPTER 9

measure of association is obtainable when the between two factors without concern for inde- more abundant of the species occurs in about pendence testing. In this case the intended meaning half of the samples in the group (Dice 1945). of association (see "Conceptual Relationships," Generally, an association analysis is applica- above), as well as the operational descriptors of ble only to the particular habitat and time of the the study situation, must be reflected in the sample. The dimensions or conditions of the choice of a measure. sample (time, place, size), therefore, must be Standardization in this section on presence- clearly delineated and described. If the same absence data cannot be extended to a delineation sampling considerations described above apply, of conditions for the selection of a single coeffi- samples taken over time can also be compared. cient. In fact, Hubalek (1982) recommended that Etespite the most scrupulous presentation, it is "three or so" alternative coefficients be used and still possible that the association of two species compared for each data set. I prefer the develop- is only accidental•that is, that it reflects a ment of an underiying rationale for selection in chance selection by the two species of the same keeping with the study objectives. To make a habitat at the moment of sampling for specific final selection of a coefficient, the definition of but independent reasons. Because the primary association must be a meaningful reflection of causes of variation in associative frequencies the study design and purposes. cannot be distinguished merely from inspection of a list of coefficients, Forbes ( 1907) and others suggested an accompanying study of species Species abundance distributions to differentiate between local population effects (e.g., abundance, rarity) and Individual Counts and Proportions general ecological factors (e.g., climate, topog- raphy, hydrography). A monitoring project can produce abundance data, or counts of individuals, n• for i = 1,... ,J SUMMARY AND RECOMMENDATIONS species. These numbers also can be presented in Significance tests and binary measures of asso- the form of proportions (when divided by the ciation are distinct. If investigation of indepen- related total number for all species). In this sec- dence is the aim, the observed frequencies of tion I describe several methods for the examina- co-occurrence of the two factors in the 2 x 2 tion of this type of data. I begin with the R x C table should be examined to determine if, in- table organization for R rows and C columns and deed, a statistically significant dependence rela- detail the specific 2x2 case, which is common tionship exists. If the relationship cannot be for analysis of amphibian data. distinguished from random occurrence, there is no basis for continued study, although extended OBSERVED SITUATIONS sampling to increase frequencies could be bene- It is often important in biology to test whether ficial. When the test shows the relationship to be the frequencies of observations in a series of significant, the size of this relationship can be categories are the same. The observed distribu- examined with a judiciously selected binary co- tion of abundances among categories defmed by efficient whose form describes the sampling sit- species and their occurrences is of interest for uation operationally and is in keeping with the amphibian sampling studies. The hypothesis of study purposes. homogeneity•for example, across k classes Alternatively, binary coefficients may be used when a set of abundances (n¡) or occurrences has simply to describe an associative relationship been observed•may be tested in different ways. Analysis of Biodiversity Data 233

The selection of an appropriate test depends sub- tional gradients could compare conditional dis- stantially on the relevant aspects of a particular tributions of 5 at various elevations. data set. Given these probabilities, we could ask if the It is usual to summarize individual counts or joint probability distribution is of some exact abundance data across some factor (e.g.. sex, form. Distributions of some exact form are un- height above ground) by using the numbers ob- usual in amphibian biodiversity work. Instead, tained directly or after adjustment for sampling, frequencies or proportions are used as unbiased in a cross tabulation for analysis by a test such as estimates of these probabilities, and a test is ap- chi-square. Factors such as height are ordered, plied to ask questions about the independence of whereas factors such as sex are unordered. In the tabled factors. some situations, the same factors or variables may be observed at two different times or in two SIZE OF THE SAMPLE related samples (e.g., juvenile and adult); this is called identical categorization. From a purely statistical point of view, the size For a general table with R rows and C col- of a sample affects only the magnitude of the umns, each cell shows the frequency of one pos- difference in proportions expected to result in sible joint event that occurred in the data (say, a a statistically significant test outcome for the shared presence at a given height on a gradient). stated type 1 error risk. A small sample may The general assumption is that & joint frequency suffice if, in the long run, the average propor- distribution represents the spread of frequencies tions are greatiy different for the two groups. in the table cells. However, if one factor is fixed If tiiey are not very different, a statistically rather than random, it is not meaningful to talk significant result is not expected unless a large of a joint distribution. The joint probability dis- sample is obtained. This is the problem of sub- tribution in the population, which corresponds to stantive significance, which is discussed in the joint frequency distribution for a sample, is Chapter 4 (see "Statistical versus Substantive described in Table 27. For a table with this type Significance"). Computations are less compli- of organization, the marginal distributions give cated for small samples, but a strong biologi- information on only a single variable. For a fixed cal association effect must exist to be termed value of factor^, one can look at the B probabil- statistically significant when samples are not ity distribution (called the conditional distribu- large, and small samples may not well repre- tion of ß at level / of A in a selected column). For sent the true population heterogeneity. Thus, a example, an mvestigator interested in eleva- small sample contains a penalty for the investi- gator in terms of the inferences to be drawn from the statistics, only if it is, in reality, a rather close Table 27. Generalized Joint call. If a significant result can be observed de- Probability Distribution spite a small sample size, then the statistical conclusion is that the finding has substantial Total merit. With any statistic whose sampling distri-

g /'(^iB,) P(A2B,) P(A\BI) bution involves the number of individuals, a sig- nificance test can ahnost always be set to any p(A,B2} PÍATBZ) P{A\B2) level of false rejection risk; that is, the usual type Tbtal íKBIAI) P(B\A2) 1.00 1 error risk (e.g., 0.05) can be used with any small number of observations, if power is sacrificed. 234 CHAPTER?

merators and the denominators of the ratios used TERMINOLOGY in biodiversity studies often have associated RATES, PROPORTIONS, AND PERCENTAGES. The variances. tenns rate and proportion are often used inter- changeably, but the terms differ. Rate means a COMPARISONS. Different types of descriptive given amount of something per unit of some- comparison should be used carefully to avoid thing else. The latter item may be a count or a confusion. For example, let us say that a particu- period of time or it may be an entity such as area lar sample includes 60 specimens of Hyla of habitat. The division of the two quantities microps and 30 of Hyla minuta and that a com- yields the rate•for example, the number of spe- parison is desired. cies per quadrat. With rate, the units of measure The type of comparison in which we say that in the numerator differ from those in the denom- there are 30 more H. microps specimens than inator. Conceptually, a rate is a dynamic measure H. minuta specimens (or 30 fewer H. minuta relating to change. Alternatively, a proportion is than H. microps) is called an absolute compari- the size of a subset, or portion, of the whole son. The absolute difference in number of speci- relative to that whole. It is essentially a static mens is 30. Absolute comparisons can be made measure or a measure of what prevails at a given only when the items compared are expressed in time or in a given place. A proportion is ex- equal units. pressed as a fraction; the subset is the numerator A statement that breeding males ofH. microps and the whole is the denominator, and the units are twice as common as breeding males of of measure in both the numerator and the de- H. minuta (or that//, minuta are half as common nominator are the same. Proportions occupy a as the H. microps) is a relative comparison. This scale of zero to one. For example, a 50-ha lake type of comparison demands not only equal might include 40 ha of open water; the propor- units but also a definite zero point or ratio scale. tion of open water in the lake would be 0.8. For example, it is incorrect to say that a temper- Percentages are rescaled proportions, that is, ature of 100° Fahrenheit is twice as wann as one proportions on a scale from zero to 100. Percent- of 50°, because the zero point on the Fahrenheit ages are obtained by multiplying proportions by scale is arbitrary and does not correspond to 100. Thus, in our example, 80% of the lake is absolute lack of warmth. open water. There is no substantive difference Relative comparisons may be expressed as between the two quantities. However, in statis- percentages. If we use the specimens of tics it is usual to perform calculations on H. minuta as the base of comparison, 30 speci- proportions. mens is equivalent to 100%, and the number of The term ratio is defined in a much more H. microps specimens is equivalent to (60/30) • general sense. It is the result of the division of 100, or 200%. The number of specimens of any two numbers. Alternative terms are quotient H. microps is 100% greater than the number of and fraction. In biodiversity work, the denomi- H. minuta specimens, which is equivalent to the nator of a percentage is often a known quantity earlier statement of the relative comparison the with no variance. For example, when density is species (i.e., that//, microps is twice as common expressed as number of individuals per area, as //. minuta). Alternatively, the specimens of area is effectively invariant and the variance of H. microps can be used as the base, and we find the percentage is merely a transformation of the that the specimens of //. minuta represent only numerator's variance. Alternatively, both the nu- 50% of those of H. microps, or number 50% Analysis of Biodiversity Data 235

fewer (i.e., H. minuta is half as common as numbers should always be given when percent- H. microps). ages or proportions are used. Consider the fol- lowing statements, USE OF PERCENTAGE COMPARISONS. The seemingly trivial distinctions between relative 1. The average number of Ololygon hayii in and absolute comparisons and between propor- the sample was 200% o/the average number tion and percentage are important, because the of Hyla poly taenia. terms are frequentiy confused and because their 2. The average number of O. hayii was 100% misuse leads to misunderstandings. Some guide- larger than tiiat of//, polytaenia. lines for use of these expressions follow. 3. The average number O. hayii in the sample Consider, for example, that any zero value is was 200% more than that of//, polytaenia. 100% smaller than any nonzero quantity, and that no positive non-zero quantity can be more Is the average number of O. hayii per sample than 100% less than another. Given these condi- two or tiiree times that of//, polytaenia? tions, a report that the average number of breed- Even when the hypothesis of interest legiti- ing males of Hyalimbatrachium valerioi fell mately involves a proportion, actual counts or 150% from one year to the next is impossible, frequencies also should be presented. For exam- and there is no way to understand the correct ple, if an investigator reports that 60% of a meaning of the statement. Confusion can be alle- sample was found to be tagged on recapture, viated by avoiding expression of relative com- whereas only 50% of the sample was tagged the parisons as percentages or by always stating last time the area was sampled, then it is difficult clearly the base that is being used. to know whether a change has occurred. In con- Suppose that some quantity is observed to rise trast, reports of the figures of 3 out of 5 (60%) from 60% to 70%. This is an absolute rise of 10% and 500 out of 1,000 (50%), provide better infor- (70% - 60%) or a relative rise of 16.7% [(70% - mation for evaluation. The sampling situation is 60%)/60%]. Quotation of the increase in absolute made clear (i.e., a sample of 5 versus as sample terms clarifies the situation; for example, "There of 1,000), and the biological significance of the was a rise of 10%, from 60% to 70%." figures can be readily assessed. Percentages should not be used to describe change unless ail figures are changing in the same direction, that is, unless changes aiç all Interrelationships of Standard Tests positive or all negative. The same tabular arrangement of data can be When relative comparisons are made between used to examine proportional and absolute abun- or among figures that differ in orders of mag- dances. In this section I discuss the equivalence nitude, percentages should be avoided. For of many of the common test statistics for both example, one could say that the breeding popu- types of abundance data. lation of males observed within an enclosure was about 2,000% larger than the sample of non- COUNTS AND PROPORTIONS breeding males. This expression might be cor- The number of counts or observations in a sam- rect, but it is not as clear as the statement that ple of size n, as well as the proportion of obser- one population was about 20 times larger tiian vations, can be seen as random variables. The the other. distribution of sample proportions can be de- Because an incorrect preposition or descrip- scribed by the binomial distribution (assuming tion can affect interpretation, absolute counts or sampling with replacement or a sufficiently large 236 CHAPTER 9

sample). The probability ip) of any proportion {P) plied to the problem of testing for a single pro- of observations being found in a sample of size n is portion. That is. exactly the same as the probability of that number of observations (r) being found. Thus,

p(X=r) = p{P = X/n) or and the only difference between the equations is :=|V7l the random variable that is being considered: X, the count, or X/n, the proportion. The basic test In addition, if the two proportions of interest of significance of association performed on the are pi = al{a + b) and pi = die + d), we can form frequencies in the contingency table is therefore related to the test of the equality of the propor- = \-q tions formed from the tabled frequencies. n

TESTS ON R X C TABLES to test the difference of these proportions. We In general, observed data are tested for signifi- use cance by calculating a particular statistic that is then compared with the tail area of some appro- P\-Pt priate theoretical distribution for determining significance. When the observed data are counts that can be arranged in tabular or cross-classified [«(^i]]' form, the underlying or comparative distribution usually is the multinomial. That is, if the identi- to calculate f, which can be compared with a cal probability distribution applies across all Student r table. For a Yates correction, we form observations, and if all n observations are inde- pendent, then the counts per species are multi- •1 + •1 nomially distributed, and the marginal distri- «1 «2 :|pr bution is binomial. • P2l - 2 The traditional chi-square tests of goodness of fit and independence for R x C tables and the H^^-U' likelihood ratio (G test) are each approximations for this multinomial distribution. They test the which is a Studentized version of the y^ statistic. goodness of fit of the observed frequencies to a Each of these statistics can be formed from theoretical distribution assumed to fit the observed the values in the single 2x2 table and can be sample or the sampled population. Although used interchangeably to test the hypothesis of treated as distina in most texts, these interrelated interest. tests are equivalent under some circumstances. Model-Based Methods TESTS ON 2 X 2 TABLES When the table is 2 x 2, the usual Pearson chi- Usually categorical data are analyzed by the square statistic yields the same value as z^ the classical analytical metiiods described above. square of the standardized normal variable, ap- However, in the early 1960s, model-based meth- Analysis of Biodiversity Data 237

ods were developed with broad analogues to ASSUMPTIONS FOR PEARSON'S general linear models (especially regression CHI-SQUARE TEST models) for continuous variables. These models do not assume a normal distribution but instead The computational simplicity of the chi- have been based primarily upon binomial, multi- square test is deceptive in that it is based on a nomial, and Poisson distributions. Dependence fairiy elaborate mathematical rationale and models usually correspond to conditional proba- therefore requires some important assump- bilities of a dependent or response variable, tions (Hays 1973). Three assumptions are of given a set of independent or explanatory vari- concern. ables. Association models correspond to joint The first assumption is that each observation probability distributions when the underlying is independent of every other. Repeated observa- variables are considered as equivalent, not caus- tion of some or all of the same individuals in the ative. The theory behind these models is well samples may negate use of the chi-square and developed, and sampling theory has been ex- related tests. The novice user should consider amined, at least in part, empirically as well as these tests only when each animal contributes to theoretically. These models provide a promis- one and only one cell. ing new approach for some of the more ad- The second assumption is that cells are well vanced analysis of amphibian data. At present, defined, so that each observation can be placed these model-based methods are being incorpo- in one and only one cell. Because a joint fre- rated into computer packages, which should quency distribution is assumed, rather than just spur their use. that the cells contain observed numbers, each amphibian or observation must be classifiable Advice about Tests for R x C Tables into only one unique row and one unique column. The apparent simplicity and clearness of tests The third assumption is that the sample size is on categorical data hide a basic problem inher- large enough for the asymptotic theory to hold. ent in their application. Starting with different However, there are no generalized rules for premises about the data, it is possible to reach knowing when this assumption is true, that is, for quite distinct numerical probability figures for knowing how well the approximation fits the the same table. It is these figures that are used multinomial. to judge the significance of the data in the table. With tests for independence, an investi- PROBLEMS IN THE APPLICATION OF gator must also be concerned about oper- CHI-SQUARE ationalization of the study design because more than one test may be appropriate, and No rules can explicitiy cover all practical situa- each test could lead to a different observed tions involving problems with fit for the chi- probability level. For example, at least seven square statistic; therefore, no rule will tell how test procedures are available for comparing close the observed level of significance (from binomial proportions and can lead to varied the computer program or the hand calculations) answers (Storer and Kim 1990). In practice, is to the correct one. probability figures derived from an analysis of An index, h, to the amount of bias can be used observed data help shape conclusions of a to identify situations in which bias is large study; however, the form of these probabilities enough to be of consequence to the test. The likely is a matter of opinion. index is calculated as follows: 238 CHAPTER 9

k Täte and Hyer (1973) noted that chi-square yields many identical outcome values when ex- •¿J Ci n ft = - pectations are small or variable, but Üiat the [32(Jt-l)] multinomial probabilities become distinct as the number of categories increases. The authors or used these observations to explain why, when the expected values per cell remain small, the k chi-square approximation is inconsistent (varies) (LA as sample size increases. Identical outcome val- i = l ues are less likely to occur with tiie log likeli- h = Í4 [32(*-l)] hood ratio (Cochran 1936). Conahan (1970) compared the x^ statistic, the for the total k cells in the table and for the e-, log likelihood function, and the exact multinom- expected values in the k cells. ial probability when expected frequencies are The x^ statistic is biased whenever h is not small and found that the likelihood ratio test is zero. Values of h that exceed 0.1 warn that bias best when the number of categories is at least may exist; values of h that exceed 1 indicate five and the expected value for each cell is at serious problems with bias, in which the statistic least 3. He concluded that (1) the likelihood is systematically distorted and not representative function is an adequate (but not perfect) approx- of the true value for the distribution. This bias imation to the exact multinomial when expected can be as large as the size of the test a even values per cell are greater tiian 10; (2) the multi- though the chi-square approximation is valid. nomial should be used directiy when there are Although it may not be possible to correct this five or more categories and the expected value bias in some cases, calculation of h will help for each cell is less than 3; and (3) tiie log likeli- prevent misinterpretation of p levels. hood function should be used with five or more In practice, we know that chi-square is an categories when the expected value for each cell approximation and, in addition, that the estima- is greater than or equal to 3. tor •^ is inconsistent, or does not converge in Agresti (1990) recommended that the log probability to the parameter it estimates, as sam- likelihood ratio be used when n/RC < 5. He also ple size increases. It is wise, therefore, when claimed Üiat when the R or C value is large, the noting the observed probability, not to consider yj- approximation can be "decent" with n/RC as as correct any obtained value near to the desired small as 1, if the table does not contain both very level of the test. large and very small expected cell values.

SAMPLE SIZE, CELL SIZE, AND BIAS. When MINIMUM CELL SIZE. An issue of constant con- sample sizes are small or expected cell sizes are cern is the minimum expected frequency for a small or highly variable, the Pearson and the cell in a table. The value of 5 appears to have likelihood ratio chi-square statistics may be become a convention, but there seems to be no substantially biased (Chapman 1976; Haber- theoretical or widespread empirical basis for this man 1988). Zelterman (1986) devised a test sta- choice. Total sample size has been suggested as tistic tiiat corrects bias when h > 1. Baglivo et a more appropriate criterion for goodness-of-fit al. (1988) provided methods to calculate cor- tests, but the distiibution of the frequencies rected tail probabilities when exact methods are within the cells is also important. Expected cell not feasible. frequencies as low as 3 or even 1 have been Analysis of Biodiversity Data 239

investigated and have proven adequate (Con- information is being ignored with these tests, ahan 1970; Agresti 1990). and a more powerful test of independence Cohen and Sackrowitz (1975) showed that the should be used. chi-square test is unbiased if each expected cell A tabled array with two rows and any number frequency equals n/k, but that is a highly un- of columns (C) greater than two is often of inter- likely event in amphibian observational studies. est in ecological studies because of the possible For these reasons, and because most studies existence of a drend. This occurs, for example, for independence are compared with a unifomi with the presence or absence of some attribute case in amphibian work, I suggest that an aver- across C samples. To reveal the presence of a age expected frequency (i.e., £ = n/RC) of 5, trend in the data, a modified form of the general which appears adequate from both biological X^ formula is used, and both exact and approxi- and statistical standpoints, be used. This average mate methods are available for parametric is considerably less restrictive and easier to ob- estimation. The C samples are binomially dis- tain, given the vagaries of amphibian sampling, tributed with expected probability of success than a minimum expected frequency of 5. constant. The test is one of homogeneity of the samples. Many procedures are available for test- FISHER'S EXACT TEST ing for trend. An alternative to the chi-square and likelihood Yates (1948) dealt specifically with group- ratio tests is Fisher's Exact Test. It is rarely used ings based on direct quantitative characters for large tables, because the computations seem- and discussed many tests for trend over cate- ingly would be impractically large, but its under- gories. Armitage (1955) discussed tests for lin- lying theory is applicable to R x C tables. ear trend. Wood (1978) provided a comparison Pagano and Halvorsen (1981) developed a com- of methods for trend detection with one quan- putational algorithm that is easily programmed titative and one qualitative factor. Lee (1980) on a desk microcomputer. Mehta and Patel provided a test in the general table array for (1983) provided what they asserted is a more multinomial data. These tests are becoming efficient algorithm for the same calculation. For more widely available in statistical packages, 2x2 tables, a form of Fisher's procedure is but, to my knowledge, have not been evalu- uniformly the most powerful and unbiased test ated empirically. Nam (1987) determined min- (see "Fisher's Exact Test" under "Advice about imal but approximate sample sizes for Test for 2 X 2 Tables," below). The same is not detecting linear trend in proportions. true when Fisher's test is used for data in R x C tables. Therefore, even though Fisher's proce- dure could yield a correct significance level for Advice about Tests for 2 x 2 Tables an R X C table by use of Mehta and Patel's method, other methods may be superior. As the CHI-SQUARE TEST STATISTIC cell expectations become large, the exact values For tests of independence with 2x2 tables, the approach those from the chi-square distribution, Pearson chi-square, with its inherent bias prob- and the %^ and Fisher's tests are equivalent. lems, is usually used even though it is only appli- cable for a large n. The statistic (Table 20: # 38) TESTS FOR TREND IN PROPORTIONS used for testing is The usual tests (x^ and G^) treat factors as having nominal scales. If data follow an ordi- 2 n{ad - bcf nal scale or have another intrinsic order, then {a + c)(Jb + d){a-\-b){c + d) 240 CHAPTER 9

Its actual distribution function is discrete and not which marginals are fixed, which appeared to easily specifiable, so the continuous chi-square solve the problem of testing small samples in 2 x distribution found in most statistics texts is still 2 tables. Fisher's Exact Test is thus actually used as an approximation for the probability based only on one possible case but is used for statements. tables described by all three cases. The power of Berkson (1978) used a normal vánate, z = the test when applied in each of these three cases \-f)^\, and found it superior for a range of sizes is quite distinct (e.g., Bennett and Hsu 1960). when the random (nonfixed) margins were of The power differs because it is related to the equal sizes (see note 3 at the end of this chapter). alternative, not the null, hypothesis, and the for- D'Agostino et al. (1988) showed that for n < 15, mer is distinct in the three cases, as might be the two-independent-sample f-test with pooled expected. variance and the uncorrected x^ were both robust The exact test devised by Fisher is the subject in the actual levels of significance and close to of some commonly held misconceptions. When or smaller than the nominal level (see note 3). Fisher's Exact Test is used, the probability level However, this pooled t is only slightly more usually is reported and interpreted in the litera- robust than the usual uncorrected y^ (Heeren and ture as being exact. However, the term exact ha:& d'Agostino 1987) under limited conditions, so at least two senses in statistics. Exact can refer to that either this uncorrected statistic or the z is the underlying distribution, meaning that it is an probably just as appropriate. exact representation of the sampling situation you intended to encounter or have encountered. FIXED AND RANDOM CLASSIFICATION Exact also can refer to the test level; for example, FACTORS an investigator may wish to perform a test with In tests for independence in 2 x 2 tables (as well as a at exactly the 0.05 level. general R x C tables), each expected cell frequency For Fisher's Exact Test, the term exact means is the product of the frequency in the column (the that the method provides the exact probability of column marginal) and the frequency in the row observing a result identical to a more extreme (the row marginal) divided by the total sample size case, according to the underlying distribution. (Barnard 1947; Pearson 1947). Forming these ex- This assumes that the given 2x2 table was pected values depends (or is conditional) on only generated by sampling from a four-variable hy- the four marginals and the total sample size. There- pergeometric distribution, which is strictly ap- fore, different methods of obtaining the samples plicable only for the fixed marginals case. This could lead to the approximation of different under- distribution is discrete, not continuous, and lying distributions and, in turn, require different therefore cannot give a test with a predetermined test statistics. There are three possibilities: (1) Both significance level of a. marginal totals (sample sizes) are fixed; (2) one Because of tíiis discreteness, the observed level marginal is fixed and the other is random or free to of significance will always be less than or equal to vary during the sampling; (3) both marginals are the mie level; the observed value will depend upon random. (This problem is discussed, and each of the fixed marginal fiequencies. That is, the test is the three sampling situations is examined in note 1 conservative. It results in an unnecessary loss of at the end of this chapter.) power, but whether the test is considered conserva- tive or destmctive depends upon whose interests FISHER'S EXACT TEST are being served (Berkson 1978). Fisher (1935) developed a test based upon the Tocher (1950) showed that if Fisher's test is hypergeometric distribution only for the case in turned into a randomized test, it is exact in both Analysis of Biodiversity Data 241

senses, as well as being the uniformly most pow- made in general terms by Pearson (1947). (See erful, unbiased test available under any of the note 2 for a discussion of the use of the continuity three described conditions of the marginals. In correction for each of the three sampling situations other words, the test, when randomized, is robust Ihat can describe data for a 2 X 2 table.) under variations in the basic model and therefore could be important for amphibian work. How- CORRELATED PROPORTIONS ever, this randomized version is not the Fisher's Observations that are not independent are also of Exact Test that Is available in texts and computer interest in amphibian work. For example, two packages, and I agree with most statisticians and observers may judge the same aspect of some developers of computer packages that randomiz- characteristic (e.g., duration of call, distance ing is not worth the effort. from call, species presence or absence) of a sam- ple of frogs. The observers' ratings are indepen- YATES'S CONTINUITY CORRECTION TO CHI-SQUARE dent, but the same frogs are observed. The question is whether the two observers rated the hi 1934, Yates proposed a so-called correction characteristics in the same way•for example, for continuity for use when the x^ is applied. whether they rated the same proportion of calls This correction of ^^ is the one most widely used as strong or weak or as near or far from the base in biological applications and is used (as are point. In this case, each sample or observed pro- other corrections, e.g., Kendall and Stuart 1973) portion is, at least partially, based upon the same to bring the obtained probabilities more in line frogs, so the proportions are dependent. with those that would be obtained with Fisher's McNemar (1947) devised an "exact" test of Exact Test. This test has become the standard the hypothesis that the two proportions are against which all other tests are compared, al- equal, using the binomial distribution. Cochran though the randomized version of Fisher's test is (1950) generalized this test to situations involv- actually the correct standard, as is discussed in ing repeated measurements in more general ta- the previous section. bles (2 X C) but in which the dependent variable The x^ statistic involves only cell frequencies remained binary. Schork and Williams (1980) that are non-negative integers. Therefore, Yates gave the exact power function and constructed subtracted the amount i/z from the cell frequency sample size tables for pairs of correlated propor- sums to adjust mathematically for the problem tions, and Paul (1975) provided an exact distri- of a discrete distribution by correcting for the bution for tiie statistic. noncontinuity in the possible outcomes. Never- theless, it is disconcerting to a researcher to ob- Summary and Recommendations tain a chi-square value that indicates hypothesis rejection and then to calculate a "corrected" CAVEATS value that indicates a nonsignificant result. For any single 2x2 contingency table, there are Plackett (1964) and Grizzle (1967) showed many choices of statistics that can lead to quite that Yates's approach corrected too much, result- distinct values for Üie final observed probabih- ing in an excessively conservative test. In con- ties. The investigator must choose how many trast, Mantel and Greenhouse (1968) argued that margins are fixed; this decision, in turn, deter- because the hypergeometric model can describe mines the number of random variables in the the 2 X 2 tabled data (both marginals and total study, the number of possible outcomes in the size fixed), the continuity correction is indeed probability space, and the form of the descriptive appropriate in this one situation, a point also underlying distribution. 242 CHAPTER 9

Nevertheless, Fisher (1935) developed an pected values. This practice does not mean that "exact" test. Some argued that this test would sample size is the best determinant for test selec- suffice for any table, but they did not prove that tion; it means that this is what someone has the test is uniformly the most powerful and unbi- decided is feasible for computer packages. ased. Rather, Fisher's Exact Test with random- The statistical approach used by an investiga- ization was the subject of the proof. However, tor who keeps these cautions in mind will always the randomized version is never available in be defensible and, with wise and cautious appli- practice and cannot be considered a practical cation, tiiese suggestions will not lead anyone choice. astray. Mathematical statisticians disagree on which test method is most appropriate. Philosophical 2x2 TABLES arguments are strong, convincing, and correct on When no margins are fixed, the uncorrected x^ many sides. As for practical advice, three points should be used regardless of sample size. When are most vital to remember when testing catego- one margin is fixed, neither Fisher's Exact Test rized data. nor Yates's corrected chi-square is appropriate. If the Í with pooled variance (d'Agostino et al. 1. There is no single correct method. The stan- 1988) is not readily available or programmable, dard for corréeme ss of the approach is deter- a program for the usual f-test may be employed mined by the problem to be solved, the with fixed sample means of Xi = at{a + c) and snidy design, and the hypothesis being X2 = bUb + d) equated to the sample proportion tested, of successes (a + c)/« in the null hypothesis, 2. Unless there is a clear winner on power and Yates's correction should not be used, especially feasibility of use, several tests may be for sample sizes greater than 30. For amphibian appropriate, samples, I am unconvinced of the need to con- 3. There may be no "exact" answer even after sider ancillary statistics. an appropriate test has been selected. It may In situations when both margins and the total not be possible to perform a test at exactly sample size are fixed, if the margins are equal. the desired level of significance or to obtain Fisher's Exact Test or Yates's correction to an answer that the investigator considers to Pearson's chi-square is the test of choice. Recall, be correct. The most one can hope to obtain however, that the hypergeometiic probability is the closest approximation; even then, the calculations are the exact values sought. These meaning of closest is different under differ- calculation are available in many statistical com- ent circumstances. If the observed value is puter packages. If the margins are unequal, I close to a decision level of the test, it indi- recommend use of the uncorrected x^ regardless cates caution, not decision. of sample size.

D'Agostino et al. (1988) pointed out that R X C TABLES major statistical packages (e.g., SAS, SPSS, Under conditions in which both n and expected BMDP) have incorporated the traditional ap- values are large, the chi-square distribution proach of ^ for large samples, -^ with Yates's should fit reasonably well, and tlie usual x^ or correction for intermediate-sized samples, and the G^ may be used with confidence. the Fisher's Exact Test for small samples. All of Under conditions of large n and small or vari- these packages issue warnings about small ex- able expected values, the x^ or G^ test statistic. Analysis of Biodiversity Data 243

available in large statistical packages for com- suring instruments and constraints imposed by puters, can be used with the simultaneous calcu- numbering systems. Thus, a continuous variable lation of h for identification of bias problems. could be defined as a variable for which a value The method of Mehta and Patel (1983) is of intermediate between any two obtained values is value for the calculation of exact probabilities. meaningful. Theoretically, an investigator could This calculation is easily programmed and should measure an amphibian's age to the nearest year, avoid some of the difficulties of the approxima- month, day, or even minute, but there is a limit to tions for the case of a hypei^eometric distribution. the fineness of the recording. Hence, the "con- When both n and expected values are small, tinuous" variable of age actually is recorded as Conahan's (1970) recommendations could be discrete measurements, yet between ages 1 and applicable. However, if conditions of sampling 2, the value of 1.75 has meaning. The obtained prevent an investigator from obtaining a sample values as well as the intermediate value can be of sufficient size, then the methods of analysis evaluated for both accuracy (closeness of the and the questions asked should be reevaluated, value to the quantity intended to be observed or and expert guidance may be necessary. used) and precision (reproducibihty). Continuously distributed variables can be subjected to a wide variety of both univariate Species density and continuously and multivariate statistical treatments, depend- distributed data ing upon the questions asked. Many texts discuss these techniques, both descriptive and inferen- species density can refer to the number of spe- tial, for the study of density and its relationship cies or to the number of individuals of a species to ancillary data collected in the field. However, found per unit of area, time, or effort. Density because of the vagaries of the data, expert advice values are derived from actual counts of individ- should be sought when dealing with questions uals or species. Because these counts and the about such relationships and associated threats standard methods for obtaining them, rather than to validity. measures derived from them, are the focus of this book, I discuss only a few approaches to the analysis of such measures. Graphical Representation Species density, as well as number of individ- Hypotheses of interest with density studies usu- uals caught from each species, may be treated as ally specify a relationship•for example, "Den- continuously distributed data for the purpose of sity 0Î Eleutherodactylus brandsfordii increases statistical tests under certain circumstances. as depth of forest litter increases." It is often of Many peripheral variables (e.g., environmental interest to characterize the form of that relation- factors such as temperature, water depth) also ship, as well as to test it. When two or three may be considered to be continuously distrib- variables or characteristics are involved, tiiere is uted. The common justification for calling a no substitute for a plot for obtaining information variable "continuous" is that between any two of on both the general shape and the scatter of the its values, it is theoretically possible to obtain field results. Such a pictorial representation is another. In fact, regardless of whether a variable indispensable for the development of the appro- under study is termed discrete or continuous, priate statistical sh-ategy and will often save most often the measurements themselves are dis- computing time. Fienberg (1979) provided a crete because of limits to the precision of mea- simple taxonomy of visual methods for data dis- 244 CHAPTER 9

play to characterize the state of graphics, in- eral, unless carried out under the discerning eye cluding computer generated plots. These same of an expert. Successful statistical description graphical displays, or innovative adaptations of demands subject matter expertise. Amphibian them, are described by Chambers et al. (1983) experts, among others, have increasingly tended and can be found in many good computer to abnegate this responsibility. Statisticians, as well as those who apply statistical methodology packages. Hypothesized relationships must always be and terminology to their studies, have intimi- examined graphically before the application of dated others into using only significant results of any model or analytical technique. The plotted hypothesis tests to define scientific merit. This is data will show whether the relationship effec- a critical defect in the process. One must rely on tively represents the situation. They also can the opinions and knowledge of experts to define show whether nonlinearity rather than lack of substance and to ensure that substantive effects substantive relatedness, is the cause of a small are under consideration. correlation coefficient. Atkinson (1985) pro- vided some illuminating examples of such problems. Species diversity The concept of species diversity is variously and Descriptive Statistics chaotically defined in the literature. It probably Once preliminary graphs have been obtained originated with Jaccaid (1908, 1912) and Glea- and examined for the existence of hypothesized son (1922), who proposed the first species diver- relationships between density and ancillary vari- sity indices. Despite considerable interest in the ables, simple descriptive measures should be ob- concept, no generally accepted definition of di- tained. Mallows (1979) provided a readable versity has emerged (Peet 1974). By 1967, com- account of robust descriptive methods, as do plaints concerning the lack of an operational most ecology texts. definition were rampant (Mclntosh 1967), and Lately, it has become fashionable to deempha- Hurlbert (1971) was of the opinion that diversity size or even eliminate elementary descriptive per se does not exist. statistics in favor of multivariate statistical treat- Basically, diversity includes two concepts: ment, which is deemed more sophisticated and richness (number of species) and evenness (dis- difficult and, therefore, "better." No knowledge- tribution of individuals among species). Thus, a able statistician would begin an analysis at the statement that the tropics are more diverse than multivariate stage, and logic precludes this tac- temperate zones implies that there are more spe- tic. The existence of an overall multivariate ef- cies in the tropics and that each such species has, fect does not indicate the nature or even on average, proportionally fewer individuals necessarily the existence of univariate effects. A than comparable species outside the tropics. Di- great portion of the most useful work done with versity measures should be clear in their incor- field sampling data consists in shnply arranging poration of these two concepts. The basic the masses of data into a comprehensible form. mathematical theme underiying diversity is the Summarization into descriptive measures, ta- pattern of allocation of a given quantity (abun- bles, and plots allows attention to be focused on dance) among a number of well-defined patterns and possible substantive interactions. divisions or categories (species), and the funda- Manipulation of field data or compilation into mental problem is the determination of the even- standard tabled summaries is fruitless, in gen- ness of this apportionment. Analysis of Biodiversity Data 245

In the following sections, I describe diversity habitats (Connell and Oral s 1964; Diamond and measures that can be used with each type of data May 1976). Even when a species count is (e.g., species lists, species abundances) normally thought to include all species in a habitat, it still obtained from inventory or monitoring studies, reflects the size of that habitat and the "density" and I discuss their limitations and advantages. of individuals (Kempton 1979). A measure of species richness is the best indica- Inventories and Richness tor of diversity that can be obtained with an inven- tory that merely identifies presence or absence of SPECIES LISTS species and provides a total species count. Because When an investigator performs an inventory, he species richness represents but one facet of the or she obtains a list of species. This basic species diversity picture, its value as a comparative index count for a taxonomic group is the simplest and is severely limited (Yapp 1979). most common measure of species diversity (Caughley 1977; Kempton 1979). The count, NUMBER OF SPECIES AND AREA first identified as species number (Lloyd and The total count of species observed without regard Ghelardi 1964) and then redefined as species to the area sampled, although clearly an important richness (Mclntosh 1967; Hurlbert 1969), is ob- aspect of species diversity, is not sufficient for served independent of area sampled or time. understanding that diversity. By adding informa- Nevertheless, diversity measured as the number tion on collection area to the basic species list, of frog species in a single pond in the midwest- species-area curves can be obtained. In the spe- em United States during a breeding season has cies-area curve, which likely was first suggested by an intrinsic meaning different from diversity Jaccard (1908, 1912) and later discussed by measured as the number of frog species in the Arrhenius (1921) and Gleason (1922), total species Serra da Mantiqueira of Brazil. Likewise, direct number is plotted against size of the area sampled. counts cannot be used to compare two popula- The shape of a species-area relationship curve tions of frogs if one is fossorial and tiie other is is roughly exponential regardless of taxonomic arboreal, because an extraneous (environmental) group (Preston 1962a; Williams 1964). For ex- factor impinges too greatly upon the species' ample, if the numbers of species of frogs inven- populations to allow for comparison. toried from increasingly larger bodies of water A species count is a simple quantitative aspect in a region are plotted against the log of the area of diversity; it cannot be related to the funda- of each, the species-area curve will have an ex- mental nature of the diversity concept, although ponential shape. If the logarithmic transforms of it is a basic component of any study of biological both the species counts and the areas are taken, diversity. In addition, this seemingly unambigu- the resultant curve approximates linearity, and ous and direct measure in fact depends on the its regression line can be used to predict values sample size (number of individuals) obtained, for areas in which observations were not ob- which in turn depends on the nature of the target tained (Gleason 1922). The parameters of the populations and the time spent searching. This fitted curve depend upon the unit of measure of dependence is clearly illustrated when different- the area. Thus, a curve based on quadrats mea- size samples from the same assemblage are com- sured in square meters yields estimates of the pared; as sample size increases, "diversity" also number of species in one square meter. increases, sometimes without apparent limit Connor and McCoy (1979) examined condi- (Taylor et al. 1976). The same is true for full tions of the species-area relationship and the fit censuses (not samples) from similar macro- of alternative models (especially semilog and 246 CHAPTER?

log-log) to the observed curves. They asserted tota! species count (s), reducing it by 1, and then that these fitted curves have litüe biological dividing it by the log of the total number of meaning and provided evidence to support their mdividuals sampled: (s - l)/logn. This index claim. Martin (1981), in contrast, provided data usually is attributed to Margalef (1958), but he for which evidence of biological pattern was only cited it. It is correctly attributed to Gleason obtained. He cautioned, however, that interpre- (1922). A second popular richness index (Men- tation of species-area curves can be compro- hinick 1964) is calculated by dividing total spe- mised by threats from correlated environmental cies count by the square root of the total number and other factors. Preston (1962b) showed that of individuals. Such indices simply and arith- the species-area curve for single sites can be metically delete the sample size effect, but in so understood comparatively easily, but that, be- doing, they assume that the two counts used are cause of the vagaries of sampling, interpretation functionally related in a specific manner in the of curves based on data from multiple sites (es- population. For example, in Menhinick's index, pecially if number of sites is limited) present the expected number of species, Eis), in the pop- greater difficulties. ulation is equal to a constant times the square The common qualitative perception of species root of the total number of individuals. Glea- diversity is often positively correlated with spe- son's index is based on the presumed linear rela- cies richness as, for example, over habitats along tion of E{s) and the log of the number of latitudinal gradients (Hurlbert 1971). Such posi- individuals. There are two fundamental condi- tive correlation is neither a biological nor a tions for the use of this type of index: (1) The mathematical necessity. Gradients can exist functional relationship between E{s) and n must along which apparent increases in the diversity remain constant over any studied assemblages; of species are accompanied by decreases in spe- and (2) the relationship must be of the exact cies richness. form stated. Peet (1974) noted that if these conditions do TOTAL COUNTS OF SPECIES AND not hold, the richness indices will vary with sam- INDIVIDUALS ple size in an unpredictable manner. Unless an Another way to eliminate the sample-size de- investigator can demonstrate these relationships pendence of direct species counts is to measure for the particular taxonomic group and sample species richness in terms of the number of indi- situation, no conclusion can be reached concern- viduals observed. Hurlbert (1971) used the term ing the merits of these indices, and 1 cannot numerical species richness to denote the number generally recommend them. of species in a collection containing a specified If an inventory provides a species count and a number of individuals or amount of biomass, total individual count, an investigator can calcu- and the term areal species richness (or density, late indices of species richness but not of overall sensu Simpson 1964), to denote the number of species diversity. species present in a given area or volume of the environment. In the same publication, however, Monitoring, Richness, and Evenness he appears to have used the terms species rich- ness and numerical species richness synony- SPECIES' ABUNDANCES AND TOTAL COUNTS mously, with the former being the preferred OF SPECIES AND INDIVIDUALS term. Another well-known index that is fre- A monitoring study can provide (1) a species list quently applied to biological richness deals with with a total species count, s, as an estimate of the the problem of sample size effect by taking the possible species in the target population S; Analysis (f Biodiversity Data 247

(2) the total number of individuals sampled, n, of specimens) decreases, the number of species as an estimate of A^; and (3) the numbers of represented would also be expected to decrease. individuals sampled or located from each The curve may be used alone as a representation species, «, (where / = 1 i), as an estimate of the parent assemblage, or it can be compared of the abundance of each species. The values with curves generated from samples taken at can be used to calculate various measures of other times or from other assemblages. diversity. Sanders (1968) suggested that the curve that is generated represents the cardinal features of the SLOPE INDICES. Whittaker (1965) devised what diversity of the parent assemblage sampled, but he called dominance diversity curves, graphical it actually is a preliminary solution with little presentations of the three quantities s, n, and n,. probabilistic basis. In addition, the rarefaction He also proposed two slope indices to measure methodology of Sanders generally overestimates in a distinct manner what he called richness or the expected number of species present in sam- species per logarithmic cycle. Peet (1974) ples of differing sizes or of highly disparate spe- pointed out that both of these indices are influ- cies abundances, especially for small samples enced by their assumed population distribution and clumped populations (Hurlbert 1971; Fager (e.g., log normal) and therefore are subject to 1972; Peet 1974). Only for samples in which the limitations similar to those of simpler richness numbers of individuals within species are ap- indices. Whittaker's indices have not achieved proximately equal do his rarefaction estimates any popularity in ecological studies, although and their correct values coincide (Peet 1974). Krebs (1989) included them on his list of recom- Hurlbert (1971) refined Sanders's method for mended diversity measures. At present, the limi- calculating the expected number to have a prob- tations of these indices appear to outweigh their abilistic basis (see note 5). advantages for amphibian work. The rarefaction procedure is based on certain problematic assumptions concerning the rela- RAREFACTION. Sanders (1968), in an attempt to tionship between the collections or assemblages obtain a richness measure that is independent of to be rarefied and their parent populations. One sample size, developed a method called rarefac- assumption is that individuals are randomly tion, which reduces the observed samples to a sampled. In fact, because most naturally occur- common size. In this procedure, the species in a ring amphibian assemblages are spatially hetero- sample are ranked according to their relative geneous (Crump 1971; Heyer et al. 1990), representation (number of individuals) in that individuals (of distinct species) are not located sample, and cumulative percentages are calcu- randomly but are sampled from discrete habitats, lated. Based on these figures, a simple scaling microhabitats, or patches. The statistical term for algorithm, the rarefaction methodology (see note this is cluster sampling. The clusters can be de- 4), is used to estimate the species richness that fined at random, but the individuals within each would be expected in a sample of individuals of cluster are not randomly sampled with respect to some designated (rarefied) size. The numbers of the population as a whole. Observed variation species that would be expected at alternative can be partitioned into components for spatial sample sizes are obtained by interpolation from variability and sampling error (Smith and the curve (number of species plotted against Grassle 1977). Nevertheless, results of the rar- number of specimens or sample size), assuming efaction procedure will be biased in an un- that the relative representation of the species predictable way if the individuals are not present is fixed. As the sample size (i.e., number sampled randomly. 248 CHAPTER 9

A second assumption is that replicate EVENNESS samples from a homogeneous assemblage, or samples from habitats or microhabitats within Species equitability, evenness, and dominance a heterogeneous assemblage, belong to the are terms that have been used synonymously to same target population in the statistical sense. designate the distribution of species' abundances Buzas (1979) and James and Rathbun (i.e., the number of individuals per species or the (1981) suggested that rarefaction methods are related proportions). Peterson (1975) stated that particularly useful for comparison of richness evenness can be indicative of relative numerical when assemblage sizes differ. Rarefaction, as dominance without regard to diversity; the spe- refined by Hurlbert (1971), is a reasonable cies with the largest number of observed individ- approach to measurement of richness and is uals is numerically dominant in the distribution. appropriate for across-study comparisons. It However, dominance in this sense has been con- should be used, however, only when sampling fused with the concept of dominance defined as and analytical methods have been consistently the degree of influence or control that one spe- applied across the samples to be compared. In cies exerts over another species of the assem- addition, samples should be compared only if blage as a result of competition or behavior they are taxonomically similar. Raup (1975) (Grieg-Smith 1957). Use of the term dominance noted that the degree of similarity between for evenness, therefore, is best avoided. collections, determined at the discretion of the investigator, could be biased. Potential bias PARAMETERS OF PROBABILITY DISTRIBUTIONS. can be avoided if similarity is operationally Many probabilistic models have been used to ex- defined. Samples to be compared should also plain distributions of species abundance, and orga- come from similar habitats. Intrahabitat com- nization patterns of species assemblages (May parisons minimize the threat to generalizabil- 1975). The most commonly cited models are the ity from habitat heterogeneity, which has been log series (Fisher 1943), the log normal (Preston shown to be correlated with diversity. As far as 1948), the negative binomial (Brian 1953), and the can be ascertained, Sanders used habitat to broken stick (MacArthur 1957). Many reviews mean some limited aspects of the substrate and discuss the uses and abuses of these models, espe- did not control for any other variables. cially for comparing abundance relationships across Arbitrarily allowing certain ecological factors assemblages. Gubert (1989) noted that it is difficult to vary and others to be restricted is not to establish that a sample belongs to one particular reasonable. statistical distribution, especially with smaU sam- When comparing diversity among several ples from heterogeneous biological populations. assemblages, the entire curve of rarefied val- Statistically, it is not reasonable to discuss best-fit ues for each assemblage should be used. The alternatives from among these abundance distribu- specific species values at each rarefied point tions based upon single amphibian samples. How- (sample size) form the richness measure (Raup ever, the application of these models is a credible 1975). One does not measure diversity of a approach to the definition of such pattems, and single entity by rarefaction, or state that the selection of one or more should be considered. rarefaction curve exemplifies high or low Buzas et al. (1982) fitted both the log series and the diversity. Finally, rarefaction should be con- log normal models to data on species abundance strained to expected values calculated within and on species occurrence (i.e., number of locali- the range of the sample. ties per species). This has not, apparently, been tried Analysis of Biodiversity Data 249

far amphibians but certainly could provide in- of these measures. However, it requires sample sight into patterns of occurrenoe and abundance. sizes in the thousands, and so I shall not discuss it further. INDICES. Because of the need to compare abun- I also omit some popular measures that as- dance distributions, researchers developed even- sume that all individuals of a finite assemblage ness indices, with the impossible task of or population have been identified and counted summarizing an underlying observed abundance (e.g., Brillouin's //•Brijlouin 1956). This as- pattern with one number. Most of the popular sumption is usually tentative for amphibian m- indices use equal abundance of all species as a ventory and monitoring, and in general such base against which to compare some manner of measures are formally related (Pielou 1975) to a divergence. Evenness is actually defined as a family of measures proposed by Hill (1973). balance point in the observed sample, because Two of the most commonly used diversity indi- there is only one balance point and an infinite ces•Simpson's and Shannon's•are included in number of ways to be unbalanced. this family. Smith and Grassle (1977) derived It is most common to describe evenness with a unbiased estimators for these indices that have single index that is based on various methods for minimum variance and allow for calculation of normahzation or scaling by minimum and maxi- confidence limits. mum values. Hurlbert (1971) showed that nor- malized indices decrease non-monotonically to REPEAT RATE (SIMPSON'S INDEX). The first di- zero from the maximum value of the fully bal- versity measure of import, developed as a single anced case (i.e., a uniform distribution of ob- index, is attributed to Simpson (1949), but actually served abundances over species). Alatalo (1981) it has a rather involved history (see note 7). As a discussed the most common measures, and Help diversity measure, it is the straightforward proba- and Engels (1974) compared the statistical be- bility that two organisms selected at random fi-om havior of seven evenness indices in low richness a population will "repeat" then- classification, that environments, (see note 6 for a discussion of the is, that they will belong to the same species. most useful and popular indices.) Whittaker (1965) stated that the Repeat Rate and the total species count together are sufficient DIVERSITY to characterize the pattern of species' abun- Richness and evenness are both facets of an op- dances in a sample. Hill (1973) showed the sense erational definition of diversity. Simpson (1949) in which this is true, on average. was the first to imply, based on Yule's (1944) semantic work, that a diversity index should en- INFORMATION INDICES. The most routinely compass both. Diversity indices that incorporate used measures of diversity are based upon infor- richness and eveimess into one numerical quan- mation theory; they equate diversity in a natural tity are termed dominance diversity indices faunal system with the amount of information in (Sanders 1968), equitability indices (Auclairand a transmitted message. If the actual message to Goff 1971), and heterogeneity indices (Good be interpreted is selected (sampled) fi-om a finite 1953). These indices can be influenced by the number of messages, then this message informa- heterogeneity and magnitude of the area sam- tion or sample, or any monotonie function of it, pled, which can lead to great confusion regard- can be taken as a measure of information. For a ing their interpretarion and use. Fisher's alpha sample, the associated entropy is not an indicator (Fisher 1943) is probably the most widely cited of the amount of knowledge available about that 250 CHAPTER?

sample, but rather it is a measure of the degree of represented in the sample. There can be a sizable randomness in that sample. The tendency for a error increase with the use of any information biological system to become less organized (i.e., based index when the proportion of total species less perfectly balanced or more shuffled) implies represented in the sample declines (Peet 1974). a high degree of entropy. Pielou (1966a) discussed the use and misuse A common information index is Shannon's of this and other information theoretic measures. index, or the H function (see note 8). The for- The most frequently cited disadvantage of these mula for H is as follows: measures is difficulty of computation, but with the current availability of desk microcomputers, this complaint is no longer valid. Routledge (1979) discussed a set of basic properties, both 1 = 1 mathematical and ecological, that should be characteristic of diversity indices. He showed Because P, (the proportion in the population be- that these properties hold only for the indices in longing to the I'th of s species), is usually esti- Hill's family of measures and recommended that mated by Pi = njn (the proportion in the sample), only Shannon's index and the Repeat Rate be used. the formula for H can be rewritten directly in The logarithmic base used in Shannon's index terms of the observed sample results (e.g., see varies across studies and investigators. This Pielou 1966a; Lloyd et al. 1968b). variation is not of biological consequence as The maximum diversity possible for n individu- long as the base used is clearly indicated, be- als occurs when each belongs to a separate species; cause a base change is effected only by a multi- the largest value for the index occurs when;?, = 1/n. plicative factor. A scale factor can be used to The entropy (information) calculated for a given convert logarithms from base 10 to any other sample can be compared with the maximal value base. For example, the index in base 10 would possible in the population, subject only to the as- be multiplied by 3.321928 to obtain base 2, or by sumption that the two situations (sample and pop- 2.302585 to work in the natural log base e. Gib- ulation) are similar. This comparison is called son and Buzas (1973) clearly stated that the low- relative entropy of the source (it reflects the rela- information example in their publication, with H tive species abundances). = 0.45, was calculated with a natural log base. If Peet (1975) illustrated the limitations of such base 10 had been used, their value would have a relativized statistic and stated that it and all decreased to 0.20. Their example with the com- similar diversity ratios are not appropriate for plete balance of species abundances yielded ecological application because they do not ap- more uncertainty and an H value of 1.6. Logio proach a constant limit with increasing sample would have reduced H to 0.69. Large and small size. When the number of species is fixed, the values of H reflect both biological reality and the information is greater and the probabilities (rela- log base used. tive abundances) of the various species are more Good (1953) generalized the Repeat Rate and balanced. Alternatively, the information index Shannon's index and determined the sampling decreases when sample size increases if samples distribution for the Shannon equation that allows of all species are not equally likely to be ob- for variance, standard error, and conñdence in- tained. Thus, Shannon's measure of the average terval calculations. Basharin (1959) showed that density per individual is reasonable when random the use of an estimate in an information index samples are taken from a very large population, formula produces a bias whose magnitude is a and all or many of Ihe species in the population are function of both the species and the individual Analysis of Biodiversity Data 251

counts. Calculation of variance using formulas all the diversity information because each gives of Good and Basharin appropriate to Shannon's different weight to specific properties of the index allows for comparison across random species' abundance distributions. When any samples. Pielou {1966b) discussed the use of index is used, sources of variability known to be alternative estimation procedures to reduce bias peripheral are controlled by transformation of for these indices based upon properties of the the raw data and weighted aggregates. In gen- assemblage sampled and the sampling method eral, indices cannot be used uncritically because selected. Most authors have ignored Pielou's incomplete knowledge of the ecological influ- procedures. Buzas (1979) determined that when ences on such relative measures precludes the the sample contains at least several hundred in- development of any effective, all-encompassing dividuals, the correction is negligible. May measure. Selection of an index tacitly involves a (1975) indicated that because the statistical dis- decision concerning the ordering of species as- tribution of this index is skewed for small num- semblages that are not intrinsically comparable bers of species, characterizing the spread by the (Patil and Taillie 1982). Peet (1975) cited the standard deviation is not exact, but it is good need for a theory, or set of rules, of index re- enough for practical purposes. sponse upon which to base a choice. I recom- If the sample individuals are not randomly ob- mend that an investigation of the species tained in a field study, altemative methods of esti- monitored include an examination of all three mation specific to the study design may be needed. types of measures when possible: richness, Pielou (1966a) devised a sequentially cumulative evenness, and diversity (the last measured by the method for estimating the Shannon measure when Repeat Rate or Shannon's index), so that the using random quadrat sampling. Liebelt (in Heyer relative effects of each component can be evalu- and Berven 1973) improved upon Pielou's meth- ated. The evaluation should be accompanied by odology and developed a standard error formula dimensions of the area from which the speci- for this case. Hutcheson (1970) defined an approx- mens were selected. imate t-test. Because Shannon's indices are nor- Across samples, the use of a single number, mally distributed (Taylor 1978), parametric such as an index, to represent a multifaceted statistical techniques can be used to compare them. situation can be highly misleading without the Mac Arthur et al. (1966) used Sharmon's formula to use of standard errors. Peet (1975) provided a compare bird samples across forest canopy heights similar warning for evenness indices. For pre- and to measure the difference in profile between sentation of results across samples, I recommend habitats; the authors provided useful formulas for that samples (equivalent areas) be randomly se- their calculations. A major advantage of informa- lected from similar habitats. This procedure ob- tion-based indices is that they are additive; this viates the need for rarefaction; increases the permits the partitioning of numbers obtained for interpretability of the selected measures of rich- larger groups into specific values for smaller ness, evenness, and diversity; and allows for groups or subgroups. maximal comparability among studies. How- ever, when such random selection is not feasible, rarefaction is an altemative. When the investiga- Summary and Recommendations tor clearly understands what is being measured When amphibian populations are compared, with a particular index, and indicates what has species counts and abundance distributions can been accomplished in space and time, the inher- be used separately or combined to describe di- ent misunderstanding surrounding species diver- versity. However, no single index will provide sity studies will be alleviated (Buzas 1979). 252 CHAPTER 9

Notes (c) Finally, both sets of marginal frequencies may be fixed in advance. This practice is rare, or perhaps 1. Obtaining and Describing Values for a 2 x 2 nonexistent in amphibian literature. It is similar to, Table. The simplest way to investigate the problem but does not exactly fit, having the total sample size of the correct reference distribution for the 2 x 2 and all four marginals of breeding, nonbreeding, table is to use a predetermined sample size. That is, males, and females fixed, predetermined, or obtained the sample size will always be n and possible distri- in advance. The null hypothesis of interest is butions are examined. Three conditions are possible. (a) The first condition is that both sets of marginal p(ll)_p(21)_ ah frequencies are random variables. This condition oc- Hç,= curs when a sample of size n is taken from a bivari- p{h) PÍ2-) a + c b+d ate distribution and subsequently classified into a double dichotomy. For example, each amphibian is The disfribution of the one cell free to vary is randomly sampled from a selected region, and both hypergeometric, a small sample description of the bi- sex and site characteristics are merely observed and nomial distribution. Kendall and Stuart (1973) gave recorded. The only constant in this case is the total the mean and variance formulas and the asymptotic sample size; no marginal totals are fixed. That is, the normality condition. The mean and variance can be number of males or females is not decided in ad- used in a /^ test (compared with a standardized nor- vance, and the samples at each of the sites are not mal distribution), and fi is proportional to the Pear- stipulated before going into the field. This test is a son y} statistic. This procedure (2x2 independence test of indep•ndence and the exact probability de- trial of Barnard 1947) tests the significance of the pends on two unknown parameters and cannot be difference between two characteristics that have evaluated (Kendall and Stuart 1973). It is assumed been observed to occur randomly in a group of « = that an individual selected at random will possess {a + c) + {b + d) individuals. The random process is characteristic A with probability p{A), and not pos- applied within the set of n, with no assumption about sess it with 1 -p(A). Corresponding probabilities are selection from a larger universe or target population. assumed for factor or characteristic B. (b) Alternatively, one set of marginal frequencies 2. Correcting for Continuity of Distribution. may be fixed. A fixed classification is merely a label- In this note, I examine the application of the continu- ing of the two samples (e.g., males and females) that ity correction to each of the three cases that can are to be compared with respect to the other classifi- describe data for a 2 x 2 table. cation (e.g., numbers in breeding condition). Prede- (a) In the first case, both the marginals and the termining the sample sizes before observing some total sample size are fixed in advance. Conover characteristic (e.g., deciding to examine gonad devel- (1974) and Starmer et al. (1974) considered two con- opment in 50 males and 50 females) results in one ditions in this situation: marginals equal and margin- fixed set of marginal frequencies and one variable or als unequal. random set. Comparison of these two (or more) sam- Marginals equal {a + b = c + d, Qtz. + c = b + d). ples (i.e., male and female in this example) with re- Under this condition. Yates's correction improves spect to a characteristic is a test of homogeneity (2 x the probability estiniates (Mantel and Greenhouse 2 comparative trial of Barnard 1947). The compari- 1968; Staimer et al. 1974), as does an alternative to son tests whether the proportions of individuals with Yates's correction developed by Kendall and Stuart characteristic A are the same in two different popula- (1973). The Kendall and Stuart correction is defined tions where a + c has been drawn at random from as the arithmetic average of the x^ observed with its the first population and b + d from the second. For next smaller possible value. The exception for both large samples, the test is equivalent to the first condi- corrections is when the distribution is extremely tion, in which both sets of marginal frequencies are asymmetric. random variables. For small samples, evaluation is Marginals unequal (a + b ^ c + d, and a + c^h + not possible (Kendall and Stuart 1973), d), Conover (1974) showed that Yates's corrected x^ Analysis of Biodiversity Data 253

and the uncorrectcd x^ statistic each improve the The t statistic can be written as a Studentized Pear- exact probability estimates about half the time, but son statistic: Conover was unable to provide general rules for their application. The Kendall and Stuart correction improved the estimate consistently. in-2) ad-be n (b) In the second case, only one marginal is fixed. [{b + d)ac + {a + c)hd\ ' In this case Fisher's Exact Test and Yates's corrected chi-square test are each too conservative and should The values of z and t differ only in their variance not be used. However, any correction should depend estimate formulas. on the degree of discreteness. Liddell (1978) substi- mted a Yates correction of VA and found it to be too 4. Rarefaction. This procedure uses the ex- liberal for asymmetric distributions. Berkson (1978) pected value of a function of samples drawn ran- compared Fisher's Exact Test with both Yates's '/i- domly from the data. Therefore, it is related to the corrected test and an uncorrected normal z-test and Bootstrap method (nonparametric technique for in- found the latter to be more powerful with an effec- ferring a statistic's distribution, derived from tive level closer to the nominal level desired. Finally, using samples from the data) of Efron (1979). Rar- d'Agostino et al. (1988) showed that the two-inde- efaction is not a sampling plan, although it may pendent-sample i-test with pooled variance is best be useful as an indicator of excessive sampling for sample sizes less than or equal to 15. Berkson (Hecketal. 1975). and d'Angostino et al. found no need to condition on the marginal. 5. Huribert's Rerinetnent of Sanders's Rar- (c) In the third case, both marginals are random. efaction Method. Hurtbert (1971) refined Many agree that the best test of independence when Sanders's method for calculating the expected spe- both marginals are random is the uncorrected y^ cies richness of a sample of individuals. (Conover 1974; Liddell 1978; but see Pirie and Ham- Hurlbert's refinement is a special case of the den 1972 for a contrary view). However, the Yates hypergeometric distribution. Plots of rarefaction correction should not be used in this case (Conover values are monotonically increasing, downwardly 1974), because the correction does not provide rea- concave curves that conveniently pass through the sonable estimates of the probabilities from the ran- points (0,0), (1,1), and (N,S). Smith and Grassle domized version of Fisher's Exact Test. Actually (1977) provided a solution to the minimum-vari- Yates's correction results in the performance of a dif- ance unbiased estimator problem of the expected ferent test. Under certain conditions, the correction number that allows for the calculation of confi- leads to values that indicate significance but are dence bounds. Heck et al. (1975) argued that bio- called nonsignificant, and vice versa. logical reasons account for a multinomial approximation for large numbers. I am unable to 3. Comparison of Tests for One-Fixed-Mar- devise a situation in which that statement could be gina) 2 X 2 Table Data. The two-independent- said to hold for either amphibians or general fau- sample f-test is very close in computation to the x^- For nal representations, and I recommend that only a comparison of the statistics of Berkson (1978) with the hypergeometric family described by Hurlbert's Öiat of d'Agostino, et al. (1988), we have the following: equations be used.

b 6. Evenness Indices. The indices discussed (WrC) (¿ + tf) below can be written as a ratio of Hill's (1973) or- ±V(xy =^ = a + b dered set. They are of the normalized form specified ri-(a + ¿)Y 1 ^ 1 n byHurlbert(1971). n \a + b b + d Pielou's / (Pielou 1975) is probably the index most widely known among ecologists, although 254 CHAPTER 9

many problems with it have been identified (see balanced one. The ultimate reference on this index Sheldon 1969; Heip and Engels 1974). This index is appears to be that of Good (1982). the log ratio of Hill's first two numbers (i.e., \og(NJNo). Buzas and Gibson ( 1969) first proposed 8. Information Indices. Shannon's information an index that is the simple untransformed ratio of index was formed when he used Boltzmann's H Hill's first two ordered numbers Aij/No and related function from statistical mechanics theory as an by exponentiation to Pielou's /. Buzas and Gibson's index of "entropy," for the evaluation of missing in- index is commonly attributed to Sheldon (1969) who formation (Shannon 1948). This information index first called this value e. Help's index (Heip 1974) is has incorrectly been termed the Shannon-Weiner the same fraction used by Buzas and Gibson but index because Weiner applied Shannon's entropy for- with the lower bound of 1 subtracted from both nu- mulas to his research. Weiner always attributed the merator and denominator to effect a limit change. entropie ideas to Shannon. The index has also been Hill (1973) suggested that the ratio of second- and called the Shannon-Weaver hidex, because first-order numbers was an evenness index (Hill's Shannon's (1949) second and most widely refer- index). When this ratio is restrained to approach a enced presentation of it appeared in an article in his limit of zero, as is preferable, it is called Hill's modi- book with Weaver (Shannon and Weaver 1949•the fied ratio. Because each of these indices is a scaled article and book have the same title). Hartley (1928) diversity index (the ratio of two indices is itself an showed that the log function was the best form for index but on a new scale), each can incorporate the amount of information. problems as well as the advantages of the original in- The additive inverse (a number that when added dices. Sheldon (1969) showed, for example, that to a second number results in zero) of Shannon's Pielou's / and the Buzas and Gibson index are af- index is called the redundancy of the population and fected to varying degrees by species count, espe- represents the fraction of the information message cially for low-richness assemblages. The same is determined by the governing stochastic process (i.e., true for numbers less than approximately 10 (Buzas unnecessary noise). Many of the objections to the 1972b). Hill (1973) stated that because of this sam- use of Shannon's index for sparse or small samples ple dependence, evenness measures cannot be re- may be overcome by jackknif ing the index, a proce- garded as measuring a property of the target dure that reduces bias in estimation and provides ap- population. proximate confidence intervals in cases where ordinary statistical distribution theory proves intrac- 7. Repeat Rate (Simpson's Index). The Repeat table (Quenouille 1949,1956; Zahl 1977). Adams Rate measure probably was first used for cryptog- and McCune (1979) devised a computer program for raphy in 1879 in a German publication by Lexis applying this procedure to Shannon's index. Hypoth- (see Keynes 1921:399; Friedman 1922). In 1941, esis tests and confidence intervals can be obtained A. W. Turing (see Good 1979) gave this quantity, for the unknown parameter using the Student's i-dis- defined by the formula £;= i pj, its most "natural tribution (Miller 1974). In theory, jackknif ing is name" (Good 1982), the Repeat Rate. Turing also valid and extremely useful. However, the appropri- derived its unbiased estimate and used it, as did ateness of the resultant confidence intervals is depen- Sacco (1951), in cryptoanalysis. Peet (1974) and dent upon the ajjproximate normality of the estimate, Bhargava and Uppuluri (1975) attributed the addi- which needs investigation under conditions of am- tive inverse (1 - £*= i pf) form of this measure, phibian sampling. which is also used as a diversity index, to Gini Several other indices are interconvertible with (1912). Weaver (1948) used a function of the re- Shannon's index. Hill (1973) showed that his N-\ equals peat rate (1 / Zf ^ i pj) as a "surprise" index, deriv- ^ and interpreted the multiplicative inverse of this ing the name from the notion that it is less (l/e") as the weighted geometric mean of the propor- surprising to locate a specimen of a rare species in tional abundances. Buzas and Gibson's evenness index, a highly heterogeneous assemblage than in a more defined as Ni/Ng, is equal to e"/*, for s species. Analysis of Biodiversity Data 255

Appendix: properties of 6. UN2 .125 28. Hamann/ .067 association measures vl2S G-Index .067 7. Kulczynski .286 29. Baroni- ^.123 Coefficients are listed in the appendix by names .286 Urbani -.123 and numbers corresponding to those in Table 20. and Buser 1 The Resemblance Equation Coefficient (Table 20: 8. Mountford .105 30. Baroni- .438 # 20) is not included because it has no general .011 Urbani .438 solution. NI and Baser 2 9. Correlation .133 31. Total .467 ratio Examples for Invariance .133 difference .467 10. Braun- .333 32. Pattern .462 In the table below, the frequency values in the Blanquet .333 difference .462 first row are multiplied by 10 to produce those in 11. Pager .161 33. Angular .051 the second row. This process in turn multiplies transfonn the total by 10 but leaves the proportions the .301 .051 NI same. Coefficients that change when proportions 12. Savage .667 34. Michael .000 remain the same but sample sizes differ are not mvariant (NI) and are indicated as such. The .66? .000 values for each coefficient as calculated using 13. Nonmetric .636 35. Faith .333 the values from each row of the table are Hsted. .636 .333 14. McCon- -.267 36. Omega .000 n naughey -.267 .000 4 ..X. .__6_ 15. Johnson .833 37. Eyraud -.010 20 40 30 60 150 .833 -.010 16. Forbes 1.000 NI A. Similarity B. Matching 1907 1.000 Coefficients Coefficients 17. Gilbert and -I.IE- 1. Simpson .400 21. Simple .533 Wells 19 matching .400 .533 -t.lE- 2. Kulczynski .367 22. UNs .231 19 .367 .231 18. Forbes .000 3. Ochiai/ .365 23. UN4 .500 1925 .000 Otsuka .365 .500 19. Tarwid .000 4. Dice .364 24. UNj 1.430 .000 .364 1.430 4a. Dice's NA 25. UN] .696 Examples for Linearity Conditions Asymmet- NA .696 ricBl A T\vo sets of four tables of frequency values are 4b. Dice's NA 26. Russell .133 [ffovided below. The marginal frequencies and Asymmet- NA and Rao .133 total frequencies in the tables are fixed for the HCAIB purposes of examining linearity conditions and 5. Jaccard .222 27. Rogersand .364 providing insight into values and ranges of the .222 Tanimoto .364 measures under this set of conditions. In the tables in set 1, the d cells have the largest values. 256 CHAPTER 9

Each table in set 2 has the same total frequency the sample data. The differences between each as in set 1, but the marginals are fixed at differ- pair of successive values, in order, are also pro- ent values. In both sets the values of a increase vided. Each set of differences is categorized ac- linearly from 0. cording to pattern of change, as follows:

Setl set 2 Linear•Differences are constant. a h c d n a b c d n Convex•When the a cell value increases, dif- 0 30 30 40 100 0 40 40 20 100 ferences increase. Concave•When the a cell value increases, dif- 1 29 29 41 100 1 39 39 21 100 ferences decrease. ND Coefficient is not defined for the chosen 2 28 28 42 100 2 38 38 22 100 values. 3 27 27 43 100 3 37 37 23 100 *•Coefficient does not detect the change in marginals when the total remains constant (the values are equal for both sets). Each coefficient is listed below with two sets of H•Coefficient does not detect this slight unit four values of that coefficient, calculated using change in the a cell value.

Setl Set 2

Value Difference Value Difference

A. Similarity Coefficients 1. Simpson .000 .000 .033 .033 .025 .025 .067 .033 .050 .025 .100 .033 Linear .075 .025 Linear 2. Kulczynski .000 .000 .033 .093 .025 .025 .067 .033 .050 .025 .100 .033 Linear .075 .025 Linear 3. Ctehiai/Otsuka .000 .000 .033 .0Î33 .025 .025 .067 .033 .050 .025 .100 .033 Linear .075 .025 Linear 4. Dice .000 .000 .033 .033 .025 .025 .067 .033 .050 .025 .100 .033 Linear ,075 .025 Linear 4a. Dice's asymmetric B | A .000 .000 .033 .033 .025 .025 .067 .033 .050 .025 .100 .033 Linear .075 .025 Linear Analysis of Biodiversity Data 257

Setl Set 2

Value Difference Value Difference

4b. Dice's asymmetric A | B .000 .000 .033 .033 .025 .025 .067 .033 .050 .025 .100 .033 Linear .075 .025 Linear 5. Jaccard .000 .000 .017 ,017 .013 .0127 .034 .018 .m .0130 .053 .018 Convex .039 .0133 Convex 6. UNj ,000 .000 .a» .0085 .006 .006 .018 .0090 .013 .007 .027 .0095 Convex .020 .007 Convex 7. Kulczynski .000 ,000 .017 .017 .013 .013 1 .036 .018 .026 .013 .056 .020 Convex .041 .014 Convex 8. Moiintford .000 .000 .001 .001 .0006 .0006 .002 .001 .001 ,0007 .004 .0010 Convex .002 .0001 Convex 9. Correlation ratio .000 .000 .001 .001 .0006 .0006 .00* .003 .003 .002 ' .010 .006 Convex .006 .003 Convex 10. Braun-Blanquet .000 .000 .033 .033 .025 .025 .067 .033 .050 .025 .100 .033 Linear .075 .025 Linear 11. Fager -.091 -.079 -.058 .033 -.054 .025 -.025 .033 -.029 .025 1 .009 .033 Linear -.004 ,025 Linear 12. Savage 1.000 1.000 .967 -.033 .975 -.025 .933 -.033 ,950 -.025 .900 -.033 Linear .925 -.025 Linear

(Continued) 258 CHAPTERS

Setl Set 2

Value Difference Value Difference

13. Nonmetric 1.000 1.000 .967 -.033 .975 -.025 ,933 -.033 .950 -.025 .900 -.033 Linear .925 -.025 Linear 14. McConnaughey -1.000 -1.000 -.933 .067 -.950 .050 -.867 .067 -.900 .050 -.800 .067 Linear -.850 .050 Linear 15. Johnson*,+ ND ND 1.000 ND 1.000 ND 1.000 .000 1.000 .000 1.000 .000 1.000 .000 16. Forbes 1907 .000 .000 .111 .111 .063 .063 .222 ,111 .125 .063 .333 .111 Linear .188 .063 Linear 17. Gilbert and Wells ND ND -.954 ND -.200 ND -.635 .301 -.903 .301 -.477 .176 Concave -.727 .176 Concave 18. Forbes 1925 -.429 -.667 -.400 .029 -.652 .014 -.368 .032 -.636 .016 -.333 .035 Convex -.619 .017 Convex 19. Tarwid -1.000 -1.000 -.800 .200 -.882 .118 -.636 .163 -.778 .105 -.500 .136 Concave -.684 .094 Concave B. Matching Coefficients 21. Simple matching .400 .200 .420 .020 .220 .020 .440 .020 .240 .020 .460 .020 Linear .260 .020 Linear 22. UNs .000 .000 .020 .020 .009 .009 .040 mi .018 .010 .061 .022 Convex .029 .011 Convex Analysis of Biodiversity Data 259

Setl Set 2

Value Dilference Value DifTermce

23. UN4 .286 .167 .310 .024 .188 .021 .333 .024 .208 .021 .357 .024 Linear .229 .021 Linear 24. UN, .667 .250 .724 .057 .282 .032 .786 .062 .316 .034 .852 .066 Convex .351 .036 Convex 25. UNi .571 .333 .592 .020 .361 .027 .611 .020 .387 .026 .630 .020 Concave .413 .026 Concave 26. Russell and Rao* .000 .000 .010 .010 .010 .010 .020 .010 .020 .010 .030 .010 Linear .030 .010 Linear 27. Rogers and Tanimoto .250 .111 ,266 .015 .124 .012 .282 .016 .136 .013 .299 .017 Convex .149 .014 Convex 28. Hamann/G-Index -.200 -.600 -.160 .040 -.560 .040 -.120 .040 -.520 .040 -,080 .040 Linear -.480 .040 Linear 29, Baroni-Urbani and Buser 1 -1.000 -1.000 -.774 .226 -.866 .134 -.668 .106 -.796 .070 -.580 .087 Concave -.735 .061 Concave 30. Baroni-Urbani and Buser 2 .000 .000 .003 .113 .067 .067 .166 .053 .102 .035 .210 ,044 Concave .133 .031 Concave 31. Total difference .600 .800 .580 -.020 .780 -.020 .560 -.020 .760 -.020 .540 -.020 Linear .740 -.020 Linear 32. Pattern difference .600 .800

{Canlinued) 260 CHAPTERS

Sell Set 2

Value Difference Value Difference

.580 -.020 .780 -.020 .560 -.020 .760 -.020 .540 -.020 Linear .740 -.020 Linear 33. Angular transformForm .043 .029 .044 .001 .031 .002 .046 .001 .032 .001 .047 .001 Concave .034 .001 Concave 34. Michael -.692 -.941 -.624 .068 -.914 .028 -.559 .070 -.896 ,030 -.477 .075 Convex -.845 .037 Convex 35. Faith .200 100 .215 .015 .115 .015 .230 .015 .130 .015 .245 .015 Linear .145 .015 Linear 36. Omega -1.000 -1.000 -.638 .362 -.790 .211 -.507 .131 -.703 .087 -.408 .099 Concave -.633 .069 Concave 37. Eyraud+ -.0002 -.0003 -.0002 .000 -.0003 .000 -.0002 .000 -.0003 .000 -.0002 .000 -.0003 .000

Range Conditions mention is never made of conditions on the two cells not affected, I also compared restrictions on For this section, the matrices were selected to these quantities. For example, restrictions versus show possible conditions under which a maxi- no restrictions were compared on a and d while mum or a minimum value of the index is at- examining conditions for the maximum when b tained. The total sample size was held constant, and/or c = 0. and extreme cell values were selected. Although

b n Conditions

500 0 0 500 1,000 f) = c = 0;£i = d 900 0 0 100 1,000 Í) = c = 0; no restrictions on a or íí 100 0 0 900 1,000 ¿> = c = 0; no restrictions on a or d Analysis cf Biodiversity Data 261

Conditions

499 0 2 499 1,000 hmc = 0;a = d 1 0 1 998 1,000 ÍJ or c = 0; no restrictions onaoid 1 1 0 998 1,000 A or £• = 0; no restrictions on a or rf 998 1 0 1 1,000 è or c = 0; no restrictions on a or d 0 500 500 0 1,000 a = d = 0;b = c 0 900 100 0 1,000 £¡f = rf = 0; no restrictions on fc or c 0 100 900 0 1,000 a = d = 0;no restrictions on ¿ or c 0 99 900 1,000 a or d = 0; no restrictions on ô or c 900 99 0 1,000 a or d = 0; no restrictions on i> or c 998 1 0 1,000 aOTd = Q; b = c 0 1 998 1,000 aOT d=0;b = c 1 1 997 1,000 incremental change from a = d = 0 and b = c,butd still largest

A. Similarity Coefficients B. Matching Coefficients C. Traditional Association Measures

1. Simpson 1.000 21. Simple 1.000 38.x' 1,000.000 matching 1.000 1.000 1,000.000 1.000 1.000 1,000.000 1.000 .998 992.030 1.000 .999 499.500 1.000 .999 499.500 1.000 .999 499.500 .000 .000 1,000.000 .000 .000 1,000.000 .000 .000 1,000.000 .000 .900 .110 .9989 .900 .110 .9990 .998 .001 .000 .998 .001 .500 .998 248.990 2. Kulczynski 1.000 22. UN5 1.000 39. Coefficient of 1.000 1.000 1.000 mean square 1.000 contingency 1.000 1.000 1.000 •998 .996 .992 .750 .707 .499 .750 flW .499 ,999 .101 .499 .000 .000 Ï.000

{Contifmed) 262 CHAPTER 9

A. Similarity Coefficients B. Matching Coefficients C. Traditional Association Measures

.000 .000 1.000 .000 .000 1.000 .000 .000 .0001 .950 .000 .0001 .999 .000 .000001 .000 .000 .000001 .500 .499 .249 a.Ochiai/Otsuka 1.000 23. UN4 1.000 40. Cramers v 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .998 .998 .996 .707 .875 .707 .707 .875 .707 .999 .875 .707 .000 .000 1.000 .000 .000 1.000 .000 .000 1.000 .948 .475 .011 .999 ,475 .011 .000 .499 .001 .000 .499 .001 .500 .749 .499 4, Dice 1.000 24. UN3 ND 41. Pearson's .707 1.000 ND contingency .707 coefficient 1.000 ND .707 .998 499.000 .706 .667 999.000 .577 .667 999.000 .577 .999 999.000 .577 .000 .000 -.707 .000 .000 -^.707 .000 .000 -.707 .000 9.000 -.010 .947 9.000 -.010 .999 499.000 -.001 .000 499.000 -.001 .500 499.000 .446 ' I

Analysis of Biodiversity Data 263

A. Similarity Coefficient B. Matching Coefficients C. Traditional Association Measures

4a. Dice's 1.000 25. UN, 1.000 42. Phi coefficient 1.000 asymmetric 1.000 1.000 B[A 1.000 1.000 1.000 1.000 1.000 .999 .996 1.000 .999 .707 .500 .999 .707 .999 .999 .707 .000 .000 -1.000 .000 .000 -1.000 .000 .000 -1.000 .000 .947 -.010 .999 .947 -.010 .999 .999 -.001 .000 .999 -.001 .500 .999 .499 4b. Dice's 1.000 26. Russell and .500 43. Yule's Q 1.000 asymmetric 1.000 Rao .900 AIB 1.000 1.000 .100 1.000 .996 .499 1.000 .500 .001 1.000 1.000 .001 1.000 1.000 .998 1.000 .000 .000 1.000 .000 .000 1.000 .000 .000 1.000 .000 .000 1.000 .901 .900 1.000 .999 .998 1.000 .000 .000 1.000 .500 .001 1.000 5. Jaccard 1.000 27. Rogers and 1.000 44. Pearce 1.000 Tanimoto 1.000 1.000 1.000 1.000 1.000 1.000 .996 .996 .997 .500 .998 .500 .500 .998 .999 .999 .998 .500 .000 .000 -1.000 .000 .000 -1.000

(Comimed) 264 CHAPTER 9

A. Similarity Coefficients B. Matching Coefficients C, Traditional Association Measures

,000 .000 -1.000 .000 .81S -.000 .900 .818 -.099 .999 .996 -.001 .a» .996 -.001 J33 .996 .499 6. UN2 1.000 28. Hamann G- 1000 45. Cl 1.000 1.000 index 1.000 1.000 1.000 1.000 1.000 .992 .996 1.000 .333 .998 1.000 .333 .998 .499 .998 .998 .499 ,000 -1.000 -1.000 .000 -1.000 -1.000 .000 -1.000 -1.000 .000 .800 -1.000 .818 .800 -1.000 .996 .996 -1.000 .000 .996 -1.000 .200 .996 .499 7. Kukzynski ND 29. Baroni- 1.000 46. C, 1.000 ND Urbani and 1.000 1.000 Buser 1 ND 1.000 1.000 249.500 .996 1.000 1.000 .940 1.000 1.000 .940 1.000 998.000 .998 1,000 .000 -1.000 -1.000 .000 -1.000 -1.000 .000 -1.000 -1.000 .000 -1.000 -1,000 9.000 .800 -1,000 499.000 .996 -1.000 .000 -1.000 -1.000 .500 .884 .499 Analysis of Biodiversity Data 265

A. Similarity Coefficients B. Matching Coefficients C. Traditional Association Measures

8. Mountford ND 30. Baroni- 1,000 ND Urbani and 1.000 Buser 2 ND 1.QQ6 1,000 .998 2.000 .970 2.000 ,970 2.000 m ,000 .000 .000 .000 .000 .000 .000 .000 .020 .900 .999 .998 ,000 .ÛOD ,500 .942 9. Correlation ratio 1.000 31. Total QOCf 1.000 difference •000 1.000 .000 .996 .oes .500 .001 .500 .(X)l .999 .001 .000 1.000 .000 1.000 .000 1.000 .000 .100 .900 .100 .998 .002 .000 .002 .230 .002 lO.Braun-Blanquet 1.000 32, Pattern .000 1.000 difference .000 1.000 ,000 .996 .000 .300 .000 .500 .000 .999 .000 .000 1.000 .000 .600

(Cominued) 266 CHAPTER 9

A. Similarity Coefficients B. Matching Coefficients C, Traditional Association Measures

.000 ,600 .000 .020 .901 .020 .999 .002 .000 ,002 .500 ,002 11. Fager .978 33, Angular ,099 .983 transform .099 .950 .099 .976 .096 .354 .097 .354 ,097 .984 ,097 -.022 .000 -.017 .000 -.017 .000 -.050 .078 .933 .078 .983 ,096 -.500 .096 .146 .096 12. Savage .000 34. Michael 1.000 .000 .360 .000 .360 .004 .999 .500 .004 .500 .004 .001 .004 1.000 -1.000 1.000 -.360 1.000 -.360 1.000 -.0005 .099 -.0005 .001 -.000 1.000 -.000 .500 .003 Analysis of Biodiversity Data 267

A. Similarity Coefficients B. Matching Coefficients C. Traditional Association Measures

13. Nonmetiic .000 35. Faith .750 .000 ,950 .000 .550 ,002 ,749 .333 JOÖ .333 .500 .001 .999 1.000 .000 1.000 .000 1.000 .000 1.000 .450 .053 .900 .001 .998 1.000 .499 .500 .4995 14. McComiaughey 1.000 36. Omega 1.000 1.000 1.000 1.000 1.000 .996 1.000 .500 1.000 .500 1.000 ,999 1.000 -l.OOQ -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 -1.000 .900 -1.000 .998 -1.000 -1.000 -1.000 .000 I : .939 15. Johnson 1.000 37. Eyraud -.00000 1.000 -.00009 1.000 -.00000 1.040 -.00000 2.000 -.00000 .500 -.00000 .999 -.49950 ND -.00000

{Coniirmed) 268 CHAPTER 9

A. Similarity Coefficients B. Matching Coefficients C. Traditional Association Measures

ND -.00001 ND -.00001 ND -.00000 1.110 -.01009 1.000 -.99900 ND .00000 1.000 -.00000 16. Forbes 1907 2.000 1.110 10.000 1.996 500.000 500.000 1.001 .000 .000 .000 .000 .999 .999 .000 250.000 17. Gilbert and .301 Wells .046 1.000 .300 2.700 2.700 .0004 ND ND ND ND .000 -.000 ND 2.400 Analysis of Biodiversity Data 269

A. Similarity Coetficients B. Matching Coefficients C. Traditional Association Measures

18. Forbes 1925 -1.000 -.111 -9.000 -.996 ^99.000 ^99.000 -.001 -1.000 -9.000 -9.000 -.110 -.0001 -.000001 -.001 1.000 19. Tarwid .333 .053 ,8m .332 .996 .996 .0005 -1.000 -1.000 -1.000 -1.000 .00005 -.0000 -1.000