A protocol to compare nestedness among submatrices
Giovanni Strona, Fabrizio Stefani, Paolo Galli & Simone Fattorini
Population Ecology ISSN 1438-3896 Volume 55 Number 1 Popul Ecol (2013) 55:227-239 DOI 10.1007/s10144-012-0343-4
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Author's personal copy Popul Ecol (2013) 55:227–239 DOI 10.1007/s10144-012-0343-4
ORIGINAL ARTICLE
A protocol to compare nestedness among submatrices Giovanni Strona • Fabrizio Stefani Paolo Galli • Simone Fattorini
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Received: 30 March 2012 / Accepted: 10 September 2012 / Published online: 28 September 2012 Ó The Society of Population Ecology and Springer 2012
Abstract Searching for nestedness has become a popular exercise in community ecology. Significance of a nestedness index is usually evaluated using z values, and finding that a matrix is nested is typically a common result. However, nestedness is not likely to be spread uniformly within a matrix of species presence/absence per site. Selected parts of the matrix may show a degree of nestedness significantly higher (or lower) than expected from the overall pattern. Here we describe a procedure to assess if a particular submatrix (i.e., a peculiar combination of rows and columns extracted from the complete matrix) is more or less nested than expected for an assortment of sites and species taken at random from the same overall matrix. The idea is to obtain several submatrices of different sizes from the same overall matrix and to calculate their z values. A regression is then performed between z values of submatrices and their sizes. A nestedness index independent of matrix size is suggested as the deviation of the z value of a particular submatrix from that expected according to the regression line. We applied our protocol to 55 matrices with different nestedness indices under various null-models and, for purpose of demonstration, we discussed in detail a single case study regarding various animal groups of the Aegean Islands (Greece). The obtained results strongly
G. Strona (&) F. Stefani P. Galli S. Fattorini Department of Biotechnology and Biosciences, Water Ecology Team (WET), University of Milano Bicocca, Piazza della Scienza 2, 20126 Milan, Italy e-mail:
[email protected] S. Fattorini Azorean Biodiversity Group (CITA-A) and Platform for Enhancing Ecological Research and Sustainability (PEERS), Departamento de Cieˆncias Agra´rias, Angra do Heroı´smo, Universidade dos Ac¸ores, Terceira, Ac¸ores, Portugal
encourage further research to focus not only on the question whether a matrix is nested or not, but also on where and why nestedness is confined. Keywords Aegean Islands Biogeography Mediterranean Null model Paleogeography
Introduction Nestedness analysis is a technique aimed at investigating species distribution patterns. In a perfectly nested distribution, species occurring at a particular site are always present in more species-rich sites, whereas species absent from this particular site never occur in less species-rich sites (Patterson and Atmar 1986). In the last 20 years, searching for nestedness has become a popular exercise in disparate fields, including community ecology, ecological networks, biogeography, conservation biology, and parasitology (Kadmon 1995; Worthen 1996; Hadly and Maurer 2001; Fleishman et al. 2002; Azeria et al. 2006; Ulrich et al. 2009; Joppa et al. 2010). A common practice to assess if a matrix of species’ presence/absence per site is significantly nested is to refer to tail distribution of z scores of nestedness indexes (Ulrich et al. 2009). For this purpose, a distribution of values is derived from randomly generated matrices, and the mean and standard deviation of that distribution is used to transform raw values into z scores. Values of z scores \-2 typically indicate significant nestedness at P = 0.05 for all nestedness indices but NODF, for which z scores [2 indicate nestedness (Ulrich et al. 2009). To compare the nestedness of different matrices, some researchers use the absolute values of the nestedness index. For example, for the matrix temperature measure of
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nestedness (Atmar and Patterson 1993), a matrix with lower temperature is considered more nested than one with higher temperature (e.g., Sfenthourakis et al. 1999; Greve and Chown 2006; Fattorini 2007a; Simaiakis and Martı´nezMorales 2010). This approach, however, has been questioned, because nestedness indices, and in particular matrix temperature, are influenced by matrix size and fill (percentage of occurrences) (Greve and Chown 2006; Fattorini 2007a; Ulrich and Gotelli 2007). This does not apply to the NC index proposed by Wright and Reeves (1992) which, differently from other nestedness metrics, is not affected by matrix size. Nevertheless, NC quantifies more a pattern of matrix-wide species aggregation than a degree of matrix nestedness (Ulrich et al. 2009). Moreover, nestedness indices do not provide information about significance of nestedness, for which z scores are needed. However, z scores of different matrices are difficult to compare (Ulrich et al. 2009). This issue is particularly problematic when the original matrix is divided into submatrices. When analysing a nested matrix, researchers are often interested in assessing whether some sectors of the matrix are more nested than others (Fattorini 2007a; Simaiakis and Martı´nez-Morales 2010; Strona et al. 2011). Every matrix containing a certain number of sites in a given geographical region is a submatrix of a larger number of sites in a wider geographical area, thus it is important that ‘matrix’ definition and subdivision is guided by biological considerations. For example, in biogeographical analyses, one can ask if a certain group of sites in the whole matrix is more or less nested than other groups of sites within the same geographical area as a possible result of different geological histories of the two site groups (Worthen 1996; Fattorini 2007a; Simaiakis and Martı´nez-Morales 2010). In spite of the interest in this topic, a method does not exist to powerfully address the problem of comparing nestedness of submatrices. Recently Saavedra et al. (2011) introduced a protocol to measure the individual contribution of each species or node of ecological networks to their nestedness, based on the randomization of single rows and columns. However, this method makes it possible to consider only one species at a time, without taking into account the possible synergistic effect of species (and site) associations. Here we provide an alternative way to address this problem, based on the construction of a regression line of z scores. This curve (which will be specific to each matrix) is constructed to model how z scores vary in response to matrix size using randomly extracted submatrices of different sizes. Thus, the curve represents the expected dependence of z scores on matrix size and its confidence intervals can be used to assess if the z score of a particular submatrix (i.e., a part of the original matrix defined by some ecological or geographical characteristics) deviates significantly from that
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predicted by the curve for a submatrix of the same size extracted at random from the complete matrix. In practice, our approach is a sort of variant of well-known resampling procedures in combination with similar use of confidence limits in species-area curves (see Fattorini 2006a). To compare matrices of different sizes Bascompte et al. (2003) used a simple method based on the calculation of a relative nestedness. Our method is conceptually different from that of Bascompte et al. (2003) because it is not addressed to compare nestedness of different matrices, but to assess if the z score of a particular submatrix is statistically different from what can be obtained by a random subdivision of the bigger matrix, thus reflecting ecological and/or biological processes not explainable by chance alone. To assess the applicability of our statistical protocol under a variety of circumstances and matrix characteristics, we used a large set of matrices taken from Atmar and Patterson (1995). We also discussed in detail a single case study of high biogeographical interest, namely the distribution of several animal groups on the Aegean Islands (Greece), to show the relevance of our approach in biogeography. With this case study, we show how our new methodology can be used to tackle two issues and the complicated interaction between them. The first issue is this: can we use nestedness analyses as a ‘‘pattern-searching device’’? Worthen (1996) states that nestedness analyses might be used as a pattern seeking, exploratory device for suggesting mechanisms potentially structuring a community. Rather than knowing a community’s paleogeographical history or interaction history and predicting that nestedness should exist, with our approach we implicate a history by searching for and finding a nestedness pattern. The second issue addressed in our study is this: how can we compare nestedness values for matrices that differ in size? Our method allows this comparison using deviation of z values from those expected by random extraction of submatrices of varying size. Finally, with a detailed analysis of the Aegean biotas, our method addresses the interaction between these issues: with the statistical protocol proposed here, we can determine which sets of sites and/or species are most likely to have interacted in the past by finding which submatrices have the most significant nestedness patterns.
Materials and methods Explanation of the procedure Our method has five steps: (1) for a given matrix we calculate the observed nestedness, the nestedness of each
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simulated matrix, their mean value, and the z scores; (2) we subdivide the initial matrix into submatrices of different random sizes, and for each of these random submatrices we calculated the nestedness value and z score as for the original matrix; (3) we regress these submatrix z scores against respective matrix size; (4) we used this regression line to calculate the expected z score for a submatrix of any size; (5) the observed z score for a particular submatrix extracted from the original matrix according to a biological criterion is finally compared to the z score predicted by the regression line for a random submatrix of the same size. Details about each step are provided below. To make our reasoning less abstract, we will refer to the NODF index as a index of nestedness. NODF (nestedness measure based on overlap and decreasing fill—Almeida-Neto et al. 2008) is the percentage of occurrences in right columns and species in inferior rows which overlap, respectively, with those found in left columns and upper rows with higher marginal totals for all pairs of columns and of rows. This nestedness index is recommended by Ulrich et al. (2009), but our basic reasoning can be applied to any other nestedness index as well. Figure 1 offers a diagrammatic representation of the procedure for a matrix from which six submatrices of various sizes are extracted at random, a regression line of their z scores is calculated, and the distance of the z score of the submatrix of interest is evaluated. The protocol to construct regression lines of z scores against matrix size of randomly extracted submatrices was applied to 55 matrices taken from Atmar and Patterson (1995). We used this large set of matrices also to assess if there is a relationship between the goodness-of-fit of the regression lines obtained with our protocol (expressed by regression R2 values) and the degree of matrix nestedness (expressed by z scores). For this purpose, we correlated the R2 values of our regression lines with the z scores of their respective matrices. Details of the procedure Computing nestedness of the original matrix A z score is defined as: z ¼ ðx lÞ=SD
ð1Þ
where x is the nestedness value observed for the examined matrix, l is the mean of the nestedness values calculated for the simulated matrices, and SD is the respective standard deviation. With reference to the NODF index, we will distinguish the nestedness observed for a given matrix (NODFo), corresponding to the x of Eq. 1, the nestedness of each simulated matrix (NODFe) (i.e., the simulated value of a single
randomly generated matrix), and their mean value NODFm (corresponding to l in the general formula of Eq. 1). Computing nestedness of random submatrices We divide a given initial matrix into several submatrices of different sizes. These submatrices are constructed by extracting at random a given number of rows and columns from the original, complete matrix. According to the definition given by Almeida-Neto et al. (2008), we use the word ‘size’ to indicate the number of cells (i.e., the product of number of sites 9 number of species) of a matrix, without any implication about matrix shape. As a possible alternative measure of matrix size we also used the number of occurrences (number of occupied cells): results with this measure are not presented because they were very similar to those obtained using the number of cells. Although our method can virtually be applied by randomly extracting only rows or columns, these alternative ways of submatrix construction are generally not applicable. In fact, random or systematic extraction of rows would likely produce empty columns, and random or systematic extraction of columns would likely produce empty rows. Note that with our procedure we do not extract cells at random, but entire rows (species) and columns (sites). Thus, nestedness analyses will consider all the occurrences of the selected species in the selected sites (a pattern that would be disrupted by extracting cells at random). For example, for an original matrix of 80 species 9 40 sites (size 3,200), we can construct submatrices of decreasing size, e.g., 400, 150, 80, 25, by sampling, at random 20, 10, 8, and 5 columns (sites) and 20, 15, 10 and 5 rows (species) respectively. Because columns and rows are taken at random, it is possible to obtain several submatrices of the same size, e.g., 400, by changing which columns and rows are extracted from the initial matrix. For simplicity, we assume here that we have constructed, by random sampling columns and rows from the original matrix, only one matrix of size 400, one matrix of size 150, one matrix of size 80, and one matrix of size 25. However, in the actual use of our protocol, the size of each constructed submatrix is random, which ensures that we will have several submatrices for each possible dimensional class of the submatrices (provided that the number of constructed submatrices is not too small). Let us refer to a certain submatrix, e.g., the matrix of 40 9 10 size. Now, we calculate the NODF value (or any other nestedness index) for this 40 9 10 submatrix. Thus, we have the observed NODF values (NODFo) for this submatrix. The same process is replicated for all the extracted submatrices. Thus, we have four values of NODFo. Note that these values of nestedness come from a process of random
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Fig. 1 Diagrammatic representation of the procedure used to compare z values of submatrices. Six submatrices of various sizes are extracted at random from the original, entire matrix; a regression line of their z scores is calculated; z score of a submatrix of interest (extracted from the complete matrix according to certain criteria) is lastly compared to the z score predicted by the regression line for a random submatrix of the same size
extraction (the random sampling of columns/rows from the original 40 9 20 matrix), but they should not be confounded with the values of NODFe, which will be obtained when each of these submatrices will be randomized according to a given null-model to calculate the expected NODFm for each submatrix. For each of the submatrices that we have constructed, we can calculate a z score as in the formula of Eq. 1, where x is the nestedness observed for the i-th matrix obtained from the original matrix (NODFoi) (i.e., the
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nestedness of the 40 9 10 submatrix in the theoretical example), and l is the mean of the NODF values obtained from the matrices constructed by randomizing that particular matrix (NODFmi) (i.e., the mean of the values of NODFe calculated from 100 matrixes constructed by randomizing the 40 9 10 submatrix extracted from the initial matrix), and SD is the respective standard deviation. Thus, we have as many z scores as the submatrices extracted from the original matrix. In our example, we have four z scores.
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Computing the regression line Now we can regress these z scores (zi values) against respective matrix ‘size’ (number of columns 9 number of rows). The estimated regression line expresses the increase in the magnitude of z (i.e., its decreasing value) with the size of the submatrices obtained by randomly sampling different numbers of columns and rows. Using the regression line to calculate expected z scores The regression line can be now used to calculate the expected z score (ze) for a submatrix of any size under the assumption that rows and columns have been extracted at random from the initial matrix. Comparing z scores of particular submatrices to the expected ones Finally, the observed z score (zs) for a certain submatrix of a given size extracted from the original matrix according to a certain criterion (e.g., ecological, biogeographical etc.) can be compared to the expected z score for a submatrix of that size constructed by randomly sampling the columns and rows of the original matrix, as given by the regression line (ze). For example, we can be interested in a particular submatrix of 10 sites containing 20 species, with a size of 20 9 10 = 200 cells. We can calculate the z score of this matrix (zs), and calculate how distant it is from a z score predicted by the regression line (ze) for a matrix of size 200 and comprised of a random assortment of species and sites from the larger matrix (the same holds if we use number of occurrences instead of number of cells). This way, we do not compare the size of z scores, which is influenced by matrix characteristics, but their residuals from predicted values, which are not influenced. Deviations between expected and observed z scores indicate that the observed sub-assemblage of sites has a nestedness that is significantly different from that expected for a random sub-assemblage of the same size. To quantify such difference, we can refer to 95 % confidence intervals of the regression line. Observed z scores outside the 95 % confidence intervals can be considered significantly larger or smaller than expected by chance (P = 0.05). Moreover, the distance of the observed z scores from the regression line (Dz = ze - zs, where zs is the observed z score for the submatrix of interest, and ze the expected z score predicted by the regression line) may be used as a measure to compare z scores, which are not comparable per se. Description of the case study To empirically evaluate the procedure described above, the complete protocol described in this paper was applied to
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various faunal groups of the Aegean Islands (Fig. 2). The faunas of the Aegean Islands have been biogeographically investigated in detail, and it has been found that these landbridge islands can be subdivided into two well differentiated districts (see Fattorini 2002, 2006b for a review): a first group of islands (the Cyclades), which were interconnected during the Pleistocene sea level declines, and which presumably acquired a typical fauna; and a second group of islands (the Anatolian islands), connected to the Anatolian mainland, which should have acquired an Asiatic fauna. Therefore, this system represents a case where a global matrix of species 9 sites should be divided, on a biogeographical basis, into two main submatrices (Cycladic islands vs. Anatolian islands). Although results varied among groups, previous nestedness analyses for the land snail, isopod, tenebrionid and chilopod faunas (nestedness in reptiles and butterflies has not been previously analysed) showed that, in spite of their generally larger areas, the Anatolian islands had typically a ‘lower’ nestedness (e.g., higher matrix temperature) compared to the Cyclades (Fattorini 2007a, b; Simaiakis and Martı´nez-Morales 2010). High nestedness is typical of relict faunas and floras (Wright et al. 1998; Millien-Parra and Jaeger 1999) and of systems thought to have undergone faunal relaxation with species having gone extinct in a non random way (Patterson and Atmar 1986; Wright et al. 1998). These explanations also stand for the Aegean Islands. The paleogeography of the Aegean area is very complex. Although various authors have expressed different opinions about the detailed paleogeographic relationships between islands (e.g., see Fattorini 2002, 2006b, 2007a, b; Simaiakis and Martı´nez-Morales 2010), there is some consensus about the general pattern. During Pleistocene sea level declines, the Cyclades were, in general, connected to each other forming a substantially single, large Cycladic landmass. This Cycladic landmass remained, during Pleistocene glacials, more or less separated from the Balkan mainland, although distance between the two landmasses was reduced. As a result of their inclusion into a single landmass, most of the Cycladic islands acquired a similar fauna composed of species coming from the Balkans. Thus, the Pleistocene Cycladic landmass probably had an inter-island homogeneous fauna, with the subsequent evolution of many Cycladic endemics, distinct from the Balkan fauna. When the islands were again definitively disconnected in the Holocene, this initially homogeneous fauna was disrupted by postglacial extinctions that were not balanced by new immigration. As a result of this fragmentation process from a common pool of Pleistocene species, these islands should show highly nested faunas. By contrast, the Anatolian islands did not form a distinct landmass during Pleistocene regressions
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Fig. 2 Map of the study area. Bold line indicates Pleistocene coast line profile at glacial maximum. The Cyclades were grouped into a single landmass, whereas most of the islands close to Turkey (Anatolian islands: east of the broken line) were connected to the Anatolian mainland
because they were, in general, directly connected to Anatolia. Thus, these islands could not evolve a distinct Pleistocene fauna but were simply a part of the Anatolian landmass. Moreover, due to their proximity to the Anatolian coasts, these islands could have received a balanced assortment of propagules also after Pleistocene glaciations, thus having a less distinct relict character, with very few endemics. Island endemics evolved by in situ speciation on groups of islands which formed Pleistocene isolated groups, and they tend to be distributed on islands which were part of the same Cycladic group. Thus, they contribute to determine nestedness between Cycladic islands. By contrast, the Anatolian islands, which did not undergo this kind of biogeographical history, should have a less pronounced nestedness (see Fattorini 2007a for details about the peleogeography and its implications for nestedness). Here we re-examine published data on the Aegean Islands to show how our method can be applied to study how nestedness varies between Cycladic and Anatolian islands and how variation among groups can be related to their dispersal ability. For this, we used the distribution of land snails, isopods, chilopods, tenebrionid beetles, butterflies, and reptiles. Data to compile presence/absence matrices (all of which are available upon request from the corresponding author) were taken from: Hausdorf and Hennig (2005) for isopods, land snails and reptiles; Simaiakis and Martı´nez-Morales (2010)
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for chilopods; Dennis et al. (2000) for butterflies; Fattorini (2002, 2007a), Fattorini and Fowles (2005) and Trichas et al. (2008) for tenebrionid beetles. For each animal group, the respective complete matrix was divided into two submatrices on the basis of the current geographical and paleogeographical settings of the archipelago (see ‘‘Introduction’’): the Cyclades and the Anatolian islands. For each group, nestedness of these two submatrices was then analyzed with reference to levels of endemism. We considered as Aegean endemics those species or subspecies which are restricted to one or more islands. We assumed that animals with lower mobility tend to be more influenced by historical (paleogeographical) events like geographical isolation, which in turn promotes speciation. Tenebrionids are wingless insects known to be very sedentary animals which can hardly cross large sea barriers. Their distribution on islands is strongly influenced by land-bridge connections and they show a high level of endemism (ca. 32 % of endemic taxa) (Fattorini 2002). Land snails, which are also very sedentary animals, have a high level of endemism too (20 %, Sfenthourakis et al. 1999). Terrestrial isopods are usually strictly associated to litter, but their level of endemism is not particularly high (16 %, Sfenthourakis et al. 1999). Although several chilopods may have poor dispersal ability they show a relatively low level of endemism on the Aegean Islands (14 %, Zapparoli 2002). Reptiles, being more vagile than isopods and chilopods, have an even lower endemism (8.6 %, Corti et al.
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1999). Finally, butterflies are mobile organisms, capable of crossing extensive sea barriers; their distribution is not influenced by paleogeographical events and they have a very low level of endemism on the Aegean Islands (5.2 %, Dennis et al. 2000). Thus, if nestedness is a result of historical process that influenced species distributions, we expect the following pattern of decreasing nestedness: tenebrionids [ land snails [ isopods [ chilopods [ reptiles [ butterflies. Implementation of the procedure For each matrix, we started with the entire matrix (in the case of the Aegean Islands, the matrices containing all sites and all species of each animal group for which data were available) and constructed 100 random submatrices of various sizes, taking at random different numbers of columns and rows (corresponding to islands and species in the matrices regarding the Aegean Islands). When each matrix was divided into 100 random submatrices of varying size, we used the following criteria: maximum size was the original matrix size, and the minimum size was 25 % of the cells of the original matrix. Submatrices of varying size were extracted from the original matrices using an R script which is available upon request to the corresponding author. For each of these submatrices extracted from the complete matrix at random, z scores were calculated using nine nestedness indices (NODF, MT, N0, N1, NC, UA, UP, UT, and BR, all reviewed by Ulrich and Gotelli 2007 and Ulrich et al. 2009) under four null-models (FF, EF, FE, and EE, also reviewed by Ulrich and Gotelli 2007) to investigate how our methods is robust under various indices and nullmodels. All nestedness analyses were performed using the software ‘‘NODF’’ (Almeida-Neto and Ulrich 2011). Because singletons are known to affect nestedness (Sfenthourakis et al. 1999; Simaiakis and Martı´nez-Morales 2010), they were alternatively included and excluded. z scores were then regressed against matrix size, and for each regression line an R2 value was then calculated. In our study based on Atmar and Patterson (1995) matrices we selected a sample of matrices from the original set imposing some restrictions to avoid too small matrices, because an excessively small matrix cannot be practically subdivided into smaller submatrices. For this, we used matrices of at least 7 rows and 7 columns, with at least 5 sites with more than one species. These restrictions did not prevent us from assembling a quite large dataset of 55 matrices accounting for a wide variability in matrix size (ranging from 77 to 5,456 cells). These matrices were then analysed with our protocol and the R2 values of the 55 regressions of each nestedness index, under a given nullmodel with included (or excluded) singletons, were averaged.
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In the case study of the Aegean islands, for each animal group, we calculated, with the same null-model, the z scores of the two submatrices defined on biogeographical basis (i.e., the Cycladic and the Anatolian submatrices), and assessed whether these z scores lie within or outside the confidence intervals of the regression line. Finally, we calculated the distances of the observed z scores from the regression line to establish which submatrix was more nested. We compared these scores among groups: (1) to assess if the Cyclades are consistently more nested than the Anatolian Islands (as expected according to the scenarios discussed above) and (2) to relate differences in nestedness with taxon response to island history, with current distributions of more sedentary animals mostly affected by landbridge connections. Finally, we used the Aegean Islands to perform a sensitivity analysis by re-allocating islands from a submatrix to another. For this, we changed the allocation of islands close to the dividing line between Cycladic and Anatolian groups, placing these borderline islands in the Cycladic and Antolian groups alternatively. Namely, we allocated Astipalea, Amorgos and Anafi alternatively in the Cycladic and Anatolian groups. To avoid spurious matrices containing sites without species, we always deleted empty rows and columns which originated during the construction of submatrices. The process of random extraction of rows and columns can produce different matrices of the same number of cells with different combination of row and column numbers. The large number of random submatrices (100) used in our analyses ensures that (1) the obtained submatrices express a wide range of rows and columns combinations; (2) that regression and hence confidence intervals cover a ‘‘complete’’ range of variation, from very small submatrices to submatrices approaching the number of cell of the whole matrix. Using the product of row and column numbers as a measure of the size of a matrix, we did not take into account matrix shape. However, we used a large number of random submatrices, which ensures that for any matrix dimensional class several submatrices are created with different combinations of row and column numbers. This makes the confidence intervals robust towards differences in matrix shape. The large variability in matrix fill of Atmar and Patterson’s (1995) matrices also indicates that our method is not influenced by this matrix characteristic.
Results Goodness-of-fit (R2 values) of the regression lines between the z scores and number of cells for 55 matrices subdivided into random submatrices of varying size varied according
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Values in parentheses indicate standard deviations. Abbreviations for null models: FF fixed–fixed, FE fixed row totals, equiprobable column totals, EF equiprobable row totals, fixed column totals, EE equiprobable row totals, equiprobable column totals. – indicates that the index is not applicable for a certain null model. Abbreviations for nestedness metrics: NODF nestedness measure based on overlap and decreasing fill, MT matrix temperature, N0 number of absences, N1 number of presences, NC number of species shared over all pairs of sites, UA number of unexpected absences, UP number of unexpected presences, UT sum of deviations from perfect nestedness (UT = UA ? UP), and BR number of discrepancies. All null models and metrics are fully described in Ulrich and Gotelli (2007)
Values of z scores were calculated using nine indices of nestedness under four null-models (Ulrich and Gotelli 2007). Singletons were alternatively included and excluded
0.443 (0.230)
0.415 (0.244)
0.534 (0.222) 0.544 (0.244) 0.580 (0.225) 0.557 (0.232) 0.497 (0.221) 0.506 (0.214) 0.494 (0.229) 0.399 (0.242) 0.205 (0.185) 0.224 (0.212) 0.492 (0.203) 0.448 (0.230) 0.371 (0.266) 0.429 (0.256) 0.499 (0.239) 0.446 (0.229) 0.571 (0.223) 0.548 (0.239) Included Excluded EE
0.429 (0.242)
0.458 (0.250) 0.460 (0.238)
0.420 (0.236) 0.296 (0.231)
0.399 (0.243) 0.167 (0.141)
0.185 (0.174) 0.514 (0.220)
0.544 (0.206) 0.376 (0.264)
0.377 (0.245) 0.354 (0.207) 0.427 (0.240) Excluded
0.407 (0.239) 0.445 (0.230) Included
0.239 (0.187) 0.337 (0.228) Excluded
0.294 (0.199)
0.215 (0.182) 0.290 (0.187) 0.321 (0.216) Included FE
0.080 (0.089)
0.086 (0.107) 0.074 (0.088) 0.076 (0.097) Excluded
0.076 (0.073) 0.069 (0.079) Included FF
EF
0.375 (0.220) 0.275 (0.201) 0.299 (0.182) 0.216 (0.164) 0.117 (0.129) –
0.043 (0.075)
0.328 (0.199) 0.227 (0.190) 0.272 (0.189) 0.196 (0.151) 0.116 (0.119) –
0.056 (0.070) 0.045 (0.077)
0.047 (0.078) 0.063 (0.074)
0.060 (0.075) 0.050 (0.059)
0.051 (0.061) 0.044 (0.058)
0.048 (0.057) –
–
MT BR UT UP UA NC N1 N0 NODF Singletones Null model
Nestedness index
to the nestedness indices and null-models (Table 1). The effects of null models and different indices on the R2 values were particularly important, whereas inclusion/ exclusion of singletons did not alter the main results (Table 2). The combination of null-models and nestedness indices which produced the regression lines with the best goodness-of-fit were, in general, EF and EE with BR (and in some cases MT and NODF). We also identified strong correlations between R2 values of regression lines and matrix nestedness. For BR and MT indices with EF and EE null-models (the models which produced the regression lines with highest goodness-of-fit), Spearman rank correlation coefficients between z scores and R2 varied between -0.738 and -0.860 (P \ 0.0001; coefficients are negative because for these indices negative z scores indicate nestedness). Spearman rank correlation coefficients between z scores and R2 of NODF for EF and EE null-models were, respectively, rS = 0.753 and rS = 0.654 (P \ 0.0001; here correlations are positive because for NODF positive z scores indicate nestedness). These strong correlations indicate that the magnitude of nestedness significance (here expressed as z scores) of a matrix affects the goodness-of-fit of the corresponding regression line obtained with our method (or, in other words, that there is a strong association between the magnitude of the z score of a given matrix and the regression line obtained from that matrix). In particular, examination of scatterplots showed that only highly significantly nested matrices (z score \ -4) produce lines with good R2 values (typically [0.7). The relationship between size or fill of the original matrix and the z scores of the random submatrices is influenced by the properties of the original matrix from which the submatrices are extracted. Thus it is important to understand what is expected when a highly nested matrix is divided into random submatrices. A matrix which is square (1), i.e., where the number of rows is the same of columns, and (2) where the species pool of each area is a subsample of that of any other richer area, is perfectly nested. Figure 3a shows a perfectly nested square matrix (herein referred to as M1). A measure of matrix nestedness for this kind of matrix is maximum, and the NODF index equals to 100. Any other possible matrix created by randomly extracting a certain number of rows from M1 (Fig. 3b–e) would respect property 2, but would not be square, thus showing a NODF inferior to 100. Additionally, the size of the random subsample of rows extracted from M1 would affect nestedness of the resulting submatrix: the larger the number of sampled rows, the closer the submatrix will be to perfect nestedness. The same happens for columns. It has to be said that, especially for small matrices, there is a sensible chance that combined extraction of random rows and columns would produce a submatrix with a NODF
Popul Ecol (2013) 55:227–239
Table 1 Mean values of R2 of regression lines between z scores and matrix size for 55 matrices subdivided into 100 submatrices of varying size (each regression included 100 z scores)
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Effect
df
MS
F
943.657 \0.0001
Null-model
3.0
99.987
319.9735
Inclusion/exclusion of singletons
1.0
0.029
33.329
Nestedness index
8.0
15.257
0.029
Null-model 9 inclusion/exclusion of singletons Null-model 9 nestedness index df Degrees of freedom, SS sum of squares, MS mean sum of squares, F Fisher F, P probability
SS
Inclusion/exclusion of singletons 9 nestedness index Null-model 9 inclusion/exclusion of singletons 9 nestedness index
a
b
c
d
3.0
0.292
1.907
24.0
14.819
0.097
P
0.828
0.363
53.998 \0.0001 2.759
0.041
17.483 \0.0001
8.0
0.432
0.617
1.527
0.142
24.0
0.464
0.054
0.547
0.964
a
b
e
f
Fig. 3 Graphical representation of a perfectly nested matrix (a) and of a set of submatrices created by random extraction of rows and columns from that matrix (b–f). Black squares represent presences. Numbers below matrices are the respective NODF values, with higher values indicating higher nestedness
larger than that of submatrices which have more rows and columns (Fig. 3f; see also dispersal of NODF values for small sized matrices in Fig. 4a). This contrasts with the fact that larger submatrices are expected to be more nested than smaller ones because they include more of the overall nestedness of the parent matrix. Use of z scores limits this problem, as demonstrated by the sensible improvement (R2) in the linear relationship between the size or fill of random submatrices and their respective nestedness values, especially for relatively large matrices (Fig. 4a, b). In general, results about nestedness indices, null models and singletons of our Aegean case study paralleled those obtained from this wider analysis. Thus, to save space and for
Fig. 4 Linear relationship between size of 100 submatrices created by random extraction of rows and columns from a 50 9 50 perfectly nested matrix and their respective: a NODF values and b z scores (computed using the null model EE). Lines are ordinary least squares regressions
sake of clarity, we will concentrate on results obtained with the BR index under the EF null model (Tables 3 and 4). In all groups except butterflies, the Cyclades appear more nested than expected for a matrix of equal size extracted at random from the respective entire Aegean matrix (Fig. 5). Differences between observed and predicted z scores decreased in the following order: land snails (highest nestedness) [ tenebrionids [ reptiles [ isopods [ chilopods [ butterflies (lowest nestedness) (Table 4). This pattern deviates from that expected on the basis of endemism for the tenebrionids, which were less nested than expected,
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and for reptiles, which were more nested. By contrast, all studied faunas are, on the Anatolian islands, significantly less nested than expected. Changing the allocation of islands at the border line between Anatolian and Cycladic groups did not alter the main results (typically, z scores changed only of a few decimal digit).
Discussion We found that our methods produce good regression lines under various combinations of nestedness indices and null models. Moreover, results obtained for the Aegean study case are consistent with what is known for the biogeography of these islands.
Table 3 Results of nestedness analyses (BR index, EF null model) performed on different animal groups from the whole Aegean archipelago Animal group
Size
Number of occurrences
z
Land snails
137 9 33
949
-47.26
Isopods
66 9 22
537
-27.77
Chilopods
52 9 24
300
-27.22
Butterflies
76 9 14
436
-23.84
100 9 30
390
-15.48
31 9 20
211
-16.22
Tenebrionids Reptiles
Our wider study, based on 55 matrices, indicates that our method performs well with EF and EE algorithms, allowing the construction of regression lines with high R2 for most indices of nestedness, and particularly for MT and BR, which are the most widely used (see Ulrich and Gotelli 2007). Regression lines, also, are substantially insensitive to inclusion or exclusion of singletons, which typically complicate the interpretation of nestedness patterns (Greve and Chown 2006; Fattorini 2007a; Simaiakis and Martı´nezMorales 2010). The choice of the null-model used to compute z scores is particularly compelling in any nestedness analysis. Restrictive null models (which create null matrices without altering some properties of the original matrix) do not suit the purpose of our demonstration, as the number of allowed row and column rearrangements decreases with matrix nestedness. For example the FF null model, which is generally recommended, cannot be used for perfect nested matrices and produces incoherent results for highly nested matrices, since all the generated null matrices will be equal or much similar to the original matrix (see Ulrich and Gotelli 2007). Consider also that NC nestedness index cannot be used with neither FF nor FE null models. By contrast, all kinds of indices can be applied with less restrictive null models which recombine matrix columns and rows with few or no structural constraints. Use of less restrictive null models is generally discouraged because they tend to be prone to type I errors and sometimes too far from biological assumptions. However, the risk of type I errors associated with less restrictive null models does not
Table 4 Results of nestedness analyses (BR index, EF null model) performed on different animal groups from, respectively, the Cycladic and the Anatolian Islands Animal group
R2
Land snails
0.872
Isopods Chilopods
0.821 0.876
Islands
Size
Occurrences
zs
ze
Dz
LL
UL
A
96 9 8
323
-9.390
-12.320
-2.930
-13.183
-11.458
C
87 9 25
653
-36.780
-27.074
9.706
-28.151
-25.996
A
51 9 6
168
-6.300
-8.420
-2.120
-9.014
-7.826
C
58 9 16
380
-23.130
-20.231
2.899
-21.211
-19.252
A
45 9 10
171
-8.930
-11.583
-2.653
-12.024
-11.142
C
36 9 14
141
-15.330
-12.747
2.583
-13.223
-12.272
-12.390 -8.430
-14.424 -7.070
-2.034 1.360
-15.149 -7.496
-13.699 -6.644
Butterflies
0.833
A C
74 9 7 34 9 7
285 152
Tenebrionids
0.669
A
67 9 8
139
-4.880
-4.798
0.082
-5.472
-4.125
C
61 9 22
268
-14.240
-9.634
4.606
-10.284
-8.984
A
27 9 5
85
-2.130
-4.407
-2.277
-4.800
-4.015
C
19 9 15
131
-11.490
-7.869
3.621
-8.324
-7.413
Reptiles
0.760
Species per locality matrices were extracted as subsets from matrices reporting the occurrences of the considered taxa in the whole Aegean archipelago R2 goodness-of-fit of regression line, A Anatolian Islands, C Cycladic Islands, Size species 9 sites, zs observed z for the submatrix, ze z scores calculated from regression line based on matrices constructed by extracting at random rows and columns from the complete matrices (i.e., those of the whole Aegean archipelago), Dz difference between ze and zs, LL lower limit of 95 % confidence intervals, UL upper limit of 95 % confidence intervals
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237
a
b
c
d
e
f
Fig. 5 Regression lines showing the relationship between the size (expressed as matrix total number of occurrences) of the submatrices randomly extracted from the complete Aegean matrices and the corresponding z scores (BR index) for land snails (a), isopods (b), chilopods (c), butterflies (d), tenebrionids (e), and reptiles (f). Squares
and triangles indicate measured z values for the Anatolian and the Cycladic submatrices, respectively. Broken lines indicate 95 % confidence intervals. R2 of regression lines are reported in Table 4. The scales of both horizontal and vertical axes differ among panels
affect our protocol, since we use null models just to standardize the value of nestedness indices for comparative purposes, and not to support any interpretation on significance of measured nestedness (which is instead derived from the deviation of the z score of the observed index from that predicted by the regression line). In particular, the dependence of confidence limits from the selected null model allowed us to take into account the risk of overestimating nestedness. For a given matrix size, a non restrictive null model which may lead to artificially high z values (type I error) would produce, on average, an upward shift of the upper confidence limit of the regression line, making our analysis much more conservative, thus reducing the risk that inflated z values of the selected submatrices are considered significant beyond reason. The null-model which performed best in the proposed method was the EF, which preserves species richness per site (column totals), but allows species occurrences among islands (row totals) to vary randomly and equiprobably. These assumptions seem to be particularly adapted for insular patterns where an island can host a fixed number of species (for example because of limitations imposed by
area size, carrying capacity, etc.), but the occurrence of certain species instead of others in a particular island can be largely stochastic. More in general, the fact that our method works very well even with little restrictive null models which are able to detect even weak nestedness patterns, ensures that it can be applied to ecological matrices with very different characteristics, as also supported by the results obtained using Atmar and Patterson’s (1995) matrices. Results obtained for the Aegean Islands strongly support previous biogeographical hypotheses about how nestedness arises in an island system. In general, all groups showed significant nestedness when the Aegean Islands were considered as a whole, but different patterns emerged when they were subdivided into two groups. Nestedness is especially well-developed in systems that experienced faunal relaxation (Patterson and Atmar 1986; Wright et al. 1998), and it is hence typically high in relict faunas and floras (Wright et al. 1998; Millien-Parra and Jaeger 1999). Differences in nestedness patterns found between the Cyclades and the Anatolian islands are consistent with the different geographical histories supposed for the Aegean
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faunas. The Cyclades were mainly colonized from the Balkans and, being connected to each other but disconnected from Greece, were subject to past inter-island faunal exchanges that increased nestedness. In these island groupings, new taxa evolved and other were extinct after islands were again disconnected, causing non-random patterns of species distribution and extinction which increase nestedness (Patterson and Atmar 1986). The high degree of nestedness found for the Cyclades supports previous conclusions that their fauna was initially homogeneous before the fragmentation of the Pleistocene Cycladic landmass into several islands. In particular, the non-random pattern of animal distribution on the Cyclades is consistent with the hypothesized importance of postglacial selective extinctions that are not balanced by colonization or local speciation processes. By contrast, thanks to their proximity to the mainland, the Anatolian islands could have been more easily colonized also after postglacial disconnection (Fattorini 2002, 2006b, 2007a, b). Although regressions of z scores against matrix size as those shown in Fig. 5 are not surprising because constructed by data which are decreasing series of a full pattern, they are useful, because deviations from the regression line are informative about the particular structure of a given submatrix and can be used to identify significant patterns (see also Worthen 1996). Also, our method is particularly useful when a certain submatrix is per se significantly nested, but not more than any other submatrix extracted at random from the same whole matrix. For example, we can imagine a situation where two submatrices have z scores with the same P values, but one is more nested than expected in respect of the overall order of the whole matrix (which suggests some ecological cause which make it ‘‘different’’ from the rest of the matrix), whereas the other one is ‘‘equivalent’’ to any other randomly extracted submatrix of the same size. Although we started with a single large matrix and then divided it into submatrices, our protocol can be also applied to the reverse situation, i.e., combining several matrices into a single larger one. For example, to test if different groups in the same study system have different nestedness one could collapse the matrices of each group into a single, multitaxon matrix, and then apply our method to search for the groups which are more nested than any random assortment of species. However, when collapsing different matrices, the obtained results will be dependent on the characteristics of the original datasets and the results will vary according to the criteria used to combine them.
Conclusions Over the last several years, the use of absolute values of nestedness indices has been severely criticised (Ulrich et al.
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2009), and most authors used z scores (as defined in Eq. 1) not only to assess whether a matrix is nested or not, but also as a measure of the degree of nestedness. However, z scores are not comparable, i.e., they do not make possible to asses if a given matrix is more or less nested than another one. Moreover, the computation of z scores is strongly influenced by the choice of the null-model used to generate the randomized matrices, thus further complicating any comparison. The method proposed here allows the researcher to establish if the observed z score of a given submatrix differs significantly from that expected for an assortment of submatrices containing the same number of sites and species taken at random from the original complete matrix. Our method is not aimed at comparing z scores of different matrices, but to assess if a particular section (submatrix) of a matrix is more nested than expected in relation to the overall order of the whole matrix. Although we applied our method to a problem of island biogeography, the possibility of assessing if a particular sector of a matrix is more nested than expected on the basis of the overall matrix nestedness may have wider applications. For example, another biogeographical application could be the use of our protocol to assess the influence of geographical barriers in determining species assemblages, using barriers to identify matrix portions. Completely different applications might involve: the study of nestedness in ecological networks, where each species would use only resources which are subsets of those used by the more generalist species (Joppa et al. 2010); mutualistic networks where each species would interact only with proper subsets of those species interacting with the more generalist species (Bascompte et al. 2003); or host–parasite networks, where generalist parasite species are widespread among available host species, but the number of parasite species found on a host is limited by host biology and ecology (Gue´gan and Kennedy 1993; Graham et al. 2009). In all these cases, it would be interesting to assess if peculiar combinations of species or of species and resources are more (or less) nested than expected in respect to the overall order of the network. By providing a protocol to identify which portion of a matrix is significantly nested, we hope to encourage further research to focus not only on the question whether a matrix is nested or not, but also on where and why nestedness is ‘nested’. Acknowledgments The authors would like to thank S. L. Pimm, J. Podani and two anonymous referees for their helpful comments on an early version of the manuscript.
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