A new non‐parametric method for analyzing replicated point patterns in ecology

Abstract

Most ecological studies that involve point pattern analyses are based on a single plot, which prevent the separation of the effects of various processes that could act simultaneously, as well as limiting the conclusions that can be extracted from these studies. However, considering the spatial distribution of individuals in several plots as replicates of the same process could help to differentiate its specific effects from those of other confounding processes. Thus, we introduce a new method for analyzing spatial point patterns that are replicated according to a two‐factorial design. By summarizing the spatial patterns as K‐functions, the proposed method computes the average K‐functions for each level of the two factors (i.e. predictors) and for each combination of levels, before estimating the sum of squared deviations from the overall mean K‐function. Inferences of the strength of the relationship between the predictors, their interaction, and the spatial structure are made based on a non‐parametric bootstrap procedure, which considers the dependency among spatial scales. We illustrate the proposed approach based on an analysis of the effects of altitude (with two levels: low and high) and slope (with two levels: flat and steep slopes) on the spatial pattern of Croton wagneri, a dominant shrub in an Andean dry scrubland. Our method detected a significant effect of the interaction between slope and altitude, which could not have been detected using current point pattern analysis methodology. The prevalence of single‐plot analysis in ecological studies may be due to a lack of familiarity with appropriate methods for replicated point patterns, as well as the greater complexity of these methods and the absence of appropriate software. Our approach can be applied to a significant number of ecological questions while maintaining a simple, understandable, and easily reportable methodological framework.

The spatial ecology paradigm (Tilman and Kareiva 1998) highlights the relevance of space for the functioning of ecosystems as well as recognizing that the assembly and functioning of plant populations and communities are also governed by spatial processes. Following the pioneering work of Watt (1947), extensive evidence has accumulated to demonstrate that the spatial distribution of plant species is not random, but instead individuals often exhibit local aggregation and/or over‐dispersion within populations and communities (Watt 1947, Pielou 1977, Kenkel 1988, Barot et al. 1999, Condit et al. 2000). In this context, point pattern analysis has emerged as a powerful tool for integrating space into ecological theory, but more specifically for connecting the actual patterns with the processes that generate them (Wiegand and Moloney 2014). A basic assumption when analyzing ecological point patterns is that the patterns conserve some signature from past processes, and thus they constitute an ‘ecological archive’, which can be used to recover information regarding the underlying processes (McIntire and Fajardo 2009, Wiegand and Moloney 2014). For example, aggregated spatial patterns can be related to limited dispersal (Seidler and Plotkin 2006) and over‐dispersed patterns to intraspecific competition (Kenkel 1988).

The links between processes and actual patterns are not always evident or direct because they may be confounded by the interactions among several processes (Wiegand et al. 2009), or by the intrinsic spatial nature of the processes that generate the patterns (e.g. environmental heterogeneity; Getzin et al. 2008, Chacón‐Labella et al. 2014, Jara‐Guerrero et al. 2015). Indeed, a methodological limitation that contributes to the prevalence of these confounding effects is the fact that most ecological studies involving point pattern analysis are based on only one plot (or on several plots that are analyzed individually). However, the spatial distribution of plants in several plots can be considered as replicates of the same process (Bagchi and Illian 2015), which could help to separate its effects from those of other confounding processes (e.g. the typical confounding between a clustering process and environmental heterogeneity; Law et al. 2009, Diggle 2014). As a consequence, the use of replicated patterns combined with an appropriate sampling design would help to generalize the insights gained from point pattern analysis, where this approach could also be employed to test specific hypotheses related to specific processes that affect the spatial structure of study populations or communities. For example, tests could determine whether there are differences in aggregation between various groups of species in a community or between life stages (i.e. adult vs juvenile) in a population (LeMay et al. 2009), or if these differences occur because the populations grow on different soil substrates (Schenk et al. 2003).

Methods for analyzing replicated point patterns are available in the statistical literature, but they have been used only rarely in ecological studies (but see Schenk et al. 2003, LeMay et al. 2009). Briefly, they can be classified into parametric and non‐parametric methods (Landau and Everall 2008). Parametric methods use maximum likelihood or pseudo‐likelihood methods to fit the parameters of a point process model. These methods have been employed to fit certain homogeneous point processes such as pairwise interactions (Diggle et al. 2000, Mateu 2001) or Gibbs point processes (Bell and Grunwald 2004). Non‐parametric methods evaluate differences in summary functions (mainly Ripley's K‐function) among experimental or observational groups. Two main alternative approaches have been developed within the non‐parametric methods: the method described by Diggle et al. (1991), which is analogous to the analysis of variance (ANOVA) for a one‐way design, and the method proposed by Landau and Everall (2008) for repeated measures (and other more complicated) designs [although the use of hierarchical Gaussian process regression was reported by Myllymäki et al. (2014)]. In general, it is considered that the performance of non‐parametric methods is more efficient compared with parametric methods and that they are less prone to model mis‐specifications (Diggle et al. 2000, Bagchi and Illian 2015). This versatility supports their use and application in ecological contexts. In order to promote their use in ecological studies, Bagchi and Illian (2015) recently summarized the non‐parametric methods, where they focused mainly on the repeated measures approach, which involves linear mixed modeling of parameters that describe the differences in spatial aggregation among groups of treatments or the effects of covariates, while considering random effects. This approach is extremely useful in some contexts (e.g. when there are many groups or levels (and replicates) for each treatment), but many important ecological questions can be answered more easily with even simpler methods (Murtaugh 2007). In fact, compared with linear mixed models, non‐parametric ANOVA‐like methods are less prone to errors and they are straightforward to explain and understand (Murtaugh 2007).

In this study, we propose an extension of the method described by Diggle et al. (1991) for replicated point patterns in the two‐way case, where we focus on the interactions between two factors that are considered in the (experimental) sampling design.

We applied this new method to a case study based on dry scrubland dominated by Croton in southern Ecuador. Croton wagneri is an endemic shrub, which comprises most of the biomass in the Ecuadorian Dry Mountain Scrub ecosystem, where it acts as a key ecosystem engineer by providing refuge and shelter from harsh environmental conditions for many other species (Valencia et al. 2000, Espinosa et al. 2014); therefore, its spatial structure governs the functioning of these communities. Observational evidence demonstrates that C. wagneri populations have an aggregated spatial distribution and the intensity of the spatial aggregation varies along an altitude gradient in response to changing environmental conditions.

Similar to any other desert or semiarid shrub, the aggregated patterns of C. wagneri could be the result of limited dispersal (Ellner and Shmida 1981, Seidler and Plotkin 2006), facilitation (i.e. the nurse effect, Flores and Jurado 2003), a preference for favorable microsites where water and nutrients are more accessible (Schlesinger et al. 1996, Rietkerk et al. 2002), or a combination of all these mechanisms. In fact, it has been suggested that the characteristic biphasic bare‐cover mosaic in dryland areas is due to the interplay between local positive interactions and competition for water, which generates scale‐dependent feedback, where this effect is locally positive but negative farther away (Rietkerk and van de Koppel 2008, Maestre and Escudero 2009). Both theoretical (Lefever et al. 2009, Siteur et al. 2014) and empirical (Deblauwe et al. 2011) studies have shown that different environmental conditions, particularly aridity and slopes (Lefever et al. 2009, Deblauwe et al. 2011), can alter or modify this feedback, thereby yielding different spatial configurations of standing biomass (Maestre and Escudero 2009).

In our case study, we aimed to test whether these mechanisms govern the spatial distribution of C. wagneri individuals in a scrubland area that extends along an altitudinal gradient (from 1000 to 2500 m), where the aridity decreases from lower to higher altitudes (Richter and Moreira‐Muñoz 2005). We hypothesized that the effect of aridity on the spatial structure of C. wagneri would be modified by the slope. We expected that populations on steeper slopes would be aggregated in a similar manner irrespective of altitude (i.e. rainfall) because of the stressful conditions (thin soils and rapid run‐off) that prevail on steep slopes. By contrast, we expected that populations on gentle slopes or flat ground would exhibit a marked effect of aridity, i.e. altitude, on aggregation. Translated into the terminology of two‐way ANOVA, we predict the existence of an interaction between slope and altitude. We applied our extension of the method proposed by Diggle et al. (1991) to the two‐way design to test for the existence of this interaction. We compared the results of our test with those obtained using a more traditional approach (Seidler and Plotkin 2006), where we applied classical ANOVA to the parameters of a spatial point process model (a Poisson cluster model) fitted to each C. wagneri pattern.

Methods

Study site

The study was conducted in southern Ecuador, on the slopes of an Andean valley (Alamala, cantón Catamayo, 03°58′29′S, 01°25′22′W), which covered an altitudinal gradient from 1400 to 1900 m a.s.l. The annual average temperature and rainfall range from 33.8°C and 423 mm respectively in the lower zone to 27.4°C and 562 mm in the upper zone. Rain is mostly distributed from January to April but almost absent for the rest of the year (Espinosa et al. 2014). Vegetation forms conspicuous perennial patches, which are interspersed with bare soil areas (occupied by annual plants during the rainy season). Croton wagneri (Euphobiaceae) is the dominant shrub that defines the vegetation patches, whereas other perennial species are represented by isolated individuals (Espinosa et al. 2014).

Data collection

In total, 16 plots (with dimensions of ca 30 × 30 m) were studied at two altitude levels: ‘low altitude’ (between 1400 and 1500 m a.s.l.) and ‘high altitude’ (between 1750 and 1900 m a.s.l.). At each altitudinal level, four plots were located on flat ground and four on steep slopes (> 27° inclination). The plots at each altitude were located ≥ 200 m from each other. In each plot, all of the Croton wagneri individuals were mapped using a compact Electronic Laser Hypsometer TruPulse 360°, which allowed us to calculate the horizontal distance, inclination, and azimuth (central angle) from a fixed common vertex to the rooting point of each individual with a precision of 1 cm. Selected maps of C. wagneri for each combination of slope and altitude are presented in Fig. 2. Maps of the spatial distribution of C. wagneri in the 16 plots are provided in Supplementary material Appendix 1, Fig. A1.

Figure 1

(A) Shrubland dominated by Croton wagneri. (B) Study site location.

Figure 2

Representative point patterns of each combination of slope and altitude. Numbers above the panels correspond to the plot numbers in Table 1, i.e. 6: low altitude, flat slope; 7: low altitude, steep slope; 14: high altitude, flat slope; 15: high altitude, steep slope.

Spatial pattern description

To summarize the spatial pattern of C. wagneri in each plot, we used Ripley's K‐function (Ripley 1977). λK(r), where λ is the intensity of the plot, is the expected number of points within in a circular area with radius r around a typical point in the pattern. The expected value for a random pattern is K(r) = πr².

The K‐functions were calculated using a standard unbiased estimator (Ripley 1976) with isotropic edge correction as follows:

(1)

where d_ij represents the Euclidean distance between individuals i and j, I(.) is the indicator function, and is an edge correction, which is defined as the proportion of the circumference of a circle centered on individual i with radius r contained within the sampling area A.

To characterize the magnitude of aggregation in each plot, we also fitted a Poisson cluster process (PCP) model to each observed K function using the minimum contrast method (Diggle 2003). A PCP describes the formation of a pattern as a two‐step process. First, a Poisson pattern of ‘parent’ points is generated with intensity ρ. Next, each parent point produces ‘offspring’, where the number of offspring per parent follows a Poisson distribution, and their locations are independent and isotropically normally distributed around the parent points, with a mean of zero and standard deviation of σ. The theoretical K‐function for a PCP is as follows.

(2)

Estimation of the K‐function for replicated data and between‐group comparisons with a two‐factorial design

A detailed description of the method for the one‐way design proposed by Diggle et al. (1991) and modified by Diggle (2003) is presented in Supplementary material Appendix 1. If we have an experimental or observational design with two crossed factors, i.e. A with levels i = 1,…, a, and B with levels j = 1,…, b, then it is straightforward to extend the approach proposed by Diggle et al. (1991). Thus, the structure of the model used to describe the data is:

(3)

where K_ijk(r) is the K‐function for the kth replicate in the ijth cell of the observational or experimental design; A_i, B_j, and A_iB_j are the effects of the corresponding levels of factors A, B, and their interaction, respectively; K₀(r) is the overall mean K‐function (defined below), and e_ijk(r) is the residual error associated with the function k_ijk(r).

In the case of a two‐factorial design, it is necessary to test for the existence of an interaction between the two factors because the main factors cannot be interpreted if one exists. A statistic analogous to the sum of squares for the interaction term in a two‐way ANOVA is:

(4)

where n_ij is the total number of points (summed up among the k replicates) in the ijth cell of the experimental design (i.e. the total number of points in the plots within the combined levels i and j), r0 is the largest scale (i.e. the maximum r) of interest in terms of spatial differences, w(r) is a function for normalizing large scale variations in the differences between K‐functions (e.g. w(r) = r^–2), is the overall weighted average K‐function, and , , and are the weighted average K‐functions estimated for different levels and combinations of levels for the factors A and B. For a particular level or combination of levels l, the average K‐functions are estimated as

(5)

where m_l is the number of replicates for that particular level or combination of levels, and n_x and are the number of points and the K‐function for each replicate x, respectively.

Following the approach described by Diggle et al. (1991), we use sampling with replacement to estimate the sampling distribution of the between treatments sum of squares (BTSS). First, we compute the residual function for the kth replicate within the combination of the ith and jth groups as follows.

(6)

Including (and subtracting) and allows the individual effects of factors A and B to be removed, which guarantees that the residual functions are exchangeable under the null and alternative hypotheses (Anderson and Ter Braak 2003).

According to the single‐factor analysis described by Diggle et al. (1991), if we sample with replacement from the set of residual functions , then we obtain resamples .: k = 1,…. m_ij; i = 1, …, a; j = ,…, b, which we can use to compute a set of resampled K‐functions under the null hypothesis of no interaction as follows.

(7)

Again, each resample is a complete set of K‐functions (i.e. as many functions as there are replicates in the study), which is used to compute a resampled BTSS value.

Inference of the test result is obtained by comparing the observed BTSS value with the distribution of s resampled BTSS values. In the case where the null hypothesis of no interaction is not rejected, we can proceed to test the effects of each of the individual factors, possibly by using the method of Diggle et al. (1991) for the one‐factor case.

Analysis of validity and power of the test for interaction effects

To analyze the validity and power of the test proposed for detecting interactions, we followed the procedure described by Diggle et al. (1991). We tested the validity of the test in the context of our study (i.e. its ability to yield an acceptable probability of making a type I error) by employing a bootstrap procedure to generate 1000 sets of 16 replicated patterns (i.e. the same number of replicates used in our study), which were simulated to conform with the null hypothesis (i.e. the absence of interaction). If the test is valid, we expect that the probability of rejecting the null hypothesis would be near to its nominal (α) value.

We simulated replicates with similar abundances and spatial patterns to those observed. Most of them exhibited clear aggregated patterns, so we first fitted a PCP (Diggle 2003) to each of the 16 observed replicates, which allowed us to select the parameters for our simulations. In order to simulate the null hypothesis of no interaction, we simulated two subsets where each had eight replicates, one of which was highly aggregated (mean intensity of clusters ρ = 0.0178, i.e. 16.1 clusters per plot) whereas the other was less aggregated (mean ρ = 0.0593, i.e. 53.4 clusters per plot). In both cases, the dispersion was set to σ = 5 m. These parameters were selected arbitrarily from among the range of parameters fitted to the original patterns. The ‘highly aggregated’ and ‘less aggregated’ patterns were assigned to the levels ‘flat’ and ‘steep’ for the factor slope, respectively, whereas the levels ‘low’ and ‘high’ for factor altitude were assigned randomly to each replicate. The numbers of points simulated for each replicate were similar to those recorded in the observed replicates.

In order to evaluate the power of the test and its dependence on the differences in clustering between replicates, we implemented 1000 more simulations (16 replicates each) for each of a series of different scenarios of the alternative hypothesis (presence of interaction). In these simulations, we assigned a fixed, less aggregated structure (PCP with ρ = 0.39, i.e. around 350 clusters per plot) to the plots using the combinations of ‘low altitude–steep slope’ and ‘high altitude–flat slope’, whereas for all of the other groups (plots with the combinations of ‘low altitude–flat slope’ and ‘high altitude–steep slope’), we simulated more aggregated PCP processes (ρ = 0.350 to ρ = 0.001, i.e. from 315 to only one cluster per plot).

Statistical analyses

Using our procedure, we tested for the existence of an interaction between the effects of altitude and slope on the spatial pattern of C. wagneri. We computed K(r) functions from r = 0 to 8 m at 0.05 m intervals and using isotropic edge correction. To compute the BTSS, we set w(r) = r^–2, according to Diggle (2003). The significance of the computed statistic was evaluated with 1000 bootstrap resamples.

To compare the performance of our approach with an alternative method, we also performed rank‐based ANOVA (Hettmansperger and McKean 2011) to analyze the variation among the observational groups for the parameter σ in the PCPs fitted to each actual C. wagneri pattern. This parameter is usually employed as a measure of the degree of aggregation of spatial patterns and it has been employed to assess differences among patterns generated by different processes (Seidler and Plotkin 2006, Jara‐Guerrero et al. 2015).

All of the calculations were conducted in R (R Core Team) using the packages ‘Rfit’ (Kloke and Mckean 2012), ‘spatstat’ (Baddeley and Turner 2005), and ‘ecespa’ (De la Cruz 2008). The functions employed to compute and test the BTSS statistic in both the one‐ and two‐way designs are included in the R package ‘replicatedpp2w’.

Data available from the Dryad Digital Repository: < http://dx.doi.org/10.5061/dryad.72193 > (Ramón et al. 2016).

Results

In all of the plots, the observed K‐functions had patterns that were compatible with PCPs (Table 1, Fig. 3), whereas some plots also exhibited small‐scale inhibition (for r < 1 m) (Fig. 3). In general, the plots of flat slopes at low altitude and the plots of steep slopes at high altitude exhibited less clustering than the rest, although the group of flat slopes at a low altitude was the most aggregated at fine scales (r < 1 m; Fig. 3 and Fig. 4).

Figure 3

K‐functions for the Croton wagneri point patterns under different altitude (number in meters) and slope conditions. We show the L(r) function for representation and interpretation purposes, where is a standardized version of K(r) (Besag 1977). For a random pattern, the function L(r) has an expected value of 0 at all distances r (Dixon 2002). The continuous black lines represent the empirical values of L(r) – r and the dashed red lines represent the L(r) – r function of a Poisson cluster process fitted to each Croton wagneri pattern.

Figure 4

Averaged K(r) functions for each combination of factor levels (transformed as to facilitate interpretation). CSR: expected value for a plot with complete spatial randomness. Global: overall average K(r) function.

Table 1. Summary of the plot attributes. Altitude: meters above sea level. n: number of points (shrubs) per plot. Canopy: mean canopy area ± 1 standard error, estimated by the area of an ellipse from the canopy diameters. Parameters of the Poisson cluster process fixed to each plot. n_c: number of clusters ( = ρ × 300 m × 300 m), μ: mean number of points per cluster, σ: dispersion parameter

Plot	Altitude (m)	Slope	n	Canopy (m^2)	nc	μ	σ (m)
1	1400	flat	859	0.2056 ± 0.0120	36.2	23.7	2.45
2	1400	flat	239	0.1379 ± 0.0137	79.8	3.0	1.14
3	1400	steep	285	0.2295 ± 0.0194	1.8	162.0	8.78
4	1400	steep	395	0.2219 ± 0.0207	20.0	19.8	2.93
5	1500	flat	815	0.1562 ± 0.0069	38.1	21.4	3.00
6	1500	flat	724	0.0960 ± 0.0067	165.0	4.4	1.24
7	1500	steep	296	0.2972 ± 0.0260	33.2	9.0	2.86
8	1500	steep	665	0.1257 ± 0.0079	18.3	36.4	3.90
9	1750	flat	650	0.1303 ± 0.0111	30.5	21.3	2.15
10	1750	flat	679	0.3328 ± 0.0190	19.2	35.4	3.30
11	1750	steep	1959	0.055 ± 0.0022	81.0	24.2	2.05
12	1750	steep	541	0.313 ± 0.0246	59.4	9.1	2.79
13	1900	flat	561	0.3397 ± 0.0319	9.6	58.4	4.19
14	1900	flat	732	0.2865 ±0.0174	1.4	536.6	22.04
15	1900	steep	879	0.0513 ± 0.0045	29.1	30.2	4.94
16	1900	steep	571	0.2430 ± 0.0187	1.7	342.9	18.82

The analysis confirmed the existence of a significant interaction effect between slope and altitude on the spatial pattern of Croton (p = 0.009; Fig. 4; Table 2).

Table 2. Results of the non‐parametric analysis of the replicated Croton wagneri point patterns. BTSS: sum of squared differences. p‐value estimated after 1000 samples with replacement (bootstrap) of the residual functions. To calculate the BTSS, we used K(r) functions estimated from r = 0.05 to r = 8 m, at intervals of 0.05 m

Factor	BTSS	p‐value
Slope	1373518.4	0.039
Altitude	259998.3	0.408
Slope–Altitude	2086014.1	0.009

In general, our validity analysis (Supplementary material Appendix 1, Table A1) showed that the test's performance was quite close to the nominal value. Given the small p value obtained in our study (0.009), we are confident that the null hypothesis of no interaction can be rejected, at least at the α = 0.05 level. In addition, and quite logically, the power of the test increased as the differences in the numbers of clusters between combinations of factor levels increased (Supplementary material Appendix 1, Table A2).

By contrast, the rank‐based ANOVA analysis could not confirm the existence of a difference in the fitted σ values among the altitude and slope groups, or of an interaction (Table 3).

Table 3. Rank‐based ANOVA results for the variation among groups of the σ parameter in the Poisson cluster process fitted to each point pattern. DF: degrees of freedom

	DF	F value	p‐value
Altitude	1	0.625	0.444
Slope	1	0.534	0.479
Altitude–Slope	1	0.663	0.431
Residuals	12

Discussion

Our method for analyzing replicated patterns aims to detect interactions between experimental or observational factors. In our case study, the spatial pattern of all the populations was aggregated. If we had employed the usual approach of using just one plot or simply obtaining a weighted average of the replicated plots to analyze the spatial pattern of Croton wagneri, we could only have concluded that the shrub exhibited aggregation, which is unsurprising given that it is a dispersal‐limited species (Jara‐Guerrero et al. 2011). By using the approach described by Diggle et al. (1991) to analyze the individual effects of slope or altitude, we could have concluded that altitude was irrelevant to the spatial organization of C. wagneri. However, using our new non‐parametric two‐way approach, we were able to determine that the spatial pattern of C. wagneri appeared to be driven by the interaction between altitude and slope (Fig. 5). This result could not have been obtained using the more traditional approach (Seidler and Plotkin 2006) based on comparing the parameters of individual fitted PCPs (Table 3). Alternatively, if spatially explicit environmental variables are available, we can use them for parametric estimation of the intensity functions in each plot (Law et al. 2009, Wiegand and Moloney 2014). The regression parameters of the intensity functions would then show how the plant density depends on the environment, while fitting inhomogeneous PCPs (Waagepetersen 2007) would demonstrate how the residual clustering depends on the environment.

Figure 5

Differences in spatial aggregation between plots. Red boxes: plots on flat slopes; white boxes: plots on steep slopes. ‘Spatial aggregation’ was measured for each plot as the sum of values of the L(r) function []. For all plots, K(r) was computed for r = 0 to r = 8 with 0.05 m intervals. The expected value of the empirical L‐function for a completely random process is L(r) = 0, whereas positive values indicate aggregation.

We hypothesized that differences in aridity would generate differences in aggregation between the low‐ and high‐altitude Croton populations growing on flat ground, and that topography would exacerbate the effect of aridity in populations growing on steep slopes, thereby promoting the greatest aggregation, irrespective of altitude. We expected that this would be detected as a significant interaction effect in our two‐way analysis. In fact, a significant interaction was detected, but it was probably caused by a different mechanism. First, the aggregation on steep slopes was not equally uniform, but instead it differed between low and high altitudes, where the plots on steep slopes at a high altitude exhibited the lowest aggregation on average. Second, in contrast to our initial hypothesis, the highest average aggregation occurred in the populations growing on flat ground at high altitude. This is surprising because this combination does not seem to be the most stressful. Nurse effects and thus more aggregated patterns are expected to be more important in the sites with the most severe environmental conditions (e.g. drier areas at low altitude; Flores and Jurado 2003, Maestre et al. 2009), but other mechanisms could explain the differences in the aggregation of C. wagneri among sites. First, C. wagneri is a barochorous species so it is intrinsically dispersal‐limited (Jara‐Guerrero et al. 2011), which could affect the expected spatial structures due to scale‐dependent feedback between plants and soil water availability (Pueyo et al. 2008). Second, Croton seeds have the potential for secondary dispersal by ants (Jara‐Guerrero et al. 2011), which could differ with altitude (Smith et al. 1989) and interfere with the formation of the spatial structure of C. wagneri via other mechanisms (Kalisz et al. 1999, Zhou et al. 2007).

We suspect that the ‘unexpected’ results obtained for the populations growing on flat ground at low and high altitudes were a consequence of a shift in the vegetation structure in response to the decrease in aridity with altitude. Deblawe et al. (2011) showed that a decrease in aridity caused a transition from isolated ‘spots’ to larger connected masses (‘labyrinths’) of standing biomass. In fact, the average σ parameter among the fitted PCPs was highest for the populations on flat ground at high altitude (σ = 7.92 m, corresponding to an average cluster size of 22.40 m). This was linked to the prevalence of heterogeneous patterns among these plots because the existence of large patches leads to the appearance of virtual aggregation (Wiegand and Moloney 2014). Previously, this has been treated as a nuisance (Schiffers et al. 2008), but the existence of virtual aggregation in our plots reflected the large scale ecological processes that actually control the spatial structure of C. wagneri. By contrast, the fitted values for σ were smallest in the populations on flat ground at low altitude (average σ = 1.96 m, corresponding to a cluster size of 5.54 m), which could help to explain why the average K‐function for this group was the highest at small scales (Fig. 4). In summary, there was a transition from small abundant spots at low altitude (i.e. more severe conditions) to a small number of large patches at high altitude. Thus, irrespective of (or in addition to) nurse effects and limited dispersal, our results suggest that the spatial organization of C. wagneri is driven mainly by eco‐hydrological feedback processes (Pueyo et al. 2008).

The effect of slope varied greatly among altitudes. In contrast to our initial hypothesis, the lowest spatial aggregation appeared on the steep slope plots at high altitude. The average size of the patches in these plots did not differ greatly from those in the high altitude plots on flat ground (σ = 7.15 m), but they exhibited the largest spatial inhibition at fine scales (Supplementary information Appendix 1, Fig. A2). We used the K‐function to measure spatial aggregation, which has ‘memory’ (Wiegand and Moloney 2014), so the low densities at small scales influenced (i.e. decreased) the value of the function at larger scales, and thus the final results at these scales were apparently ‘less aggregated’. In fact, if we had employed the approach of Wiegand et al. (2007, 2009) to remove the confounding effects of small scale inhibition on the K‐function and refitted the Poisson cluster models, we would have obtained results compatible with greater aggregation (e.g. average refitted σ = 2.56 m). Regardless of the method employed, the spatial inhibition reached up to 0.8 m, which excludes the existence of nurse effects at this location.

In addition, the increased spatial aggregation with respect to flat ground populations at low altitude was caused by an increase in the average size of the patches (σ = 4.62 m; average size = 13.01 m) and a parallel decrease in their number (average number of clusters per plot = 18.2). According to the same reasoning employed previously, this suggests that the hydrological environment was better on the slopes than that on the flat ground, which appears odd. An alternative or complementary hypothesis may be related to the fact that Croton was at its uppermost local limit, which suggests that the climate conditions were far from its optimum at these sites. Therefore, the physiological performance of Croton would have been limited under these milder conditions and thus the species may have been severely stressed. This paradoxical situation where the worst performance occurred on flat ground is compatible with the patterns found in the steep, high altitude scenarios simply because more xeric conditions are physiologically favorable for this species.

In conclusion, our new method combined with replicated point patterns obtained from an appropriate sampling or experimental design can yield insights that are difficult to obtain using other spatial techniques. Our method is easy to apply and the results can be interpreted in a straightforward manner in ecological terms. In addition, the reliability and power of our method can be assessed easily in any specific context. This method could be extended to a more general model‐based approach, which could include a number of factors and covariates (i.e. continuous variables) obtained from observational or experimental studies. We consider that the use of replicated spatial point pattern analyses by ecologists should be encouraged in the future.

Acknowledgements

This study was partially supported by the Islas‐Espacio CGL2009‐13190‐C03‐02 and Mountains CGL2012‐38427 projects funded by the Spanish Ministerio de Ciencia, the REMEDINAL3 project financed by Comunidad de Madrid, and a UPM‐SANTANDER scholarship 2012. We thank MALCA S.A. for access to the study area and the following people who assisted with field sampling: Omar Cabrera, Ana Arevalo, Israel Gutierrez, and Ronny Luzuriaga. We thank Belinda Ashe for English language revision. Thorsten Wiegand provided useful ideas and suggested some corrections, which improved the content of this paper.

Supplementary material (Appendix ECOG‐01848 at < www.ecography.org/appendix/ecog‐01848 >). Appendix 1.