Ecological Resemblance. Ecological Resemblance. Modes of Analysis. - Outline - Welcome to Paradise

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1 Ecological Resemblance - Outline - Ecological Resemblance Mode of analysis Analytical saces Association Coefficients Q-mode similarity coefficients Symmetrical binary coefficients Asymmetrical binary coefficients Symmetrical quantitative coefficients Asymmetrical quantitative coefficients Probabilistic coefficients Q-mode distance coefficients Metric distance Semimetrics R-mode coefficients of deendence Non-abundance measures Secies abundance measures Choice of a coefficient ANOSIM Welcome to Paradise Ecological Resemblance A quantitative measure of resemblance (i.e., similarity) between either objects (e.g., sites) and the variables describing them (e.g., secies) can be an end in itself, or the standard recursor to subsequent ordination and classification rocedures. Association between objects: Association between descritors: Q-mode analysis R-mode analysis Often times, an examination of the association hemimatrix (derived from rimary matrix) suffices to elucidate the basic structure of the data, and no additional analysis is needed. Modes of Analysis Catell (1952, 1966) was the first to recognize that EEB data could be studied from at least 6 viewoints from 3-D data matrices comosed of descritors, objects, and times. He defined 6 ossible modes of analysis. The two main viewoints used in EEB are R-mode or Q-mode. The imortant oint here is that these two modes of analysis are based on different measures of association. Whether a articular analysis is R- or Q-mode is often confused in the literature & textbooks due to a lack of clarity. 1

2 Q- and R-Mode In order to revent confusion, any study starting with the comutation of an association matrix among objects should be called a Q-mode analysis. Any study starting with the comutation of an association matrix among descritors should be called an R-mode analysis. Analytical Sace Following the terminology of Williams & Dale (1965), the sace of descritors (attributes) will be called A-sace. In this sace, objects may be reresented along axes that corresond to the descritors. Symmetrically, the sace of reference in which the descritors are ositioned relative to axes corresonding to objects (or individuals) is called I-sace. Descritor y 2 A-Sace - Examle of 5 Objects with 2 Descritors - y 22 y 21 y 24 y 23 y x 1 x x 3 y 11 y 12 y 13 y 14 y 15 Descritor y 1 x 4 x NB: The thickness of the lines joining objects is roortional to their degree of resemblance based on the two descritors. 2

3 Analytical Sace The number of dimensions that can be reresented on aer is obviously limited to two or three. However, we will see soon, that this simle analog can be extended to multile dimensions. The A- and I-saces are called metric or Euclidean because the reference axes are quantitative and metric. Ordination rocedures by definition are restricted to these saces, clustering rocedures are not. Ecological Resemblance Mode of analysis Analytical saces Association [Resemblance] Coefficients Q-mode similarity coefficients Symmetrical binary coefficients Asymmetrical binary coefficients Symmetrical quantitative coefficients Asymmetrical quantitative coefficients Probabilistic coefficients Q-mode distance coefficients Metric distance Semimetrics R-mode coefficients of deendence Non-abundance measures Secies abundance measures Choice of a coefficient ANOSIM Association Coefficients The most usual aroach to assess the resemblance among objects or descritors is to start with a rectangular data matrix and condense all of the information in to a square hemi-matrix of association values. The structure resulting from the numerical association analysis may not necessarily reflect all of the information that was originally contained in the rimary matrix. Thus, there is considerable imortance to choosing the aroriate measure of association. 3

4 Selection of Association Coefficient The basic considerations for selecting an aroriate association coefficient fall under the following: (1) the nature of the study determines the structure of the data to be evaluated with an association matrix, (2) the various measures available are subject to different mathematical constraints (deends uon whether one continues with ordination or clustering), (3) comutational asects such as what measures are available or can be rogrammed in articular software (less of a roblem in recent years). Selection of Association Coefficient Because there are few mathematical constraints, biologists are free to define and use any measure of association suitable to the henomenon under study hence why there are so many coefficients in the literature. We will use the general term association coefficient to describe any measure used to quantify resemblance; more secifically, R-mode studies generally use deendence coefficients. Q-mode studies tyically use similarity coefficients. Q-Mode: Similarity Coefficients This is the largest grou of coefficients in the literature. All of these coefficients are used to measure the association between objects. Similarity measures are never metric since it is always ossible to find two objects, A & B, that are more similar than the sum of their similarities with another more distant object C. Thus, similarities can NOT be used to osition objects in metric sace (they must be converted to distances) such as ordinations; however, they can be used in clustering analysis. 4

5 Similarity Coefficients Similarity coefficients were first develoed for binary (resence/absence) data and later became generalized for multi-state descritors with the advent of comuters. NB: Similarity coefficients for binary data are used widely with current molecular data! A major dichotomy in these descritors exists regarding how the coefficient handles the double-negative or double-zero situation. Double-Negative Problem From classical niche theory, we know that secies are distributed unimodally along environmental gradients. The abundance of a secies reaches an otimum at some central set of conditions and is minimized near the minimum and maximum of the gradient sector. The double-zero situation is a roblem in this context because if a secies is resent at two sites, this is an indication of similarity of these sites. However, if a secies is absent from both sites, it may be because both sites roduce an environment that is above the otimal niche value, below the otimal, or one above and one below. One cannot tell. Asymmetrical vs. Symmetrical Coefficients Thus, it is [generally] referable to abstain from drawing biological conclusions from double-negative situations (excet under those conditions that ermit accurate interretation). Numerically, this means that you should ski double zeros when comuting similarity or distance coefficients when using binary data. Coefficients of this tye are called asymmetrical because they treat zeroes in a different way than the rest of the data. With symmetrical coefficients, the zero state for two objects is treated exactly the same way as for all other airs of values, when comuting similarity... 5

6 Binary Coefficients In the simlest case, the similarity between two sites is based on resence-absence (binary) data. Observations are often summarized in a 2 2 table: Where: Object x 1 Object x a b 0 c d a + b c + d a + c b + d a + b + c + d a = the no. of descritors for which the two objects are coded as 1 d = two objects coded as 0 b & c = two objects coded differently = the sum of all descritors Simle Matching Coefficients An obvious way to comute the similarity between two objects is to count the number of descritors that code the objects in the same way and divide a+ d this by the total: S1( x1, x2) = Coefficient S 1 is called the simle matching coefficient (Sokal and Michener 1958). When using the coefficient, one assumes that there is no difference between double-0 or double-1. Rogers and Tanimoto (1960) roosed a variant that gives more weight to differences: a+ d S2( x1, x2) = a + 2b + 2c + d Simle Matching Coefficients Sokal & Sneath (1963), Sneath & Sokal (1973) 2a+ 2d S3( x1, x2) = 2a+ b+ c+ 2d a+ d S4( x1, x2) = b+ c 1 a a d d S5( x1, x2) = a+ b a+ c b+ d c+ d S 3 counts resemblances as being twice as imortant as differences S 4 comares resemblances to differences, measure: 0 to S 5 comares resemblances to marginal totals S ( x, x ) = a d ( a+ b)( a+ c) ( b+ d)( c+ d) S 6 roduct of the geometric means of the terms relative to a & d in coefficient S 5 6

7 Simle Matching Coefficients Coefficients S 1 & S 3 have generally been the most oular; however, there may be times where the others are aroriate. Three additional measures are available in the NT-SYS comuter software (oular among systematists): a + d b c S = ad bc S = ad + bc Hamann coefficient Yule coefficient φ = ad bc ( a+ b)( c+ d)( a+ c)( b+ d) Pearson s hi MVSP is a oular software ackage that comutes a large number (20) of similarity coefficients. htt:// Similarity Coefficients - MVSP Software - Consider the examle of 5 Panamanian cockroach secies scored for the resence/absence (0/1) in 6 habitats: 7

8 MVSP Similarity Coefficients are rovided under cluster analysis otions Simle Matching Coefficient - Similarity Hemi-matrix - NB: diagonal = 1 Simle Matching Coefficients - MVSP Software - So what formula does MVSP use to calculate the simle matching coefficient? SMc ij = S 1 (Sokal & Michener 1958) Confirm for yourself by comaring two samles: BCI vs. LC ( a+ d) SMcij = ( a+ b+ c+ d) a = 3 b = 1 c = 0 d = 1 SM = 4/5 =

9 Similarity Coefficients - R - Use a sreadsheet and create a file named roach.csv Similarity Coefficients - R - If you look around in R, you will find that it does not suort an otion for a simle matching coefficient like S 1. Q: What to do? A: Write it yourself slacker! With a bit of oking around, you will see that there is a secial function in the vegan ackage called designdist. This function is designed to use the abcd notation that we have develoed thus far and write any coefficient you wish! Virtually all of the coefficients of Legendre & Legendre (1998) can be written using designdist. Similarity Coefficients - R - 9

10 Similarity Coefficients - R - 10

11 Asymmetrical Binary Coefficients Coefficients aralleling the ones just resented are available for comaring sites using secies resence-absence data, where the comarison must exclude double-zeros. The best known measure is Jaccard s (1900) coefficient of community, or more simly Jaccard s coefficient: a S7 ( x1, x2) = a+ b+ c Sørenson (1948) made an imortant modification by giving double weight to double resences. Sørenson s coefficient: 2a S8( x1, x2) = 2a+ b+ c Asymmetrical Binary Coefficients Another variant of S 7 gives trile weight to double resences: 3a S9( x1, x2) = 3a+ b+ c The weights seem to be most imortant when dealing with rare secies. You may wish to exlore resonse atterns over 2 or more weightings. The asymmetrical analog to S 2 rovides double weight to differences in the denominator: S10 a+ d ( x1, x2 ) = a + 2b + 2c Asymmetrical Binary Coefficients Russell & Rao (1940) suggested a measure that allows the comarison of the number of double resences, in the numerator, to the total number of secies found at all sites, including secies that are absent (d) from the airs of sites considered: a S11 ( x1, x2 ) = 11

12 Asymmetrical Binary Coefficients While Kulcynski (1928) roosed a coefficient oosing double-resences to differences: S12 a ( x1, x2 ) b c Sokal and Sneath (1963) rovide a modification of Kulcynski s index where double-resences are comared to the marginal totals (a + b) and (a + c): S13 1 a a ( x1, x2 ) = 2 + a+ b a+ c Asymmetrical Binary Coefficients Ochiai (1957) used, as a measure of similarity, the geometric mean of the ratios of a to the number of secies in each site, i.e., the marginal totals (a + b) and (a + c): S14 ( x1, x2 ) = a ( a+ b)( a+ c) S 14 is the same as S 6 excet for the art re double-zeros. Faith (1983) suggested a coefficient in which disagreements (0/1s) are given a weight oosite that of double resences: a+ d/2 S26 ( x1, x2 ) = b+ c Symmetrical Quantitative Coefficients EEB descritors often have more than two states. The binary coefficients we just discussed can frequently be extended to accommodate multi-state descritors. For examle, the simle matching coefficient may be used as follows: agreements S1( x1, x2) = where the numerator contains the number of descritors for which the two objects are in the same state. 12

13 Symmetrical Quantitative Coefficients - Examle: Simle Matching - For examle, if a air of objects was described by the following 10 multi-state descritors: Descritors Object x Object x Agreements = 4 agreements 4 agreements S1( x1, x2) = = = descritors Symmetrical Quantitative Coefficients In a similar fashion, it is ossible to extend virtually all of the binary coefficients to create multi-state coefficients. However, coefficients of this tye often result in the loss of valuable information, esecially in the case of ordered descritors for which two objects can be comared on the basis of the amount of difference between states. Gower s Coefficient Gower (1971) roosed a general coefficient of similarity which can combine different tyes of descritors and rocess each according to its own mathematical tye. This coefficient can be VERY useful when you need to record data on binary, multi-state, and even quantitative variables all in the same rimary matrix! Gower s coefficient takes 1 the general form: S15 ( x1, x2 ) = s12 j = 1 j 13

14 Gower s Coefficient The similarity between two objects is the average, over the descritors, of the similarities calculated for all descritors. For each descritor j, the artial similarity value s 12j between objects x 1 and x 2 is comuted as follows: For binary descritors, s j =1 (agreement) or 0 (disagreement); double-zeros are treated as agreement. Qualitative and semi-quantitative descritors are treated following the simle matching coefficient rule above. Quantitative descritors (real numbers) are treated in an interesting way... Gower s Coefficient - Quantitative Descritors - For each descritor, one first comutes the difference between the states of the two objects y 1j - y 2j. This value is then divided by the largest difference (R j ) found for the descritor across all sites of the study or if one refers, in a reference oulation. Since the ratio is actually a normalized distance, it is subtracted from 1 to transform it into a artial similarity: s = 1 y / j j y j Rj Gower s Coefficient Gower s coefficient may be rogrammed to include an additional element of flexibility: no comarison is comuted for descritors where information is missing for one or the other object. This is obtained by a value w j, called Kronecker s delta, describing the resence or absence of information: w j = 0 when the information about y j is missing one or the other object, or both; w j = 1 when info is resent for both objects. Thus, the final form of Gower s coefficient is: (ranges from 0 to 1) S ( x, x ) = j = 1 w j = 1 12 j 12 w 12 s j j 14

15 Gower s Coefficient - Worked Examle - Two sites, eight quantitative descritors of the environment. Descritors j Object x Object x Sum w 12j = 7 R j y 1j -y 2j y 1j -y 2j /R j w 12j s 12j = 4.63 Thus, S 15 (x 1,x 2 ) = 4.63/7 = (NB: R j, range of values among all objects for each y j, was re-calculated.) Gower s Coefficient - Using R - Use a sreadsheet to create a comma-delimited data file (CSV) containing the 8 observations for each of 2 objects. In this case, I used CALC in oenoffice.org and saved the files as: gower.csv Gower s Coefficient - Using R (Gower Scrit File) - 15

16 Gower s Coefficient - Using R - So, R gives us 0.75, but hand calculation gives WHY? Modification Estabrook and Rogers (1966) made a modification (S 16 ) of the Gower coefficient. The general equation is the same as S 15, but differs in the comutation of the artial similarities s j. Here the artial similarity between two objects for a given descritor j is comuted using a monotonically decreasing function of artial similarity. The following function can be used for two numbers d and k: 2( k+ 1 d) s12 j = f ( d12 j, kj) = when d k 2k+ 2+ dk s = f ( d, k ) = 0 when d> k 12 j 12 j j 16

17 Modification For the two values d and k, d is the distance between the states of the two objects x 1 and x 2 for descritor j (i.e., the same value y 1j -y 2j as in Gower s coefficient) and k is a arameter determined a riori by the users for each descritor. Parameter k is equal to the largest difference d for which the artial similarity s 12j (for descritor j) is allowed to be different from 0. Values of k for the various descritors may be quite different from each other. For examle, for a descritor coded 1 to 4, one might use k = 1; for a descritor coded 1 to 50, erhas k = 10 could be used. Similarity Calculation S 16 -Modification Examle- Descritors j S 16 (x 1, x 2 ) Object x Object x k j s 12j = f (d 12j, k j ) = 2.9 / 6 = Possible values of k are used for examle. NB: if k = 0 for all descritors, S 16 is identical to the SM coefficient. Asymmetrical Quantitative Coefficients Just as we did in the revious section, we extend asymmetrical binary coefficients to accommodate multi-state descritors. For examle, Jaccard s coefficient becomes: agreements S7( x1, x2) = double-zeros Where the numerator is the number of secies with the same abundance state at the two sites. This coefficient is useful when secies abundances are coded to a small number of classes and you wish to strongly contrast the abundances. 17

18 Bray-Curtis Coefficient Some coefficients lessen the effect of the largest differences and may therefore be used with raw secies abundances. The best known coefficient here is one frequently referred to as the Bray-Curtis coefficient. The origin is a bit unclear as the coefficient has been discovered numerous times in the literature. This coefficient comares two sites (x 1, x 2 ) in terms of the minimum abundance of each secies: S17 W 2W ( x1, x2 ) = = ( A+ B)/2 ( A+ B) Bray-Curtis Coefficient In this formula, W is the sum of the minimum abundances of the various secies; A & B are the sums of the abundances of all secies at each of the two sites. For examle: Secies Abundances Site x A B W Site x Minimum S17 2W 2(11) ( x1, x2 ) ( A+ B) Bray-Curtis Coefficient Obviously, Excel can be used to calculate all of these similarity measures (many are not available in re-ackaged statistical software as). 18

19 Building Excel Macros For real data sets, it is usually much easier to write a macro Building Excel Macros Building Excel Macros May need to edit VB macro 19

20 Kulczynski Coefficient Kulczynski s coefficient (1928) also belongs to this grou of measures that are suited to raw abundance data. The sum of the minima is first comared to the grand total at each site; then the two values are averaged: S18 1 W W ( x1, x2 ) = + 2 A B For binary data, S 18 becomes S 13. For the numerical examle just comleted (BC): S18 1 W W ( x1, x2 ) = + = + = A B Normalized Data Often, secies distributed across an ecological gradient are strongly skewed. Several coefficients have been adated to handle normalized abundance data. Data can be normalized through transformation or scaling (e.g., 0-7, rare-abundant). For examle, Gower s coefficient (S 15 ) can be easily adated: Modified Gower s Coefficient S ( x, x ) = j = 1 w j = 1 12 j 12 w 12 s j j, where s 12j = 1 - [ y 1j - y 2j /R j ], as in S 15, and w 12j = 0 when y 1j or y 2j = absence of information, or when y 1j or y 2j = absence of secies (y 1j or y 2j = 0), while w 12j = 1 in all other cases. 20

21 χ2 Similarity The last quantitative coefficient that excludes doublezeros is called χ2 Similarity. It is the comlement of the chi-square metric (D 15 ) discussed subsequently (and we will defer a fuller discussion of how to aly this measure until then). S ( x, x ) = 1 D ( x, x ) Ecological Resemblance - Outline - Ecological Resemblance Mode of analysis Analytical saces Association Coefficients Q-mode similarity coefficients Symmetrical binary coefficients Asymmetrical binary coefficients Symmetrical quantitative coefficients Asymmetrical quantitative coefficients Probabilistic coefficients Q-mode distance coefficients Metric distance Semimetrics R-mode coefficients of deendence Non-abundance measures Secies abundance measures Choice of a coefficient Probabilistic Coefficients Probabilistic coefficients form a secial category. These coefficients are based on statistical estimation of the significance of the relationshi between objects. One of the better known coefficients in this category is Goodall s robabilistic coefficient. This coefficient takes into account the frequency distribution of the various states of each descritor in the whole set of objects. This coefficient was originally develoed for lant taxonomy, has been used in aleontology, and has good alication in ecology. 21

22 Goodall s Probabilistic Coefficient This coefficient is nice because like Gower s coefficient, it ermits the use of binary and quantitative descritors together. Goodall (1966) first conceived of this index and it was later imroved by Orlóci (1978). NB: the use of this coefficient is limited to cluster analysis only! There are five comutational stes which we will review first, then do an examle. Goodall s Coefficient - Ste 1 - A artial similarity coefficient s j is first calculated for all airs of sites and for each secies j. With n sites, there are n(n-1)/2 calculations. If the abundances have been normalized, choose the s 12j function of S 19. Double-zeros MUST be excluded. This is accomlished by multilying the artial similarities s j by the Kronecker delta w 12j, whose value is 0 uon occurrence of double-zero. For raw abundance data, S 17 may be used, comuted for a single secies at a time. The result is a artial similarity matrix containing as many rows as there is secies, and n(n-1)/2 columns. Goodall s Coefficient - Ste 2 - In a second matrix of the same size, for each secies j and each of the n(n-1)/2 airs of sites, one comutes the roortion of artial similarity values belonging to secies j that are larger than or equal to the artial similarity of the air of sites being considered; the s j value under consideration is itself included in the calculation of the roortion. The larger the roortion, the less similar are the two sites with regards to the given secies. 22

23 Goodall s Coefficient - Ste 3 - The roortions (robabilities) of ste 2 are combined into a site site similarity matrix, using Fisher s method, i.e., by comuting the roduct Π of the robabilities relative to the various secies. Since none of the robabilities can be zero (from revious ste) there is no roblem in finding roduct. But, there is an assumtion that secies are non-correlated. If they are not, the rocedure requires that you use rincial comonent scores instead of secies abundances. Goodall s Coefficient - Ste 4a - There are two ways to define Goodall s similarity index. In the first aroach (4a), the roducts Π are ut in increasing order. Following this the similarity between the sites is calculated as the roortion of the roducts that are larger than or equal to the roduct of the air of sites considered: d d= 1if Π Π airs ( x1, x2 ) = where nn ( 1)/2 d= 0 if Π < Π12 S Goodall s Coefficient - Ste 4b - In the second aroach (4b), the χ 2 value corresonding to each roduct is comuted under the hyothesis that the robabilities of the different secies are indeendent vectors: 2 χ = 2lnΠ which has 2 degrees of freedom ( = no. of secies). the similarity index is the comlement of the robability associated with this χ 2 : S ( x, x ) = 1 rob ( χ )

24 Goodall s Coefficient - Examle - 5 onds, 8 hytolankton secies, rel. abundance 0-5 Ponds S Range Goodall s Index - Examle: Ste 1 - Gower s matrix of artial similarities Pairs of Ponds S Goodall s Index - Examle: Ste 2, Part a - Determine the roortion of artial similarity in each row that are of the air of sites being considered S For examle, for the ond air (214,233), the s-3 has a S of In the third row, there are 3 values out of ten (including the value itself) that are Thus, the associated ratio for the new table is

25 Goodall s Index - Examle: Ste 2, Part b - Build a new matrix based uon the roortions of artial similarity ratios determined in Part-a S Σ Goodall s Index - Examle: Ste 3 - Assemble a symmetrical site site hemi-matrix (roducts of the terms in each column from revious matrix) Ponds Ponds e.g., = (1) (0.1) (1) (0.3) (1) (1) (0.4) (1) Goodall s Index - Examle: Ste 4a - Construct site site similarity hemi-matrix (based on the roortions of the roducts that are larger the roduct corresonding to each air of sites) Ponds Ponds e.g., Product of (212, 431) is 0.28; 3 of 10 values are >, hence the similarity S 23 (212, 431) =

26 Goodall s Index - Examle: Ste 4b - If the chosen similarity measure is the comlement of the robability assoc. with χ 2 (alternative aroach): Ponds Ponds e.g., For (212, 431), χ 2 = -2ln(0.28) = , df = 2 = 16 with a corresonding P = ; S 23 (212, 431) = = Ecological Resemblance Mode of analysis Analytical saces Association Coefficients Q-mode similarity coefficients Symmetrical binary coefficients Asymmetrical binary coefficients Symmetrical quantitative coefficients Asymmetrical quantitative coefficients Probabilistic coefficients Q-mode distance coefficients Metric distance Semimetrics R-mode coefficients of deendence Non-abundance measures Secies abundance measures Choice of a coefficient Q-mode Distance Coefficients Distance coefficients are functions that take their maximum values (usually 1) for two objects that are comletely different, and 0 for objects that are identical over all descritors. Note that all of the similarity coefficients that we just reviewed can be transformed in to distances, usually as the comlement; i.e., D = (1 - S). Some simle transforms include, D = (1 - S). Distances, like similarities, are used to measure the association between objects. Distance coefficients can be divided in to 3 grous: 1) metrics 2) semimetrics 3) nonmetrics 26

27 Metric Distance Coefficients Metric distance coefficients share the following roerties: 1) minimum 0: if a = b, then D(a, b) = 0; 2) ositiveness: if a b, then D(a, b) > 0; 3) symmetry: D(a, b) = D(b, a) 4) triangle inequality: D(a, b) + D(b, c) D(a, c) Semimetric & Nonmetric Distance Coefficients Semimetric: These measures do not follow the triangle inequality axiom. These measures cannot directly be used to order oints in a metric or Euclidean sace because, for three oints (a, b, and c), the sum of the distances from a to b and from b to c may be smaller than the distance from a to c. Nonmetric: These coefficients can take negative values, thus violating the roerty of ositiveness of metrics. Metric Distances The most common metric measure is the Euclidean distance. It is comuted using Pythagora s formula, from site-oints ositioned in a -dimensional sace called a metric or Euclidean sace: 2 D1( x1, x2) = ( y1j y2j) When there are only two descritors, this exression becomes the measure of a right-angled triangle: j = 1 D ( x, x ) = ( y y ) + ( y y )

28 Euclidean Distance The square of D 1 may also be used for clustering uroses. One should notice though that D 1 2 is a semimetric, which makes it less aroriate than D 1 for ordination. 2 D ( x, x ) = ( y y ) j 2 j = 1 j Note that ED does not have an uer limit, its value increases indefinitely with the number of descritors. The value also deends uon the scale of the descritors. Standardization may be used to reduce scale effects (instead of using raw data). Euclidean Distance The Euclidean distance, used as a measure of resemblance among sites on the basis of secies abundances, may lead to the following aradox: two sites without any secies in common may be at a smaller distance than another air of sites sharing secies. Orloci (1978) rovides an examle: Sites Secies Sites Sites y 1 y 2 y 3 x 1 x 2 x 3 x x x x x x Euclidean Distance From the revious examle, we see that the ED between x 1 and x 2, which have no secies in common, is smaller than x 1 and x 3 which share secies y 2 and y 3. In general, double-zeros lead to reduction in distances. This situation must be avoided (but occurs frequently with community data, less so with morhometric data). Most argue that ED should NOT be used with secies abundance data. The main difficulty in ecology is that a major method (PCA) orders objects in multidimensional sace using D 1. 28

29 Average Euclidean Distance Various modifications have been roosed to deal with the drawbacks of the Euclidean distance alied to secies abundances. The effect of the number of descritors may be temered by comuting an average distance: 1 D y y D D ( x1, x2) = ( 1j 2 j) or 2( x1, x2) = 2 j = 1 Chord Distance Another modification of ED was roosed by Orloci (1967) and named the chord distance, which has a maximum value of 2 for sites with no secies in common and 0 when two sites share the same roortions (without it being necessary for the same absolute abundances). This measure is the ED comuted after scaling the site factors to length 1 (vector normalization). The chord distance may also be calculated directly from nonnormalized data: This solves the y1j y2 j roblem of using j = 1 D3 ( x1, x2) = 2 1 = 2(1 cos θ ) s. abundance 2 2 y1j y data. 2 j j= 1 j= 1 Geodesic Metric The geodesic metric is a transformation of the chord rocedure. It measures the length of the arc at the surface of the hyershere of unit radius: D D4( x1, x2) = arccos 1 ( x, x ) In the numerical examle we did, airs of sites (x 1, x 2 ) and (x 2, x 3 ), with no secies in common are at an angle of 90º, whereas airs of sites (x 1, x 2 ), which share two of the three secies, are at a smaller angle (88º). 29

30 Mahalanobis Distance Mahalonobis (1936) develoed a generalized distance that takes in to account the correlations among descritors and is indeendent of the scales of the various descritors. This measure comutes the distance between two oints in a sace whose axes are not necessarily orthogonal. In ractice, the Mahalanobis generalized distance is only used for comaring grous of sites. Mahalanobis Distance For two grous of sites, w 1 and w 2, containing n 1 and n 2 sites, resectively, and described by the same variables, the square of the generalized distance is given by the following matrix formula: 2 1 D5 ( w1, w2) = d12 V d' Where, 12 d 12 is the vector (length = ) of the differences between the means of the variables in the two grous of sites. V is the ooled within-grou disersion matrix of the two grous of sites estimated from the matrices of SS&CP between grou-centered descritors for each of the two grous, added u term by term (as in MANOVA). In other words, Mahalanobis Distance 1 V = [( 1 1) 1 ( 2 1) 2] n n 2 n S + + n S 1 2 Where S 1 and S 2 are the disersion matrices for each of the two grous. Whereas d-bar measures the difference between the - dimensional means of the two grous ( descritors), V takes into account the covariance among descritors. 30

31 Mahalanobis Distance The nice feature of this latter method is that the result can be tested for significance. One must first meet the assumtion of matrix homogeneity by alying Kullback s test: g ( n 1) 2 j V χ = ln 2 V j = 1 j with df = (g-1)m(m+1)/2, n j the number of objects in the grou, and V is the determinant of the ooled within-grou disersion matrix of grou j. Mahalanobis Distance - Testing for Significance - To erform the test of significance, the generalized distance is transformed into Hotelling s T 2 (1931) statistic: T nn = D5 ( n1+ n2) Then comute the aroriate F-statistic as: n + n ( + 1) F = T ( n + n 2) With df =, [n 1 + n 2 - ( + 1)] Manhattan Metric D ( x, x ) = y y j 2 j j = 1 The Manhattan metric, city-block metric, or taxicab metric all refer to the same distance measure. It refers to the fact, that for two descritors, the distance between two sites is the distance on the abcissa lus the distance on the ordinate (much like the orthogonal distances traveled by taxicabs in NYC). This metric resents the same roblems with double zeros as in ED and leads to the same aradox. 31

32 Mean Character Difference The mean character difference was originally roosed by Czekanowski, an anthroologist, in 1909: 1 D ( x, x ) = y y j 2 j j = 1 It has the advantage over D 7 of not increasing with the number of descritors (). It may be used for secies abundance analysis if you exclude double zeros from the absolute value of the differences in y by relacing with ( - no. double-zeros). Whittaker s Index of Association WIA is well adated to secies abundance data, because each secies is first transformed into a fraction of the total number of individuals at the site, before the subtraction: 1 y D9( x1, x2) = 2 y 1j 2 j j = 1 y1j y2 j j= 1 j= 1 The difference is zero for a secies when its roortions are identical in the two sites. Alternative to Whittaker s Index An identical result to the WIA is obtained by comuting, over all secies, the sum of the smallest fractions calculated for the two sites: y j D9( x1, x2) = 1 min j = 1 y 32

33 Alternative to Manhattan Metric Likewise, Australians Lance & Williams (1967) rovide the Canberra metric as an alternative to the Manhattan metric: y1j y 2 j D10 ( x1, x2 ) = j = 1 ( y1j + y2 j) A scaled version of D 10 was devised by Clark (1952): 1 y1j y 2 j D11 ( x1, x2 ) = j = 1 y1j + y 2 j 2 Alternative to Manhattan Metric Another index with some good roerties, which is related to D 11, was develoed by an anthroologist under the name Coefficient of Racial Likeness. Using this coefficient, it is ossible to measure a distance between grous of sites, like with the Mahalanobis distance (D 5 ), but without eliminating the effect of correlations among descritors: D 1 ( y y ) 2 = ( / ) ( / ) 2 1j 2 j 12 ( w1, w2 ) 2 2 j = 1 s1j n1 + s2 j n2 where w 1 & w 2 contain n 1 & n 2, y-bar is mean of descritor j in grou i, s ij is the variance. Chi-Square Metrics The last grou of common metrics are the χ 2 distance measures. The most general form is known as the χ 2 metric. In order to calculate the χ 2 metric, the data matrix must first be transformed into a matrix of conditional robabilities. The elements of the matrix become the new terms y ij /y i+ where y i+ is the sum of the frequencies in row i. An examle may be the easiest way to understand... 33

34 Chi-Square Metric [ ] [ y ] i Y = [ y ] j = 10/ yij / yi = Chi-Square Metric The distance between the first two rows of the right-hand matrix could be comuted using the formula for Euclidean distance (D 1 ), but, the most abundant secies would contribute redominantly to the sum of squares. Instead, the χ 2 metric is comuted using a weighted exression: 1 y1j y2 j D15 ( x1, x2 ) = j = 1 y+ j y1 + y2+ Where y +j is the sum of the frequencies of the column j. While this measure has no uer limit, most values < 1. 2 Chi-Square Metric For the numerical examle, comutation of D 15 between the first two sites yields: D 15 (x 1, x 2 ) = ( ) ( ) ( ) (0 0) ( ) = NB: The 4th secies, which is absent from the first two sites, cancels itself out; thus how χ 2 metric deals with double-zeros. 1/2 34

35 Chi-Square Distance The χ 2 distance (D 16 ) is related to the χ 2 metric (D 15 ). It differs from the metric in that the terms of the sum of squares are divided by the robability of each row in the table instead of it absolute frequency. Thus, 1 y1j y2 j D16 ( x1, x2 ) = y++ j = 1 y+ j y1 + y2+ The χ 2 distance is the distance reserved in corresondence analysis (CA), when comuting similarity between sites (as we ll see later). 2 Hellinger Distance The last distance measure in this category is the Hellinger distance. This is often recommended rior to a rincial coordinates analysis (PCO): y D17 ( x1, x2 ) = y y 1j 2 j y j = Q-mode Distance Coefficients: Semimetrics Some distance measures do not follow the fourth roerty of metrics, i.e., the triangle inequality axiom. As a consequence, they do not ermit a roer ordination of oints in Euclidean sace. They may, however, be used for ordination by PCO after correction for negative eigenvalues. One of the first semimetrics was derived from the Sørenson coefficient (S 8 ) which was used to form the nonmetric coefficient: 2a b+ c D13( x1, x2 ) = 1 = 2 a+ b+ c 2 a+ b+ c 35

36 Percentage Difference Among the measures for secies abundance data, the coefficients of Steinhaus (S 17 ) and Kulczynski (S 18 ) are semimetrics when transformed in to distances. In articular, D 14 =1-D 17 was first described by Odum (1950) and later by Bray and Curtis (1957) who called it the ercentage difference: y1j y2 j j = 1 2W D14 ( x1, x2 ) = = 1 A B ( y1j y2 j) + + j = 1 Contrary to the Canberra metric (D 10 ), differences between abundant secies contribute the same as rare secies. This is often a desirable roerty, articularly when using normalized data. Ecological Resemblance Mode of analysis Analytical saces Association Coefficients Q-mode similarity coefficients Symmetrical binary coefficients Asymmetrical binary coefficients Symmetrical quantitative coefficients Asymmetrical quantitative coefficients Probabilistic coefficients Q-mode distance coefficients Metric distance Semimetrics R-mode coefficients of deendence Non-abundance measures Secies abundance measures Choice of a coefficient R-mode: Coefficients of Deendence The main urose of R-mode analysis is to investigate the relationshis among descritors, and are sometimes used in PCA or DA to order objects. Most deendence coefficients are amenable to statistical testing. For such coefficients, it is thus ossible to associate a matrix or robabilities with the R-matrix, if required by subsequent analyses. If you do statistical testing, the data must follow all of the regular assumtions for the data (e.g., normality, etc.). 36

37 Descritors Other Than Secies Abundances The resemblance between quantitative descritors can be comuted using arametric measures of deendence; i.e., measures based on arameters of the frequency distributions of descritors. These measures are the covariance and the Pearson correlation coefficient. They can ONLY be adated to descritors whose relationshis are linear. Covariance Recall that the covariance between descritors j and k is comuted from centered variables ( y y ) and ( y y ) ij ij ik k The range of values of the covariance has no a riori uer or lower limits. The variances and the covariances among a grou of descritors form their disersion matrix S: 1 S = ' n 1 y y y y Recall: multily matrix of centered data w/its transose. Correlation Pearson s correlation coefficient r jk is their covariance of descritors j and k comuted from standardized variables. The coefficients of correlations among a grou of descritors form the correlation matrix R. Correlation coefficients range in value from -1 to +1. The significance of individual coefficients (i.e., H o : r = 0) can be statistically tested. 37

38 Correlation: R vs. Q Some authors have used Pearson s r for Q-mode analyses after transosing the rimary matrix. There are, however, a number of objections to doing this, some of which include: (1) r is dimensionless and may be hard to interret (2) In R-mode, the value of r remains unchanged after rescaling, but may change dramatically in Q-mode UPSHOT: measures that are designed for one mode of analysis should not be analyzed in the other mode! Nonarametric Correlation The resemblance between semi-quantitative descritors, and more generally between any air of ordered descritors whose relationshi is monotonic may be determined using nonarametric measures of deendence. Searman s r (continuous or ordinal variables) and Kendall s τ (ordinal variables) are aroriate to use under these circumstances, and like Pearson s r, can be subjected to statistical testing. Secies Abundances: Biological Associations Analyzing secies abundance descritors causes the same roblem in the R as in the Q mode: what to do with doublezeros? This roblem surfaces regularly in community data because biological assemblages usually contain a small number of dominant secies and a large number of rare secies. The literature is relete with incorrect aroaches to this roblem. Double-zeros need to be neutralized or not included in the analysis. 38

39 Aroaches to Minimizing the Double-Zero Problem 1) Eliminate less frequent secies from the rimary data matrix. They will be of little use in assessing ecological secies associations. 2) Eliminate all zeros from the comarisons by declaring that zeros are missing values. 3) Eliminate double-zeros only from the comutation of the correlation or covariance matrix (this must generally be rogrammed searately). The resulting disersion matrix can then be directly analyzed (e.g., PCA). Aroaches to Minimizing the Double-Zero Problem Note that this is not a full list of otions. For examle, Corresondence Analysis (CA) is a secial form of PCA which reserves the χ 2 distance (D 16 ) instead of the Euclidean distance (D 1 ). Because D 16 excludes double-zeros, whereas D 1 includes them, CA is usually better adated to the study of secies associations than is PCA. Other Aroaches Biological associations may also be defined on the basis of co-occurrence of secies instead of the relationshis between fluctuations in abundances. In fact, quantitative data may not accurately reflect the roortions of the various secies in the environment (usually because of samling or identification roblems). There are many aroaches to this in the literature, but by far, the most common is the 2 2 frequency table. 39

40 2 2 Frequency Table Secies y 2 Secies y 1 Presence Absence Presence a b a+b Absence c d c+d a+c b+d n=a+b+c+d Where a and d are numbers of sites in which the two secies are resent and absent, resectively; whereas b and c are the numbers of sites in which only one of the two secies is resent; n is the total number of sites. Binary Coefficients As already discussed, many binary coefficients exclude doublezeros. Jaccard s coefficient of community (S 7 ) has been oular a S7 ( x1, x2) = a+ b+ c along with its corresonding distance measure: b+ c D = 1 S7( y1, y2) = a+ b+ c Binary Coefficients Dice s coincidence index (S 8 ) (a.k.a. Sørenson s coefficient) was in fact originally designed to secifically study secies associations: 2a S8( x1, x2) = 2a+ b+ c A more elaborate coefficient was roosed by Fager & McGowan (1963) to make minor corrections to S 14 (es. for small samle sizes): a 1 S24 ( y1, y2 ) = ( a+ b)( a+ c) 2 a+ c 40

41 Choice of a Coefficient Given that multivariate statistics is exloratory in nature, there are not the same hard and fast rules as one might often see in inferential statistics. There are, however, imortant guidelines. We have seen how the choice of coefficient can have a major influence on the outcome and interretation of resemblance. Thus, considerable care should be exercised in choosing a resemblance coefficient. Please refer to class handout for selection criteria. R suorts a variety of coefficients used in different libraries. The R STATS library contains the dist () function which contains euclidean, manahattan, binary, and Canberra. The vegan library suorts the vegdist () function and LabDSV rovides the dsvdis() function, both of which suort many more coefficients. In all cases, the first argument to the function refers to the rimary matrix and the second argument is the index. For examle: dis.bray <- vegdist(veg, method= bray ) Which creates a dissimilarity hemi-matrix called dis.bray using the vegdist rocedure and bray-curtis coefficient. OCCAS Analysis One way of heling to assess resemblance coefficients is to construct artificial data reresenting contrasting situations that a S or D value should be able to discriminate. OCCAS (ordered comarison case series; Hajdu 1981) involves constructing just such a series, corresonding to linear changes in the abundance of two secies along a simulated gradient. The method is straightforward and easy to aly. In order for a coefficient to erform well, it MUST rovide a linear result. Gower and Legendre (1986) used this aroach to evaluate 15 binary coefficients and 10 coefficients for quantitative data. 41

42 Consider two secies. OCCAS Analysis -Examle- Site 1 has frequencies were y 11 = 100 and y 12 = 0; Site 2 has frequencies y 21 = 50 and then y 22 was varied from 10 to 120, in stes of 10. The results for three coefficients are: Distance D 17 D 14 (1-S 15 ) Abundance of secies y 22 ANOSIM Analysis of similarities (ANOSIM) rovides a way to directly test whether there is a significant difference between two or more grous of samling units. The function ANOSIM oerates directly on a dissimilarity matrix roduced by VEGDIST. The test statistic is R. The statistical significance of R is assessed by ermuting the grouing vector to obtain the emirical distribution of R under a null-model. ANOSIM - Examle - Create two data sets that share a common dimension. One data set contains the secies/stand info, the other is an indexing set. 42

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