APPENDIX S3
Reciprocal causal modeling: (partial) Mantel correlations between isolation by
distance (IBD), landscape resistance (IBR) and clustering (IBB) hypotheses
We used reciprocal causal modelling (Cushman et al. 2013) to compete the alternative
hypotheses governing gene flow (i.e. IBD, IBB and IBR), and identify the hypothesis in the set
that was uniquely supported relative to the others.
First, the Rousset’s ar inter-individual genetic distance (Rousset 2000) was computed
using the program SPAGeDI (Hardy & Vekemans 2002) since this parameter of relatedness
does not rely on a reference population (Vekemans & Hardy 2004) and has been successfully
applied to infer the effect of landscape on genetic structure of continuously distributed
vertebrates (Coulon et al. 2004; Broquet et al. 2006; Blair & Melnick 2012; Dudaniec et al.
2013).
Later, the pairwise genetic distances matrix (Rousset’s ar) was correlated with different
distances matrices representing the different hypothesis potentially driving gene flow (i.e. IBD,
IBR and IBB). The correlation between distance matrixes was calculated by means of the
Mantel test (Mantel 1967) and partial Mantel tests (Smouse et al. 1986) as implemented in the
ECODIST package (Goslee & Urban 2007) in R version 2.7 (R Development Core Team 2008)
with 10 000 permutations. Given the potential sensitivity of Mantel tests to non-linear
relationships between genetic and cost-distances (Rousset 1997), we compared results between
two sets of analyses, one log transforming the effective and Euclidean distances, and one using
the original untransformed cost-distance matrices.
We used reciprocal causal modeling (Cushman et al. 2013c) to compete the alternative
hypotheses governing gene flow (i.e. IBD, IBB and IBR), and identify if any hypothess in the
set that was uniquely supported relative to the others. Cushman & Landguth (2010) found that
the inherent high correlation among alternative models results in a high risk of spurious
correlations using simple Mantel tests. Several refinements, including causal modeling
(Cushman et al. 2006), have been developed to reduce the risk of affirming spurious
correlations and to assist model selection. However, Cushman et al. (2013c) showed these still
suffer from elevated Type I error rates. Consequently, Cushman et al. (2013c) proposed
“reciprocal causal modeling”, which they showed greatly lessens Type I error rates observed by
Meirmans (2012) and Amos et al. (2012) in landscape genetic analysis, allowing rigorously
testing for the joint and independent effects of alternative resistance models (e.g. Yang et al.
2013; Castillo et al. 2014; Ruiz-Gonzalez et al. 2014).
The method of reciprocal causal modeling directly competes all alternative hypotheses
based on relative support. The approach uses a pair of "reciprocal" Mantel tests for each pair of
alternative resistance hypotheses (Fig. S1). The first is a partial Mantel test which calculates the
partial Mantel correlation between the first hypothesis and genetic distance, partialling out the
second (G ~ A|B). The second is a partial Mantel test which calculates the partial Mantel
correlation between the second hypothesis and genetic distance, partialling out the first (G ~
B|A) (Fig. S1). The difference between the partial correlations of these two tests (A|B - B|A) is a
measure of the relative support for hypothesis A relative to hypothesis B (e.g. Cushman et al.
2013b,c; Yang et al. 2013; Castillo et al. 2014) (Fig. S1). Specifically, if hypothesis A is correct
then the difference A|B - B|A should be positive. Conversely, if hypothesis B is correct then
(A|B - B|A) should be zero or negative. The reciprocal causal modeling approach works by
calculating a full matrix of all the A|B - B|A differences between each pair of alternative
hypothesis. For a model that is fully supported all values in the column associated with that
hypothesis would be positive, while all values in the row associated with that hypothesis would
be negative. This would indicate that that hypothesis is supported independently of all
alternative hypotheses (positive values down the column) and that no alternative hypotheses are
supported independently of it (negative values across the row).
In addition to the reciprocal causal modelling matrix (Cushman et al. 2013c), we calculated the
raw form of the partial Mantel matrix that provide further insights when competing hypothesis
could be jointly true (Castillo et al. 2014).
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