## Introduction

Ecologists have long distinguished between different components of species diversity. Traditionally, three are recognized, alpha or local diversity (α), beta diversity or differentiation (β) and gamma or regional diversity (γ). Beta diversity, the spatial turnover or change in the identities of species, is a measure of the difference in species composition either between two or more local assemblages or between local and regional assemblages. For a given level of regional species richness, as beta diversity increases, individual localities differ more markedly from one another and sample a smaller proportion of the species occurring in the region. As has long been recognized, it thus captures a fundamental facet of the spatial pattern of biodiversity (Whittaker 1960, 1972; Cody 1975; Wilson & Shmida 1984; Shmida & Wilson 1985; Gaston & Williams 1996). None the less, it has received surprisingly limited attention, particularly when contrasted with the vast number of studies of local or alpha diversity; arguably this is increasingly true also when contrasted with the attention that is being paid to regional or gamma diversity in the context of broad-scale patterns of variation in species richness and their determinants (for reviews see Brown 1995; Gaston & Blackburn 2000; Gaston 2000; Blackburn & Gaston 2003).

Notwithstanding, several generalizations have been made about the form that spatial patterns in beta diversity are likely to take. Foremost among these is the assertion that spatial turnover in species identities increases towards lower latitudes, providing a proximate mechanism for the high levels of tropical species richness (Gaston & Williams 1996; Mourelle & Ezcurra 1997; Brown & Lomolino 1998). A number of recent studies have sought empirically to test such generalizations (Harrison, Ross & Lawton 1992; Willig & Sandlin 1992; Blackburn & Gaston 1996; Mourelle & Ezcurra 1997; Clarke & Lidgard 2000). Their findings have been inconsistent, suggesting that any patterns may be strongly contingent on taxon or geographical region. However, comparisons between the results of different studies may be hindered severely by the variety of measures that have been used to quantify beta diversity and by the variety of ways in which these measures have been applied. These problems are seldom acknowledged and the results of different analyses are injudiciously, and not infrequently, treated as if they were strictly comparable (Gaston & Williams 1996).

Since Whittaker's (1960, 1972) original suggestion that beta diversity should be measured as the proportion by which the species richness of a region exceeds the average richness of a single locality within that region, numerous measures have been proposed that constitute variations on this theme (Cody 1975; Routledge 1977, 1984; Wilson & Shmida 1984; Magurran 1988; Harrison *et al*. 1992; Williams 1996a; Harte & Kinzig 1997; Mourelle & Ezcurra 1997). Indeed, arguably, almost for each new application of the concept of beta diversity a fresh measure has been derived, with studies employing measures of beta diversity to address such issues as the changes in species composition along spatial or environmental gradients (MacArthur 1965; Whittaker 1960, 1972; Cody 1975; Routledge 1977; Wilson & Shmida 1984; Harrison *et al*. 1992; Blackburn & Gaston 1996; Mourelle & Ezcurra 1997; Willig & Gannon 1997) and between patches of similar habitats (Whittaker 1977; Harrison 1997); the degree of association or similarity of sites or samples (Magurran 1988; Price *et al*. 1999); and the identification of biogeographical boundaries or transition-zone patterns and their implications for conservation planning (Poynton & Boycott 1996; Williams 1996a; Ruggiero, Lawton & Blackburn 1998; Williams, de Klerk & Crowe 1999; Gaston *et al*. 2001). Efforts have been made to compile and compare selected measures of beta diversity (see Routledge 1984; Wilson & Shmida 1984; Magurran 1988; Williams 1996a; Krebs 1999; Southwood & Henderson 2000). Some measures are clearly closely correlated, at least under some conditions, while others give wildly different patterns of results for the same data sets (Wilson & Shmida 1984; Blackburn & Gaston 1996; Williams 1996a; Williams *et al*. 1999; Clarke & Lidgard 2000). In general, however, the reasons for using, or preferring, any particular measure rather than another remain unclear. Around half of the studies of beta diversity have used more than one measure (Fig. 1a), although Whittaker's original measure (β_{w}) has been the most frequently employed (Fig. 1b).

In this study, we collate the 24 published measures of beta diversity based on presence/absence data of which we are aware (the vast majority of explicit studies of beta diversity have focused on presence/absence data). These are then, often for the first time, expressed in common terms, and some of their basic properties are compared. The main issues addressed are what beta diversity measures actually measure, how measures differ, and the importance of the differences. This exercise reveals that existing measures divide into a few basic groups, each of which captures a different facet of spatial turnover in the identities of species. We do not enter into debate as to the relative merits of multiplicative and additive approaches to the relations between alpha, beta and gamma diversity measures, while acknowledging that recent discussion of the latter addresses a potentially exciting avenue of investigation (Lande 1996; Loreau 2000; Veech *et al*. 2002).

### the measures

The 24 measures of beta diversity considered are listed in Table 1. These comprise those measures that have been suggested or employed in recent analyses of patterns in beta diversity based on species presence/absence data. Other similarity or distinctness measures, of which there are many, that have not been used or proposed as suitable for this purpose are ignored; some of these may yet prove useful in the context of spatial turnover in species identities. Following the conventions established previously, the 24 measures are identified in terms of a subscripted ‘β’ using, where possible, those subscripts employed previously and indicating fresh ones where appropriate.

Original formulation | Measure re-expressed | Source |
---|---|---|

*S*= total number of species recorded for both quadrats (*S*=*a + b + c*); = average number of species found within the quadrats; α_{1}= total number of species found in the focal quadrat; α_{2}= total number of species found in the neighbouring quadrat; α_{j}= total number of species found in the quadrat*j*; α_{max}= maximum value of species richness for the two quadrats;*N*= number of quadrats;*r*= number of species pairs whose distributions overlap; g = cumulative gain in species; l = cumulative loss in species;*H*= range of habitat gradient;*e*= number of quadrats under comparison in which species_{i}*i*is found;*T*= Σ*e*= Σα_{i}_{j};*C*= species in common between two censuses;*T*_{i}= total number of species in census*i*;*r*= number of cases in which there is no species overlap (number of unique pairwise species comparisons without co-occupancy of any quadrat); SAR = species–area relationship,_{s}*S*=*kA*^{z}, where*S*is species number,*A*is area and*z*and*k*are constants. The parameter*z*is a beta diversity measure based on species gain. See Appendix for derivation of β_{z}.- 1
β _{–}_{1}= β_{–}_{2}when*a*= α_{max}. - 2
Originally formulated for five, four or three adjacent quadrats, for which β _{g}of the focal quadrat is the percentage of ‘transition species’ in relation to the total number of species found in the sequence of adjacent quadrats. The transitional species for a pair of quadrats were considered here as*b*and*c*. - 3
From definition of ‘turnover = 1 − commonality (the number of species in common divided by the average number of species in the two patches)’. - 4
Measure of assemblage similarity between quadrats calculated as ‘the proportion of species at this central point present at every one of the remaining cells’. - 5
Not intended to be a measure of ‘beta diversity’ *per se*, this was originally used to represent differences in species richness between quadrats*,*i.e. to provide a measure of local alpha diversity gradients.
| ||

1 | Whittaker (1960), see alsoMagurran (1988), Southwood & Henderson (2000) | |

2 | Harrison et al. (1992) | |

3 | Cody (1975) | |

4 β_{wb} (a + b) + (a + c) − 2a | b + c | Weiher & Boylen (1994) |

5 | Routledge (1977), see also Magurran (1988), Southwood & Henderson (2000) | |

6 | Routledge (1977), Wilson & Shmida (1984) | |

7 β_{e} | exp(β_{I}) − 1 | Routledge (1977) |

8 | Wilson & Shmida (1984) | |

9 | Mourelle & Ezcurra (1997) | |

10 | Jaccard (1912), see also Magurran (1988), Southwood & Henderson (2000) | |

11 | Sørensen (1948) based on Dice (1945); see also Whittaker (1975), Magurran (1988), Southwood & Henderson (2000) | |

12 β_{m} (α_{1} + α_{2})(1 − β_{j}) | Magurran (1988) | |

13 | Harrison et al. (1992)^{1} | |

14 | Cody (1993) | |

15 | Colwell & Coddington (1994; ‘complementarity’ measure), see also Pielou (1984) | |

16 β_{g} | Gaston et al. (2001)^{2} | |

17 | Williams (1996a) | |

18 | Lande (1996) | |

19 | Williams (1996a), 1999Williams et al. (1999) | |

20 | Harte & Kinzig (1997)^{3} | |

21 β_{rlb} | Ruggiero et al. (1998)^{4} | |

22 | Lennon et al. (2001), based on Simpson (1943) | |

23 β_{gl} | Lennon et al. (2001)^{5} | |

24 β_{z} From SAR | Lennon et al. (2001), see also Harte & Kinzig (1997) & Appendix |

In each case, the original equations for the measures of beta diversity have also been re-expressed in terms of the pairwise matching/mismatching components used in similarity/dissimilarity coefficients, and usually denoted as *a*, *b* and *c* (e.g. Krebs 1999). Considering the simplest case, that of two equal-area quadrats, one can be regarded as the ‘focal quadrat’, and its species composition can be compared with that of the other, the ‘neighbouring quadrat’. The matching component *a* is the total number of species shared by the two quadrats (i.e. the number occurring in both), *b* is the number of species present in the neighbouring quadrat but not in the focal quadrat, and *c* is the number of species present in the focal quadrat but absent from the neighbouring quadrat (Fig. 2). The *a* component is thus the species in common for a pair of quadrats, while the *b* component measures species gain and the *c* component species loss relative to the focal quadrat. The re-expressed formulae in Table 1 are valid only for pairwise comparisons. Although the approach could be extended to multiple comparisons, pairwise ones remain the basis of the vast majority of published analyses of patterns in beta diversity and thus this constraint is not overly restrictive.

Expressed in terms of matching/mismatching components and this simple scenario, it is clear that a wide range of the possible simple ways of combining *a*, *b* and *c* have been proposed at some time as measures of beta diversity (Table 1). Four pairs of the 24 measures are precisely the same for the case of a pair of quadrats [β_{w} = β_{−1} (if the former is formulated with subtraction of a constant), β_{t} = β_{me}, β_{c} = β_{l}, β_{cc} = β_{g}], and β_{j} = 1 − β_{cc}. In addition, β_{wb} is the same as β_{c} multiplied by a constant. Our analysis is therefore restricted to the remaining 18 measures of beta diversity.

### the simplest case

For the purpose of comparing the properties of the 18 measures, we consider first the contribution to each of the three matching components for a pair of quadrats, the focal and the neighbouring (the most common basis in the literature for calculating beta diversity). In order that this would be independent of variation in the overall numbers of species, we conducted the comparisons in terms of *a*′, the percentage of species shared by the pair of quadrats, *b*′, the percentage of species present exclusively in the neighbouring quadrat, and *c*′, the percentage of species present only in the focal quadrat. The total number of species for the pair of quadrats is *a + b* + *c*, and therefore *a*′ *+ b*′ *+ c*′ = 100%. The species richness for the focal quadrat (i.e. alpha diversity, α) equals the total number of species common to both quadrats plus those present exclusively in this quadrat, that is, *a*+ *c*, which is represented by *a*′ + *c*′ (this being negatively correlated with *b*′, *r* = −1); the equivalent expression for the neighbouring quadrat is *a*′ + *b*′ (negatively correlated with *c*′, *r*=−1); of course, values of *a*, *b* and *c* are not necessarily correlated in this manner (Lennon *et al*. 2001). We analyse all possible combinations of integer percentages of species among the three components, and use ternary plots to examine the relationship between these and beta diversity values (Figs 3 and 4). The degree of shading in each plot represents (on an equal interval scale) the value of a given beta diversity measure (darker shading for larger values for dissimilarity measures, lighter shading for similarity measures, see below), relative to the values of *a*′, *b*′ and *c*′ (each of which increase in the direction of the appropriate letter at the corners of the plot). The magnitude of *a*′, the degree of species continuity between the two quadrats increases from the base of the plot towards its apex, and for a given value of *a*′ the lateral position of a point reflects the relative contribution of species gains (*b*′) and losses (*c*′) (Fig. 3). The displacement of a point from a vertical line between the base and the apex of the triangle (i.e. *b*′ −* c*′) reflects the difference in alpha diversity between the two quadrats.

Similar empirical spatial patterns of variation in beta diversity have been observed for β_{−1} and β_{t} when these are applied to the same data sets (Wilson & Shmida 1984; Blackburn & Gaston 1996). Under the above conditions, β_{−1} and β_{t} were not identical for a pair of samples (Wilson & Shmida 1984); however, by definition they are obviously linearly dependent and using *a*′, *b*′ and *c*′ values were perfectly positively correlated (*r* = 1). Such linear relationships are also expected by definition (Table 1) between other measures, such as β_{t} and β_{sor} (Mourelle & Ezcurra 1997) and β_{sor} and β_{hk}, which are perfectly negatively correlated (*r* = −1), as are therefore β_{sor} and β_{−1}. Other pairs of measures showing perfect negative correlations are β_{j}−β_{c}, and β_{j} − β_{cc}, while β_{t} − β_{hk}, and β_{c}−β_{cc} are perfectly positively correlated (*r* = 1).

Intuitively, when viewed as indices of similarity, measures of beta diversity for presence/absence data might be expected to exhibit minimum and maximum values of 0 and 1, respectively, or some simple function thereof. As some measures are cast in terms of dissimilarity and some in terms of similarity, an increase in the value of a measure can mean either an increase in turnover (dissimilarity) or a decrease (similarity). Thus, if such an intuitive property were to hold, beta diversity should have minimum possible values of zero (dissimilarity measures) or maximum possible values of one (similarity measures), when there is no similarity between the species composition of the two quadrats, and maximum possible values of one (dissimilarity measures) or minimum values of zero (similarity measures) when their species composition was identical using *a*′, *b*′ and *c*′. Several of the measures considered here scale rather differently from this, both in that they do not attain these maximum and minimum values, and in that they do not do so under these particular conditions (Table 2; see Wolda 1981 for a similar finding for similarity indices). Both β_{w} and β_{r} vary from 0 to 1 or from 1 to 2, depending on whether they are corrected by −1, providing the same scale of variation, as does β_{M}, which varies from 0 to 100. However, there are other cases when the minimum value is zero but the maximum value is not 1 (for β_{gl}, β_{c}, β_{I}, β_{e} and β_{−3}), and one in which neither is so (β_{rs}) (Table 2). As long as limits are finite, maximum and minimum values can be normalized to the unit interval (this re-scaling and comparability is achieved in effect by the graphical scaling system in Williams 1996a; Williams *et al*. 1999). However, as formulated originally (or in terms of *a*, *b* and *c*), for some of the measures finite limits may be difficult to define.

Measure | Minimum | a′ | b′ | c′ | Maximum | a′ | b′ | c′ |
---|---|---|---|---|---|---|---|---|

β_{rlb} | 0 | 0 | < 100 | > 0–100 | 1 | > 0–100 | < 100 | 0 |

– | 0 | 100 | 0 | |||||

β_{gl} | 0 | 100 | 0 | 0 | 2 | 0 | >> 0 | >> 100 |

>> 100 | >> 0 | |||||||

β_{j} | 0 | 0 | 0–100 | 0–100 | 1 | 100 | 0 | 0 |

β_{sor} | 0 | 0 | 0–100 | 0–100 | 1 | 100 | 0 | 0 |

β_{w} | 1 (0) | 100 | 0 | 0 | 2 (1) | 0 | 0–100 | 0–100 |

β_{c} | 0 | 100 | 0 | 0 | 50 | 0 | 0–100 | 0–100 |

β_{t} | 0 | 100 | 0 | 0 | 1 | 0 | 0–100 | 0–100 |

β_{hk} | 0 | 100 | 0 | 0 | 1 | 0 | 0–100 | 0–100 |

β_{z} | 0 | 100 | 0 | 0 | 1 | 0 | 0–100 | 0–100 |

β_{m} | 0 | 100 | 0 | 0 | 100 | 0 | 0–100 | 0–100 |

β_{co} | 0 | 100 | 0 | 0 | – | 0 | 100 | 0 |

– | 0 | 0 | 100 | |||||

1 | 0 | 0–< 100 | 0–< 100 | |||||

β_{r} | 1 (0) | 0–100 | 0 | 0–100 | 2 (1) | 0 | 50 | 50 |

0–100 | 0–100 | 0 | ||||||

β_{I} | 0 | 100 | 0 | 0 | 0·30103 | 0 | 50 | 50 |

– | 0 | 100 | 0 | |||||

– | 0 | 0 | 100 | |||||

β_{e} | 0 | 100 | 0 | 0 | 0·35125 | 0 | 50 | 50 |

– | 0 | 100 | 0 | |||||

– | 0 | 0 | 100 | |||||

β_{rs} | 0·0002020 | 0–100 | 0 | 0–100 | 0·50525 | 0 | 50 | 50 |

0–100 | 0–100 | 0 | ||||||

β_{−2} | 0 | 0–100 | 0 | 0–100 | 1 | 0 | 50 | 50 |

0–100 | 0–100 | 0 | ||||||

β_{−3} | 0 | 0–100 | 0–100 | 0 | 0·5 | 0 | 50 | 50 |

0–100 | 0 | 0–100 | ||||||

β_{sim} | 0 | 0–100 | 0 | 0–100 | – | 0 | 100 | 0 |

0–100 | 0–100 | 0 | – | 0 | 0 | 100 | ||

1 | 0 | 0–100 | 0–100 |

On the basis of their responses to variation in the percentages of the three matching components (*a*′, *b*′ and* c*′), four groups of measures of beta diversity can be distinguished. For convenience, we term these (1) measures of continuity and loss –β_{rlb}; (2) measures of species richness gradients –β_{gl}; (3) measures of continuity –β_{j}, β_{sor}, β_{w}, β_{c}, β_{t}, β_{hk}, β_{z}, β_{m}; and (4) measures of gain and loss *–*β_{co}, β_{r}, β_{i}, β_{e}, β_{rs}, β_{−2}, β_{−3}, and β_{sim}.

#### (i) Measures of continuity and loss

One of the 24 measures analysed, β_{rlb}, depends solely on the values of *a* and *c* (Fig. 4a). This measure returns values of zero (total dissimilarity) when no species are shared between two quadrats (*a*′ = 0), and values of 1 (total similarity) when *c*′= 0 (Table 2). When values of *a*′ and *c*′ are low, then the value of β_{rlb} is very sensitive to small changes in *c*′ (Fig. 4a).

#### (ii) Measures of species richness gradients

Values of β_{gl} depend on the difference in species richness between the two quadrats under consideration (Fig. 4b). This measure has been employed previously to test if the values of other measures of beta diversity serve simply to recover patterns in local species richness gradients (Lennon *et al*. 2001). When the matching component *a*′ equals zero, minimum values of β_{gl} are observed when the two quadrats have the same species richness, that is *b*′ and *c*′ = 50 (Fig. 5d). When either component *b* or *c* is zero there is a curvilinear increase in the value of β_{gl} (Fig. 5b).

#### (iii) Measures of continuity

The values of several measures of beta diversity are dependent essentially on variation in the matching component *a*, the level of continuity in species composition between two quadrats. Such measures will always tend to assume extreme values (high or low depending on the direction of scaling with *a*) when differences in the richness between the focal and the neighbouring quadrat are large − because *a* will then tend to be small, i.e. *b* or *c* are large relative to *a*. This group of measures reflects what we term ‘broad-sense’ turnover, which implicitly incorporates differences in composition attributable to diversity gradients, but ignores the relative magnitude of species gains and species losses.

There are two groups of measures whose values are related directly to variation in the matching component *a*. Values for the first group all scale positively with increases in *a*. These are β_{j} and β_{sor} (Fig. 4c,d), which are well known as similarity indices, because of this dependence on *a*. High values are interpreted as reflecting low beta diversity (high similarity), and low values as reflecting high beta diversity (high dissimilarity).

The second group of measures whose values show simple relations to variation in *a* all scale negatively with increases in this matching component. These are β_{c}, β_{w}, β_{hk}, β_{t}, β_{m} and β_{z} (Fig. 4e–h). β_{w} is the most widely used measure of beta diversity in ecology (Fig. 1b, e.g. Routledge 1977, 1984; Wilson & Shmida 1984; Weiher & Boylen 1994; Blackburn & Gaston 1996; Harrison 1997; Mourelle & Ezcurra 1997; Clarke & Lidgard 2000; Koleff & Gaston 2001). However, following the criticisms of this measure by Wilson & Shmida (1984; see below), increasing use has been made of β_{t} (Fig. 1b, e.g. Willig & Sandlin 1992; Blackburn & Gaston 1996; Mourelle & Ezcurra 1997; Willig & Gannon 1997; Koleff & Gaston 2001). β_{z} is derived from the species–area relationship (Table 1), which implicitly considers information about species turnover (Harte & Kinzig 1997).

The values of the nine measures of beta diversity based on species continuity show rather different patterns of change when either components *b* or *c* are zero. β_{j} and β_{c}, which are based on the average change in species composition between quadrats, have linear relationships (Fig. 5a), while curvilinear increases are shown by β_{w}, β_{hk}, β_{t} and β_{z} (Fig. 5b), and concave relationships by β_{sor} and β_{m} (Fig. 5c).

#### (iv) Measures of gain and loss

Values of the final group of measures of beta diversity, β_{co}, β_{r}, β_{I}, β_{e}, β_{rs}, β_{−2}, β_{−3} and β_{sim}, depend on *a* and on the relative magnitude of *b* and *c*. These can be considered as ‘narrow-sense’ measures of turnover, in that they focus on compositional differences more than differences in species richness. While values of these measures all increase with declines in component *a*′, they also peak at intermediate values of *b*′ and *c*′ (Fig. 4i–o), and attain their maximum values when the percentage of species shared in common between the focal and neighbouring quadrats is small and the percentages gained and lost in moving from the focal to the neighbouring quadrat are similar.

The values taken by β_{r}, β_{I}, β_{e} and β_{rs} reflect the degree of overlap of species distributions, and thus the ‘*r*’ parameter proposed by Routledge (1977; Table 1). These all exhibit a curvilinear pattern of change in their values, with changes in the relative magnitudes of *b*′ and *c*′. The values of β_{−2} and β_{−3} exhibit a triangular pattern of change, while those of β_{sim} follow a similar pattern at high values of *a*′ but become highly sensitive to small changes either in *b*′ or *c*′ when values of *a*′ and either *b*′ or *c*′ are low.

When there are no species in common between the two quadrats (*a*′ = 0), the pattern of change in the value of beta diversity with variation in the other two matching components differs between measures. It is hump-shaped for β_{r} (Fig. 5e) and β_{I}, β_{e} and β_{rs} (Fig. 5g), and exhibits a triangular pattern for β_{−2} (Fig. 5h) and β_{−3} (Fig. 5f). Under this circumstance, β_{sim} assumes a constant value of 1, regardless of variation in the matching components *b* and *c*.

### other simple properties

We tested three further simple properties that beta diversity measures might be expected to possess (Table 3).

Measure | Symmetry | Homogeneity | Nested quadrats | Additivity |
---|---|---|---|---|

β_{rlb} | – | √ | √ | – |

β_{gl} | √ | √ | – | – |

β_{j} | √ | √ | – | – |

β_{sor} | √ | √ | – | – |

β_{w} | √ | √ | – | – |

β_{c} | √ | – | – | – |

β_{t} | √ | √ | – | – |

β_{hk} | √ | √ | – | – |

β_{z} | √ | √ | – | – |

β_{m} | √ | – | – | – |

β_{co} | √ | √ | – | – |

β_{r} | √ | √ | √ | – |

β_{I} | √ | √ | – | – |

β_{e} | √ | √ | – | – |

β_{rs} | √ | – | √ | – |

β_{−2} | √ | √ | √ | – |

β_{−3} | √ | √ | √ | – |

β_{sim} | √ | √ | √ | – |

#### (i) Symmetry

For many purposes, for two quadrats *x* and *y*, a beta diversity measure β (*x*,*y*) must be equal to β (*y*,*x*). That is, beta diversity should be symmetric with regard to variation in the matching components *b* and *c* (Janson & Vegelius 1981). In other words, if we exchange the neighbouring and focal quadrat identities the measure should remain unchanged if symmetric. All measures satisfy this criterion, except for β_{rlb} (Table 3).

#### (ii) Homogeneity

If all the matching components, *a*, *b* and *c* are multiplied by the same constant, this should not affect the value of the measure of beta diversity. That is, beta diversity should be independent of the total number of species as long as the proportions comprising the different components are constant (Janson & Vegelius 1981). This criterion is fulfilled by all of the measures when re-expressed in terms of the matching components, except for β_{co}, β_{c} and β_{m}, although for the last two multiplying the matching components by a constant multiplies the measure by the same constant (Table 3).

#### (iii) Nested quadrats

Quadrats may be nested physically, such that the focal quadrat lies entirely within the neighbouring quadrat, and all the species occurring in the former also occur in the latter. If all the species in the neighbouring quadrat also occur in the focal quadrat (*b* = *c* = 0), then there should be no turnover, and beta measures should attain either their minimum or maximum values depending on the direction in which they scale (Wilson & Shmida 1984). This condition is met by all of the measures considered here. If all of the species in the focal quadrat are also found in the neighbouring quadrat (*c* = 0), but the neighbouring quadrat has species not found in the focal quadrat (*b* > 0), then one would expect that there should be no turnover if this equates to gain and loss, independently of how of large or small is the percentage of species that the two quadrats have in common. For all the measures whose values depend on *b* and *c*, minimum species turnover is detected if either of those components become zero, except for β_{co}, β_{I}, and β_{e} (Table 3).

### increasing complexity

Thus far, with the exception of nestedness, we have considered the simplest case, that of a single pair of adjacent equal-area quadrats. In practice, there are many other situations in which measures of beta diversity are applied. Although typically they remain based on pairwise comparisons, they commonly bring with them additional considerations.

#### (i) Transects

When beta diversity is calculated along transects several concerns arise. The first is the sensitivity of measures to local gradients in species richness. This is a particular problem for measures that scale essentially with variation in *a*, the number of species in common to a given pair of quadrats (measures of continuity). As observed above, these will tend to assume extreme values (high or low depending on the direction of scaling with *a*) when differences in the richness between the focal and the neighbouring quadrat are large − because *a* will then tend to be small.

Secondly, the additivity of measures of beta diversity may become important in the context of transects (Wilson & Shmida 1984). For the simplest case of three quadrats in the spatial sequence *x*, *y*, *z*, a measure will be additive if the sum of the values of beta diversity between *x* and *y* and between *y* and *z* equals the value of beta diversity between *x* and *z*. The additivity exhibited by some measures depends on the patterns of species richness in the quadrats along the transect (Wilson & Shmida 1984; Magurran 1988) and also on whether or not the species distributions are nested. Here we consider whether the additivity criterion is met for four different hypothetical situations along a transect of three contiguous quadrats (Fig. 6).

None of the 18 measure of beta diversity is completely additive for all of the cases tested (Table 3; see Wilson & Shmida 1984; Magurran 1988); most measures fail the test of additivity under most of the cases examined.

To study the pattern of spatial turnover along a gradient, a number of different approaches have been used, and the distinctions between them are important, irrespective of the actual measures of beta diversity employed. First, beta diversity has been calculated between adjacent pairs of quadrats along a transect, and the relationship sought between these values and the position on the transect (Wilson & Shmida 1984; Cody 1986), or a single value has been obtained from the cumulative gain and loss of species from the pairwise comparisons (Willig & Sandlin 1992; Blackburn & Gaston 1996; Mourelle & Ezcurra 1997). Secondly, beta diversity has been calculated between all pairs of quadrats along the transect, and the relationship sought between these values and the distance separating each of the pairs (Whittaker 1960, 1972; Cody 1986; Harrison *et al*. 1992; Mourelle & Ezcurra 1997; Clarke & Lidgard 2000) or the beta diversity between the pairs is expressed graphically, grouping similar areas (Magurran 1988; Price *et al*. 1999). Species turnover often depends on the distance between the quadrats compared, with low to moderate values typically expected over short distances and higher levels over longer distances (Cody 1986; Harte & Kinzig 1997). Thirdly, beta diversity is estimated for the whole transect on the basis of the relationship between the species richness of the quadrats and the overall richness of the whole transect (Whittaker 1960; Wilson & Shmida 1984; Harrison *et al*. 1992; Blackburn & Gaston 1996).

A fourth possible approach involves direct comparison of the extremes of the transect, effectively assuming that a species present at both extremes is continuously present between them, even though it may actually be absent from intermediate quadrats. However, such assumptions may produce unsatisfactory results (Whittaker 1960; Wilson & Shmida 1984).

#### (ii) Grid systems

In the case of grid systems, beta diversity can be evaluated for a focal quadrat relative to that of its multiple neighbours. Consider a grid of equal area quadrats with a central focal cell and the eight surrounding neighbours. Values of the matching components *a*, *b* and *c* could be calculated between the focal cell and each neighbour, and then the average of these values used to calculate beta diversity. Alternatively, the matching components could be used to calculate eight values of beta diversity and these could then be averaged (Williams 1996a; Lennon *et al*. 2001). In both cases, corrections can be made readily where data are missing, or unavailable, for some neighbours.

These two methods will give the same results, except for measures of beta diversity that are based on maximum or minimum values of any component, such as β_{−2}, β_{−3} and β_{sim}. We contrasted both approaches for β_{sim}, using data for the breeding birds of South-east Scotland (Murray *et al*. 1998), and based on the occurrences of 125 species across 1756 grid cells of 2 × 2 km (using Worldmap in the calculation of values of beta diversity; Williams 1996b). The values for the two methods of analysis were positively correlated (*r* = 0·849, *n*= 1756, *P* < 0·0001); none the less, because the minimum value of *b* or *c* is considered in the formulation of β_{sim} (see Table 1), higher values of beta diversity result using the first method (Fig. 7).

An alternative approach may be to estimate beta diversity from the matching components for the focal quadrat compared with the composition of the entire set of neighbours, treating them as a single unit (Williams 1996a). This results in the comparison of focal and neighbouring areas of different size, which is often undesirable. It builds a strong asymmetry into the values of *b* and *c*; species gains are magnified relative to species losses, compared with the case of equal-sized quadrats. Using the South-east Scotland breeding birds data a weak relationship exists between the values of β_{sim} obtained with this methodology and those generated by averaging across each of the pairwise comparisons between the focal and neighbouring quadrats (*r* = 0·135, *n*= 1756, *P* < 0·001). Lower values were generally observed when treating the neighbourhood as a single unit and different focal quadrats were highlighted as areas of low or high spatial turnover using the different approaches (Fig. 7).

A novel approach to evaluating beta diversity (although not a novel measure) for data on grid systems, was suggested recently by Gaston *et al*. (2001). β_{g} was estimated across transects in four different directions (W–E, N–S, NW–SE and NE–SW), and the maximum value was attributed to the central focal quadrat. This methodology could be used for other measures. However, it was intended to detect areas with high species replacement when data are fragmented because of variable sampling effort (see also Williams *et al*. 1999). The main problem for this approach is that it will find high beta diversity when differences in alpha diversity are large, i.e. choosing the maximum biases the measure towards pairs of quadrats differing in alpha diversity. A minor weakness is also that the area of contact among the focal and the neighbouring cells is larger for the vertical (N–S) and the horizontal (W–E) comparisons than for the two diagonals (NW–SE, NE–SW); the Euclidean distance between the centres of the focal and each surrounding quadrat is different in the diagonal directions than in the vertical or horizontal directions. This might increase the probability of finding a species on two contiguous quadrats for the N–S and W–E lines.

#### (iii) Unequal numbers of areas

It is not unusual for a single value of beta diversity to be calculated across multiple areas, and this value compared with those generated from other such sets of areas (e.g. values are calculated across quadrats within each of several latitudinal bands and used to assess whether beta diversity changes with latitude; Willig & Sandlin 1992; Blackburn & Gaston 1996; Clarke & Lidgard 2000). Seldom in such cases is the number of areas used to calculate each value of beta diversity constant (e.g. latitudinal bands differ in their longitudinal span). This means effectively that beta diversity is being calculated over total areas of different sizes, which can strongly bias the resulting values.

Some of the measures of β-diversity listed in Table 1 were formulated with the explicit intention of correcting for biases caused when comparing different numbers of areas. Thus, the division by the number of quadrats (*N* − 1) in the formulation of β_{−1} and β_{−2} by Harrison *et al*. (1992), and of β_{me} by Mourelle & Ezcurra (1997) (Table 1) was suggested to address the problem of variation in transect length. Unfortunately this constitutes an over-correction, because the exponent of the species–area relationship is typically much less than 1 and may lead to beta diversity becoming negatively correlated with the number of quadrats.

Without correction, the comparison of beta diversities calculated for different numbers of quadrats or different-sized areas will inevitably bias results. The extent to which this effect is sufficient to obscure real biological patterns remains unknown, but it is clearly undesirable.

#### (iv) Unequal spatial scale

Measures of beta diversity have been used to evaluate spatial turnover at many different scales. Units of study vary from a few square metres (e.g. Whittaker 1960; Routledge 1977; Wilson & Shmida 1984; Pharo, Beattie & Binns 1999) to tens and hundreds of square kilometres (e.g. Cody 1986; Harrison *et al*. 1992; Willig & Sandlin 1992; Blackburn & Gaston 1996; Poynton & Boycott 1996; Williams 1996a; Price *et al*. 1999; Clarke & Lidgard 2000; Koleff & Gaston 2001). Changing spatial scale (quadrat size) is likely to change spatial patterns in beta diversity, with the matching component *a* often tending to increase as spatial resolutions become coarser (Lennon *et al*. 2001). In consequence, the values of some measures of beta diversity are likely to increase with spatial scale and those of others to decrease. This may create difficulties for the comparison of patterns in turnover reported by studies performed at different scales, although the extent to which differences in the behaviour of turnover are solely a result of scale is unknown.

#### (v) Comparing assemblages with different rank–occupancy distributions

Finally, measures of beta diversity have been employed to examine patterns of turnover in assemblages with different underlying occupancy structures; some may have many restricted species and few widespread ones, and others the converse (differences in the shape of species–occupancy distributions is partly a function of spatial scale; Gaston 1994; Gaston & Blackburn 2000). Assemblages with high proportions of widely and narrowly distributed species will, respectively, tend to exhibit low and high levels of beta diversity. Thus, splitting the breeding birds of South-east Scotland into two groups, the 50% with the lowest and the 50% with the highest levels of tetrad occupancy, reveals that the former sample a much broader spectrum of potential values of turnover than do the latter (Fig. 8). Hence, in the case of an assemblage of relatively restricted species, turnover will depend more strongly on the behaviour of the measure of beta diversity across the entire range of the matching components, and is likely to be influenced more by differences in species richness between quadrats if these are not explicitly taken into account. Moreover, comparative studies of beta diversity between different assemblages or different study areas may, for the most part, detect differences in the overall commonness and rarity of species. It may be desirable, in some instances, to develop ways of allowing for this effect, such that the element of turnover not solely attributable to commonness and rarity differences can be studied.