Zeleny (2008) has shown that the co-occurrence based estimation of species specialization introduced by Fridley *et al*. (2007) is influenced by variation in the size of the species pool along an environmental or compositional gradient. To correct for this sensitivity to species pool variation, he suggests replacing the original measure of beta diversity that is based on additive partitioning of diversity components (e.g. Lande 1996; Veech *et al*. 2002) with Whittaker's beta (Whittaker 1960). Zeleny's observation is an important one that we have independently uncovered in recent cross-continental comparisons of realized niche widths of temperate trees (Manthey 2007). Because variation in species pool size is a common characteristic in community data that cover a wide variety of habitats, and co-occurrence based estimation of species specialization should be independent of species pool attributes, we here briefly expand on Zeleny's (2008) analysis to provide additional guidance to potential users of co-occurrence based niche width estimates.

**Journal of Ecology**

# Beta diversity metrics and the estimation of niche width via species co-occurrence data: reply to Zeleny

*Correspondence author. E-mail: manthey@uni-greifswald.de

## Summary

- 1Zeleny (2008) demonstrated that the co-occurrence based assessment of species habitat specialization (introduced by Fridley
*et al*. 2007) depends on the size of the species pool. To correct for the effect of the species pool on the estimation of species niche width, Zeleny suggested a modification of the original algorithm by replacing additive partitioning as a measure of beta diversity with Whittaker's beta. - 2We used simulated data to show that the alternative index proposed by Zeleny (2008) will poorly represent the niche widths of species inhabiting a set of plots with a highly skewed distribution of local richness values. We therefore expand on Zeleny's (2008) analysis by considering two additional metrics of beta diversity based on compositional similarity and by testing the performance of these indices under different local-regional richness relationships.
- 3
*Synthesis.*None of the four tested metrics of beta diversity produced unbiased estimates of niche width under curvilinear local-regional richness relationships. In this context, we provide additional guidance to potential users of co-occurrence based niche width estimates by specifying the conditions under which certain indices of beta diversity best represent niche width information.

## Alternative measures of beta diversity

The calculation of the generalist-specialist metric proposed by Fridley *et al*. (2007), originally expressed as ‘theta’, is based on a measure of beta diversity calculated from a fixed number of randomly chosen plots in which a focal species occurs. Since the origin of the concept (Whittaker 1960) ecologists have used a large number of different measures to investigate the variation in beta diversity (Wilson & Shmida 1984; Koleff *et al*. 2003). This variety reflects differences in the basic properties of each statistic and researchers have repeatedly warned that the relevance of any metric depends on the research objective as well as properties of the community data (Koleff *et al*. 2003). Fridley *et al*. (2007) applied additive partitioning as a measure of beta diversity (Lande 1996):

where γ is the total number of species encountered in a random sample of plots and µ(α) is mean alpha diversity (mean plot richness). Zeleny (2008) replaced this measure by Whittaker's beta (Whittaker 1960):

Zeleny showed that Whittaker's approach of multiplicative partitioning is independent of variation in species pool size under the condition that plot richness is a linear function of the species pool. Although we agree that there are circumstances where β_{w} is a better generalist-specialist metric than β_{a}, we demonstrate here that use of β_{w} is not appropriate in some cases. In particular, β_{w} will poorly represent the niche widths of species inhabiting plots with a highly skewed distribution of local richness values. For example, in the data set from the south-eastern US used in Fridley *et al*. (2007), the flooded-habitat specialist *Taxodium ascendens* was unrealistically highly ranked in the habitat generalist-specialist list when β_{w} was used. The species is common to the south-eastern Coastal Plain, where it typically grows in species-poor swamp forests (Christensen 2000). In most of the plots that contain *T. ascendens* the species co-occurs with a few other specialized trees adapted to continuous high water levels. Additionally, it occurs rarely in low abundance in nearby species-rich floodplain forests that are flooded only periodically. Thus, the plot richness distribution of the species is strongly positively skewed. Without these scattered occurrences in adjacent habitats, γ and µ (α), as well as the resulting β_{w}, would be very low. The few occurrences in the rich floodplain forests, however, contribute strongly to γ, whereas mean alpha is hardly affected by the few rich species plots, which leads to a strong increase in β_{w}. Because we expect that specialists from species-poor habitats will often have positive skewness in plot richness distribution, the resulting correlation between plot richness and a generalist-specialist metric based on β_{w} in empirical data can be negative and decouple the beta diversity metric from actual niche width.

Because neither β_{a} nor β_{w} are universal solutions to estimating niche widths from co-occurrence data, we consider here two additional metrics of beta diversity based on compositional similarity. The simplest such approach involves the average of pairwise similarities (e.g., Lennon *et al*. 2001; Condit *et al*. 2002) of plots containing a focal species. Here we evaluate the suitability of Jaccard's similarity index, noting that results using the Sørensen index are qualitatively identical. In addition, we use the more recent advance of ‘multiple-site’ similarity measures that take into account species shared by more than two sites and avoid the problem of covariance between pairwise similarities (Diserud & Ødegaard 2007). Of two recent suggestions of multiple-site similarity measures, we consider only the multiple Simpson index (Baselga *et al*. 2007), as the multiple Sørensen index is a simple linear function of β_{w} (Diserud & Ødegaard 2007).

## Performance of different beta diversity metrics under simulated species pool variation

We simulated a gradient in species pool size by constructing a coenocline of 300 species whose optimum locations followed an exponential distribution along a single dimension of 5000 units (Fig. 1a–c). Niche shapes were constructed randomly using a Beta function as in Fridley *et al*. (2007). To focus on the sensitivity of the co-occurrence based estimation of species specialization to local richness gradients we used fixed niche widths of 1000 gradient units and restricted species optimum locations between 500 and 4500 gradient units. Species whose curves were truncated by gradient ends were excluded from further analysis. We randomly located 1000 plots along the gradient and ‘sampled’ species from each plot according to either a linear or curvilinear relationship between local and regional richness (Fig. 1d). While a linear relationship might be more common (e.g. Cornell & Lawton 1992; Caswell & Cohen 1993; Cornell & Karlson 1996), a curvilinear relationship may arise simply from small sampling scales or if per-species stochastic extinction rates are higher than colonization rates (He *et al*. 2005). For the linear local-regional relationship we randomly sampled 50% of the species pool based on the simulated probability of occurrences; for the curvilinear relationship we fixed the random sample at 50 individuals.

To demonstrate the effect of strongly skewed plot richness distributions on the estimation of niche width, we added five species with symmetrical, bimodal response curves to the gradient. While the total niche widths are equal to the other species, the parameters for the two modes vary regularly to produce different plot richness distributions (Table 1, Fig. 1c). For each sampling design we applied the four selected measures of beta diversity (β_{a}, β_{w}, multiple Simpson and Jaccard's similarity), expressing the two similarity indices as dissimilarities. All calculations were performed in the R statistical environment (R Development Core Team 2008). The modified algorithm for the simulation and the estimation of habitat specialization including all measures of beta diversity can be found in Supporting Appendices S1 and S2.

Species | Left mode | Right mode | Local-regional richness relationship | |||||
---|---|---|---|---|---|---|---|---|

Linear | Curvilinear | |||||||

Skewness of richness distribution | Mean plot richness | Skewness of richness distribution | Mean plot richness | |||||

Width | Amplitude | Width | Amplitude | |||||

1 | 900 | 30 | 100 | 10 | –0.784 | 61.7 | –1.125 | 28.0 |

2 | 700 | 25 | 300 | 15 | –0.203 | 47.1 | –0.030 | 23.4 |

3 | 500 | 20 | 500 | 20 | 0.141 | 45.0 | 1.004 | 16.5 |

4 | 300 | 15 | 700 | 25 | 0.839 | 34.9 | 0.742 | 21.0 |

5 | 100 | 10 | 900 | 30 | 2.485 | 14.1 | 1.343 | 12.2 |

Figure 2 presents the correlation between mean plot richness and estimated niche widths, expressed by the scores of the respective measures of beta diversity, for the two sampling designs. Because niche widths are fixed in the simulated coenocline there should be no correlation between niche widths and mean plot richness. None of the four selected measures of beta diversity could reproduce such a pattern under the condition of a curvilinear local-regional species richness relationship (Fig. 2b,d,f,h). The lowest correlation coefficients, as well as the flattest slopes, are realized with β_{w} and multiple Simpson (Fig. 2b,f). However, for β_{w}, the distortion of the bimodal species from the regression line is relatively high (Fig. 2b). With a linear local-regional richness relationship the correlation coefficients are low for the relation between mean plot richness and scores for β_{w}, multiple Simpson, and Jaccard (Fig. 2a,e,g), while additive partitioning (β_{a}) is strongly affected by the richness of plots (Fig. 2c). With increasing positive skewness of the plot richness distribution and decreasing mean plot richness, the β_{w} scores of the bimodal species rise strongly (Fig. 2a, Table 1).

## Conclusion

As shown by Zeleny (2008), additive partitioning is strongly affected by species pool variation (resulting in local richness gradients) under both linear and curvilinear local-regional richness relationships and should generally be replaced by other measures of beta diversity when using co-occurrences to estimate niche widths. Zeleny's (2008) proposed alternative, β_{w}, performs well with continuous species richness gradients but is sensitive to species from species-poor habitats that have strongly skewed plot richness distributions (filled dots in Fig. 2a), consistent with our experience of using β_{w} to describe the niche widths of south-eastern US tree species of well-known habitat affinities. In contrast, co-occurrence metrics based on compositional similarity were robust to richness gradients and species of strongly skewed or bimodal distribution (Fig. 2e,g).

Our recommendations for choosing an appropriate beta diversity index in the estimation of relative niche widths from co-occurrence data can be summarized as follows. First, no beta diversity metric is immune to problems that arise when a community is either highly saturated ecologically (e.g., when competition strongly decouples realized and fundamental niches) or ‘appears’ saturated due to a very small sample of individuals (see Fridley *et al*. 2006; Palmer *et al*., in press). In such cases co-occurrence data are a poor descriptor of habitat affinities, and experimental data are required to delimit niche widths. Second, where plot richness distributions among species are non-skewed, β_{w}, along with similarity indices, provide a more robust description of relative niche widths than the original β_{a} used by Fridley *et al*. (2007). A more usual case, however, is that at least some species will exhibit strongly skewed richness distributions (see the example of *T. ascendens* mentioned above) that β_{w} cannot handle but are relatively inconsequential to metrics based on similarity. We therefore advocate that the original formulation of co-occurrence based estimates of niche widths based on β* _{a}* be replaced with similarity indices like pairwise Jaccard or more recent multi-site similarity techniques. Like the original Simpson index the multiple Simpson approach ignores complete nestedness as a difference between sites (Lennon

*et al*. 2001; Koleff

*et al*. 2003, Baselga

*et al*. 2007). Thus, the choice of similarity index will depend in part on whether researchers choose to interpret nestedness patterns as ‘turnover’ in the sense of habitat variation.