## INTRODUCTION

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*=*cA ^{z}* (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 km^{2}; 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 σ^{2} (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.