A peer-reviewed open-access journal

NeoBiota 38: 77—96 (2018)

doi: 10.3897/neobiota.38.235 18 @) NeoBiota

http:/ / neob iota. pen soft. net Advancing research on alien species and biological invasions

Models of alien species richness show moderate
predictive accuracy and poor transferability

César Capinha'?, Franz Essl?,; Hanno Seebens’,

Henrique Miguel Pereira'>*®, Ingolf Kiihn>*’

| CIBIO/InBIO, Centro de Investigagdo em Biodiversidade e Recursos Genéticos, Universidade do Porto,
Campus Agrario de Vairéio, 4485-661 Vairao, Portugal 2 Zoologisches Forschungsmuseum Alexander Koenig,
Adenauerallee 160, D-53113 Bonn, Germany 3 Division of Conservation Biology, Vegetation and Landsca-
pe Ecology, University Vienna, Rennweg 14, 1030 Wien 4 Senckenberg Biodiversity and Climate Research
Centre (SBiK-F), Senckenberganlage 25, 60325 Frankfurt am Main, Germany 5 Martin Luther University
Halle-Wittenberg, Institute of Biology/Geobotany and Botanical Garden, Am Kirchtor 1, 06108 Halle, Ger-
many 6 German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Deutscher Platz 5e,
04103 Leipzig, Germany 1 Helmholtz Centre for Environmental Research — UFZ, Dept. Community Ecology,
Theodor-Lieser-Str. 4, 06120 Halle, Germany

Corresponding author: César Capinha (ccapinha@cibio.up.pt)

Academic editor: Petr Pyiek | Received 9 January 2018 | Accepted 23 May 2018 | Published 6 June 2018

Citation: Capinha C, Essl F, Seebens H, Pereira HM, Kihn I (2018) Models of alien species richness show moderate
predictive accuracy and poor transferability. NeoBiota 38: 77—96. https://doi.org/10.3897/neobiota.38.235 18

Abstract

Robust predictions of alien species richness are useful to assess global biodiversity change. Nevertheless,
the capacity to predict spatial patterns of alien species richness remains largely unassessed. Using 22 data
sets of alien species richness from diverse taxonomic groups and covering various parts of the world, we
evaluated whether different statistical models were able to provide useful predictions of absolute and rela-
tive alien species richness, as a function of explanatory variables representing geographical, environmental
and socio-economic factors. Five state-of-the-art count data modelling techniques were used and com-
pared: Poisson and negative binomial generalised linear models (GLMs), multivariate adaptive regression
splines (MARS), random forests (RF) and boosted regression trees (BRT). We found that predictions
of absolute alien species richness had a low to moderate accuracy in the region where the models were
developed and a consistently poor accuracy in new regions. Predictions of relative richness performed in a
superior manner in both geographical settings, but still were not good. Flexible tree ensembles-type tech-
niques (RF and BRT) were shown to be significantly better in modelling alien species richness than para-
metric linear models (such as GLM), despite the latter being more commonly applied for this purpose.
Importantly, the poor spatial transferability of models also warrants caution in assuming the generality of

the relationships they identify, e.g. by applying projections under future scenario conditions. Ultimately,

Copyright César Capinha et al. This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

78 César Capinha et al. / NeoBiota 38: 77—96 (2018)

our results strongly suggest that predictability of spatial variation in richness of alien species richness is
limited. The somewhat more robust ability to rank regions according to the number of aliens they have
(i.e. relative richness), suggests that models of aliens species richness may be useful for prioritising and

comparing regions, but not for predicting exact species numbers.

Keywords

biological invasions; clamping; model evaluation; predictive modelling; transferability

Introduction

Knowing the distribution patterns of alien species richness is increasingly crucial for as-
sessing and monitoring global biodiversity (Dornelas et al. 2014, Tittensor et. al. 2014,
Capinha et al. 2015, Latombe et al. 2017). Knowledge about alien species richness is
also required by environmental managers and researchers to assist in decision-making
of tasks such as ecosystem restoration (Catford et al. 2012), the identification of points
of entry for introduced species (e.g. Seebens et al. 2013), the quantification of impacts
posed by invasive alien species (i.e. the subset of alien species that have harmful effects
on the recipient ecosystems; Blackburn et al. 2014) or the assessment of the ecological
degradation of habitats (Vandekerkhove et al. 2013).

Despite substantial progress being made in the availability of alien species distribu-
tions, there still are numerous regions worldwide for which alien species richness data
are lacking or highly incomplete (Dawson et al. 2017, Pysek et al. 2017). These gaps
occur at multiple spatial scales and, although often related to lower levels of socioeco-
nomic development (McGeoch et al. 2010), they can also be found in generally well-
studied taxonomic groups and regions (Pysek et al. 2008, 2010). A further challenge
for preparing an inventory of alien species richness is the highly dynamic nature of
alien species spread, which requires regular updates as time progresses (McGeoch et al.
2010, Tittensor et al. 2014, Seebens et al. 2017).

Despite the relevance of information on alien species richness, little work has been
done to assess whether alien species richness can be accurately predicted for areas where
such data are lacking. If this metric is possible to predict with accuracy, then available
data can be used to geographically broaden the current knowledge on biodiversity pat-
terns and to support conservation and alien species management decisions in areas that
are currently not surveyed. Further, high reliability of predictive models would enable
the integration of predictive modelling in alien species mapping.

Two main lines of modelling approaches are used for predicting alien species rich-
ness. The first consists of the use of stacked species distribution models (e.g. Bertels-
meier et al. 2015). Here, alien species distributions are modelled individually as a
function of environmental factors and the predictions are turned into binary maps of
species’ presence/absence. Predictions of alien species richness in regions are obtained
by stacking the individual maps. The second approach consists of statistical modelling
of alien species richness directly as a function of environmental factors (e.g. Jarnevich

Models of alien species richness show moderate predictive accuracy and poor transferability 7s)

et al. 2006, Nobis et al. 2009). The framework for this approach is similar to that of de-
scriptive models relating alien species richness to spatial factors (e.g. Kiihn et al. 2003,
Westphal et al. 2008, Pysek et al. 2010, Blackburn et al. 2016, Capinha et al. 2017,
Dawson et al. 2017), but it goes one step further by using the identified relationships
to make predictions.

Species distribution modelling has been intensively applied to alien species in re-
cent years and modelling practices are increasingly refined (e.g. Calabrese et al. 2014).
Thus, one might expect that the stacked modelling approach would be a preferred
means for predicting alien species richness. However, stacked species distribution
models, which imply developing models for each species individually, are data- and
resource-demanding and may not be applicable when a high number of species or
regions are involved. For instance, one would need to have at least 10 times as many
presence points as one has predictor variables (Babyak 2004) — and ideally at least as
many absence points (but see MaxEnt; Elith et al. 2011). Further, in species distribu-
tion models, one of the most important fundamental assumptions, i.e. the species
modelled being in equilibrium with the environment, is violated. This would lead to
underestimating the potential niche space of the species and would result in biased
models, usually leading to incorrect predictions and inflated turnover rates in projec-
tions (Pompe et al. 2008, Pompe et al. 2011). In this context, statistical models directly
relating alien species richness with spatial drivers (hereafter referred to simply as “spe-
cies richness models”) become particularly relevant as they are less data~-demanding.
However, benchmarking the accuracy and performance of these models using typical
datasets of alien species distributions has rarely been done, which hampers the assess-
ment and comparison of the model predictions.

Here, we perform a formal evaluation of the ability of species richness models to
predict the richness of alien species. We measure and compare the predictive accura-
cies of five modelling techniques extensively used in ecology: 7) a Generalised Linear
Model (GLM) using a Poisson distribution (GLM-P), iz) a GLM using a negative
binomial distribution (GLM-NB), iz) boosted regression trees (BRT), iv) multivariate
adaptive regression splines (MARS) and v) random forests (RF). We assess the ability
of the modelling techniques to predict within the geographical range of the model’s
calibration data (i.e. “geographical interpolation”) and in new, spatially independent
regions (i.e. transferability or “geographical extrapolation”; Wenger and Olden 2012).
We perform this assessment using a collection of 22 datasets of alien species richness
analysed in previous studies.

Methods

Alien species richness data

We collected 22 typical data sets of alien species richness from previous studies
(Table 1). Each dataset provides the total number of established alien species in distinct

César Capinha et al. / NeoBiota 38: 77-96 (2018)

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Models of alien species richness show moderate predictive accuracy and poor transferability 81

geographical units (hereafter referred to as ‘regions’), such as countries, sub-national
administrative regions (e.g. federal states, provinces) or islands. Ten taxonomic groups
are covered by the data. Spatial coverage (extent) differs widely, ranging from the whole
world, the European continent, temperate and subtropical regions to oceanic islands.

In order to have a common geographical basis for the assignment of values of the
predictor variables (below), we matched each region with the corresponding polygon
of the Global Administrative Areas Database v.2.8 (GADM; http://www.gadm.org/).
The GADM is the most detailed delimitation of worldwide administrative divisions
available. We excluded all regions that we could not identify unambiguously, that had
no geographical match in GADM and also regions that were smaller than 1 km? - the
highest spatial resolution provided by gridded predictor variables; see below. In some
cases, this resulted in reduced numbers of records compared to the original datasets
(62% to 100% of the records in the original datasets kept for our analyses, average =
92% + 10.2%). Most data sets in our collection contain a relatively low number of re-
gions with 15 datasets consisting of less than 100 regions and 7 of less than 50 regions

(Table 1).

Predictor variables

We selected nine explanatory variables representing factors that have been shown in
previous studies to explain the variation in alien species richness (Kithn et al. 2003,
Lambdon et al. 2008, McGeoch et al. 2010, PySek et al. 2010, Essl et al. 2011, Essl
et al. 2013, Blackburn et al. 2016, Dawson et al. 2017, Kiihn et al. 2017). These vari-
ables were: geodesic area (log-transformed); insularity (island/mainland); mean annual
temperature; mean annual precipitation; diversity of bioclimatic types; geographical
isolation; GDP per capita; human population density and proportion of area covered
by urban land use.

Geodesic area (km*) was measured using the spatial polygon of the region after
re-projection to a Mollweide equal area projection. Insularity was a binary variable (is-
lands or mainland regions). Mean annual temperature and mean annual precipitation
represent region-wide averages of the corresponding climatic conditions (at ca. 1x1
km) derived from WorldClim (http://www.worldclim.org/). We defined bioclimatic
diversity as the total number of distinct bioclimatic types delimited by Metzger et al.
(2013) that are found within a region. ‘The bioclimatic types defined by Metzger et al.
(2013) consist of 125 divisions that group relatively homogeneous environmental con-
ditions at the global scale. Geographical isolation corresponded to the shortest travel
time possible in the region to reach a populated place with 50,000 or more people as
mapped by Nelson (2008). For each region, we extracted the minimum travel value
found in the region. Gross Domestic Product (GDP) per capita in 2005 — in pur-
chasing power parity-constant 2005 US dollars — and human population size were
retrieved mainly from the World Development Indicators (http://data.worldbank.org/
data-catalog/world-development-indicators) and Gennaioli et al. (2014) but also from

82 César Capinha et al. / NeoBiota 38: 77-96 (2018)

a few other sources including national and regional statistical agencies, Index Mundi
(http://www.indexmundi.com/) and online reports. When data for the year 2005 was
not available, we used data for the closest available year from the first decade of the
21* century. Human population density was calculated by dividing population size of
the region in 2005 by its area. The proportion of area covered by cities was calculated
by dividing the area of urban land cover of the region — as measured from GlobCov-
er2009 (http://due.esrin.esa.int/page_globcover.php) — by region size. For each data
set, we tested for redundancy amongst the predictors using pairwise Pearson correla-
tion and selected only predictors that were moderately correlated (|7|<0.7, Dormann et
al. 2013). The final set of predictors considered for each dataset is shown in the Suppl.
material 1: Table Al.

We performed all data processing in R (v. 3.4.1) (www.R-project.org/). The extrac-
tion of values from the source gridded datasets was done using the ‘extract’ method of

the RASTER (v. 2.3-40) package.

Modelling techniques used for predicting species richness

We tested five techniques for modelling alien species richness: i) GLM-P using a Pois-
son distribution, ii) GLM-NB using a negative binomial distribution, iii) boosted re-
gression trees using a Poisson distribution (BRT), iv) multivariate adaptive regression
splines (MARS) and v) random forests (RF). These methods were selected because they
fall into different positions along the spectrum of statistical assumptions and mod-
elling architectures, allowing a number of relevant comparisons. These include 7) a
comparison of GLMs having a restrictive (GLM-P) and a more relaxed distribution-
al assumption (GLM-NB); iz) comparison of GLMs with machine learning models
(BRT, MARS and RF) and 777) comparison of a linear regression-type machine learn-
ing model (MARS) with tree ensembles-type machine learning models (BRT, RF). We
briefly describe each of the modelling techniques used in the Suppl. material 1: Text
Al. Their implementation is described below.

Generalised linear models using Poisson and negative binomial distributions

We implemented GLM-P using the standard ‘glm()’ function of R and GLM-NB
using the ‘glm.nb()’ function of the MASS (v. 7.3-37) package. The theta parameter,
which represents the dispersion of the data in the calculation of the variance of the
NB distribution, was estimated by means of maximum likelihood. An important step
in the application of GLMs is to identify the ‘best’ combination of predictors (Brewer
et al. 2016), especially when multiple “best models” can be found due to collinearity
(Dormann et al. 2013). We adopted a multi-model inference approach (Burnham and
Anderson 2002), in which we identified the combination of predictors that were best

Models of alien species richness show moderate predictive accuracy and poor transferability 83

supported by the Akaike information criteria corrected for small sample sizes (AICc).
We tested two options to select the models used for comparison with other meth-
ods. First, we simply used the model receiving the highest support from AlCc. Sec-
ond, we accounted for the possibility of multiple models receiving strong support, i.e.
A AlCc < 2 and corresponded to the use of the average of their coefficients. To avoid
overfitting the models by including more predictor variables than warranted by the
number of observations available (Babyak 2004), the maximum number of predictor
variables considered simultaneously in each multi-model comparison was set to one
tenth of the number of observations being used for model calibration (Babyak 2004).
The multi-model assembly and calculation of AlCc were performed using the R pack-
age MUMIN (v. 1.13.4). We found the accuracy of predictions from these two options
to be virtually identical (Suppl. material 1: Table A2) and used the results for the best
supported model by AlCc for comparison with other methods. For GLM-P, over-dis-
persion was calculated and detected for several of the datasets used in this study (Suppl.
material 1: Table A3). For these cases, we also performed multi-model comparisons
using the quasi Akaike Information criterion corrected for small sample size, QAICc, a
criterion more suitable for model selection in the presence of over-dispersion. Adopt-
ing QAICc, however, did not improve the models nor change the general outcome
(results not shown).

Hurdle models (Potts and Elith 2006) using GLM-P and GLM-NB were also
implemented for a subset of datasets with zero inflation (for details on implementa-
tion see Suppl. material 1: Text Al). We found that small sample sizes impeded model
convergence for some datasets and that predictions from converging models were not
significantly superior to those from ‘plain’ GLM’s (Suppl. material 1: Text Al). Accord-
ingly, we refer no further to the results from hurdle models in our work.

Multivariate adaptive regression splines

Multivariate adaptive regression splines (MARS; Friedman 1991) were implemented
using the EARTH (v. 4.2) package of R. Interactions amongst predictor variables were
not considered (i.e. only additive models), as preliminary tests revealed that interac-
tions did not improve predictive performances (results not shown). Exhaustive prun-
ing, a method that considers all candidate model terms, was used for selecting those to

keep in the final MARS model.

Random forests

Random forests (RF; Breiman 2001) were implemented using the ‘randomForest’
(v. 4.6-10) package for R. All models corresponded to an ensemble of 1000 trees and
used a random selection of 4 predictors as candidates for branch splitting.

84 César Capinha et al. / NeoBiota 38: 77-96 (2018)

Boosted r egr ession trees

To implement Boosted regression trees (Elith et al. 2008), we used the ‘dismo’ (v. 1.1-
4) package for R with an assumed Poisson distribution for the response variable. For
all models, we used a learning rate of 0.001, which implies a fairly low contribution of
each tree added to the model and imposes a desirably high number of total trees in the
ensemble (often > 1000) (Elith et al. 2008). Remaining parameters were kept at default
values. Following the fitting of models using all predictors, we performed a stepwise
variable selection procedure, in which any predictor not contributing to the decrease
in model deviance was removed (Elith et al. 2008).

Assessment of predictive performance

The performance of the previous techniques in predicting alien species richness for each
of the 22 datasets was assessed using two distinct approaches. ‘The first was a leave-one-out
cross-validation (Wenger and Olden 2012). In leave-one-out cross-validation, the model
is fitted and used to predict one hold-out observation at a time. This is repeated until all
observations are used for validation. This approach provides unbiased estimates of in-sam-
ple predictive accuracies, but does not allow assessing the model performance for predic-
tions outside the spatial range covered by the data set (Wenger and Olden 2012). The sec-
ond approach was a k-fold regional cross-validation (Jiménez-Valverde et al. 2011). This
approach, which relies on the use of geographically distinct subsets of the data, provides a
reliable assessment of the accuracy of the predictions made to new, unrelated, geographi-
cal domains — i.e. it assesses the spatial transferability and generality of the relationships
identified by the models (Wenger and Olden 2012). Here we used a 4-fold regional cross-
validation in which each dataset was divided into four geographical quadrants based on
the centroids of all regions. Then, regions in three quadrants were used for model fitting
while one was left-out for model evaluation. The procedure is repeated four times, so that
all possible three quadrant-combinations are used for model calibration.

The assessment of validation accuracy was made for two criteria: (1) agreement be-
tween reported and predicted absolute values of alien species richness and (2) agreement
between the rank order of reported and predicted alien species richness. For the first
criterion, we calculated the ‘relative absolute error’ (RAE). An RAE of zero represents a
perfect match between predicted and observed values, while 100% corresponds to the
level of error that is obtained if all predictions simply represent the average of the alien
species richness values used for evaluation (Witten et al. 2016). For the second criterion,
we compared the Spearman rank correlation (g) between predictions and left-out obser-
vations. Both criteria were calculated using the RMINER (v. 1.4) package.

Two distinct evaluation assessments were made for GLM-P, GLM-NB and MARS,
in order to account for their greater susceptibility to errors when predicting beyond the
sampling space of the calibration data (i.e. when extrapolating). While BRT and RF ‘do

not extrapolate’ because they use the closest known subspace of the calibration data as

Models of alien species richness show moderate predictive accuracy and poor transferability 85

target for the prediction (Elith and Graham 2009), GLM-P, GLM-NB and MARS as-
sume that the trend of fitted responses continues outside the sampling space, which may
lead to predictions of unrealistically high values of richness. The first evaluation con-
sisted of performing the predictions from GLM-P, GLM-NB and MARS without any
constraints. The second evaluation consisted in constraining these models to mimic the
behaviour of BRT and RF by means of ‘clamping’ (Elith and Graham 2009). Clamping
corresponds to setting the range margins of values used for prediction to the range mar-
gins found in the model calibration data. That is, if (x < min) then x = min or if (x > max)
then x = max; where x is the value of the predictor in the test data and min and max are
the minimum and maximum values of the predictor in the calibration data, respectively.

Multiple pairwise Wilcoxon tests were used to test for significant differences in
RAE and g between the five modelling techniques. The differences were assessed by
comparing the performances achieved by each method in the 22 datasets of alien spe-
cies richness tested. In the case of GLM-P, GLM-NB and MARS, pairwise Wilcoxon
tests were also used to compare the accuracy from clamped versus the not clamped
versions of the predictions.

Spatial autocorrelation

Spatial autocorrelation (SAC) in the distribution of alien species richness may lead to
incorrect model parameter estimates (Dormann et al. 2007), even resulting in chang-
ing the sign of the relationship (Ktihn 2007). Testing for transferability is particularly
important when the autocorrelation structure of the areas of calibration and prediction
differ (as in the case of cross-validation applied here). We tested for SAC in the residu-
als in all models during the regional cross-validation process by means of correlograms
showing the correlation amongst Pearson residuals of regions over a range of uniformly
distributed distance classes. The statistical significance of the correlations was based
on 1000 permutations for the values of the residuals. The package NCF (v. 1.1-7) for
R was use to perform the analyses. We found no or limited SAC in the residuals for
the majority of models. A few significant autocorrelations occurred, but these were
generally of low magnitude and did not tend to be conserved across the four-folds of
regional cross-validation or across the different modelling techniques. These results led
us not to consider the application of predictive models explicitly accounting for SAC.

Results
Predictive performance of alien species richness using leave-one-out cross-validation
Results for the leave-one-out cross-validation — which assesses the accuracy of predic-

tions made for within the geographical range of the data - show that RF and BRT
provided the comparably best performances (Figure 1a; Suppl. material 1: Table A4).

86 César Capinha et al. / NeoBiota 38: 77-96 (2018)

Relative absolute error (RAE,%o)

GLM.P GLM.NB BRT RF MARS GLM.P GLM.NB BRT RF MARS

°
° w
£oa
Cc
a
£
a. 2°
a Oo
a
Ww
wW
9
GLM.P GLM.NB BRT RF MARS GLM.P GLM.NB BRT RF MARS
Leave-one-out cross-validation Four-fold regional cross-validation

Figure |. Detailed legend: Accuracy was measured for generalised linear models using Poisson (GLM-P)
and negative binomial distributions (GLM-NB), boosted regression trees (BRT), random forests (RF) and
multivariate adaptive regression splines (MARS) for predictions of the total number of alien species per
region (relative absolute error, RAE; lower is better) (a, b) and the rank order of each region (Spearman’s
tho; higher is better) (c, d). Boxplots represent variations in accuracy across 22 datasets of alien species
richness for GLM-P, GLM-NB, RF and MARS, but not for BRT. Due to model convergence issues, re-
sults for BRT comprise only a subset of datasets and are thus not directly comparable with the results of
the other techniques. Panels in the right left (a, C) refer to predictions evaluated using a leave-one-out ap-
proach, which measures the accuracy of predictions within the geographical range of the model calibration
data. Panels in the right (b, d) refer to predictions evaluated using a four-fold regional cross-validation
approach, which assesses the spatial transferability of the models. A few outliers lie outside the ranges of
the Y-axes, see Tables A2 and A3 for the complete list of values.

Still, these two techniques had a median RAE of about 76% and five or less datasets
had a RAE higher than 90% (Figure 1a; Suppl. material 1: Table A4). The predictive
performance of these two techniques was not significantly different from one another
and both performed significantly better than GLM-P, GLM-NB and MARS (p < 0.05;
Wilcoxon rank-sum test) (Table 2A). For GLM-P and GLM-NB, about half or more
datasets had a RAE of 90% or higher.

The application of data ‘clamping’ in predictions of GLM-P, GLM-NB and MARS
resulted in improvements in predictive performance for nearly all datasets (Suppl. ma-
terial 1: Table A5). However, this improvement was only statistically significant for
GLM-P (p < 0.05; Wilcoxon rank-sum test).

Regarding the predictions for the relative order of regions in alien species rich-
ness, these were more accurately predicted by RF (median g = 0.63), closely followed
by BRT (median g = 0.62). The higher performance of RF was significantly different

Models of alien species richness show moderate predictive accuracy and poor transferability 87

Table 2. Results of pairwise Wilcoxon tests of significant differences for the performance of the tech-
niques for predicting absolute richness (as measured by relative absolute error) using leave-one-out cross-
validation (A) and regional cross-validation (B). Predictions of GLM-P, GLM-NB and MARS refer to
models using ‘clamped’ data (see main text). Significant differences (at a = 0.05) are shown in bold.

NARS rr
cia =
GLM-NB 0.622

MARS :

RF 0. i 0.011 0.04 .
BRT 0.263 0.561 0.159 0.004

from the performances achieved by GLM-P, GLM-NB and MARS (p < 0.05; Wil-
coxon rank-sum test), but not for BRT (p > 0.05). BRT was also significantly better in
predicting the relative order of regions in terms of alien species richness than the two
GLM-type models (p < 0.05). The application of clamping to GLMs and MARS did
not significantly alter their accuracy (p > 0.05; Wilcoxon rank-sum test). A reasonable
number of data sets achieved high ( > 0.6) degrees of correlation between the predict-
ed and observed order of regions, particularly for the two best performing techniques
(RF and BRT), whereas weak (9 < 0.25) correlations were less common (Figure 1C;
Suppl. material 1: Table S4).

Predictive performance of alien species richness using regional cross-validation

Results for the 4-fold regional cross-validation, which assesses the transferability of model
predictions, showed a consistently worse predictive accuracy than the one evaluated by
leave-one-out cross-validation. All modelling techniques showed substantially higher
medians of RAE (Figure 1B; Suppl. material 1: Table AG) and, in the case of MARS,
RF and BRI, the lower performance than their counterparts, evaluated by leave-one out
cross-validation, was statistically significant (p < 0.05; Wilcoxon rank-sum test).

The least inaccurate technique was RF (median RAE = 95.1%; interquartile range
= 25.5%) (Figure 1B; Suppl. material 1: Table S6) which showed significant (p < 0.05;
Wilcoxon rank-sum test) to marginally significant (p < 0.1) differences in all pairwise
comparisons with the other methods (Table 2B). Despite performing best, RF still
delivers a substantial amount of error. For ten datasets, the predictions of alien species

88 César Capinha et al. / NeoBiota 38: 77-96 (2018)

richness were worse than the ones obtained if using simply the mean absolute value
of alien species richness (i.e. RAE > 100%) and, only for eight data sets, the accuracy
of RF was clearly superior (RAE < 90%) to this benchmark (Suppl. material 1: Table
AO). All remaining techniques performed worse, with median performance above the
100% RAE benchmark (Figure 1B; Suppl. material 1: Table AG). In addition, BRT is
demanding in terms of sample size and could not be fitted for 7 of the smallest datasets
(mean n = 38, S.D. = 10). The remaining techniques were able to fit models for all
datasets. No significant differences in performance were found between any two tech-
niques other than with RF (Table 2B).

Similarly to what was verified for leave-one-out cross validation, the application of
data ‘clamping’ in predictions of absolute alien species richness by GLM-P, GLM-NB
and MARS resulted in clear increases in predictive performances for nearly all datasets
(Suppl. material 1: Tables A6 and A7), but this improvement was only statistically
significant for GLM-P (p < 0.05; Wilcoxon rank-sum test).

For predictions of the relative order of regions in alien species richness (g), no
method emerges as best performing (p > 0.05; Wilcoxon rank-sum test). The applica-
tion of clamping did not significantly alter the results (p > 0.05; Wilcoxon rank-sum
test). Most datasets showed moderate to low (0 < 0.45) degrees of correlation between
the predicted and observed order of regions (Figure 1D; Suppl. material 1: Table S6).

Discussion

Our results show that values of alien species richness can be predicted with reasonable
to moderate accuracy within the geographical range of the model calibration data, but
only poorly in regions outside this range. This drop in predictive power was verified
across modelling techniques and concerned the capacity to predict both absolute alien
species richness and relative alien species richness.

The poor transferability of statistical models is not unexpected because the rela-
tionships they identify are not functional (mechanistic) and may thus be limited in
their realism outside the space of the calibration data. Issues related to transferability
have been well documented and examined for species distribution models (e.g. Ran-
din et al. 2006, Heikkinen et al. 2012, Wenger and Olden 2012, Bahn and McGill
2013). Our results extend these findings to models of alien species richness. Given the
similarity of the two model approaches, we argue that the causes for these congruent
findings could be largely shared. First, poor transferability can be a consequence of
overfitted models, which have relationships overly adjusted to the calibration data (e.g.
also expressing random noise), reducing their generality (Wenger and Olden 2012).
However, we expect this potential source of error to be of minor importance in our
models because RF, which are known to be susceptible to overfitting (Heikkinen et al.
2012, Wenger and Olden 2012, Bahn and McGill 2013), showed consistently better
transferability than GLMs selected by AlCc, a modelling approach that is expected to
provide models robust to overfitting (Randin et al. 2006; Wenger and Olden 2012).

Models of alien species richness show moderate predictive accuracy and poor transferability 89

Another possibility for the poor transferability of the aliens species richness models
could be that the relationships (i.e. the covariance structure) between predictor and
response variables and amongst response variables are not conserved in the areas that
lie beyond the spatial range of the calibration data (Bahn and McGill 2013). These
sorts of changes can result from real differences in the way factors influence alien spe-
cies richness in the new areas (e.g. temperature may be an important limiting factor at
high latitudes, but not near the tropics) or from changes in the correlation structure of
the predictors (Elith et al. 2010). Importantly, these changes are more likely for pre-
dictors that are proxies of the causal process, because relationships for them should be
less robust to the effects of confounding factors or to changes in predictor’s correlation
structures (Austin 2002). In our models, several predictors are actually proxies of the
putative relevant mechanisms rather than direct measures, making them particularly
susceptible to this type of problem. We stress however, that this is frequently the case
in models of alien species richness, as better, proximal information is often not avail-
able. One particular example refers to colonisation pressure, i.e. the number of species
introduced into a region (Lockwood et al. 2009), which is a strong determinant for the
number of alien species that become established (Dawson et al. 2017). In the absence
of better information, colonisation pressure was represented by variables depicting
variation in levels of human activity (e.g. per capita GDP; human population den-
sity; proportion of urban areas). The assumption is that higher human activity should
translate into higher colonisation pressure e.g. due to a higher purchase of pets and
plants or to higher volumes of imported cargo potentially carrying alien species. While
this relationship should hold some degree of generality (Dawson et al. 2017), it is also
likely that the shape and relative importance of the relationship changes to some extent
across space, given the local influence of regional-scale factors such as regulations for
the importation of living animals and plants (e.g. Reino et al. 2017).

A third possible cause for the observed poor transferability of models concerns
extrapolation, which is also related to the information content in the model calibra-
tion data. Predictions made for conditions out of the range of the calibration data are
extremely challenging, no matter the modelling technique used (Elith and Graham
2009, Elith et al. 2010). We applied ‘clamping’ to GLMs and MARS when confronted
with extrapolating conditions, mimicking the behaviour of BRT and RE This substan-
tially improved the overall accuracy of these models, providing circumstantial evidence
for a substantial prevalence of extrapolation in the predictions. It is worth emphasising
that, while clamping avoids ‘off the chart’ predictions under extrapolation, it does not
‘add’ extra information to the models and any extrapolating prediction, clamped or
not, should generally be less accurate than a prediction made for conditions sampled
by the data (i.e. interpolation). Our 4-fold regional cross-validation, which holds-out
one geographical quadrant of the data at a time, should imply a substantial number of
extrapolating data points, hence also likely contributing to the verified sharp drop in
predictive accuracy when models are spatially transferred.

A good transferability of models of alien species richness may not be required if
predictions or model-based inferences are intended for the geographical range of the

90 César Capinha et al. / NeoBiota 38: 77—96 (2018)

sampling data. Our results show that, under these settings, models of alien species
richness can achieve moderate (e.g. RAE ~ 75% and @ = 0.6) predictive accuracy.
However, one of our most prominent results was that predictions from RF and BRT
were significantly better, despite not being good, than those from GLM-type mod-
els and MARS. This occurred even after data clamping being applied to GLMs and
MARS, allowing the effect of the higher susceptibility of these models to extrapola-
tion errors to be discarded.

It is not unexpected to find RF and BRT outperforming GLMs in non-transferred
predictions. The model fitting process of the former techniques consists in iteratively
fitting the data and testing the ability of the fitted relationships for prediction using
portions of data left-out from the fitting. The leave-one-out cross-validation mimics
this procedure, differing mainly in that the error levels measured are not used to retune
the model. Hence, machine learning techniques are specifically optimised to predict
well, based on the patterns sampled by the data. Besides, the capacity of machine
learning techniques to fit complex functions could be particular relevant for models
of alien species richness, because the relationships between variables in these models
are often fitted along wide gradients (such as for global-scale environmental and socio-
economic variation; e.g. Dawson et al. 2017), where the persistence of simple linear or
monotonic relationships (as it is assumed by GLMs) should be less common. MARS,
another machine learning method, was also substantially outperformed by RF and
BRT. Although also able to fit non-linear functions, the complexity of MARS models
is usually considerably lower than that of BRT and RF (Merow et al. 2014) which
could explain its comparatively lower predictive performance.

Similarly to what was verified in the predictions of transferred models, extrapola-
tion also severely harmed predictions made for in-sample geographical ranges. Ideally,
extrapolation should be overcome by the use of additional data, sampling the extrapo-
lating predictors’ space. However, when that is not possible, our results show that the
use of clamping is strongly recommended. Further benefits could also be expected
from the examination of the conditions leading to extrapolation, such as the identity
of the predictors involved and of how far the model has to extrapolate in the predic-
tors space. This has been assessed in SDMs previously (see, for instance, Elith et al.
2010) and allows a further refinement of the predictions by, for instance, identifying
extrapolating regions of ‘low novelty’ and for which prediction could be appropriate or
inversely, regions where conditions are far beyond what is sampled by the data and for
which predictions should be avoided.

Overall, our results suggest that accurate predictions of regional alien species rich-
ness from correlative models are beyond the scope of the models we used. This is par-
ticularly the case for absolute values of richness, whereas relative richness, despite not
achieving overall good accuracy, showed to be more robust to errors. Here we analysed
the transferability of models on species richness between regions. A complementary
analysis, recognising species identity explicitly and, hence, also allowing for the analy-
sis of species turnover, are models of compositional similarity (e.g. Hui and McGeoch
2014, Capinha et al. 2015). A promising way forward would be testing the transfer-
ability of such models.

Models of alien species richness show moderate predictive accuracy and poor transferability 91

Conclusions

We showed that regional alien species richness cannot be predicted with reliability
using the data and methods typically found in literature. Given that these data and
methods already reflect best available possibilities to modellers, in the near future the
coverage of information gaps on alien species richness is likely to remain entirely de-
pendent on the publication and updating of alien species inventories, which reinforces
recent calls for the publication of this information (Pysek et al. 2017). A comple-
mentary approach to model species richness and by some, regarded as a potential way
forward, is to model and perhaps predict, patterns of alien biodiversity using measures
of compositional differentiation and similarity between regions.

Two of our results are also of relevance for descriptive models of alien species rich-
ness. First, we found that tree ensembles-type modelling techniques (RF and BRT) are
consistently superior in predicting non-transferred values of richness than GLM and
MARS. This supports the fact that flexible, non-linear, models are better able to capture
information from the data than GLM, a more commonly used technique. The common
justification for the use of GLM-type models for analysing alien species richness concerns
their ease of interpretation. However, a number of methods have recently been developed
to improve the interpretability of tree ensembles (e.g. Fokkema 2017), which may be
worth considering in order to further refine our understanding about the relationships
between alien species richness and spatial factors. Second, our results also warrant a call
for caution in making inferences beyond the geographical (and likely also temporal)
range of the data used to calibrate the models, e.g. for future projections. We recommend
performing a transferability assessment of the models, as allowed, for instance, by re-
gional cross-validation, in order to confirm the generality of the relationships identified.

Acknowledgements

We acknowledge the comments from one anonymous reviewer and Cang Hui
who helped to improve the manuscript. CC was supported by a postdoctoral grant
from FEDER Funds through the Operational Competitiveness Factors Programme
“COMPETE? and by National Funds through the Foundation for Science and Tech-
nology (FCT) within the framework of project “PTDC/AAG-GLO/0463/2014-PO-
CI-01-0145-FEDER-016583”. FE acknowledges funding from the Austrian Science
Fund (FWF, grant 12086-B16). HS was supported by a grant from the Deutsche
Forschungsgemeinschaft (DFG, grant SE 1891/2-1).

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Supplementary material |

Table A1—A5

Authors: César Capinha, Franz Essl, Hanno Seebens, Henrique Miguel Pereira,

Ingolf Kihn

Data type: ecology data

Copyright notice: This dataset is made available under the Open Database License
(http://opendatacommons.org/licenses/odbl/1.0/). The Open Database License
(ODDbL) is a license agreement intended to allow users to freely share, modify, and
use this Dataset while maintaining this same freedom for others, provided that the
original source and author(s) are credited.

Link: https://doi.org/10.3897/neobiota.38.23518.suppl1