Péter Sólymos, Alberta Biodiversity Monitoring Institute, Department of Biological Sciences, CW 405, Biological Sciences Bldg., University of Alberta, Edmonton, AB T6G 2E9, Canada. E-mail: firstname.lastname@example.org
Aim We conducted a meta-analysis of species–area relationships (SARs) by combining several data sets and important covariates such as types of islands, taxonomic groups, latitude and spatial extent, in a hierarchical model framework to study global pattern and local variation in SARs and its consequences for prediction.
Location One thousand nine hundred and eighteen islands from 94 SAR studies from around the world.
Methods We developed a generalization of the power-law SAR model, the HSARX model, which allows: (1) the inclusion of multiple focal parameters (intercept, slope, within-study variance), (2) use of multiple effect modifiers based on a collection of SAR studies, and (3) modelling of the between- and within-study variability.
Results The global pattern in the SAR was the average of local SARs and had wide confidence intervals. The global SAR slope was 0.228 with 90% confidence limits of 0.059 and 0.412. The intercept, slope and within-study variability of local SARs showed great heterogeneity as a result of the interaction of modifying covariates. Confidence intervals for these SAR parameters were narrower when other covariates in addition to area were accounted for, thus increasing the accuracy of the predictions for species richness. The significant effect of latitude and the interaction of latitude, taxa and island type on the SAR slope indicated that the ‘typical’ latitudinal diversity gradient can be reversed in isolated systems.
Main conclusions The power-law relationship underlying the HSARX model provides a good fit for non-nested SARs across vastly different spatial scales by taking into account other covariates. The HSARX framework allows researchers to explore the complex interactions among SAR parameters and modifying variables, to explicitly study the scale dependence, and to make robust predictions on multiple levels (island, study, global) with associated prediction intervals. From a prediction perspective, it is not the global pattern but the local variation that matters.
The species–area relationship (SAR) is one of the best-documented repeating patterns in ecology (McGuinness, 1984; Rosenzweig, 1995). The rate of increase in species richness (S) with area (A) is often described by the power-law relationship S=cAz (Arrhenius, 1921; Preston, 1960), which translates to the linear model (LM) ln(S) = ln(c) +z ln(A), on the log–log scale. Theoretical arguments modelling SARs based on the lognormal species-abundance distribution suggest that the exponent z has a value of 0.25 (May, 1975). It has also been noted in the past that the slope estimates vary greatly due to the spatial scale, latitude, degree of isolation and the type of organism investigated (Connor & McCoy, 1979; Rosenzweig, 1995; Drakare et al., 2006). Unlike the slope parameter, no regularly recurring values for these regression coefficients have been reported and no ‘canonical’ value is hypothesized for the intercept parameter ln(c) (Connor & McCoy, 1979).
Meta-analysis is an important tool for providing a formal and quantitative synthesis of a collection of studies (Osenberg et al., 1999; Stewart, 2010) and it has increasingly been used in macroecology. For example, Hillebrand (2004) studied the latitudinal diversity gradient, Soininen et al. (2007) inspected the distance decay of similarity and Drakare et al. (2006) analysed the SAR based on meta-analytical techniques. These studies quantified the effect of covariates on slope or intercept but not both. In case of a two-parameter model, such as the log–log power-law model for the SAR, modelling how the covariates affect the slope can only provide a qualitative description of the patterns. Qualitative description in the SAR context can reveal, for example, scale dependence (Scheiner et al., 2000). If the latitudinal diversity gradient reverses with spatial scale, i.e. if the SAR curves cross each other as a function of latitude within the range of observations, the pattern is considered not to be rank invariant (Scheiner et al., 2000). A quantitative description is required to determine whether the pattern is rank invariant. To make such quantitative predictions, it is necessary to model the intercept as well as the slope (Lyons & Willig, 2002), and ‘we need both parameters, z and c, to describe species–area curves’ (Rosenzweig, 1995, p. 13). The SAR intercept is also expected to vary with respect to spatial scale, taxonomic group, isolation (MacArthur & Wilson, 1967) and latitude (Lyons & Willig, 2002) because it is related to the expected number of species in a unit area. But the fact that the intercept of the log–log SAR model is influenced by the choice of the unit for area (m, ha or km2; Rosenzweig, 1995; Drakare et al., 2006) and that traditional analysis requires homogeneous slopes and balanced designs to compare intercepts (Connor & McCoy, 1979) has precluded any large-scale synthesis of SAR intercepts so far.
The joint modelling of the SAR intercept and slope, on the other hand, is common in trivariate models aiming to describe the relationship between species richness, area and a third variable. Such LMs with the main effect of log area, the main effect of the additional covariate and an interaction of the two are commonly used in the SAR literature (Tjørve, 2003). For example, Adler et al. (2005) used such a model for the species–area–time relationship, Qian et al. (2007) for the species–area–latitude relationship, Kallimanis et al. (2008) for the species–area–habitat diversity relationship, Storch et al. (2005) and Hurlbert & Jetz (2010) used it for the species–area–energy relationship. It is well known (Connor & McCoy, 1979; Rosenzweig 1995; Drakare et al., 2006) that the shape of the SAR itself can be modified by more than one covariate. Such trivariate models, however, have failed to describe the complexity of the relationships affecting the SAR by marginalizing the problem to a single covariate besides area.
In addition to the slope and intercept, one can view the within-study variance (incorporating the effects of unmeasured covariates and observation error) as a third parameter of the log–log power-law model, ln(S) = ln(c) +z ln(A) +ε, where ε follows a normal distribution with mean 0 and within-study variance σ2 (Fig. 1). In the past, the within-study variability of individual SAR studies has been used as a goodness-of-fit measure to select among competing models describing the functional form of the SAR (e.g. Connor & McCoy, 1979; Dengler, 2009). Drakare et al. (2006) used the correlation between species richness and area (Fisher r-to-z transformed for meta-analysis) as a goodness-of-fit measure and found that the fit of the SAR varied with habitat type, taxa, spatial scale and latitude. Because of the stochastic and dynamic nature of island systems, one might expect that the study of variability within and among archipelagos is at least as important as the functional form of the SAR, especially if the aim is not only to describe the form of the relationship but to quantify the predictive power of the relationship. Knowing the degree of uncertainty in the predictions (Berger, 1985) and whether the uncertainty is mainly within versus between studies is crucial for effective application of SAR models. We are not aware of any meta-analysis on SARs that systematically inspects the within-study variance as a function of covariates and determines its implications for prognosis.
We suggest the appropriate meta-analytical solution to the problem of jointly modelling the intercept, slope and the within-study variance of SARs when multiple covariates influence overall patterns is to use a hierarchical mixed model (McCulloch & Searle, 2002; Cressie et al., 2009). In contrast to the traditional meta-analysis (Drakare et al., 2006), this approach uses not just the estimates but the complete data from various studies which are combined in a full likelihood approach, thus improving its statistical efficiency. We use SAR studies of (non-nested) islands, taking into account study-specific covariates such as taxonomic group, island type, latitude and spatial extent. We demonstrate the usefulness of the proposed hierarchical SAR model to further understand the complexity of SAR patterns and its benefits for conservation through improved predictive performance. Using latitude as an example, we also demonstrate how this model is useful for studying scale dependence in macroecological patterns.
MATERIALS AND METHODS
Conceptual overview of SAR models
The power-law SAR is recommended for use in describing and comparing SARs based on theoretical considerations and empirical results (Dengler, 2009). We assumed that the power-law SAR relationship, S=cAz, holds in our analysis. This model easily translates into a LM on the log–log scale using ln(S) = ln(c) +z ln(A). Use of this model also made our results comparable with previous studies (Connor & McCoy, 1979; Drakare et al., 2006). To account for measurement error and unexplained variation, a random component is added making this into a stochastic model, ln(S) = ln(c) +z ln(A) +ε, where ε is a normal random variable with mean 0 and variance σ2[ε∼N(0, σ2) in compact notation].
A trivariate model (Tjørve, 2003; Adler et al., 2005; Storch et al., 2005; Qian et al., 2007; Kallimanis et al., 2008; Hurlbert & Jetz, 2010) describes the relationship between species richness, area and a third variable. We will call this model SARX, indicating that it describes a trivariate species–area–X relationship. This model studies how an additional covariate, X, affects the intercept and slope of the SAR through the LMs, ln(c) =α0+α1X, and z=β0+β1X. By substituting these into the log–log model of the SAR, we get the model ln(S) = (α0+α1X) + (β0+β1X) ln(A). This can be rearranged as ln(S) =α0+α1X+β0 ln(A) +β1X ln(A), which is a LM with two main effects and an interaction term. Similar to the SAR, one can add a random component to SARX, ln(S) =α0+α1X+β0 ln(A) +β1X ln(A) +ε.
Now we generalize the SAR model for combining data from multiple studies using the following hierarchical model. Studies are indexed by i= 1, 2, . . . , n. Islands within a study are indexed by j= 1, 2 , . . . , mi. The SAR model for the ith study is given by ln(Sij) = ln(ci) +zi ln(Aij) +εij, where εij∼N(0, σi2). We have n study-specific intercept (ln(ci)), slope (zi) and within-study variance (σi2) parameters. These study-specific parameters vary from study to study. We model this between-study variation by further assuming that ln(ci) ∼N(α0, τα2), zi∼N(β0, τβ2) and ln(σi) ∼N(θ0, τθ2). This leads to the classic linear mixed model (LMM) (McCulloch & Searle, 2002). We call this model the hierarchical SAR (HSAR) model. We are interested in estimating the global means (α0, β0, θ0) of the study-specific estimates and the between-study variances (τα2, τβ2, τθ2). The between-study variances determine how variable the study-specific estimates are around their global values.
We can modify the above model to account for the effect of covariates on the HSAR parameters so that ln(ci) ∼N(α0+α1X, τα2), zi∼N(β0+β1X, τβ2) and ln(σi) ∼N(θ0+θ1X, τθ2). We call this model the HSARX model. The HSARX model can be extended to accommodate more than one covariate in a straightforward fashion. In Fig. 1, we depict graphically the relationships between these different models.
Island area (in km2) and species richness values were compiled from 94 studies. The combined data set comprised data from 1918 different islands; for 79 islands we had observations for two taxa, so altogether we had 1997 observations. Island areas ranged over ten orders of magnitude, from 1 m2 to 757,770 km2, species richness ranged from 0 to 1666. We also included islands with 0 species, to avoid biased estimates of z-values (Williams, 1996; Dengler, 2010). Islands were grouped according to studies; the number of islands per study ranged from 5 to 86 with an inter-quartile range of 11–25 islands per study. Studies were grouped according to taxonomic groups and island types. Taxonomic groups included birds (migratory types varied among studies), non-volant mammals and vascular plants. Island type involved three categories. Non-marine islands were isolated habitats situated on continents (islands in inland waters were excluded), including sky islands in a desert matrix or forest fragments in agricultural landscapes. Landbridge islands contained marine islands situated on the continental shelf, while oceanic islands included islands on oceanic plates. Taxonomic groups were selected in order to have at least three studies per island type. In addition to study categorization, we determined the latitudinal position of the study, by determining the latitudinal band of its extent. We then used the latitudinal middle band of the studies without differentiating between hemispheres, hereafter referred to as latitude. We also calculated the latitudinal extent of the studies as the latitudinal difference between the endpoints of the latitudinal band. Extent was log-transformed due to its skewed distribution. The log of extent is hereafter referred to as extent. The database used for the studies along with the characteristics of the covariates can be found in Appendix S1 in the Supporting Information.
We used the raw data rather than the estimates of SAR parameters to obtain empirical Bayes estimates. Such pooling of information is known to lead to better estimates than any single study estimate (e.g. Efron & Morris, 1973). To see how study-specific SAR parameter estimates can be improved with the empirical Bayes approach, we first fitted study-specific SAR models. In the following, i denotes the study index that goes from 1 to 94. The number of islands in the ith study is indicated by mi. The number of species on the jth island in the ith study is indicated by Sij and the corresponding area is Aij. We used the individual studies with mi islands (j= 1, 2, . . . , mi) to fit the LM, ln(Sij+ 0.5) = ln(ci) +zi ln(Aij) +εij, where εij∼N(0, σi2) is the within-study variance. This corresponds to the basic SAR conceptual model (Fig. 1). Fitting this model to each of the 94 studies resulted in estimates of ln(ĉi), , , (i= 1, 2, . . . , 94). We added a small constant (0.5) to S to ensure a valid logarithm for all values (Digby & Kempton, 1987, p. 14). We used natural logarithms throughout.
We combined species richness data from all 1997 observations from the n= 94 studies in a single LMM corresponding to the HSARX conceptual model (Fig. 1): ln(Sij+ 0.5) = ln(ci) +zi ln(Aij) +εij. Dependence of study-specific intercepts (ln(ci)), slopes (zi) and within-study variances (σi2) on covariates were further modelled by the equations ln(ci) =α0+XiTα+εαi, zi=β0+XiTβ+εβi, and ln(σi) =θ0+XiTθ, +εθi, where the error terms (εαi, εβi, εθi) followed a normal distribution with mean 0 and variances (τα2, τβ2, τθ2), respectively. Xi is the covariate matrix with study characteristics (taxonomic group, island type, latitude, log of latitudinal extent, and all second-order interactions), α0, β0 and θ0 are intercepts and α, β and θ are vectors of parameters influencing the ln(ci) and zi parameters of the power-law relationship and the log of the within-study standard deviation (ln(σi)), respectively. Spatial scale is known to influence SAR parameter estimates (Rosenzweig, 1995; Crawley & Harral, 2001; Dengler, 2009), thus we checked if there remained any residual variation in the estimates of SAR parameters (ln(ĉi), , ) that is related to spatial scale (range of island sizes). We plotted the residuals against the means of the log areas within studies, and found no pattern (see Appendix S2), which implies that we have taken into account important covariates.
The variance components τα2, τβ2 and τθ2 are the between-study variances corresponding to the variability unexplained by the linear predictors in Xi. We quantified the proportion of the variance explained by the covariates in Xi by comparing the variance estimates to the model with only the intercept included (null model, corresponding to the HSAR conceptual model; Fig. 1). We used the log–log version of the power-law SAR, because it is most straightforward for mixed effect modelling. Nonlinear mixed models are available (Pinheiro & Bates, 2000) where one has to specify the form of the variance function to be able to model the within- and between-study variability. We are currently unaware of any attempt that describes the form of the variance function applicable for SAR-related mixed models. Hence we used the well-established log–log form of the power-law SAR. Extending the mixed modelling framework for other functional forms of the SAR would be interesting but is outside of the scope of this paper.
We used the data cloning algorithm (Lele et al., 2007, 2010) to obtain maximum likelihood estimates and asymptotic variances for the LM (ln(ĉi), , , i= 1, 2, . . . , 94), and LMM parameters (, , , , , , , , ) (see Appendix S3). Hypothesis tests for parameters (H0: estimate is zero, no effect) were based on asymptotic Wald-type confidence intervals. Data analysis was done in the R statistical environment (R Development Core Team, 2009), data cloning and tests of convergence were performed by the ‘dclone’ R package (Sólymos, 2010a) built on the JAGS software (Plummer, 2009). The data sets used in this paper and code to fit the various SAR models presented here are made available in the ‘sharx’ R package (Sólymos, 2010b).
The inference on the model parameters helps in understanding how different ecological mechanisms affect the shape and variability in the power-law SARs. The next step in the analysis is prediction. Prediction has two purposes. First, prediction for the observations gives feedback about the model performance and can serve as a measure of goodness-of-fit. Second, it has profound applied relevance for predicting unobserved cases. Applied aspects of prediction include decision making in conservation that must rely heavily on the mechanistic understanding of the modifiers of the SAR pattern. The mixed model approach can serve this purpose well as opposed to a black box approach where mechanistic details are unimportant and hidden (e.g. Breiman, 2001). For the single study SAR models, we wished to predict species richness for a new island of a known area within the ith study system for which SAR data (Sij and Aij) are available for mi locations. The usual prediction based on linear regression (e.g. Ramsey & Schafer, 2002, Ch. 7) uses parameter estimates based on the data for that study only, whereas we used the parameter estimates obtained from the LMM meta-analysis, where information across all studies was pooled together using the likelihood formulation described above. We used the conditional distribution of the species richness of the new island given the observed data to obtain predictions (Lele et al., 2010; Sólymos, 2010a). We used the mean of this distribution as the point prediction value and the 90% middle section as the prediction interval. We call this level of prediction ‘new island’ prediction. We calculated mean squared error based on LM and LMM ‘new island’ predictions and the observations to express the fit of prediction to the data. We also calculated the length of the prediction interval at the midpoint of the ln(Aij) range based on LM and LMM results.
Similar to the ‘new island’ prediction, we wanted to conduct ‘new study’ prediction when we had no observations on ln(Sij+ 0.5) from a particular island system except for the study-specific covariate values Xi (see Appendix S3). This prediction usually results in wider prediction intervals than the ‘new island’ prediction because the ‘new study’ prediction is not conditional on the observations within the study, but depends only on Xi. The ‘new study’ prediction represented the case where we were interested in knowing possible species richness values for different ln(Aij) values given a particular taxon, island type, latitudinal position and study extent. We studied the performance of LMM meta-analysis approach using cross-validation where we excluded one study from the observations, and calculated expected values and prediction intervals of ln(Sij+ 0.5) for the omitted study using its ln(Aij) and Xi values. To express the performance of cross-validation, we calculated the proportion of observations falling outside of the prediction interval at different α levels (the cross-validation error rate). If the error rate is less than or equal to the α-value, we can say that the predictive ability of the LMM approach is acceptable.
The third level of prediction is the ‘global prediction’, when we have neither observations on ln(Sij+ 0.5) nor study covariates in Xi. Thus, in this case, we are integrating over all possible taxa, island types, latitudes and extents. We only use estimates of (, , , , , , , , ) from the LMM model. For this prediction we resampled the studies with replacement 100 times to keep covariates in Xi random and calculated expected values and prediction intervals for ln(Sij+ 0.5) at various ln(Aij) values. We used the same resampling-based ‘global prediction’ approach to study the effect of covariates by systematically changing one or some covariates while keeping the values of other covariates random.
To help understand scale dependence with the HSARX model, we hypothesized different scenarios for the effect of latitude on the SAR as summarized in Appendix S6. Then, we used the ‘global’ and ‘new study’ prediction approaches to explore the effect of latitude on SAR. When evaluating the results, we compared entire SAR lines and not slope or intercept values individually. The value of the intercept depends on the basic unit of area used, but the relationship of the lines in the log richness–log area plots is not affected.
All the covariates, including taxonomic group, island type, extent and latitude, and their interactions significantly influenced one or more SAR parameters (intercept, slope, within-study variability) based on the LMM estimates (Table 1). The LM and LMM estimates of the study-specific intercept (ln(ĉi)) and slope () values were similar (Appendix S4), although some of the LM estimates were extreme. The absolute difference between the LM and LMM estimates was negatively correlated with sample size mi (Spearman's rank correlation; intercept, r=−0.19, P= 0.07; slope, r=−0.2, P= 0.55) reflecting the fact that the shrinkage factor for LMM is larger for smaller sample sizes than for larger sample sizes. LM estimates often had higher variances than the corresponding LMM estimates, and the absolute deviation between the variances of the LM and LMM estimates was significantly negatively correlated with sample size mi (Spearman's rank correlation; intercept, r=−0.46, P < 0.001; slope, r=−0.51, P < 0.001). Estimates of the within-study variances () were similar based on the LM and LMM models. Standard errors of the variance estimates were higher for single-study LM than for LMM (Appendix S4). Consequently, the LMM approach resulted in better estimates and standard errors for small sample sizes compared with the single study LM. This was to be expected because LM estimates were based on study-specific data whereas LMM pooled information across studies and provided empirical Bayes estimates.
Table 1. Estimates and 90% confidence limits (CL1, CL2) of the hierarchical species–area relationship (HSARX) model parameters based on the linear mixed model. Intercepts include the reference categories of birds and landbridge islands. Area was measured in km2, extent in degrees of latitude. Study-specific intercept and variance values are log transformed in the HSARX model. A colon indicates interaction and bold numbers indicate effect sizes different from 0 based on 90% confidence limits.
The LM and LMM approaches performed equally well for prediction of ‘new islands’ within a study. Mean squared error of the prediction was sometimes higher for LMM, because the estimates reflected the shrinkage due to the information from other studies, while LM-based predictions were based only on a particular study (Appendix S4). Deviation between expected values and observations was sometimes higher for LMM, for example, in the case when LM estimates were unstable due to small sample size (Appendix S4). The accuracy (measured as mean squared error; Appendix S4) of LM-based prediction in this case outperformed the LMM-based predictions, but the relationship might not be ecologically sensible, e.g. slope may be negative due to small sample sizes and the influence of uncontrolled island level covariates indicating the decrease in species richness with increasing area (cf. Fig. 2; study of Hice & Schmidly, (2002) with six islands). According to the cross-validation results, the ‘new study’ prognosis based on the LMM was accurate and the actual coverage is comparable to the nominal prediction level (Appendix S5). ‘New study’-based prediction intervals were naturally wider, but when the number of islands was low, the additional uncertainty compared with the ‘new island’ prediction was small (Fig. 2).
We used the ‘global prediction’ approach to calculate SAR parameters for different covariate settings. We found that the ‘global’ SAR pattern (all covariates are random), inevitably has large uncertainty (Figs 2 & 3): the mean intercept (ln(c)) was 1.838 with 90% confidence limits of −0.562 and 4.705, the mean slope (z) was 0.228 with 90% confidence limits of 0.059 and 0.412. The mean of the within-study variability (ln(σ)) was −0.972 with 90% confidence limits of −1.764 and −0.180. Deviation from the ‘global’ average became more pronounced and confidence limits became shorter as we used specific values of the covariates to model intercept, slope and within-study variability (Fig. 3). For example the mean slope value for birds on non-marine islands at the equator (latitude = 0°) and with study extent 1° was 0.166 with confidence limits of 0.020 and 0.310 (Fig. 3). This reflects the joint effect of multiple covariates on the SAR parameters.
Effect of latitude
When we fixed only latitude and kept the other covariates random, we found that the intercept was lower at the equator than at higher absolute latitudes and the slope changed accordingly without intersecting within the range of observed area values (Fig. 3). When we fixed both latitude and extent, we observed that the intercept was higher at the equator for small extent (≤ 1°), while it was lower at the equator for larger extent (5°) (Fig. 3). We plotted expected SARs for all combinations of taxa and island type, with extent being 1° and latitude being 0° and 45° (Fig. 4). We observed that intercept was higher at the equator on non-marine islands, but not on marine (landbridge, oceanic) islands. The effect of latitude on SARs was scale invariant (lines were parallel) for plants and birds on non-marine islands and for mammals on landbridge islands (Fig. 4).
Generality of the power-law SAR
The most serious objection to the use of the pure power-law SAR when comparing different studies is that the slope varies with spatial extent (Crawley & Harral, 2001; Fridley et al., 2005; Drakare et al., 2006). We included latitudinal extent as a measure of spatial scale (extent) and found that it had a negative but not significant main effect on SAR slope, although its interaction with taxonomic group and island type was significant, indicating the importance of modelling the interactions among covariates. We found no residual pattern that could be attributed to unexplained variation caused by spatial scale. The power law is a bivariate linear relationship in the log richness–log area plane. A trivariate relationship between log richness, log area and an additional continuous covariate including the interaction of log area and the covariate stretches a plane as the SAR relationship. The projection of the observations onto the log richness–log area plane can appear as nonlinear if the covariate is correlated with log area. This simple explanation based on multiple linear regression can account for nonlinearities, i.e. the small-island effect (Lomolino & Weiser, 2001), found among SARs if the continuous covariate has a modifying effect and is correlated with log area, as in the case of habitat diversity (Triantis et al., 2006).
Many functional forms have been proposed to describe the SAR (Tjørve, 2003; Dengler, 2009). We did not make comparisons with competing forms of the SAR due to a lack of knowledge regarding the variance components necessary to fit nonlinear mixed models. But prediction accuracy of the HSARX model showed that the power law can be used to make robust predictions. Our results on the variance explained by covariates indicate that model fit can be substantially improved by additional covariates without changing the functional form of the SAR. Alternative functional forms for a bivariate relationship cannot account for all the variability caused by multiple interacting factors, because area cannot be a perfect approximation for all unaccounted for covariates. Additional covariates that are not correlated with area (e.g. latitude) can add noise to the projection of the observations onto the log richness–log area plane. Both nonlinearities and residual variation in the log richness–log area plane can be accounted for by choosing adequate covariates.
The ‘global’ prediction of the SAR revealed an intercept of 1.838 and slope of 0.228, both associated with huge variation (left plot in Fig. 2; see also Fig. 3). This slope value is close to the theoretical value of 0.25 and within the range most often found empirically (e.g. Drakare et al., 2006, found 0.27). It is important to note that the ‘magic’ number 0.25 is not one of the estimated parameters of the LMM, but it is the mean of the study-specific slope estimates, or equivalently, the slope of the average line of the global SAR prediction when study characteristics are treated as random (Fig. 2). Consequently, it is likely that the 0.25 value is a result of many processes, and not one single or a few mechanisms are responsible for it. It is probably a result of the interaction of multiple processes (Lawton, 1999; Nekola & Brown, 2007). It must be emphasized that our results, as in any other analysis, are conditional on the data, and validity depends on how representative the sample is to the population in question, here the SAR studies over the globe. Reducing possible biases in the data is only possible by increasing their representativeness by adding new studies. To this end, we share our data base (Sólymos, 2010b) in the hope that it will serve as a transparent and extendable resource for further understanding of SARs.
Confidence intervals for the expected SAR parameter estimates were shorter when some or all covariates were known (Fig. 3). The intercept of the SAR was significantly influenced by taxonomic groups, it increased in the mammals, birds, plants direction (‘global prediction’ values 0.787, 2.31 and 3.719, respectively; Fig. 2) that corresponds well to the overall species diversity of these taxa. Taxonomic group significantly influenced the slope of the SAR also, and it increased in the mammals, birds, plants direction (‘global prediction’ values 0.191, 0.221, 0.273, respectively; Fig. 2), indicating that species accumulation was fastest for mammals and slowest for plants. This corresponds to expectations of beta diversity in these groups, higher SAR slope indicating higher beta diversity due to the higher rate of decline in community similarity (Soininen et al., 2007). The within-study variability was lowest for birds and highest for plants, indicating that other unaccounted for factors influenced richness the most in plants and less in birds.
Island type had a significant effect on SAR intercepts and the ranking of island type groups based on their intercept estimates (‘global prediction’ values; non-marine, 2.064; landbridge, 1.842; oceanic, 1.546) coincided well with the expectations based on the equilibrium theory of island biogeography (MacArthur & Wilson, 1967). The number of species in equilibrium decreased with isolation for islands of similar size because of decreased colonization rates. Drakare et al. (2006) found only habitat (similar to our island type factor) to significantly affect the slope of the SAR on true islands. But they used more habitat types for non-marine islands than we did. We did not find the main effect of island type on SAR slopes to be significant, but its interactions with all other covariates were significant. Slope values based on ‘global prediction’ for non-marine (0.239) and landbridge islands (0.269) were consistent with expectations based on the equilibrium theory of island biogeography (Preston, 1960; MacArthur & Wilson, 1967). This pattern reflects the effect of isolation on species dispersal and consequently on species turnover. MacArthur & Wilson (1967) hypothesized that slope should increase with increasing isolation, while Schoener (1976) hypothesized that slope should be higher for near isolates and lower for non-isolates and far-isolates. Schoener (1976) argued that this pattern can be explained by the smaller size of the species pool for highly isolated archipelagos. We found that ‘global prediction’ of slope for landbridge islands (near-isolates) was higher than for oceanic islands (far-isolates; 0.163). The within-study variability was highest for landbridge islands, indicating that non-equilibrium processes might make richness values deviate from what is expected in equilibrium more often than in the case of non-marine and oceanic islands.
The effect of extent on SAR intercept was significant and negative, indicating that as the extent of the studies increased, the biotic surveys could have been less exhaustive. This finding may indicate the possibility of observation error in the measurement of total species richness (Borges et al., 2009). The main effect of extent on SAR slopes was negative but not significant, indicating that SAR slope is slightly decreasing with increasing spatial scale (latitudinal extent). The interaction of extent with island type and taxonomic group was significant, indicating considerable heterogeneity in the scale dependence of SAR slopes with respect to these covariates. Within-study variability increased with extent (Fig. 3), indicating that studies with a larger latitudinal extent can represent higher residual variation due, for example, to increased chance of including islands that are not in equilibrium or due to the unaccounted effects of isolation.
Latitudinal gradient of SAR
Our modelling approach focused on how SARs vary with latitude, and not on how the latitudinal diversity gradient varies with spatial scale. Nevertheless, discussing the implications in the latitudinal diversity gradient context is useful. Our results suggest the expected latitudinal diversity gradient with a peak of richness at the equator for non-marine islands and, unexpectedly, a reversed latitudinal gradient for marine islands. A reversed latitudinal gradient, when diversity is lower at the equator, is not impossible. On relatively small spatial scales it can be a result of the mid-domain effect (Colwell & Lees, 2000) and the latitudinal variation of range sizes (Rappaport effect). But the reversed pattern has been reported on larger scales also. Rabenold (1979) found a reversed latitudinal gradient for eastern deciduous forest bird faunas in North America, where the reversed gradient was due to summer migration of birds to northern territories. Buckley et al. (2003) also reported a reversed latitudinal diversity gradient for pitcher plant communities due to southern distribution of a generalist predator. The fact that reversed patterns can occur as a result of geometric constraints (Colwell & Lees, 2000) and appear in highly isolated systems (Buckley et al., 2003) might mean that the nature of truly isolated areas and their dynamism can explain why patterns found here disagree with latitudinal gradient patterns observed on continuous areas.
Previous studies investigating how the slope of the SAR is changing with latitude (Connor & McCoy, 1979; Martin, 1981; Drakare et al., 2006) cannot help to solve this puzzle, because without knowing the latitude–intercept relationship, it is impossible to differentiate between the non-rank-invariant and rank-invariant manifestations of the scale dependence (Scheiner et al., 2000). We collected studies that used the trivariate SARX model for latitude, or a closely related measure, energy availability, and compared those findings with ours (Appendix S6). Lyons & Willig (2002), Rodríguez & Arita (2004), Qian et al. (2007) and the South African data set of Storch et al. (2005) represented non-marine systems for birds, plants and mammals. All these have found the expected latitudinal diversity gradient in the range of observations, but the latitudinal diversity gradient showed scale dependence with intersecting SAR lines. An interesting exception was the data set of Storch et al. (2005) for birds of the UK. Estimates of the UK data set for the species–area–energy relationship indicated a reversed latitudinal diversity gradient at the range of observations for birds in a country that is a landbridge island system. Although none of these studies have used non-nested areas to establish SARs, these findings of previous studies match our results: latitudinal diversity gradient is present at large spatial scales (grain sizes, i.e. island area) on non-marine islands for all taxa, and we can observe a reversed gradient for birds on landbridge islands. We acknowledge that our present data set might not be fully representative for all islands of the globe. But the fact that SAR slope is significantly affected by latitude and by the interaction of latitude, taxa and island type together with the external validation of our modelling approach indicated that the interaction of space and latitude requires further attention to reassuringly unify two basic macroecological phenomena: the SAR and the latitudinal diversity gradient.
Prognosis: an important feature of HSARX model
Unlike single-study SAR models the hierarchical SAR approach resulted in multiple levels of prediction with associated prediction intervals due to modelling of the between- and within-study variability. By using the hierarchical SAR approach, it is possible to make prognoses for unobserved island systems or unobserved states of already observed island systems. We can make prognoses for changes in study covariates (variables in Xi) and changes in area (Aij) at the same time. For example, invertebrates making up large proportions of total biodiversity compared with plants and higher vertebrates are usually underrepresented in conservation decisions (Brummitt & Lughadha, 2003). By including SAR studies of underrepresented taxa in the full likelihood model, it is possible to predict their SAR parameters for other study locations where data on well-studied taxa are available. Decision making should consider not only the point estimates (e.g. ranks of hotspots) but also the associated uncertainty (McCarthy et al., 2010), which the hierarchical approach can provide on various spatial scales.
We have developed a generalization of the power-law SAR model, the HSARX model that allows the inclusion of multiple focal parameters (intercept, slope, within-study variance), the use of multiple effect modifiers based on a collection of SAR studies and modelling the between- and within-study variability. The global pattern in the SAR was the average of local SARs and had wide confidence intervals. The intercept, slope and within-study variability of local SARs showed great heterogeneity as a result of the interaction of modifying covariates. Confidence intervals for these SAR parameters were narrower when other covariates besides area were accounted for, thus predictions for species richness were more accurate. The significant effect of latitude and the interaction of latitude, taxa and island type on the SAR slope indicated that the ‘typical’ latitudinal diversity gradient can be reversed in isolated systems. The power-law relationship underlying the HSARX model provided good fit for non-nested SARs across vastly different spatial scales by taking into account other covariates. The HSARX framework allows researchers to explore the complex interactions among variables, to explicitly study the scale dependence, and to make predictions on multiple levels (island, study, global) with associated prediction intervals. We demonstrated that from a prediction perspective, it is not the global pattern but the local variation that matters.
Péter Sólymos is a post-doctoral researcher interested in community ecology and biodiversity assessment, and is developing quantitative techniques to study the effect of human development on biodiversity.
Subhash Lele is a professor of statistics interested in applying state-of-the-art statistical methods to ecological problems.