Measures of geographic range size: the effects of sample size

ECOGRAPHY 19: 259-268. Copenhagen 1996

Measures of geographic range size: the effects of sample size

Kevin J. Gaston, Rachel M. Quinn, Simon Wood and Henry R. Arnold

Gaston, K. J., Quinn, R. M., Wood, S. and Arnold, H. R. 1996. Measures of geographic range size: the effects of sample size. - Ecography 19: 259-268.

A number of methods have been used for quantifying the sizes of the geographic ranges of species. The consequences of different levels of sampling (the proportion of actual spatial occurrences) are explored for eight of these, using data on the occurrences of butterfly species on a 10 x 10 km grid across Britain. For all methods, the percentage error of estimation (PEE) decrrases with the number of 10 x 10 km squares which a species occupies, most rapidly for extent measures. and more rapidly for area measures than for measures of numbers of units occupied. The rate of decline in PEE itself falls as sampling effort increases. At a given sampling level, rank correlations between range sizes measured by different methods are generally high, but there is no consistent change in the magnitude of these correlations as the level of sampling increases. The composition of the set of species with the smallest range sizes changes with the level of sampling.

K. J . Gaston (correspondence) Dept of Animal and Plant Sciences, Univ. of Shefield. Shefield, U.K. SIO 2TN ([email protected]). - R. M. Quinn. NERC Centre for Population Biology, Imperial College at Silwood Park. Ascot, Berkshire, U.K. SL5 7PY. - H . R. Arnold, Inst. of Terrestrial Ecology, Monks Wood. Abbots Ripton, Cambridgeshire. U.K. PEI 7 2LS. - Simon WoQd, School of Mathrmatical and Computational Sciences, Univ. of St. Andrew, Fife, U.K. KY16 9SS.

The sizes of the geographic ranges of species vary. A number of methods have been uskd to quantify this variation. These include, for example, measures of the latitudinal extent of occurrences, the area of the minimum convex polygon enclosing all occurrences, the numbers of grid squares containing occurrences, and the numbers of sites occupied (e.g. Reaka 1980, Rapo- port 1982, Juliano 1983, McAllister et al. 1986, Ander- son and Marcus 1992; for reviews see Gaston 1994a, b, in press). Justification is seldom provided for the use of a particular method in a given study, and the consequences of applying alternatives rarely are explored. Nonetheless, in deciding between methods there are several important considerations (Gaston 1994a). We would draw particular attention to two. First, there is the feature of the spatial distribution of a species which is to be quantified. Although often treated as though it were otherwise, methods quantify a variety of different characteristics of the large scale spatial distributions of

species. In particular, distinction can be drawn between a group of methods which quantify the extent of occurrence of a species, and a group which quantify its area of occupancy (Gaston 1991). The former is the distance or area between the outer-most limits to the occurrence of a species. The latter is the area over which the species is actually found, and will tend to be smaller because species do not occupy all areas (or habitats) within the limits to their occurrence. Although rank correlations between the range sizes of species, measured using different methods, may often be significant, they are variable and do not always translate into linear relationships between different pairs of measures (McAllister et al. 1986, Brown and Briggs 1991, Ander- son and Marcus 1992, Maurer 1994, Quiiiii et al. 1996).

The second consideration is the form and quality of the data available. Particular methods will be more appropriate for some kinds of data than for others. In

Accepted 15 December 1995

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ECOGRAPHY 1 9 3 (1996) 259

general, cruder techniques tend to be applied when data are poor. It remains unclear, however, how robust the different methods of measurement of range size are to low levels of sampling effort, or how estimates of range size generated by any particular method may change with levels of effort. In this paper we examine the relationship between sampling effort and observed geographic range size for each of eight methods used for measuring range sizes, using data for butterfly species in Britain.

Methods Data The spatial distributions of 53 species of butterfly resi- dent in Britain were derived from information collected by volunteer recorders, collated by the Biological Records Centre (Harding and Sheail 1992), and summarised by Heath et al. (1984). Only recent records were used, collected between 1960 and 1982. As is usual for data on species distributions in Britain, these were summarised on a grid of 10 x 10 km squares, as defined by the British National Grid. A species is considered to be present within a 10 x 10 km square if it has been recorded from the square at least once during the specified time period. Because the Occurrences of butterfly species across this grid have been particularly well

The following methods of measunng range sizes were examined: a) Latitudinal extent (lat.) - distance (in km) between the two latitudinally most distantly separated occupied 10 x 10 km squares. b) 95%) latitudinal extent (95% lat.) - distance (in km) between the two latitudinally most distantly separated occupied 10 x 10 km squares after exclusion of 2.5% of all occupied 10 x 10 km squares from each latitudinal extreme. This measure reduces the influence of outliers on measure (a). c) Longitudinal extent (long.) - distance (in km) between the two longitudinally most distantly separated occupied 10 x 10 km squares. d) 95% longitudinal extent (95% long.) - distance (in km) between the two longitudinally most distantly separated occupied 10 x 10 km squares after exclusion of 2.5u/0 of all occupied 10 x 10 km squares from each longitudinal extreme. This measure reduces the influence of outliers on measure (c). e) Latitudinal extent x longitudinal extent (longlat.) - the area (in km3 obtained by multiplying measures a) and c). t) Minimum convex polygon (mcp) - the area (in km’) within the minimum polygon containing all the records, and in which no internal angle exceeds 180 degrees. g) Numbers of 100 x 100 km squares (100 km) - as defined by the British National Grid. h) Numbers of vice-counties occupied (v-c) - Vice-counties are ir- regularly sized geopolitical units, ranging in approxi- mate area from 381 km2 to 5423 km2. They provided the basis for many early species distribution maps in Britain (e.g. Taylor 1907).

documented, these distributions can be treated as close to reality. By sampling subsets of the squares compris- ing the overall grid, an assessment can be made of the differences between the range sizes of species measured Results when their presence is known only for some squares and when their presence is known for all squares in which they occur.

AnalySeS

The range size of each species was measured using each method for samples of 5, 10, 20, 40 and 80% of the squares actually occupied. Sample squares were chosen at random, and each level of sampling effort repeated I500 times for each species to generate a mean estimated range size and associated variance. The results of analyses did not change markedly with > 1500 itera- tions.

Following Russell and Lindberg (1988), results are expressed in terms of percentage error of estimation (PEE) of range size, where PEE= ( b o w n range- ‘observed’ range]/known range} x 100. For each method of measurement the known range is that measured on the basis of all occupied 10 x 10 km squares, and the observed range that based on the sample of occupied squares.

PEE decreases with the number of 10 x 10 km squares occupied for all eight of the methods used for measuring range size. This decrease is greatest at low levels of sampling effort. The trajectories are quite variable in some cases, because of differences in the patterns of spatial distribution of occupied 10 x 10 km squares for different species (see Fig. 1 for examples). Use of five- point moving averages enables readier interpretation of the underlying patterns (Fig. 2; standard errors of observed range sizes are so small that they are not shown).

PEE decreases most rapidly with increasing numbers of 10 x 10 km squares occupied for extent measures of range size (latitudinal extent, longitudinal extent, and the related 95% extent measures), and more rapidly for area measures (latitude x longitude, minimum convex polygon) than for measures of numbers of units occupied (100 x 100 km squares, vicecounties) (Fig. 2). The range of values of PEE and the value of PEE of the most widely distributed species, for a given level of sampling effort, tend to follow similar sequences. The former tend to be highest for measures of extent and least for measures of numbers of units occupied, and

ECOGRAPHY 193 (1996) 260

the latter vice-versa. Thus, for example, when only 5% of the squares a species actually occupies are sampled, the ranges of the most widely distributed species have PEE values of < 15% for extent measures, of 20-30"!n for area measures, and of ca 30-40?/0 for measures of numbers of units occupied.

Using 95% latitudinal extent and 95% longitudinal extent measures it is possible for the sues of the geo-

graphic ranges of restricted species to be over-estimated at low levels of sampling. This produces an associated negative PEE. When only small numbers of squares are being sampled the 2.5% of squares which would be excluded by these methods from the extremes of the recorded distributional extent will be very few in number or may amount to less than one whole square. Failure to exclude some squares from the range size

100 -I a

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0 250 500 750 lo00 1250 1500 1750 2000 2250 2500

Number of 10 km squares occupied

Fig. 1 . Changes in percentage error of estimation (PEE) of range size for species occupying different numbers of 10 x 10 km squares, at three different sampling levels (solid circles - 5%, open squares - 20%. solid triangles - 800h)). using a) latitudinal extent, b) minimum convex polygon, and c) 100 x 100 km squares.

26 1 ECOGRAPHY 1 9 3 (1996)

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Number of 10 krn squares occupied

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estimate can produce a larger range sue than would be found at higher levels of, or complete, sampling.

The rate of decline in PEE with increasing numbers of 10 x 10 km squares occupied steadily declines as sampling effort increases, for all methods of range size measurement (Fig. 2). However, there are strong rank correlations between the range sizes of species measured at each sampling level and their known range sizes (Table 1). Unsurprisingly, these increase with sampling level.

Spearman rank correlations between the range sizes of species measured by different methods at a given level of sampling are generally high, in excess of 0.9 when 5% of occupied squares are sampled (Table 2). There is no consistent pattern of change in the magnitude of the correlation between the range sizes measured by a pair of methods and the level of sampling at which the range sizes are determined (Table 3). In some instances the correlation increases (e.g. vice-counties and minimum convex polygons), in some it declines (e.g. all relationships with longitudinal extent), and in others there is no significant interaction.

Similarity in the composition of rare species (defined as the 20% of species having the smallest range sues, similar to many other studies; see Gaston 1994b) delineated at different levels of sampling for a given method of range size measurement declines with the intensity of sampling (Table 4). Fewer of the species identified as rare at lower sampling levels were also identified as such at a sampling level of 100%.

Discussion Our understanding of the spatial distributions of species, and hence often also of species richness, can be affected profoundly by sampling considerations (Koch 1987, Koch and Morgan 1988, Nelson et al. 1990, Wright 1991, Anderson and Marcus 1992, Rich and Woodruff 1992, Hanski et a]. 1993, Prendergast et al. 1993, Gaston 1994b, Hiigmander and Msller 1995). These considerations are of two principal kinds, the effects of low levels of sampling effort on patterns observed for individual species, and the effects of those levels on between-species comparisons. The analyses reported here primarily concern the former, but have implications for the latter. In discussing the results we focus on a situation in which the distributions of different species are subject to random sampling of a similar level. In reality, of course, patterns of sampling may be biased with respect to certain geographical areas and particular species (e.g. Rich and Woodruff 1992).

PEE declines with the size of the geographic range of a species using all of the methods of measuring range sizes examined here. A similar result was reported by Russell and Lindberg (1988) using data for I80 species of marine molluscs, random sampling of 10 km seg- ments of the overall linear distance spanned by all species, and linear extent as a measure of range size. This outcome is not surprising, because any single locality is likely to contribute a greater proportion to a measure of geographic range size for species which are more poorly distributed. The decline is steepest and

262 ECOGRAPHY 1 9 3 11996)

attains lower levels of PEE for widely distributed species using those methods of measuring range size, such as measures of extent, which depend least on particular occupied squares being sampled. This is well illustrated by comparison of latitudinal and longitudinal extent, with 95% latitudinal and 95% longitudinal extent. Re- moval of the potential effect of outliers causes a more rapid reduction in PEE. Counts of numbers of areas

occupied (i.e. 100 x 100 km squares, vice-counties) are strongly influenced by individual data points, PEE does not decline rapidly, and at low levels of sampling PEE remains high even for species occupying large numbers of 10 x 10 km squares. The differential influence of a small proportion of occupied squares in part results from the spatial clumping of individuals, populations and occurrences (and therefore records), such that most

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Fig. 2. Changes in percentage error of estimation (PEE) of range size for species occupying different numbers of 10 x 10 km squares. at three different sampling levels (symbols as in Fig. I ) , with PEE and numbers of squares occupied given as a five-point moving averages. Methods of measurement are a) latitudinal extent, b) 95% latitudinal extent, c) longitudinal extent, d) 95% longitudinal extent. e) latitudinal extent x longitudinal extent, f) minimum convex polygon, g) 100 x 100 km squares, and h) vice-coun ties.

ECOGRAPHY I 9 3 (1996) 263

areas at almost any scale have relatively rather few of these entities and a few areas have a relatively large number (the frequency distribution of the numbers of areas containing different numbers of individuals, populations or occurrences is strongly right-skewed; e.g. Taylor et al. 1978, Gaston 1994b).

The differential impact of low levels of sampling on poorly distributed species may potentially have a significant effect on analyses of patterns in the range sizes

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of species. It will tend to accentuate the right-skew in the frequency distributions of species geographic range sizes commonly documented at moderate to large spatial scales (Anderson 1984, Schoener 1987, Gastun 1990, 1994b). and may affect observed inter-specific relationships between ranges sizes and other variables (see Gaston 1990, 1994b for summaries of the relevant literature). Attempts to explore patterns in absolute rather than relative range sizes, and to interpret the

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Fig. 2. (c), (d).

264 ECCGRAPHY 1 9 3 (1996)

slopes of relationships between range sizes and other variables may be seriously confounded.

I t is unclear to what extent the severity of the effects of sampling will change with spatial scale (the 10 x 10 km grid is the smallest scale at which distribution information is available consistently for butterflies throughout Britain, though there are some regional schemes mapping species at finer scales; e.g. Thomas

and Webb 1984, Mendel and Piotrowski 1986, Sawford 1987). Much may depend on whether the spatial distributions of species can be viewed as fractal, as some have suggested (Williamson and Lawton 1991, Virkkala 1993, Maurer and Heywood 1993, Gautestad and Mys- terud 1994, Maurer 1994), and, if so, how variable changes in the patterns of spatial occurrence with changing scale are for different species.

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ECOGRAPHY 193 (1996) 265

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Table I . Spearman rank correlations between species range sizes measured at each sampling level and their known range sizes (looo/), using each of the eight methods of measuring range sizes.

lat 95% lat long 95% long longlat mcp 100 km V C

5% 0.965 0.968 0.938 0.9 12 0.970 0.977 0.969 0.951 I P/o 0.981 0.987 0.971 0.962 0.983 0.986 0.977 0.959 2w/o 0.987 0.992 0.984 0.978 0.99 I 0.994 0.988 0.98 1 W / O 0.995 0.996 0.992 0.983 0.995 0.997 0.993 0.991 8P/O 0.999 0.999 0.996 0.993 I .Ooo 1 .ooo 0.998 0.999

ECOGRAPHY 1 9 3 (1996) 266

Table 2. Spearman rank correlations between species range sizes measured using pairs of methods at sampling levels of a) 5% and b) IOO%

~ ~~

95% lat long 95% long longlit mcP 100 km V C

lat 0.989 0.886 0.864 0.980 0.972 0.931 0.88 1 95'% lilt 0.853 0.838 0.961 0.950 0.901 0.843 long 0.988 0.950 0.957 0.967 0.964 95% long 0.929 0.936 0.944 0.946 longlat 0.996 0.97 1 0.936 mcP 0.979 0.946 100 km 0.985

b)

95% lat long 95% long longlat mcp 100 km V C

lat 0.957 0.808 0.744 0.977 0.969 0.936 0.91 I YY/U lilt 0.756 0.696 0.927 0.92 I 0.872 0.846 long 0.914 0.898 0.890 0.916 0.897 95% long 0.832 0.829 0.847 0.826 longlat 0.993 0.973 0.953 mcP 0.98 I 0.968 100 km 0.990

Table 3. Spearman rank correlation coefficients between the Spearman rank correlation coefficients for the relationships between range sizes measured at a given sampling level (e.g. Table 2) and the proportion of data points sampled. For example. r = 0.886 for the interaction between the rank correlations for range sizes measured using 100 x 100 km squares and numbers of vicecounties, and the proportion of data points sampled (5 . 10, 20, 40, 80 and 100%). * = p < 0.05 and ** = p < 0.01. [Note, relationships are not independent and hence probabilities will tend to be inflated].

95% lat long 95% long longlat mcP 100 km V C

lat - I .OOo** - 1 .m** -0.943.' -0.829. - 0.029 0.371 0.829. 95% lat - 1.000.. -0.943** - l.OOo** - 1.000.' - 0.600 0.543 long - I.OOo** -1.000** - 1.000** - I .OOo** - 1 .000** 95% long -0.943** -0.943.' - I .000** - 1.000.. longlat - 0.086 0.371 0.771 mcP 0.829. 0.886. 100 km 0.886.

Table 4. Proportions of those species delineated as rare (20% of species) at lower levels of sampling (5-SOOh). which are also rare at a sampling level of IW/U (e.g. at a 5% level of sampling 82% of the species recopnised as rare using the latitudinal extent measure are also rare at a sampling level of 1Wh).

5'%1 10% 2Ph 40% 80%

lat 0.82 0.91 1.00 1.00 1.00 95% la1 0.82 0.91 0.91 1.00 1.00 long 0.73 0.91 0.91 1.00 1.00 95% long 0.73 0.91 1.00 1.00 1.00 longlat 0.82 0.91 0.91 0.91 1.00 mcp 0.82 0.91 0.91 1.00 1.00 100 km 0.91 0.91 0.91 1.00 1.00 V-C 0.82 0.82 0.82 0.82 1.00

Quinn et al. (1996) documented significant rank correlations between the range sizes of s w e s measured using different methods, for butterflies and freshwater molluscs in Britain. The analyses here suggest that the strength of such relationships can be a function of the proportion of occurrences of species which have been recorded. Moreover, these effects vary widely depend-

ing on the pair of methods under consideration. Whilst the magnitudes of the correlations between range sizes measured by several pairs of methods show increases as sampling levels increase, there is a consistent decline in the correlation between other range size measures and longitudinal extent and between some other pairs of measures where at least one is a measure of extent (Table 3).

Estimates of range sizes have frequently been applied as criteria in priotitising species and areas for conservation, often as measures of rarity (e.g. Daniels et al. 1991, ICBP 1992, Mace et al. 1992, Mace 1994a, b). The rigour with which the set of species having the smallest range sizes is identified correctly is a function of the level of sampling (Table 4). When levels of sampling are poor, a higher proportion of the rarest species are not distinguished as such. Although the numbers of species are small, in some cases this effect is sufficient that nearly a third of the species distinguished as rare at a sampling level of 100% are not distinguished as such at low levels of sampling effort. Given that sampling levels are low (c 5%) for most species of

ECOGRAPHY 193 (1996) 261

most higher taxa in most areas of the world, this effect could have a substantial impact on the recognition of species of major concern to conservation.

rlcknodedgernenfs - We thank volunteer recorders from the Butterfly Recording Scheme and staff at the Biological Records Centrc for collection of, and access to, the buttertly distribution data. P. Harding kindly commented on the manuscript. K.J.G. is a Royal Society Univ. Research Fellow. R.M.Q. is supported by a NERC Special Topics studentship from the Esmee Fairbairn Charitable Trust, a CASE award with the Inst. of Terrestrial Ecology.

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Measures of geographic range size: the effects of sample size