Christy M. McCain. 2007. Area and mammalian elevational diversity. Ecology 88:76–86.

Appendix C. Comparison and discussion of species-area correction techniques, including examples of methodologies.

The aim of this appendix is to compare area-correction methods for elevational diversity and delineate the most rigorous methods to apply. Several methods for producing area-corrected diversity curves on elevational gradients are published. Rahbek (1997) used the power function model for species-area curves: S = cA^z, which is inherently curvilinear. Vetaas and Grytnes (2002) simply divided species richness in each elevational band by log area of that elevational band, which assumes a semi-logarithmic area function (S = logA). Bachman et al. (2004) used a GIS to delineate bands of elevation with equal area. In this case, the bands differ in elevational extent (in some cases <1 m) but are equal in area. Lastly, linear correction methods could be employed by adjusting the diversity of each elevational band by a correction factor equal to the difference in area (e.g., Fu et al. 2004).

Each method has its drawbacks. Methodologically the fine delineation of area into equal area bands in a GIS (Bachman et al. 2004) assumes the diversity data are as accurately measured. For instance, three equal area bands had elevational extents of less that 1 m, 10 less than 50 m, and 16 less than 100 m whereas most elevational diversity data would be measured at 100 m bands at the minimum and very rarely at 50 m or below. The main drawback with the linear method is the assumption that the species-area relationship is linear, which is often not the case (Table 1; Preston 1962, Conner and McCoy 1979). The species-area relationship tends to be curvilinear (Preston 1962, Connor and McCoy 1979, Rosenzweig 1995). Therefore, the power function model or the semi-logarithmic area function offer more promise for accurately assessing the area effect on elevational gradients. The semi-logarithmic area function tends to be used for species-area relationships mainly in the plant literature, so has been tested less. Because the power function is the basis for much of the species-area theory and is the pattern most supported in the literature and among the elevational diversity patterns tested here (Table 1), this appears to be the appropriate model. The difficultly with applying the power model is determining the appropriate z value.

The z value determines how rapidly diversity increases with increasing area (in log-log space; Rosenzweig 1995, Lomolino 2000 and references therein). Unfortunately, Rahbek’s method for estimation of z is not widely applicable for several reasons. First, robust calculations of z values are quantified with a large number of regression points spread across a large range of area values (Conner and McCoy 1979, McGuinness 1984, Williamson 1988, Rosenzweig 1995, Lomolino 2000), both of which are limited in the Rahbek method (4–8 points). Second, Rahbek’s method assumes that the change in diversity with area is more similar for bands of area at the same elevations from different mountain regions than it is across various elevational bands on the same mountain. The veracity of this assumption is unknown and untested. Third, for this method to work diversity must decrease with elevation on all mountains, which is rarely the case for most taxa and not the case for the elevational gradients of mammals considered here.

There are several alternative ways to estimate a z value for elevational gradients. The most obvious would be to use the z value calculated from the species-area regression for that particular mountain. This suffers from some of the same drawbacks as the Rahbek method: few points in the regression and a small spread of area values reduce the reliability of the z value estimation. This is illustrated by the spread in z values calculated for the 23 significant species-area gradients of NVSM and bats which range from 0.03–0.86 (Appendix D; mean = 0.40, SD = 0.25), 58% of which are well above or below values normally associated with z values (e.g., Rosenzweig 1995: 0.12–0.35). The spread in area values is twice as large in NVSM as that of bats, and area spread for individual mountains is low compared to the overall spread for the taxon (average NVSM = 36%, bats = 39%). In fact, for NVSM, z values are negatively related to the spread of area values; small z values are found on mountains with a large range in area between elevations (r² = 0.29, P = 0.02).

Another z value that could be applied is the “canonical” value of z = 0.25 proposed by Preston (1962). But this seems a bit arbitrary given that we know empirically that z values can vary systematically for islands, continents, and taxonomic groups. I propose a taxon method: all the elevational area and diversity measures for each mammal group (NVSM, bats) that show significant log-log species area relationships are combined in a species-area analysis to estimate a taxon-specific, global z value for elevational gradients. This procedure covers the widest available set of area values and includes hundreds of data points in the regressions (NVSM = 399; bats = 140). Such a composite z value also eliminates the influence of extreme values resulting in a more conservative estimate than other potential estimators. To compare different species-area correction methods, I calculated curves using the power model with different z values, the semi-logarithmic method and the linear method (Appendix D; for details of methods see below). The z values tested with the power model included the taxon-specific z value (average, upper 95% confidence limit, and lower 95% confidence limit), the mountain z value, and the canonical value of 0.25 (Preston 1962).

The taxon-specific z value for NVSM was 0.22 with 95% confidence limits of 0.19–0.25 (Fig. 5A; log Species = -0.7152 + (0.2223) log Area), and 0.38 for bats with 95% confidence limits of 0.32–0.44 (Fig. 5B; log Species = -2.3803 + (0.3767) log Area). No trends in mountain-specific z values were detectable with species richness of the mountain (r² = 0.0047, P = 0.7558), mountain size by total area (r² = 0.0260, P = 0.4625), latitude (r² = 0.0506, P = 0.3019, or mountain type, for instance mountain range versus geopolitical elevational gradient (F ratio = 0.77, P = 0.3891).

There was little difference in the area-corrected diversity curves with the four different z values, as the location of the diversity peak was the same as the taxon value for 79% of data sets (Fig. 4; Appendix D). Most deviations occurred with the mountain z values, which as stated above were highly variable. Diversity peak shifts among the z values averaged ~750 m for the mountain z values and ~400 m for all other z values. There were larger differences in the location of peak diversity when comparing curvilinear methods to either semi-logarithmic or linear correction methods. For instance, 48% of the semi-logarithmic and 35% of the linear had different locations of peak diversity with an average shift of ~525 m away from the peak of the curvilinear methods using the taxon z value (Fig. 4; Appendix D). The semi-logarithmic correction methods altered the empirical diversity curves the least. In only four cases was semi-logarithmic diversity peak different from the empirical peak, whereas about half were changed with the other correction methods. Nonetheless, the species diversity curves resulting from all area correction methods and the multiple z values were highly correlated (Fig. 4; Appendix E): taxon-specific value with lower 95% CL (r = 0.994), with upper 95% CL (r = 0.994), with canonical value (r = 0.988), with semi-logarithmic (r = 0.840), and linear correction (r = 0.858). The only non significant correlations were NVSM of the Great Smoky Mountains (linear, mountain specific z, and semi-logarithmic), and NVSM of the Uinta Mountains (mountain specific z).

Regardless of the area-correction method, the fit to the mid-domain effect did not improve appreciably after removing the area effect (Table 1). All correction methods (best model fit (Table 1), curvilinear with taxon z, mountain specific z, or canonical z; semi-log; and linear) found similar numbers of MDE fits that increased (9–13), decreased (6–8) or did not change (4–6). The magnitudes of increases and decreases were also very similar (average increase: 0.17–0.26; average decrease: 0.12–0.23), except for the semi-logarithmic correction which were much lower (increase 0.03, decrease 0.06). Thus, the resulting average fits to MDE were very similar: best fit (r² = 0.27); taxon z (r² = 0.29); mountain z (r² = 0.33); canonical z (r² = 0.29); semi-logarithmic (r² = 0.26); and linear (r² = 0.30).

I advocate using the power model, S = cA^z, first used by Rahbek (1997) and used for latitudinal gradients (Lyons and Willig 1999, Romdal et al. 2004), since it accounts for the strong curvilinear shape of most species-area relationships and because it is the basis of the species-area theory (Rosenzweig 1995 and references therein). I suggest that a taxon-specific z value (and is confidence limits) is the most comprehensive and conservative estimation method for elevational data. Since the taxon z value is estimated from hundreds of points it limits the effects of extreme values resulting in an average species-area correction (Fig. 3).

Nonetheless, my comparison of various values of z found the resulting area-corrected curves to be very similar and highly correlated (r = 0.84–0.99). In general, z values ranging from 0.1–0.4 tend to produce very similar diversity curve shapes with little variation in the location of the peak in diversity (≤250 m, Appendix D and E; not all analyses shown). The magnitude of the z value has the least effect on patterns of decreasing diversity paired with a strongly decreasing area profile, since all values of z caused peak diversity to shift to the same mid-elevation (e.g., 4G). Thus, a range of z values appears to be robust for use in mammalian area-corrected diversity curves, and most likely for other taxonomic groups. Therefore, I would advocate using the power model with (1) a taxon-specific z value and its upper and lower confidence limits if the data is available, or (2) a spread of possible z values including the canonical value (i.e., 0.15, 0.25, 0.35). Using a range of z values concedes that there is error in estimation of the true z value and therefore examines the possible error effects on the diversity curves within a range of probable z values. Such an error analysis seems the most robust method for using the power model since a single estimated value is unlikely to be the true value. The suggested range of z values (and those from the taxon specific calculations) is also consistent with research on continental and island species-area curves which usually range between 0.12–0.35 (Rosenzweig 1995: pg. 17) and with the z values calculated by Rahbek (1997) for birds on South American mountains.

The linear and semi-logarithmic methods of constructing area-corrected diversity curves are the simplest methods to calculate. The inconsistency with the linear method is that most species-area relationships are not linear (Table 1; Preston 1962, Conner and McCoy 1979). The drawback seen with the semi-logarithmic corrections is that it is relatively insensitive to area effects (see above; Appendix D), and is less supported by theory. Nonetheless, both types of area-corrected curves are still highly correlated with the power model in most cases (r = 0.84–0.86; Fig. 4, Appendix E). Thus, previous analyses using linear and semi-logarithmic methods are most likely robust, but I would advocate these methods only if the power method with a variety of z values gives highly conflicting results or if the species-area relationships are strongly linear or semi-logarithmic.

Examples of Methodology:

(1) Curvilinear Method:

The empirical number of species (S) and the estimated area (A) for each 100m elevational band plus the appropriate z value are entered in the power model and solved for the constant c (c = S / A ^z). C then becomes the area-corrected diversity estimate for that particular elevational band. In order to rescale c to similar values as the empirical diversity each c estimate is multiplied by a constant.

For example, if you had three elevational bands with species richness values of 40, 40, 10, and area estimates of 100, 50, 25 million m² with an estimated z value of 0.25 then the calculations would be the following:

(2) Linear Method:

This method assumes that if each 100m elevational band had the same area then diversity would increase by a factor equal to the lesser amount of area. Thus, each elevational band is compared to the elevational band of greatest area. The proportion less is multiplied by the diversity in the band with the greatest area, and this additional diversity is added to the empirical diversity for the band.

For example, using the same three areas in the curvilinear example, the calculations would be the following:

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