Alexis Criscuolo
email: alexis.criscuolo@pasteur.fr

Brian Ondov

Burkhard Morgenstern

Guy Perrière

On the transformation of MinHash-based uncorrected distances into proper evolutionary distances for phylogenetic inference [version 1; peer review: 3 approved]

Keywords: MinHash, p-distance, evolutionary distance, substitution model, phylogenetics, genome, simulation Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Software, Validation, Visualization, Writing -Original

Recently developed MinHash-based techniques were proven successful in quickly estimating the level of similarity between large nucleotide sequences. This article discusses their usage and limitations in practice to approximating uncorrected distances between genomes, and transforming these pairwise dissimilarities into proper evolutionary distances. It is notably shown that complex distance measures can be easily approximated using simple transformation formulae based on few parameters. MinHash-based techniques can therefore be very useful for implementing fast yet accurate alignment-free phylogenetic reconstruction procedures from large sets of genomes. This last point of view is assessed with a simulation study using a dedicated bioinformatics tool.

Introduction

To estimate the level of proximity between two non-aligned genome sequences x and y, recent methods (e.g. 1-7) have focused on decomposing the two genomes into their respective sets K x and K y of non-duplicated nucleotide k-mers (i.e. oligonucleotides of size k). A pairwise similarity may then be easily estimated based on the Jaccard index j = |K x ∩ K y | / |K x ∪ K y | [START_REF] Jaccard | Nouvelles recherches sur la distribution florale[END_REF] . The Jaccard index between two sets of k-mers is a useful measure for two main reasons. First, it can be quickly approximated using MinHash-based techniques (MH [START_REF] Broder | On the resemblance and containment of documents[END_REF] ), as implemented in e.g. Mash 2 , sourmash 3 , Dashing 4 , Kmer-db [START_REF] Deorowicz | Kmer-db: instant evolutionary distance estimation[END_REF] , FastANI 5 , or BinDash [START_REF] Zhao | BinDash, software for fast genome distance estimation on a typical personal laptop[END_REF] . Such techniques select a small subset (of size σ) of hashed and sorted k-mers (called sketch) from each K x and K y , and approximate j by comparing these two subsets (for more details, see 2,9-12). Second, the proportion p of observed differences between the two aligned genomes (often called uncorrected distance or p-distance) can be approximated from j (therefore without alignment) with the following formula (e.g. 13,14):

1 / 2 1 , 1 k j p j   = -  +   (1)
provided that both sizes σ and k are large enough, and j is not too low (see below).

As a consequence, a pairwise evolutionary distance d can be derived from the Jaccard index j using transformation formulae of the following form:

1 2 log (1 / ), e d b p b = - - (2)
where p is obtained using Equation (1). Parameters b 1 and b 2 can be defined according to explicit models to estimate the number d of nucleotide substitutions per character that have occurred during the evolution of the sequences x and y, e.g. 15-24. When b 1 = b 2 = 1, Equation (2) corresponds to the Poisson correction (PC; e.g. 21) distance. Although it is based on a simplistic model of nucleotide substitution 1,[START_REF] Dickerson | The structure of cytochrome c and the rates of molecular evolution[END_REF][START_REF] Zuckerkandl | Evolutionary divergence and convergence in proteins[END_REF][START_REF] Jukes | Comparison of polypeptide sequences[END_REF] , PC is the p-distance transformation implemented in many MH tools (e.g. Mash, Dashing, FastANI, Kmer-db, BinDash). However, more accurate distance estimates may be obtained by using substitution models based on more parameters. Among these models, equal-input (EI, sometimes called F81 [START_REF] Tajima | Biases of the estimates of DNA divergence obtained by the restriction enzyme technique[END_REF][START_REF] Tajima | Estimation of evolutionary distance between nucleotide sequences[END_REF][START_REF] Kumar | MEGA7: Molecular evolutionary genetics analysis version 7.0 for bigger datasets[END_REF][START_REF] Felsenstein | Evolutionary trees from DNA sequences: a maximum likelihood approach[END_REF][START_REF] Mcguire | Improved error bounds for genetic distances from DNA sequences[END_REF][START_REF] Criscuolo | A fast alignment-free bioinformatics procedure to infer accurate distance-based phylogenetic trees from genome assemblies[END_REF] ) takes into account the equilibrium frequency π r of each residue r in Σ = {A, C, G, T}. An EI distance can be estimated using Equation (2) with where π r x and π r y are the frequencies of r in the two sequences x and y, respectively [START_REF] Tamura | Evolutionary distance estimation under heterogeneous substitution pattern among lineages[END_REF] . Further assuming that the heterogeneous replacement rates among nucleotide pairs and sites can be modelled with a Γ distribution, an EI distance d can be derived from p using the following formula:

1/ 1 2 [(1 / ) 1], a d ab p b - = - - (3)
where a > 0 is the (unknown) shape parameter of the Γ distribution, e.g.

Results and discussion

MinHash-based p-distance approximation

Varying d from 0.05 to 1.00 (step = 0.05), a total of 200 nucleotide sequence pairs with d substitution events per character were simulated under the models GTR and GTR+Γ. Each model was adjusted with three different equilibrium frequencies: equal frequencies (f 1 ;

π A = π C = π G = π T = 25%), GC-rich (f 2 ; π A = 10%, π C = 30%, π G = 40%, π T = 20%), and AT-rich (f 3 ; π A = π T = 40%, π C = π G = 10%
). The GTR substitution rates and the Γ shape parameters were obtained based on a maximum likelihood (ML) analysis of 142 real-case phylogenomics datasets. Overall, ML estimates of Γ shape parameters were quite low (i.e. varying from 0.162 to 0.422, with an average of 0.314), confirming that the heterogeneity of the substitution rates across sites is a non-negligible factor when studying evolutionary processes. Every simulation was completed with indel events, resulting in sequences > 3 Mbs with relative lengths (i.e. longer/shorter) varying from 1.0196 (d = 0.05) to 1.1117 (d = 1.00), on average.

For each of the 2 (GTR, GTR+Γ) × 3 (f 1 , f 2 , f 3 ) × 20 (d = 0.05, 0.10, ..., 1.00) × 200 = 24,000 simulated sequence pairs x and y, the corresponding p-distance was estimated using three MH tools: Mash, BinDash and Dashing. Of note, the accuracy of a MH estimate ĵ of the Jaccard index between K x and K y is mainly dependent on two parameters: the k-mer size k and the sketch size σ. The size k should be large enough to minimize the probability q of observing a random k-mer shared by x and y by chance alone. Such a value can be obtained from q by k = ⌈log |Σ| (g(1-q)/q) -0.5⌉, where g is the length of the largest sequence 2,[START_REF] Fofanov | How independent are the appearances of n-mers in different genomes?[END_REF][START_REF] Criscuolo | A fast alignment-free bioinformatics procedure to infer accurate distance-based phylogenetic trees from genome assemblies[END_REF] . The size σ should be large enough to minimize the error bounds of ĵ 2 , but also to avoid the inconvenient estimate ĵ = 0. Following 29, σ was set by the proportion s of the average sequence length.

To investigate the impact of both parameters σ and k on the accuracy of the MH estimates, each MH tool was used with s = 0.2, 0.4, 0.6, 0.8 and q = 10 -3 , 10 -6 , 10 -9 , 10 -12 . As in simulated sequences, g ranges from 4.99 Mbs (d = 0.05) to 3.38 Mbs (d = 1.00) on average, s translates into moderately to very large sketch sizes σ, and q into k-mer sizes k = 16, 21, 26, 31.

Two statistics were calculated to assess the linear relationship between the MH estimate p (derived from ĵ ≠ 0) and the 'true' p-distance p: the coefficient of determination R 2 and the slope β of the linear least-square regression p = βp. Let Ψ(p max ) be the subset of pairs (p, p) such that p ≤ p max . Varying p max from 0.10 to 0.55, R 2 and β were estimated from Ψ(p max ) (Figure 1). The cumulative proportions ˆ0 j f = of MH Jaccard index ĵ = 0 within [0, p max ] were also measured (Figure 1). Finally, every value p r>0.99 was estimated, where p r>0.99 is defined as the highest p-distance such that the subset Ψ(p r>0.99 ) provides a coefficient of correlation r > 0.99 (as assessed by a Fisher transformation z-test with p-value < 1%; Figure 1). The highest values p r>0.99 were obtained with parameters k = 26 (q ≤ 10 -9 ) and s = 0.8 (illustrated in Figure 2). One important result (Figure 1) is that current MH implementations return suitable estimates of p as long as p ≤ 0.25, provided that k is sufficiently large. Indeed, when k ≥ 21 (and any s ≥ 0.2), the statistics p r>0.99 are higher than 0.25 (Figure 1), therefore showing that p and p are highly linearly correlated when p ≤ 0.25 (see e.g. Figure 2). Interestingly, when p ≤ 0.25, the worthless estimate ĵ = 0 was almost never observed with the different selected parameters s and q (Figure 1). Points corresponding to ĵ = 0 are not represented. Each scatter plot is completed with the least-square regression line through the origin (dashed black line) estimated from the subset of points (p, p ) such that p ≤ p r>0.99 , where p r>0.99 = 0.345 (Mash), 0.335 (BinDash) and 0.330 (Dashing).

Furthermore, when p > 0.25, large k-mers are required to obtain satisfactory estimates, i.e. k > 21 or q < 10 -6 (Figure 1). However, dealing with k > 21 involves using large sketch sizes to minimize the cases ĵ = 0 (see ˆ0 j f = in Figure 1). Simulation results suggest that k = 26 (i.e. q = 10 -9 ) and s > 0.4 yield suitable estimates of p, obtained from sequences of lengths > 4 Mbs with pairwise p < 0.35 (see Figure 1 and Figure 2). Indeed, when p ranges between 0.25 and 0.35, small sizes k (e.g. k ≤ 21 or q ≥ 10 -6 ) always provide underestimated p (with any s). Large size k (e.g. k = 31 or q = 10 -12 ) results in the same trend, but also in high numbers of useless estimates ĵ = 0 (even with large σ; see ˆ0 j f = in Figure 1).

When p ≥ 0.35, MH tools always underestimate the p-distances between the sequences simulated for this study (Figure 1 and Figure 2). One could suggest that more accurate MH estimates p will be expected with larger sketch sizes σ.

Nevertheless, results represented in Figure 2 (i.e. q = 10 -9 and s = 0.8, providing the highest p r>0.99 ) are based on average σ varying from ∼2. PC and EI p-distance transformations (2) result in improper underestimates as the expected distance d increases. Indeed, when compared with realistic GTR-based distances d, PC and EI transformations (2) give distance estimates that are always lower than d, especially under GTR+Γ and when d is large (e.g. d > 0.1; Figure 3). This downward bias is somewhat expected, knowing that PC and EI transformations (2) are based on less parameters than both models GTR and GTR+Γ. However, the additional parameter a in Equation (3) may help dealing with heterogeneous substitution rates among residue pairs (e.g. 36). Hence, the relationship between GTR distances d and the corresponding p-distances p can be approximated by the EI transformation (3) with a = 4.590 (Figure 3). Moreover, as d returned by Equation ( 3) is inversely proportional to a (for any fixed p), the relationship between d and p under the model GTR+Γ (with Γ shape parameter of 0.314, on average) can also be approximated by the EI transformation (3) with a = 0.291 (Figure 3).

These results show that complex distance measures can be approximated by simple analytical formulae based on few parameters. In practice, nucleotide frequencies (four parameters) can be trivially computed and p-distances (a fifth parameter) can be estimated using MH tools (see above). Therefore, the evolutionary distance d between two sequences that have evolved under the parameter-rich model GTR+Γ can be approximated from these only five parameters using (3) with a ≤ 4.590 (Figure 3).

At this point, it should be stressed that MH p tends to be overestimated. Indeed, MH estimates are of the form p ≈ βp with slope β varying from 1.08 (BinDash, k = 31, s = 0.2) to 1.15 (Dashing, k = 26, s = 0.2) when p ≤ p r>0.99 (Figure 1). This has a direct impact on the derived distances: using PC and EI transformations (2) on p = βp with β = 1.15 and p ≤ 0.35 reference phylogenetic trees were obtained based on a ML analysis of real-case phylogenomics datasets. Sizes of the reference trees ranged from 10 to 154 taxa (31 on average), with diameters (i.e. maximum distance between any two leaves of a tree) varying from 0.204 to 2.883 (0.975 on average). Sequence lengths and indel events were simulated in the same way as the previous sequence pair simulations.

The script JolyTree v2.0 was used to reconstruct phylogenetic trees from the simulated sequences. For each pair of unaligned sequences, this script estimates the MH p-distance using Mash, and transforms it into an evolutionary distance. Using these MH-based distances, JolyTree next reconstructs a minimum evolution phylogenetic tree with confidence supports at branches, based on a ratchet-based hill-climbing procedure (for more details, see 29). To obtain accurate MH p-distance estimates, JolyTree was run with parameters q = 10 -9 and s = 0.5 (see above). Evolutionary distances were estimated using the PC and EI transformations (2), as well as the EI transformation (3). To observe the impact of the parameter a, the EI transformation (3) was computed with a varying from 0.05 to 10.0. The accuracy of each p-distance transformation for phylogenetic inference was assessed by the percentage of recovered reference trees, i.e. identical topologies (Figure 5).

Using JolyTree with EI transformations improves the percentage of recovered reference trees (Figure 5). In spite of their limitations, PC distances result in the recovery of 75.3% of the 142 reference trees, but EI transformation (2) increases this percentage to 76.7% (Figure 5). Furthermore, the EI transformation (3) generally provides better results in a large range of a, i.e. up to 83.1% of recovered reference trees (Figure 5). Low a-values (e.g. a ≤ 0.3) translate into many incorrect tree topologies, whereas high ones (e.g. a > 6) tend to provide the same reference tree recovering percentage as the EI transformation (2) (Figure 5). Most suitable values of a (corresponding to the highest reference tree recovering percentages, e.g. 80%) seem to range in the interval [1.0, 2.0] (Figure 5).

These simulation results are consistent with two views which are somehow contradictory. On the one hand, accurate (parameter-rich) distance estimates are required, because biased ones (i.e. corresponding to a concave or convex function of the actual evolutionary distances) may result in incorrect phylogenetic trees [START_REF] Mctavish | Twisted trees and inconsistency of tree estimation when gaps are treated as missing data -The impact of model misspecification in distance corrections[END_REF][START_REF] Susko | On inconsistency of the neighbor-joining, least squares, and minimum evolution estimation when substitution processes are incorrectly modeled[END_REF] . On the other hand, simple (underparameterized) distance estimates should often be preferred, because they frequently result in more accurate tree topologies [START_REF] Nei | Evolutionary distance: Estimation[END_REF][START_REF] Zharkikh | Inconsistency of the maximum-parsimony method: the case of five taxa with a molecular clock[END_REF][START_REF] Russo | Efficiencies of different genes and different tree-building methods in recovering a known vertebrate phylogeny[END_REF][START_REF] Takahashi | Efficiencies of fast algorithms of phylogenetic inference under the criteria of maximum parsimony, minimum evolution, and maximum likelihood when a large number of sequences are used[END_REF][START_REF] Rosenberg | Traditional phylogenetic reconstruction methods reconstruct shallow and deep evolutionary relationships equally well[END_REF][START_REF] Yoshida | Efficiencies of the NJp, Maximum Likelihood, and Bayesian Methods of Phylogenetic Construction for Compositional and Noncompositional Genes[END_REF] .

Here, the simple PC and EI transformations (2) (one and five parameters, respectively) enable many reference trees to be recovered (Figure 5). However, the EI transformation (3) is able to approximate realistic distance measures (e.g. GTR+Γ) by using only one supplementary parameter a (Figure 3). It therefore enables more reference trees to be recovered (Figure 5).

In line with 43, most suitable values of a (e.g. between 1.0 and 2.0) are all higher than the Γ shape parameter values used for simulating the sequence datasets (i.e. varying from 0.162 provide distance estimates that are quite comparable to the ones returned by Equation ( 3) with a ranging from 1.000 to 4.590 (see Figure 4 for the equilibrium frequencies f 2 ; similar results were observed with f 1 and f 3 -not shown). The PC transformation (2) on the upward biased MH p returns distances that are then comparable to some complex distance measures (e.g. derived from a GTR model), therefore justifying its use by many MH tools. Nevertheless, the EI transformation (3) remains necessary when dealing with distantly related sequences (e.g. p > 0.2) and strong heterogeneity of the substitution rate across sites (e.g. often observed Γ shape parameter < 1.000). In such cases, the value of the parameter a should always be slightly increased to compensate the MH upward bias. For instance, EI transformation (3) on p with a = 0.291 (i.e. GTR+Γ distance least-square fitting in Figure 3) can be approximated by the same equation on p = βp with β = 1.15 and a = 0.431.

Phylogenetic reconstruction from MinHash-based evolutionary distances

To assess whether MH p-distance transformations may translate into reliable phylogenetic trees, additional simulations were performed. A total of 142 sets of sequences was simulated under the model GTR+Γ along reference phylogenetic trees.

Representative GTR+Γ model parameters (same as above) and to 0.422, with an average of 0.314). This can be explained by the MH upward bias (see above), but also by the large variance of the estimate (3) when a becomes low. Reminding that the Γ shape parameters (and the reference trees) used in these simulations were inferred from real-case datasets, these results suggest that using EI transformation (3) with a ≈ 1.5 may be suitable to infer genus phylogenetic trees. In light of this, it should be stressed that the article [START_REF] Criscuolo | A fast alignment-free bioinformatics procedure to infer accurate distance-based phylogenetic trees from genome assemblies[END_REF] describing the first distributed version of JolyTree (v1.1) incorrectly stated that the Mash output is the MH estimate p (instead of its PC transformation). As JolyTree v1.1 uses the EI transformation (2), this misinterpretation translates into the odd transformation formula δ = -b 1 log e (1 + log e (1 -p) / b 2 ). However, as δ can be approximated on p ≤ 0.35 by the EI transformation (3) with a = 1.208, this explains the overall accuracy of JolyTree v1.1 despite its use of δ [START_REF] Criscuolo | A fast alignment-free bioinformatics procedure to infer accurate distance-based phylogenetic trees from genome assemblies[END_REF] .

Conclusions

Alignment-free phylogenetic inference from pairwise MH-based distance estimates is a promising approach. It enables phylogenetic trees to be quickly reconstructed from a large number of genomes without the burden of multiple sequence alignments (see e.g. 2,29, 44-52). This report confirms this view by showing that proper evolutionary distances can be easily derived from MH p-distance estimates, therefore enabling accurate phylogenetic inferences.

First, although implemented to approximate nearest neighbors in sequence sets, current MH tools (e.g. Mash, BinDash, Dashing) were shown to be able to conveniently estimate pairwise p-distances p up to p ≈ 0.35. In practice, as p is very similar to the one-complement of the Average Nucleotide Identity (ANI; e.g. 29,53,54), MH estimates of p can then be obtained between genomes gathered from many bacteria, archaea or eukaryota genera, i.e. with pairwise ANI > 65%.

Second, the EI p-distance transformation (3) was proven efficient to approximate complex distance measures, e.g. derived from GTR model with heterogeneous substitution rates across sites. Because of an upward bias observed in MH p-distance estimates, simpler transformations (based on few parameters, as the commonly used PC) still provide distance measures that are comparable to GTR ones, but with (unrealistic) homogeneous substitution rates across sites. However, thanks to its supplementary parameter a, EI transformation (3) remains necessary to approximate distance measures between distantly related sequences that have arisen from more realistic substitution events.

Third, as proper evolutionary distances can be derived from MH p-distance estimates, their efficiency in phylogenetic inference was established using the dedicated tool JolyTree [START_REF] Criscuolo | A fast alignment-free bioinformatics procedure to infer accurate distance-based phylogenetic trees from genome assemblies[END_REF] . In particular, the EI transformation (3) with a ≈ 1.5 enables accurate phylogenetic trees to be inferred.

Methods

Model parameter estimation

To simulate the evolution of nucleotide sequences according to realistic substitution processes, the 187 genus datasets compiled in 29 (available at https://doi.org/10.3897/rio.5.e36178. suppl2) were first considered to infer a representative range of GTR parameter values. For each of the 187 genera, the associated genome assemblies were processed using Gklust v0.1 to obtain one representative genome assembly for each putative species. This analysis provided 142 sets of representative genome assemblies after discarding genera containing < 10 putative species. For each of these 142 sets, coding sequences were clustered using Roary v3.12 [START_REF] Page | Roary: rapid large-scale prokaryote pan genome analysis[END_REF] . Each cluster with at least four coding sequences was used to build a multiple amino acid sequence alignment using MAFFT v7.407 [START_REF] Katoh | MAFFT multiple sequence alignment software version 7: improvements in performance and usability[END_REF] . Multiple sequence alignments were back-translated at the codon level and concatenated, leading to 142 supermatrices of nucleotide characters. A phylogenetic tree was inferred from each supermatrix of characters using IQ-TREE v1.6.7.2 [START_REF] Nguyen | IQ-TREE: a fast and effective stochastic algorithm for estimating maximum-likelihood phylogenies[END_REF] with evolutionary model GTR+Γ. All data related to these analyses are publicly available as Extended data at https://doi.org/10.5281/zenodo.4034244 [START_REF] Criscuolo | Phylogenomic analyses of 142 prokaryotic genera[END_REF] .

Sequence simulation

To assess the accuracy of different pairwise distance estimates, a simulation of sequence pairs was performed under both models GTR and GTR+Γ with three different sets (π A , π C , π G , π T ) of equilibrium frequencies: was used to simulate the evolution of 200 sequence pairs with d substitution events per character. Initial sequence length was 5 Mbs, and an indel rate of 0.01 was set with indel length drawn from [1, 50000] according to a Zipf distribution with parameter 1.5. For each simulated sequence pair, model parameters (i.e. GTR: six relative rates of nucleotide substitution; GTR+Γ: six rates and one Γ shape parameter) were randomly drawn from the 142 sets of estimated ones (see above). All simulated sequences are publicly available as Extended data at https://doi.org/10.5281/zenodo.4034461 [START_REF] Criscuolo | Simulated pairs of nucleotide sequences for testing (alignment-free) genome distance estimate methods[END_REF] .

f 1 = (0.
To compare the efficiency of p-distance transformations for phylogenetic reconstruction, the program INDELible v1.03 was also used to simulate the evolution of a sequence along each of the 142 phylogenetic trees previously inferred from different genera (see above). For each of the 142 genera, sequence evolution was simulated under the model GTR+Γ with the corresponding parameters (i.e. four nucleotide frequencies, six relative rates, and one Γ shape parameter). Sequence length and indel events were simulated as described above. The 142 simulated sequence sets are publicly available as Extended data at https://doi.org/10.5281/zenodo.4034643 [START_REF] Criscuolo | Model trees and associated simulated nucleotide sequences for testing phylogenetic inference methods[END_REF] .

Sequence and phylogenetic analyses MH p-distances were estimated with Mash v2.2, BinDash v1.0, and Dashing v0.

Guy Perrière

Laboratoire de Biométrie et Biologie Evolutive, CNRS, UMR5558, Université Claude Bernard Lyon 1, Villeurbanne, France I have no special criticism on the methodology and my comments on that part are really minor. First, what is the justification to the choice of a value equal to 1.5 for the shape of the Zipf distribution used to simulate the indels. As a user of INDELible myself I know very well how this value can be of importance so I would like to know how this value was chosen (arbitrary choice?). Second, I am not sure that the right reference for the computation of missing distances from the triangle inequality property is Guénoche and Grancolas (1999). The first paper I have seen on that topic is Lapointe and Kirsch (1995 1 ). Also, I have no concerns on the results presented and everything seems ok for me.

The conclusion of the manuscript presents the approach of phylogenetic inference from pairwise MH-based distance estimates as a possible alternative to classical methods of phylogenetic reconstruction (especially in bacteria). Indeed, a classical phylogenomic approach involves the chaining of a complex set of procedures: the identification of orthologous genes in multiple species; the alignment of these genes; and finally, the building of a tree from the concatenation of those alignments. I think that the inclusion in the paper of a comparison of two phylogenies obtained with: i) the MH-based distance and ii) a concatenation would strengthen this claim. A lot of concatenation datasets extracted from complete bacterial genomes are available and it would be not too difficult to do the experiment. I think this addition is of importance because I have no idea on the influence horizontal gene transfers can have on MH-based reconstructions when classical approaches try to minimize this influence.

In the last few years, a number of alignment-free methods have been developed to calculate distances between genomic sequences. While earlier alignment-free methods used rough measures of sequence (dis)similarity, some methods have been proposed recently that accurately estimate distances -the number of substitutions per position since two sequences evolved from their last common ancestor -based on stochastic models of evolution (see below). These methods are restricted, however, to the simplest possible model, the Jukes-Cantor model.

In recent papers, MinHash techniques were proposed as an attractive way of estimating the pdistance between DNA sequences, i.e. the number of mismatches per position in an (unknown) alignment of the compared sequences.

The present paper shows how distances based on complex models of evolution (GTR, GTR + \Gamma) can be approximated using MinHash-based p-distances. The approach is carefully evaluated based on simulated sequences.

The proposed method is a welcome and useful addition to existing alignment-free methods, extending them to more realistic models of evolution. The approach is novel, the paper is very well written and clear, and suitable references are given to the literature for more details. Therefore, I support indexing of the manuscript.

A certain limitation is that the manuscript is restricted to MinHash distances. According to the author, these methods are accurate for p-distances roughly < 0.25 (for reasonable k-mer length), but are less accurate for larger distances. However, a number of other alignment-free methods have been proposed that accurately estimate distances for much larger distances, e.g. Kr (Haubold et al., 2009 1 ), FSWM (Leimeister et al., 2017 2 ), Phylonium (Klötzl and Haubold, 2019 3 ), Slope-SpaM (Röhling et al., 2020 4 ). All these methods estimate distances based on the simple Jukes-Cantor model; the program Co-phylog estimates non-corrected p-distances (Yi and Jin, 2013 5 ). It should be straight-forward to transform these distances to more complex models, as done in the present paper, and to compare them to the MinHash methods evaluated in the paper. of pure k-mer counting a la Fan et al., rather than the MinHash approaches, which trade some accuracy of the Jaccard estimation for speed. This is still useful, but could be somewhat misleading for those that need the speed benefits of running MinHash with more typical parameters. Additionally, since the sweep of s is linear, but MinHash error bounds relate to an exponential of s, the sweep ends up being less informative than it could be, which is likely why the columns of Figure 1 are nearly identical.

As a simple remedy, I would suggest sweeping the parameter s exponentially rather than linearly, starting closer to the default sketch size for the tools. It could still end closer to the genome size to get a sense of the behavior of MinHash as it degenerates to the actual k-mer Jaccard score. Along these lines, it would also make sense for the later phylogenetic experiments to either use a smaller s or do another sweep of values.

As a minor point when listing relevant software in the introduction: Dashing is based on HyperLogLog sketching, which is similar in concept to MinHash but algorithmically distinct.

Is the work clearly and accurately presented and does it cite the current literature? Yes

Is the study design appropriate and is the work technically sound? Yes

Are sufficient details of methods and analysis provided to allow replication by others? Yes

If applicable, is the statistical analysis and its interpretation appropriate? Yes

Are all the source data underlying the results available to ensure full reproducibility? Yes

Are the conclusions drawn adequately supported by the results? Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: High-Performance Computing, Locality Sensitive Hashing, Sequence Alignment I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Figure 1 .

 1 Figure 1. Accuracy of MH tools for estimating p-distances from unaligned nucleotide sequences. For each sketch size (columns; set by s = 0.2, 0.4, 0.6, 0.8) and each k-mer size (rows; k = 16, 21, 26, 31), three line charts represent different statistics determined with Mash (green), BinDash (red), and Dashing (blue). For p max ranging from 0.10 to 0.55 (x-axes), represented statistics are (i) the coefficient of determination R 2 (up; y-axis ranging from 0.85 to 1.00) and (ii) the slope of the linear least-square regression through the origin (middle; y-axis ranging from 0.8 to 1.2) computed from estimated p and corresponding 'true' p-distances p ≤ p max , as well as (iii) the cumulative proportion ˆ0 j f = of estimated Jaccard index ĵ = 0 within [0, p max ] (bottom; y-axis ranging from 0.0 to 0.3). Circles in R 2 line charts (up) indicate the largest value p r>0.99 such that the subset of pairs (p, p ) defined by p ≤ p r>0.99 provides a coefficient of correlation r > 0.99.

Figure 2 .

 2 Figure 2. MH p-distance estimates between 24,000 pairs of unaligned nucleotide sequences. The p-distances p estimated by Mash (up left), BinDash (up right) and Dashing (bottom left) with k = 26 (q = 10 -9 ) and s = 0.8 are plotted against the 'true' p-distances p between 24,000 pairs of nucleotide sequences simulated under six scenaria of evolution: GTR with equilibrium frequencies f 1 = (0.25, 0.25, 0.25, 0.25) (green points), f 2 = (0.10, 0.30, 0.40, 0.20) (red) and f 3 = (0.40, 0.10, 0.10, 0.40) (blue), and GTR+Γ with f 1 (cyan), f 2 (orange) and f 3 (magenta).

  7 × 10 6 (d = 1.00) to ∼3.6 × 10 6 (d = 0.35), which are larger than some real genomes. Transformation of p-distances into evolutionary distances When a pairwise p-distance p can be estimated from unaligned nucleotide sequences, it may be transformed into an evolutionary distance d, based on Equation (2) or Equation (3). The relationship between p and d was represented in Figure 3 for different distance estimators: PC transformation (2) (b 1 = b 2 = 1), and EI transformations (2) and (3) with equilibrium frequencies f 1 (b 1 = b 2 = 0.75 under homogeneous substitution

Figure 3 .

 3 Figure 3. Relationships between p-distances and different evolutionary distance estimates under various models of nucleotide substitution. The six charts represent the evolutionary distance d (y-axis ranging from 0.00 to 1.00) against the p-distance p (x-axis ranging from 0 to 0.55). Gray points (p, d) are derived from the simulation of sets of 4,000 sequence pairs, each under six different scenaria of evolution: GTR (top) and GTR+Γ (bottom) with equilibrium frequencies f 1 = (0.25, 0.25, 0.25, 0.25) (left), f 2 = (0.10, 0.30, 0.40, 0.20) (middle) and f 3 = (0.40, 0.10, 0.10, 0.40) (right). PC and EI versions of Equation (2) are represented with red and green curves, respectively. EI version of Equation (3) is represented with blue curves for a = 1, and with black curves for the values a = 4.590 and a = 0.291 determined by leastsquare regression from the gray points derived from the models GTR (top) and GTR+Γ (bottom), respectively.

Figure 4 .

 4 Figure 4. Impact of the MH p-distance upward bias on PC and EI transformations. The relationship between the p-distance p (x-axis ranging from 0.00 to 0.35) and the corresponding evolutionary distance d (y-axis ranging from 0.00 to 0.70) is represented when using PC (red dots) and EI (with equilibrium frequencies f 2 ; green dots) transformations (2) on p = β p with β = 1.15. For ease of comparison with Figure 3, EI (f 2 ) transformation (3) on p are represented with a = 1.000 (blue curve) and a = 4.590 (black curve).

  25, 0.25, 0.25, 0.25), f 2 = (0.10, 0.30, 0.40, 0.20), and f 3 = (0.40, 0.10, 0.10, 0.40). For each of these six scenaria and for each d varying from 0.05 to 1.00 (step = 0.05), the program INDELible v1.03[START_REF] Fletcher | INDELible: a flexible simulator of biological sequence evolution[END_REF]

Figure 5 .

 5 Figure 5. Accuracy of different p-distance transformations for phylogenetic inference. The percentage of recovered reference trees (y-axis ranging from 50% to 100%) is represented (light blue dots) in function of the parameter a (x-axis ranging from 0.0 to 10.0) in EI formula (3). The overall trend of these dots is illustrated using a moving average (dark blue curve). Dashed lines represent the percentages of recovered reference trees obtained with the PC (red) and EI (green) transformations (2).

3 . 4 -

 34 11-gb44a. BinDash and Dashing were used with the MH b-bit flavor with b = 18. Of note, as Mash and Bindash directly return the PC distance d, the corresponding p-distance was computed by p = 1e -d . Phylogenetic tree reconstructions from simulated sequences were performed with the script JolyTree v2.0. This version implements the PC and EI transformations (2) and (3) of the pairwise p-distances estimated by Mash. If any, missing evolutionary distances d uv = ∅ (i.e. corresponding to ĵ = 0 or p ≥ b 2 ) between sequences u and v are approximated by JolyTree from the other non-missing evolutionary distances by is derived from the triangle inequality property d uv ≤ d xu + d xv expected from the triplet of evolutionary distances induced by any sequence triplet u, v, x (see e.g. 62).

  The aim of this study is to assess the accuracy of Equation (2) and Equation (3) in transforming a MH p-distance p, where p is derived from the MH Jaccard index ĵ using Equation (1).

	In the following, analyses of large sets of simulated nucleotide
	sequences show three complementary results. First, current
	MH implementations enable p-distances to be conveniently esti-
	mated under several conditions. Second, PC and EI transfor-
	mations (2) and (3) of MH p-distance estimates can suitably
	approximate evolutionary distances derived from general time
	reversible (GTR; e.g. 34) models of nucleotide substitution.
	Third, PC and EI distances derived from MH estimates enable
	accurate phylogenetic tree reconstruction from unaligned nucle-
	otide sequences.

22,24,30-33.

It is worth noticing that when a is high, Equation (2) and Equation (3) yield very similar distance estimates (for any fixed b 1 and b 2 ).

F1000Research 2020, 9:1309 Last updated: 03 DEC 2020

Acknowledgements

The author thanks Pascal Campagne for his meaningful comments on the manuscript. The author is also obliged to Sylvain Brisse and to the Hub de Bioinformatique et Biostatistique, Institut Pasteur, Paris (France), for their support. This work used the computational and storage services (TARS cluster) provided by the IT department at Institut Pasteur, Paris.

The author(s) declared that no grants were involved in supporting this work.

Data availability

Source data A list of the 14,244 genome assemblies used to build the 187 genus datasets (Supplementary material of 29). https://doi.org/ 10.3897/rio.5.e36178.suppl2.

Extended data

Zenodo: Phylogenomic analyses of 142 prokaryotic genera. https://doi.org/10.5281/zenodo.4034244 [START_REF] Criscuolo | Phylogenomic analyses of 142 prokaryotic genera[END_REF] .

Zenodo: Simulated pairs of nucleotide sequences for testing (alignment-free) genome distance estimate methods. https://doi. org/10.5281/zenodo.4034461 [START_REF] Criscuolo | Simulated pairs of nucleotide sequences for testing (alignment-free) genome distance estimate methods[END_REF] .

Zenodo: Model trees and associated simulated nucleotide sequences for testing phylogenetic inference methods. https://doi. org/10.5281/zenodo.4034643 [START_REF] Criscuolo | Model trees and associated simulated nucleotide sequences for testing phylogenetic inference methods[END_REF] .

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

Yes

Is the study design appropriate and is the work technically sound? Yes

Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes

Are the conclusions drawn adequately supported by the results? Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Phylogenetics, comparative genomics and bioinformatics I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Not applicable

Are all the source data underlying the results available to ensure full reproducibility? Yes

Are the conclusions drawn adequately supported by the results? Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Software development for sequence comparison and phylogeny. Alignmentfree sequence comparison.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Brian Ondov

Genome Informatics section, National Human Genome Research Institute, USA, Bethesda, MD, USA

The manuscript describes a series of simulations assessing various parametric transformations of MinHash-based genomic distance estimates into more robust evolutionary distances. It concludes that (1) MinHash methods accurately estimate nucleotide distances, (2) that accurate estimates of corrected evolutionary distances can be derived from these, with the more parameterized of two models performing better, and (3) that accurate phylogenies can be inferred from those estimates.

The manuscript is well written and generally clear, although some of the methods are described so technically as to be hard to follow (e.g. p max and p r>0.99 in "MinHash-based p-distance approximation").

My main concern, which is not large, is the choice of the sketch size, s (or σ). The MinHash family of algorithms assumes s << G (genome size). The smallest sketch size tested here, however, is 1/5 of the average genome size. This size, described as "moderately large," is actually two or three orders of magnitude larger than Mash's default, for example. The justification for this is to lower the error bounds. As a consequence, however, what is being tested here is, in a sense, the viability The benefits of publishing with F1000Research: Your article is published within days, with no editorial bias • You can publish traditional articles, null/negative results, case reports, data notes and more •

The peer review process is transparent and collaborative • Your article is indexed in PubMed after passing peer review • Dedicated customer support at every stage • For pre-submission enquiries, contact research@f1000.com


Laurent Cordonier
email: laurent.cordonier@gmail.com

Florian Cafiero

Gérald Bronner

Why are conspiracy theories more successful in some countries than in others? An exploratory study on Internet users from 22 Western and non-Western countries

Keywords: conspiracy theories, distrust of authorities, international comparison, public sector corruption, social threat comparaison internationale, corruption du secteur public, défiance envers les autorités, menace sociale, théories du complot

This exploratory study aims at identifying macro-social factors associated with the international variance of belief in conspiracy theories. We computed a Conspiracy Index for 22 Western and non-Western countries based on the results of an online survey on conspiratorial beliefs. Stepwise regression analyses show that more than 70% of the international variance of this Conspiracy Index is explained by the following three national variables: the level of democracy, the unemployment rate, and the perceived level of public sector corruption. Conspiracy theories thus appear to be more commonly endorsed in countries where people cannot take an active part in the political life of their country (low level of democracy), where they may feel socially threatened (high unemployment rate), and where institutions and authorities are perceived as untrustworthy (high perception of public sector corruption).

Introduction

Conspiracy theories are narratives claiming that malevolent groups secretly manipulate the functioning of societies, and that authorities, the media, and/or scientists lie to the public about the real causes of significant events. Psychologists Robert Brotherton and Silan Eser (2015: 1) note that conspiracy theories have the following main characteristics: '... the label 'conspiracy theory' [...] typically refers to claims of conspiracy which are less plausible than alternative explanations, contradict the general consensus among epistemic authorities, are predicated on weak evidence, postulate unusually sinister and competent conspirators, and are ultimately unfalsifiable.' Several national and international surveys have shown that a significant proportion of the population in Western countries believes in various conspiracy theories (see, for example, YouGov, 2018). One reason for concern about the success of these theories is that there is a link between the endorsement of some of them and an inclination towards political and/or religious violence [START_REF] Bronner | La Démocratie des crédules[END_REF][START_REF] Bronner | Pourquoi les théories du complot se portent-elles si bien ? L'exemple de Charlie Hebdo[END_REF]. This link can be illustrated by recent tragedies, such as the 2019 attacks on the Muslim community in Christchurch, New Zealand, committed by a terrorist who referred to the 'Great Replacement' conspiracy theory to justify his actions, or the current wave of violence targeting Jews in the New York area, which is motivated by anti-Semitic conspiracy theories [START_REF] Levenson | Le complotisme à l'origine de l'antisémitisme noir aux États-Unis[END_REF]. Other conspiratorial beliefs can have adverse public health consequences. For example, it has been shown that parents who believe in conspiracy theories according to which the government and pharmaceutical companies cover up the alleged dangers of vaccines are less willing than other parents to vaccinate their children [START_REF] Shapiro | Validation of the vaccine conspiracy beliefs scale[END_REF].

Belief in conspiracy theories

There has been a considerable amount of research focusing on the psychological factors associated with belief in conspiracy theories. Cognitive and social psychologists have shown, for example, that conspiratorial beliefs are related to a variety of cognitive biases (e.g., [START_REF] Brotherton | Belief in conspiracy theories and susceptibility to the conjunction fallacy[END_REF][START_REF] Wagner-Egger | Creationism and conspiracism share a common teleological bias[END_REF], to intuitive (v. analytical) thinking style (e.g., [START_REF] Swami | Analytic thinking reduces belief in conspiracy theories[END_REF], and to paranormal beliefs (e.g., [START_REF] Bruder | Measuring individual differences in generic beliefs in conspiracy theories across cultures: Conspiracy mentality questionnaire[END_REF]. It has also been argued that conspiracy beliefs form part of a 'monological belief system' in which a conspiratorial idea serves as evidence for other conspiracist ideation [START_REF] Goertzel | Belief in conspiracy theories[END_REF]. This hypothesis is corroborated by numerous studies showing that the strongest predictor of belief in particular conspiracy theories is often belief in other conspiracy theories and that conspiracy beliefs tend to correlate strongly with each other (e.g., [START_REF] Brotherton | Measuring belief in conspiracy theories: The generic conspiracist beliefs scale[END_REF][START_REF] Goertzel | Belief in conspiracy theories[END_REF][START_REF] Swami | Conspiracist ideation in Britain and Austria: Evidence of a monological belief system and associations between individual psychological differences and realworld and fictitious conspiracy theories[END_REF].

In comparison, less research has been done on the social conditions that underlie conspiratorial beliefs [START_REF] Digrazia | The social determinants of conspiratorial ideation[END_REF]. Some studies indicate that conspiracy theories are particularly likely to attract individuals belonging to disadvantaged social groups, as well as those who feel socially threatened -for example, people worried about losing their job [START_REF] Digrazia | The social determinants of conspiratorial ideation[END_REF][START_REF] Goertzel | Belief in conspiracy theories[END_REF][START_REF] Mazzocchetti | Sentiments d'injustice et théorie du complot. Représentations d'adolescents migrants et issus des migrations africaines (Maroc et Afrique subsaharienne) dans des quartiers précaires de Bruxelles[END_REF].

These individuals can probably find in conspiracy theories an alternative worldview that makes sense of their situation and designates an unequivocal cause for the social injustices they believe (rightly or wrongly) to be victims of [START_REF] Abalakina-Paap | Beliefs in conspiracies[END_REF][START_REF] Hofstadter | The Paranoid Style in American Politics and Other Essays[END_REF][START_REF] Mazzocchetti | Sentiments d'injustice et théorie du complot. Représentations d'adolescents migrants et issus des migrations africaines (Maroc et Afrique subsaharienne) dans des quartiers précaires de Bruxelles[END_REF][START_REF] Sullivan | An existential function of enemyship: Evidence that people attribute influence to personal and political enemies to compensate for threats to control[END_REF]. More broadly, political scientists Joseph E. Uscinski and Joseph M. Parent (2014) argue that 'conspiracy theories are for losers': these theories appeal to people who perceive themselves to be on the margins of society (the 'losers') because they would allow them to regain a sense of power.

It has also been shown that distrust of authorities, institutions, and the media is a factor strongly associated with conspiratorial beliefs [START_REF] Abalakina-Paap | Beliefs in conspiracies[END_REF][START_REF] Barkun | Apocalyptic Visions in Contemporary America[END_REF][START_REF] Cordonier | Le succès des théories du complot. Flambée d'irrationalité ou symptôme d'une crise de confiance[END_REF]. Such mistrust could be both a consequence and a cause of belief in conspiracy theories. Indeed, in many conspiracy theories, the authorities and the media are presented as protagonists of the conspiracy. This could lead individuals convinced by these theories to distrust their government and the journalists. At the same time, individuals who distrust the authorities and the media will logically tend to reject their 'official explanations' of significant events. These individuals are thus more likely than others to be attracted to conspiratorial 'alternative explanations' of these events.

The propensity to endorse conspiratorial beliefs is also linked to individuals' political orientation. Indeed, as several studies have shown, people at the extreme ends of the political spectrum are more likely than others to believe in conspiracy theories (e.g., [START_REF] Van Prooijen | Political extremism predicts belief in conspiracy theories[END_REF]. Finally, some studies have found that conspiratorial beliefs are negatively related to educational level [START_REF] Douglas | Someone is pulling the strings: Hypersensitive agency detection and belief in conspiracy theories[END_REF][START_REF] Mancosu | Believing in conspiracy theories: Evidence from an exploratory analysis of Italian survey data[END_REF][START_REF] Van Prooijen | Why education predicts decreased belief in conspiracy theories[END_REF]. However, this relationship is not systematically found in correlational studies (see for e.g., [START_REF] Goertzel | Belief in conspiracy theories[END_REF].

Except for these few factors -sense of social threat, mistrust of authorities, institutions, and the media, political extremism and, probably, low educational attainment -little is known about the social variables that may explain, at the national level, why some groups of individuals are more receptive to conspiracy theories than others. Even less research has been done on the macro-social factors associated with the international variance of belief in conspiracy theories. To our knowledge, only one study has attempted to identify these factors on the basis of an international survey on conspiratorial beliefs [START_REF] Drochon | Who believes in conspiracy theories in Great Britain and Europe?[END_REF]. This survey, conducted in 2016, covered six European countries (Great Britain, Germany, Italy, Poland, Portugal, and Sweden). Without detailing the methodology followed, the author of the study comes to the following conclusions:

Poorer and less democratic countries, as measured by indices such as GDP, the Democracy Index, Transparency International, and the Gini coefficient […] return higher levels of conspiracy thinking than those that do better. […] So countries in which inequality is higher and democracy is considered not to be functioning as well as it should -that is, where citizens feel excluded politically and economically -will exhibit higher levels of conspiracy thinking. (Drochon, 2018: 337-338) In countries with little or no democracy, citizens cannot take an active part in the political life of their country, nor can they express themselves freely or access trustable information from independent media sources. Such a situation could increase the feeling of political powerlessness among the population, which could in turn lead to high levels of conspiratorial thinking. Studies have indeed shown that people who feel powerless are particularly receptive to conspiracy theories [START_REF] Abalakina-Paap | Beliefs in conspiracies[END_REF][START_REF] Van Prooijen | The influence of control on belief in conspiracy theories: Conceptual and applied extensions[END_REF].

Other studies point out that the popularity of certain conspiracy theories varies from country to country depending on cultural and geopolitical factors (e.g., [START_REF] Mashuri | The role of identity subversion in structuring the effects of intergroup threats and negative emotions on belief in anti-West conspiracy theories in Indonesia[END_REF][START_REF] Swami | Social psychological origins of conspiracy theories: The case of the Jewish conspiracy theory in Malaysia[END_REF]. For example, a case study in Indonesia shows that '[...] Muslims believe in anti-West conspiracy theories because they perceive the West as posing a symbolic and realistic threat to Muslims' group entity' (Mashuri et al., 2016: 2). In addition, the governments of some countries themselves encourage the dissemination of conspiracy theories for propaganda purposes, as it is for example the case in Russia (e.g., [START_REF] Yablokov | theories as a Russian public diplomacy tool: The case of Russia Today (RT)[END_REF].

The present study

The aim of this study is to identify macro-social factors associated with the international variance of belief in conspiracy theories. To do so, we first computed a Conspiracy Index for 22 Western and non-Western countries based on the results of an online survey on conspiratorial beliefs conducted in 2019. Second, we conducted correlation and stepwise regression analyses to assess the relationships between the Conspiracy Index of the 22 countries and a set of national variables. The following macro-social variables were selected on the basis of the existing literature:

1) Level of democracy: It has been suggested that the level of democracy is negatively related to the level of conspiracy belief [START_REF] Drochon | Who believes in conspiracy theories in Great Britain and Europe?[END_REF]. To test this hypothesis, we took into account the following national variables in our analyses: the Democracy Index, the level of freedom of the press, and the level of freedom of discussion. The levels of freedom of the press and of discussion are relevant as proxy for a country's level of democracy.

2) Distrust of institutions and authorities: Studies have shown that distrust of institutions and authorities is positively related to conspiratorial beliefs [START_REF] Abalakina-Paap | Beliefs in conspiracies[END_REF][START_REF] Barkun | Apocalyptic Visions in Contemporary America[END_REF][START_REF] Cordonier | Le succès des théories du complot. Flambée d'irrationalité ou symptôme d'une crise de confiance[END_REF]. To account for this factor, we included the following national variables in our analyses: the perceived level of public sector corruption and the perceived level of politicians' ethics. The rationale is that if people perceive their institutions as corrupt and/or their politicians as behaving unethically then, by definition, they view them as untrustworthy.

3) Social threat: Populations that feel socially threatened tend to be more conspiratorial than others [START_REF] Digrazia | The social determinants of conspiratorial ideation[END_REF][START_REF] Goertzel | Belief in conspiracy theories[END_REF]. To account for this factor, the following national variables were included in our analyses: the unemployment rate, the intentional homicide rate, and the level of life satisfaction. These indicators of social threat were chosen because they are computed according to the same methodology for all countries and therefore allow for a valid international comparison.

4) Income, inequality, and education: Low levels of economic inequality and high levels of income and education appear to protect against conspiratorial beliefs [START_REF] Douglas | Someone is pulling the strings: Hypersensitive agency detection and belief in conspiracy theories[END_REF][START_REF] Drochon | Who believes in conspiracy theories in Great Britain and Europe?[END_REF][START_REF] Mancosu | Believing in conspiracy theories: Evidence from an exploratory analysis of Italian survey data[END_REF][START_REF] Van Prooijen | Why education predicts decreased belief in conspiracy theories[END_REF]. To account for these factors, we incorporated the following national variables into our analyses: the Gini Index, which measures income inequality within a country, and the Human Development Index, which is a summary measure of average standard of living, educational level, and life expectancy at birth in a country.

Material and methods

Conspiracy Index

We computed a Conspiracy Index based on an online survey of the YouGov Institute conducted in 22 countries from February 28 to March 26, 2019. This survey is part of the YouGov-Cambridge Globalism Project.1 It should be noted that we took no part in the design or execution of this survey and that we used the data made publicly available by YouGov on its website.2

In the survey, respondents, all aged 18 and over, were asked to evaluate 8 conspiratorial statements by answering the following question: 'Which of the following statements, if any, would you say are true? (Please select all that apply).' Participants could also select 'None of these' or 'Prefer not to say'. The 8 items tested were the following:

-The US Government knowingly helped to make the 9/11 terrorist attacks happen in America on 11 September 2001 (item called 'September 11 conspiracy' thereafter).

-The truth about the harmful effects of vaccines is being deliberately hidden from the public (item called 'Vaccines conspiracy' thereafter). -Regardless of who is officially in charge of governments and other organizations, there is a single group of people who secretly control events and rule the world together (item called 'Secret group conspiracy' thereafter).

-The idea of man-made global warming is a hoax that was invented to deceive people (item called 'Global warming conspiracy' thereafter).

-Humans have made contact with aliens and this fact has been deliberately hidden from the public (item called 'Hidden Aliens conspiracy' thereafter). The Conspiracy Index was computed as follows. First, we discarded the item on Russian interference in the 2016 US elections (last item listed above). Indeed, this item does not correspond to the definition of a conspiracy theory. It does not, for example, fit into the criteria of a conspiracy theory retained by [START_REF] Brotherton | Bored to fears: Boredom proneness, paranoia, and conspiracy theories[END_REF], because there is no consensus among the epistemic authorities on the exact extent and nature of Russian interference in the 2016 US elections. Due to this lack of consensus among the epistemic authorities, it is possible that reliable media sources published contradictory information on this topic. Therefore, we discarded this last item to avoid introducing unnecessary noise in the construction of the Conspiracy Index.

Second, we averaged for each country the percentage of positive assessments (i.e., percentage of 'true' answers) received by the 7 remaining conspiratorial statements. These averages correspond to the Conspiracy Index of the 22 countries (see Figure 1 and Table 1). Aggregating various conspiratorial items in this way is justified by the monological nature of conspiracist ideation [START_REF] Goertzel | Belief in conspiracy theories[END_REF] and is a common procedure in conspiracy theory research (see, for example, [START_REF] Brotherton | Measuring belief in conspiracy theories: The generic conspiracist beliefs scale[END_REF][START_REF] Bruder | Measuring individual differences in generic beliefs in conspiracy theories across cultures: Conspiracy mentality questionnaire[END_REF][START_REF] Goertzel | Belief in conspiracy theories[END_REF][START_REF] Lantian | Measuring belief in conspiracy theories: Validation of a French and English single-item scale[END_REF][START_REF] Swami | Conspiracist ideation in Britain and Austria: Evidence of a monological belief system and associations between individual psychological differences and realworld and fictitious conspiracy theories[END_REF].

Note that YouGov did not test the Holocaust conspiracy item in Germany. To compensate for this missing data, we computed the mean value of this item among the 21 other countries and attributed it to Germany. The 7-item Conspiracy Index computed for the 22 countries has a very good internal consistency (Cronbach alpha test, N = 22, 7 items, α = 0.89) and its distribution between the 22 countries does not significantly differ from normality (Shapiro-Wilk test, N = 22, W = 0.93, p = 0.15; see Figure 2). Unfortunately, we were unable to conduct these analyses at the individual level within each country, because YouGov did not give us access to detailed data by country. However, prior research has systematically found satisfactory internal consistency at the individual level between various conspiracy items tested in a given country (e.g., [START_REF] Brotherton | Measuring belief in conspiracy theories: The generic conspiracist beliefs scale[END_REF][START_REF] Bruder | Measuring individual differences in generic beliefs in conspiracy theories across cultures: Conspiracy mentality questionnaire[END_REF][START_REF] Goertzel | Belief in conspiracy theories[END_REF][START_REF] Lantian | Measuring belief in conspiracy theories: Validation of a French and English single-item scale[END_REF][START_REF] Swami | Conspiracist ideation in Britain and Austria: Evidence of a monological belief system and associations between individual psychological differences and realworld and fictitious conspiracy theories[END_REF]. It should be noted that the YouGov survey we used to compute the Conspiracy Index was conducted online, which probably limits its demographic representativeness in countries where the Internet is used by only a minority of the population. We therefore take the Conspiracy Index as an indicator of the level of conspiratorial beliefs among the Internet users of the 22 countries rather than as an indicator of the prevalence of conspiratorial beliefs among the overall population of these countries.

While the demographic representativeness of the YouGov survey is limited by the fact that it has been conducted online, this also represents an advantage. By definition, this survey measured the conspiratorial beliefs of individuals who all have access to the conspiracy theories circulating on the web. As the Internet is an important vector of conspiracy theories (e.g., [START_REF] Bronner | La Démocratie des crédules[END_REF], it is not unjustified to compare the success of an identical set of conspiracy theories among these individuals, even though they live in societies that differ significantly from each other in many ways.

In addition to its lack of demographic representativeness, another potential limitation of the YouGov survey lies in the fact that, at the national level, the success of some of the conspiracy theories tested might depend more on cultural and geopolitical factors than on the actual level of conspiratorial thinking among the population. For example, a strong anti-American sentiment in a given country could increase respondents' tendency to endorse an item such as the September 11 conspiracy, not necessarily because they believe in this specific theory but to express their hatred for the US.

In our study, the level of the Conspiracy Index is significantly lower in the 11 Western countries covered by the YouGov survey (Australia, Canada, Denmark, France, Germany, Italy, Poland, Spain, Sweden, UK, US; mean Conspiracy Index = 11.88) than in the 11 non-Western countries (mean Conspiracy Index = 18.57; bilateral t-test: t(20) = 3.79, p = 0.001; see Figure 3). This could suggest that our index is culturally biased. The possibility of such a bias is a common concern in studies using results of international surveys on conspiratorial beliefs. In an attempt to overcome it, psychologists have developed standardized conspiracy scales using generic items -that is, items that avoid mentioning any specific alleged conspiracy and that do not name any specific groups potentially responsible for a conspiracy [START_REF] Brotherton | Measuring belief in conspiracy theories: The generic conspiracist beliefs scale[END_REF][START_REF] Bruder | Measuring individual differences in generic beliefs in conspiracy theories across cultures: Conspiracy mentality questionnaire[END_REF].

In the YouGov survey we used to compute the Conspiracy Index, only one item is generic in this sense: the Secret group conspiracy ('Regardless of who is officially in charge of governments and other organizations, there is a single group of people who secretly control events and rule the world together'). This item is close to one of the 5 generic, cross-culturally validated items of the Conspiracy Mentality Questionnaire by Bruder and colleagues (2013): 'I think that there are secret organizations that greatly influence political decisions'. It is also similar to the 3 items termed 'malevolent global conspiracies' in the Generic Conspiracist Beliefs Scale by Brotherton and colleagues (2013) (e.g., 'A small, secret group of people is responsible for making all major world decisions, such as going to war').

As the Secret group conspiracy is a generic item, responses to this item are unlikely to be biased by cultural or geopolitical variables [START_REF] Bruder | Measuring individual differences in generic beliefs in conspiracy theories across cultures: Conspiracy mentality questionnaire[END_REF]. In order to control whether or not the Conspiracy Index as a whole is affected by cultural or geopolitical factors, we computed its value for each country with 6 items only, excluding the generic Secret group conspiracy one. Next, we compared the level of this 6-item Conspiracy Index with that of the Secret group conspiracy item among the 22 countries. A correlation analysis shows that the 6-item Conspiracy Index and the Secret group conspiracy item are highly correlated (R = 0.80, p < 0.001; N = 22; see Figure 4). This clearly suggests that the Conspiracy Index itself is not fundamentally biased by cultural or geopolitical variables. Moreover, if the responses to the YouGov survey were influenced to a large extent by geopolitical cleavages, items reflecting those cleavages should be those for which the rate of positive evaluations varies the most between the countries. An examination of the evaluations received by each item tested in the 22 countries shows that this is not the case. Indeed, it is the generic Secret group conspiracy item that shows the greatest variability between the 22 countries (SD = 9.52; see Table 1). The September 11 conspiracy item -potentially the weakest item in the YouGov survey, since it is the only one to explicitly mention a national actor -is only in third place out of seven according to this criterion (SD = 7.25; see Table 1).

Finally, as mentioned above, the level of the Conspiracy Index is lower in Western than in non-Western countries. But this difference is not due to a stronger endorsement of some items in particular in the group of non-Western countries, which could have indicated that the Conspiracy Index is culturally biased. On the contrary, there is a quasi-linear relationship between the two groups of countries in the endorsement of each of the 7 items composing the Conspiracy Index (see Figure 5).

Overall, the computed Conspiracy Index has a very good internal consistency and is unlikely to be strongly biased by cultural variables or geopolitical cleavages. This index thus allows a credible comparison of the level of conspiratorial beliefs between Internet users of the 22 countries included in this study.

Independent variables

As mentioned in the introduction, we hypothesized that the following variables may contribute to explain the international variance of the Conspiracy Index. The descriptive statistics of the independent variables of the 22 countries taken together are exposed in Table 2.

Freedom of the press (reversed World Press Freedom Index)

To assess the level of Freedom of the press in each of the 22 countries of this study, we used the 2019 World Press Freedom Index (WPFI). The WPFI is an index published annually by Reporters Without Borders (RSF). It is computed as follows: 'The degree of freedom available to journalists in 180 countries is determined by pooling the responses of experts to a questionnaire devised by RSF. This qualitative analysis is combined with quantitative data on abuses and acts of violence against journalists during the period evaluated. The criteria evaluated in the questionnaire are pluralism, media independence, media environment and self-censorship, legislative framework, transparency, and the quality of the infrastructure that supports the production of news and information.' 4 Note that the higher the WPFI value, the less freedom of the press. In our study, we reversed the polarity of the index using the following transformation for each country's WPFI: 'Reversed WPFI' = (100 -WPFI). Thus, the higher the value of the 'Reversed WPFI' (called Freedom of the press thereafter), the greater the freedom of the press. 5

Freedom of discussion (mean of 'Freedom of discussion for women' and 'Freedom of discussion for men') To assess the level of Freedom of discussion in each of the 22 countries of this study, we used the 2019 'Freedom of discussion for women' and 'Freedom of discussion for men' indicators made available by the V-Dem Institute. The description of these indicators is the same for women and men: 'This indicator specifies the extent to which women [or men] are able to engage in private discussions, particularly on issues, in private homes and public spaces (restaurants, public transportation, sports events, work, etc.) without fear of harassment by other members of the polity or the public authorities. We are interested in restrictions by the government and its agents but also cultural restrictions or customary laws that are enforced by other members of the polity, sometimes in informal ways' (Coppedge, Gerring, Knutsen, et al., 2019: 165). The range of values for these indicators goes from 0 = freedom of discussion not respected to 4 = freedom of discussion fully respected.

In our study, we averaged for each country the 'Freedom of discussion for women' and 'Freedom of discussion for men' indicators (in the 22 countries of this study, these two indicators are very highly intercorrelated: R = 0.97; p < 0.001; N = 22). These averages correspond to the Freedom of discussion values of the 22 countries. 6

Democracy Index

The Democracy Index is an index compiled by the Economist Intelligence Unit: 'The Democracy Index is based on five categories: electoral process and pluralism; the functioning of government; political participation; political culture; and civil liberties' (Economist Intelligence Unit, 2019: 3). The higher a country's score, the more democratic is this country.

Corruption perception (reversed Corruption Perception Index)

To assess the level of Corruption perception in each of the 22 countries of this study, we used the 2019 Corruption Perception Index (CPI). The CPI is an index published annually by Transparency International indicating populations' 'perceived levels of public sector corruption, as determined by expert assessments and opinion surveys'. Corruption is defined as 'the misuse of public power for private benefit '. 7 Note that the higher the CPI value, the lower the perception of corruption. In our study, we reversed the polarity of the index using the following transformation for each country's CPI: 'Reversed CPI' = (100 -CPI). Thus, the higher the value of the 'reversed CPI' (called Corruption perception thereafter), the stronger the perception of corruption.8

Politicians' ethics perception

To assess the public perception of politicians' ethics (called Politicians' ethics perception thereafter), we used the following item from the Global Competitiveness Index established by the World Economic Forum (index component 1.04): 'In your country, how would you rate the ethical standards of politicians?' Answers range from 1 = 'extremely low' to 7 = 'extremely high' (World Economic Forum, 2017).

Unemployment

The unemployment rate for each of the 22 countries of this study is that indicated by the World Bank. It is expressed in % of total labor force (modeled ILO estimate).9 6 Source: V-Dem -Varieties of Democracy, V-Dem Dataset Version 9; https://www.vdem.net/en/data/archive/previous-data/data-version-9/. Data year: 2018. 7 https://www.transparency.org/

Intentional homicides

The intentional homicides for each of the 22 countries of this study is expressed as the number of homicides per 100 000 people. The World Bank's description of this indicator is the following: 'Intentional homicides are estimates of unlawful homicides purposely as a result of domestic disputes, interpersonal violence, violent conflicts over land resources, intergang violence over turf or control, and predatory violence and killing by armed groups. Intentional homicide does not include all intentional killing; the difference is usually in the organization of the killing. Individuals or small groups usually commit homicide, whereas killing in armed conflict is usually committed by fairly cohesive groups of up to several hundred members and is thus usually excluded.' 10

Life satisfaction

Life satisfaction is a self-reported measure corresponding to the average answer to the Cantril Ladder question in the Gallup World Poll: 'Please imagine a ladder, with steps numbered from 0 at the bottom to 10 at the top. The top of the ladder represents the best possible life for you and the bottom of the ladder represents the worst possible life for you. On which step of the ladder would you say you personally feel you stand at this time?' 11

Gini Index (inequality index)

The US Central Intelligence Agency (CIA) description of this indicator is the following: 'Gini index measures the degree of inequality in the distribution of family income in a country. The more nearly equal a country's income distribution, the lower its Gini index […]. The more unequal a country's income distribution, the higher its Gini index […]. If income were distributed with perfect equality the index would be zero; if income were distributed with perfect inequality, the index would be 100.' 12

Human Development Index

The United Nations Development Programme's description of this indicator is the following: 'The Human Development Index (HDI) is a summary measure of average achievement in key dimensions of human development: a long and healthy life, being knowledgeable and have a decent standard of living. The HDI is the geometric mean of normalized indices for each of the three dimensions. The health dimension is assessed by life expectancy at birth, the education dimension is measured by mean of years of schooling for adults aged 25 years and more and expected years of schooling for children of school entering age. The standard of living dimension is measured by gross national income per capita.' 13

Results

The correlogram in Figure 6 shows the relationships (Pearson's correlations) between all the variables presented in the previous section. It can be observed that, among the 22 countries of this study, the Conspiracy Index is positively associated with Corruption perception (R = 0.81, p < 0.001; N = 22; see Figure 7.A), Intentional homicides (R = 0.50, p < 0.05; N = 22), and Gini Index (R = 0.63, p < 0.01; N = 22).

The positive correlation between the Conspiracy Index and Unemployment is only tendentially significant (R = 0.42, p = 0.05; N = 22). In order to assess which of the macro-social factors we have taken into account affect the level of conspiratorial beliefs across the 22 countries, we conducted a stepwise multiple linear regression analysis with, as the dependent variable, the Conspiracy Index and, as independent variables, Freedom of the press, Freedom of discussion, Democracy Index, Corruption perception, Politicians' ethics perception, Unemployment, Intentional homicides, Life satisfaction, Gini Index, and HDI. A stepwise regression analysis was used because of the high level of intercorrelation between the independent variables (see Figure 6) and the exploratory nature of this study.

The Conspiracy Index is negatively associated with

Results show that the best-fitted model includes three independent variables out of ten: Democracy Index, Corruption perception, and Unemployment. This model explains 73% of the variance of the Conspiracy Index across the 22 countries (Adjusted R 2 = 0.731, F(3,18) = 20.03, p < 0.001; see Table 3).

Human_Development_Index

To evaluate whether the of this regression analysis is driven by a specific conspiracy item or a specific country, we conduced robustness analyses by testing the best-fitted model (dependent variable: Conspiracy Index; independent variables: Democracy Index, Corruption perception, and Unemployment) by omitting, first, one of the 7 items comprised in the Conspiracy Index at the time and, second, one of the 22 countries at the time:

1) Inspection of Table 4 shows that the result of the initial regression analysis is not driven by a specific item: the weakest adjusted R 2 , obtained by omitting the generic Secret group conspiracy item, is still of 0.679, p < 0.001. Table 5 exposes the complete regression analysis with this item removed from the Conspiracy Index.

2) Inspection of Table 6 shows that the result of the initial regression analysis is not driven a specific country: the weakest adjusted R 2 , obtained by omitting Denmark, is still of 0.699, p < 0.001. Table 7 exposes the complete regression analysis without this country.

Discussion

The aim of this exploratory study was to identify macro-social factors associated with the international variance of belief in conspiracy theories. Using the outcomes of a recent online survey on conspiratorial beliefs covering 22 countries, we found that the level of belief in conspiracy theories is higher among Internet users from countries (1) with weak democracies, (2) where politicians are perceived as behaving unethically and the public sector is perceived as corrupt, (3) where the level of social threat is high, and (4) where income and education levels are low, while the level of economic inequality is high.

Regression analyses suggest that it is not the levels of wealth, education, and economic inequality that directly modulate the success of conspiracy theories among the 22 countries of this study. Indeed, we found that more than 70% of the international variance in conspiratorial beliefs among these countries is explained by the following three national variables: the level of democracy, the unemployment rate, and the perceived level of public sector corruption.

Since the present study is correlational in nature, it does not allow to conclude with certainty that these macro-social factors play a causal role in the endorsement of conspiracy theories at the individual level. However, as indicated in the introductory section of this paper, previous research has shown that individuals who feel powerless, socially threatened, and who distrust the authorities and institutions are more likely than others to endorse conspiracy theories. Therefore, a country with a low level of democracy, a high unemployment rate, and where the public sector is perceived as highly corrupt offers fertile ground for conspiracy theories. Indeed, in such a country many citizens may feel powerless, as they cannot take an active part in the political life of their country, they also may feel socially threatened, as the risk of losing one's job is tangible and, finally, they are likely to perceive their country's institutions and authorities as untrustworthy, as they see them as corrupt. In other words, in this country, the social and psychological conditions favoring the endorsement of conspiracy theories are present for a larger part of the population than in more democratic and less corrupt countries, where the unemployment rate is also lower.

In conclusion, it should be noted that this research has several limitations. The Conspiracy Index we have computed for the 22 countries of this study is based on an opinion survey that was conducted online, which probably limits its demographic representativeness in countries where the Internet is used by only a minority of the population. It also cannot be totally excluded that this Conspiracy Index is partly biased by cultural variables or geopolitical cleavages, although the controls we have carried out suggest that this is not the case. Finally, the number of countries covered by our study is small. This necessarily affects the robustness of our results, which should be replicated with a larger set of countries.

Despites these limitations, this study is, to our knowledge, the first to identify macro-social factors associated with the international variance of belief in conspiracy theories among a sample of more than twenty Western and non-Western countries. Results of this study, conducted at the international level, corroborate and extend those of existing research at the national level.

  -The AIDS virus was created and spread around the world on purpose by a secret group or organization (item called 'AIDS conspiracy' thereafter). -The official account of the Nazi Holocaust is a lie and the number of Jews killed by the Nazis during World War II has been exaggerated on purpose (item called 'Holocaust conspiracy' thereafter). -(Members of Donald Trump's election team knowingly worked with the Russian Government to help him win the 2016 US Presidential Election.) 3 Data from this survey are available for the following 22 countries: Australia (N of respondents = 1006), Brazil (N = 1006), Canada (N = 1006), Denmark (N = 1010), Egypt (N = 1537), France (N = 1021), Germany (N = 1479), India (N = 1035), Indonesia (N = 1001), Italy (N = 1005), Japan (N = 1143), Mexico (N = 1009), Nigeria (N = 644), Poland (N = 1019), Saudi Arabia (N = 828), South Africa (N = 1002), Spain (N = 1014), Sweden (N = 1011), Thailand (N = 1043), Turkey (N = 506), United Kingdom (UK) (N = 1949), United States (US) (N = 2012).

Figure 1 .

 1 Figure 1. Conspiracy Index of the 22 countries

Figure 2 .

 2 Figure 2. Density plot of the Conspiracy Index (N = 22). The vertical dashed line indicates the mean value of the Conspiracy Index.

Figure 3 .

 3 Figure 3. Level of the Conspiracy Index in the non-Western and Western countries

Figure 4 .

 4 Figure 4. Pearson's correlation between the 6-item Conspiracy Index and the Secret group conspiracy item. Each dot represents a country (N = 22). Grey surface: confidence interval (95%).

Figure 5 .

 5 Figure 5. Comparison of the of endorsement of the 7 items composing the Conspiracy Index between the Western and non-Western countries (Pearson's correlation: R = 0.89, p < 0.01; N = 7). Grey surface: confidence interval (95%).

4

  https://rsf.org/en/detailed-methodology 5 Source: Reporters Without Borders; https://rsf.org/en/ranking. Data year: 2019.

  Freedom of the press (R = -0.66, p < 0.001; N = 22), Freedom of discussion (R = -0.55, p < 0.01; N = 22), Democracy Index (R = -0.73, p < 0.001; N = 22; see Figure 7.B), Politicians' ethics perception (R = -0.60, p < 0.01; N = 22), Life satisfaction (R = -0.66, p < 0.001; N = 22), and HDI (R = -0.72, p < 0.001; N = 22).

Figure 6 .

 6 Figure 6. Pearson's correlations all the variables (N = 22 countries). Correlation coefficients (R) not crossed out are statistically significant (p < 0.05).

Figure 7 .

 7 Figure 7. Pearson's correlations between the Conspiracy Index and (A) Corruption perception, (B) Democracy Index. Each dot represents a country (N = 22). The color of the dots indicates countries' HDI. Grey surfaces: confidence intervals (95%).

Table 1 .

 1 Descriptive statistics of the Conspiracy Index and of its 7 items (N = 22 countries)

	Min.	Max.	Mean	Median	SD

Table 2 .

 2 Descriptive statistics of the independent variables (N = 22 countries)

	Min.	Max.	Mean	Median SD

Table 3 .

 3 Multiple linear regression analysis, best-fitted model. Dependent variable: Conspiracy Index; Independent variables: Democracy Index, Corruption perception, and Unemployment.

		Estimate	Std. Error	t-value	p-value
	(Intercept)	13.471	4.653	2.895	0.009 **
	Democracy Index	-0.853	0.430	-1.984	0.063 °
	Corruption perception	0.135	0.046	2.940	0.009 **
	Unemployment	0.259	0.106	2.444	0.025 *
	Residual standard error: 2.748 on 18 degrees of freedom; Multiple R 2 = 0.770; Adjusted R 2 = 0.731; F-
	statistic: 20.03 on 3 and 18 DF; p < 0.001.

Table 4 .

 4 Robustness analysis: multiple linear regression analyses (best-fitted model) omitting one of the 7 items comprised in the Conspiracy Index at the time

	Conspiracy Index	Multiple R 2	Adjusted R 2	F-statistic*	p-value
	Omitting September 11 conspiracy	0.772	0.734	20.32	< 0.001
	Omitting Vaccines conspiracy	0.762	0.723	19.22	< 0.001
	Omitting Secret group conspiracy	0.725	0.679	15.83	< 0.001
	Omitting Global warming conspiracy 0.786	0.750	22.01	< 0.001
	Omitting Hidden Aliens conspiracy	0.787	0.752	22.17	< 0.001
	Omitting AIDS conspiracy	0.764	0.725	19.44	< 0.001
	Omitting Holocaust conspiracy	0.730	0.685	16.25	< 0.001
	* On 3 and 18 degrees of freedom

Table 5 .

 5 Multiple linear regression analysis, best-fitted model. Dependent variable: Conspiracy Index without the Secret group conspiracy item

		Estimate	Std. Error	t-value	p-value
	(Intercept)	14.101	4.607	3.061	0.007 **
	Democracy Index	-0.936	0.426	-2.199	0.041 *
	Corruption perception	0.107	0.045	2.349	0.030 *
	Unemployment	0.186	0.105	1.774	0.093 °
	Residual standard error: 2.721 on 18 degrees of freedom; Multiple R 2 = 0.725; Adjusted R 2 = 0.679; F-

statistic: 15.83 on 3 and 18 DF; p < 0.001.

Table 6 .

 6 Robustness analysis: multiple linear regression analyses (best-fitted model) omitting one of the 22 countries at the time

	Countries	Multiple R 2	Adjusted R 2	F-statistic*	p-value
	Omitting Australia	0.769	0.728	18.84	< 0.001
	Omitting Brazil	0.807	0.773	23.65	< 0.001
	Omitting Canada	0.775	0.736	19.53	< 0.001
	Omitting Denmark	0.744	0.699	16.47	< 0.001
	Omitting Egypt	0.757	0.714	17.67	< 0.001
	Omitting France	0.767	0.726	18.66	< 0.001
	Omitting Germany	0.797	0.761	22.22	< 0.001
	Omitting India	0.772	0.731	19.14	< 0.001
	Omitting Indonesia	0.812	0.779	25.54	< 0.001
	Omitting Italy	0.782	0.743	20.28	< 0.001
	Omitting Japan	0.759	0.717	17.89	< 0.001
	Omitting Mexico	0.787	0.749	20.92	< 0.001
	Omitting Nigeria	0.749	0.705	16.94	< 0.001
	Omitting Poland	0.769	0.729	18.89	< 0.001
	Omitting Saudi Arabia	0.781	0.743	20.22	< 0.001
	Omitting South Africa	0.748	0.703	16.80	< 0.001
	Omitting Spain	0.770	0.729	18.92	< 0.001
	Omitting Sweden	0.753	0.709	17.23	< 0.001
	Omitting Thailand	0.778	0.739	19.90	< 0.001
	Omitting Turkey	0.759	0.716	17.81	< 0.001
	Omitting UK	0.757	0.715	17.69	< 0.001
	Omitting USA	0.768	0.727	18.78	< 0.001
	* On 3 and 17 degrees of freedom

Table 7 .

 7 Multiple linear regression analysis, best-fitted model. Country excluded: DenmarkResidual standard error: 2.760 on degrees of freedom; Multiple R 2 = 0.744; Adjusted R 2 = 0.699; Fstatistic: 16.47 on 3 and 17 DF; p < 0.001.

		Estimate	Std. Error	t-value	p-value
	(Intercept)	14.120	4.726	2.988	0.008 **
	Democracy Index	-0.860	0.432	-1.991	0.063 °
	Corruption perception	0.124	0.047	2.614	0.018 *
	Unemployment	0.257	0.106	2.416	0.027 *

https://yougov.co.uk/topics/international/articles-reports/2019/05/01/about-yougov-cambridge-globalismproject

2 https://d25d2506sfb94s.cloudfront.net/cumulus_uploads/document/2c6lta5kbu/YouGov%20Cambridge%20Gl obalism%20Project%20-%20Conspiracy%20Theories.pdf

This item was not included in our Conspiracy Index.

Source: Transparency International; https://www.transparency.org/cpi2019#downloads. Data year: 2019.

Source: https://data.worldbank.org/indicator/SL.UEM.TOTL.ZS. Data year: 2019.

Biographies

Laurent Cordonier holds a Ph.D in social sciences from University of Lausanne. He is currently associate researcher at Paris-Diderot University. His work focuses on children's socialization process, social cognition, collective beliefs, and on exploring how some methods and knowledge of cognitive science can be integrated into classical sociological paradigms. In 2018 he published La Nature du social. L'apport ignoré des sciences cognitives (The Nature of the Social. The Overlooked Input of Cognitive Science) at the Presses Universitaires de France.

Florian Cafiero is a research engineer at the Groupe d'étude des méthodes de l'analyse sociologique de la Sorbonne (GEMASS). A specialist in quantitative sociology and computational social science, he is a lecturer in quantitative methods for social science and humanities and in social networks at PSL University. His work has focused on the politicization and fragmentation of social movements around vaccine controversies, but also on conspiracy theories and radicalization.

Gérald Bronner is professor of sociology at Paris-Diderot University and member of the National Academy of Technologies of France (NATF) and the French Academy of Medicine. He works in particular on collective beliefs and cognitive sociology and published in 2013 La Démocratie des crédules (The Democracy of the Gullible) at the Presses Universitaires de France.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: this study was conducted as part of a project funded by the French Agence Nationale de la Recherche (ANR) entitled 'CONSPIRACY -Responding to the spreading of conspiracy theories' (ANR-17-CE39-0010-01).