Gambling-Specific Cognitions Are Not Associated With Either Abstract or Probabilistic Reasoning: A Dual Frequentist-Bayesian Analysis of Individuals With and Without Gambling Disorder

Background Distorted gambling-related cognitions are tightly related to gambling problems, and are one of the main targets of treatment for disordered gambling, but their etiology remains uncertain. Although folk wisdom and some theoretical approaches have linked them to lower domain-general reasoning abilities, evidence regarding that relationship remains unconvincing. Method In the present cross-sectional study, the relationship between probabilistic/abstract reasoning, as measured by the Berlin Numeracy Test (BNT), and the Matrices Test, respectively, and the five dimensions of the Gambling-Related Cognitions Scale (GRCS), was tested in a sample of 77 patients with gambling disorder and 58 individuals without gambling problems. Results and interpretation Neither BNT nor matrices scores were significantly related to gambling-related cognitions, according to frequentist (MANCOVA/ANCOVA) analyses, performed both considering and disregarding group (patients, non-patients) in the models. Correlation Bayesian analyses (bidirectional BF10) largely supported the null hypothesis, i.e., the absence of relationships between the measures of interest. This pattern or results reinforces the idea that distorted cognitions do not originate in a general lack of understanding of probability or low fluid intelligence, but probably result from motivated reasoning.


INTRODUCTION
Gambling is a leisure activity, practised non-problematically by a large share of the population, but that can generate substantial harm to the community (Shannon et al., 2017). The severity of potentially problematic gambling lies on a continuum in which gambling disorder is placed at its highest end (Shaffer and Martin, 2011;Rai et al., 2014). However, from a public health perspective, gambling-related harms go beyond the individual, and are not exclusively driven by the severity of disordered gambling (Wardle et al., 2019).

A Brief Review of the Literature on the Link Between Domain-General Reasoning and Gambling Cognitions
Studies on domain-general reasoning skills in gamblers fall into three broad categories. In the first one, intelligence or domain-general reasoning is recorded only for control purposes, in case-control designs with problematic vs. non-problematic gambling (so that domain-general reasoning measures were not the main variables of interest). This category is heterogeneous and the studies in it do not systematically report associations between domain-general reasoning and gambling cognitions. With regard to the association between domain general reasoning and gambling disorder symptoms or diagnosis, results are mixed: in some studies, the group with disordered or problematic gambling obtained lower scores than controls in domaingeneral reasoning constructs (Martínez-Pina et al., 1991;Toplak et al., 2007;Forbush et al., 2008), whereas, in others, the groups did not show significant differences (Brevers et al., 2012). It is important to take into account, however, that in part of these studies, domain-general reasoning scores were intentionally matched across groups (groups were sampled a priori to show no differences in general reasoning ability), so the absence of differences in reasoning abilities between groups is not always informative. For that reason, studies in which matching in general reasoning measures was forced are not included in this review.
A second category of studies has intentionally investigated the putative associations between gambling severity (or presence of gambling disorder/problem gambling) and domain-general reasoning (Templer et al., 1993;Fernández-Montalvo et al., 1999;Delfabbro et al., 2006;Lambos and Delfabbro, 2007;Kaare et al., 2009;Hodgins et al., 2012;Rai et al., 2014;Primi et al., 2017) in broad community or convenience samples, using regression or correlation techniques. These show that individuals with low domain-general reasoning abilities show more severe gambling problems or are in a higher risk of presenting disordered or problematic gambling, with few exceptions [(Fernández-Montalvo et al., 1999); in Primi et al. (2017), gambling problems' severity was found to correlate positively with fluid intelligence, but negatively with probabilistic reasoning]. Again, however, gambling-specific cognitions were not central variables of interest. With the exception of Lambos and Delfabbro (2007), the moderating role of gambling-related cognitions in the association between general reasoning and gambling problems was not assessed either.
Studies of these two categories, primarily or supplementarily estimating the association between domain-general reasoning abilities and presence or severity of gambling problems, are summarized in Table 1.
A third category of studies, more directly relevant to the aims of the present study, has directly investigated whether gambling-related cognitions are underpinned in some way by domain-general reasoning processes. Most of the studies in this category are also observational or correlational, but they do straightforwardly focus on the relationship between domaingeneral and gambling-related reasoning. For instance, using a card-guessing task, Xue et al. (2012a) found that students with higher cognitive abilities (intelligence and executive function) were more prone to show the gambler's fallacy. i.e., the erroneous belief that streaks of bad luck are bound to end in a win. In a similar vein, Perales et al. (2017) found gamblers with stronger biases to perform better than gamblers with weaker biases on non-gambling related causal learning tasks [for a different, although compatible, result, see Orgaz et al. (2013)]. The abovementioned study by Lambos and Delfabbro (2007), beyond the association between gambling problems and general understanding of odds, also found such a measure of odds understanding to be unpredictive of gambling-related irrational beliefs. However, in a recent study by Delfabbro et al. (2020), participants who reported greater illusory control in nongambling-related everyday tasks (in a self-report questionnaire) scored higher on standardized measures of gambling-specific illusory control. To our knowledge, only one study in this last category has directly intervened on general-domain reasoning abilities in an attempt to reduce gambling-related biases. Donati et al. (2018) showed that a preventive intervention to modify erroneous cognitions by shaping probabilistic and superstitious thinking in adolescents, reduced their erroneous gamblingrelated cognitions, suggesting that gambling-related cognitions could related to domain-general reasoning.

Present Study
The present study is aimed at directly testing the association between domain-general reasoning abilities and gambling cognitions, in two samples of (a) individuals from the community who present a detectable level of gambling but do not present gambling problems (henceforth, individuals with nonproblematic gambling, NPG), and (b) treatment-seeking patients with gambling disorder (PGD).
Reasoning abilities (i.e., the independent variables in our study) were assessed using the matrices task of the WAIS-IV intelligence scale (Wechsler, 2008), and the Berlin Numeracy Test [BNT (Cokely et al., 2012)], for abstract and probabilistic reasoning, respectively, mirroring the two mechanisms described earlier. These two measures have good validity and reliability. The BNT is a sound index of probabilistic reasoning in practice (Cokely et al., 2018), namely individuals' easiness to deal with basic probabilistic operations from real-life problems (Lipkus and Peters, 2009;Cokely et al., 2012). The matrix reasoning subtest of the WAIS-IV assesses non-verbal perceptual reasoning abilities, and is considered to be a reliable measure of fluid intelligence (Bugg et al., 2006;Wechsler, 2008;Stephenson and Halpern, 2013;Gignac, 2014;Green et al., 2017;Kim and Park, 2018). This mostly overlaps with the g-factor (Spearman, 1927;Tranel et al., 2008;Jaeggi et al., 2010).
On the side of dependent measures, gambling-related cognitions were assessed using the GRCS, described earlier.
Complementarily, severity of potentially disordered gambling was assessed with the South Oaks Gambling Screen [SOGS, Spanish version (Echeburúa et al., 1994)].
In view of the evidence briefly reviewed earlier, we expect participants in the PGD sample to present a small-to-moderate disadvantage in the matrices and BNT tests, and much stronger dysfunctional/distorted gambling-related cognitions, relative to participants in the NGD sample. Yet, our main hypotheses, specifically regarding the relationships between BNT/matrices scores and gambling-related cognitions, remain open. Firstly, across samples, we will estimate the independent contribution of domain-general reasoning scores to the five domains of gambling-related cognition. Secondly, associations (or their absence) between reasoning and gambling-related cognitions will be tested in the two samples separately. Support for the existence (H 1 ) or inexistence (H 0 ) of such links will be assessed using Bayes factors.

Participants
The study sample comprised 135 participants, divided in 77 treatment-seeking patients with gambling disorder (PGD) and 58 participants with non-problematic gambling involvement (NPG). Characteristics of the two samples are reported in Table 2. Participants in the PGD group had a diagnosis of gambling disorder, as established by their therapist based on DSM5 criteria, and they had abstained from gambling for 15 days or more. The NPG group consisted of individuals with different degrees of involvement in gambling activities (with the minimum being "having ever gambled"). A specific exclusion criterion for NPG was presenting a gambling pattern severe enough to be classified as a disordered gambler [i.e., ≥5 in SOGS; (Stinchfield, 2002)]. The rest of exclusion criteria were similar for both groups, i.e., having ever been diagnosed or treated for any psychopathology (beyond gambling disorder in the case of PGD), and any history of neurological disease or brain trauma causing unconsciousness for 10 min or longer. Common exclusion criteria were assessed with a semi-structured interview.

Procedure
Patients with gambling disorder were recruited from different associations of rehabilitated gamblers in Andalucía (Spain), whereas NPG were recruited using convenience and snowball sampling methods among researchers' and patients' acquaintances, and using advertisements.
All participants were recruited across different phases of a more ambitious multi-stage research project (GBrain, and GBrain-2, see section "Funding"), with the different stages having slightly different aims and assessment protocols (with some measures being common to all phases and others present in only some of them). The participants included in the present study were the ones from all the phases of the project that were assessed with both the Matrices test for abstract reasoning, and the BNT for probabilistic reasoning (i.e., the two main independent variables involved in the hypotheses articulated earlier). Across phases, PGD and NPG participants were sampled from similar social milieus, and groups were intendedly matched in sociodemographics, including gender, age and education years (but not psychological/cognitive characteristics; please see complementary information about matching in the section "Preliminary Analyses").
In all phases, the protocol consisted of a set of questionnaires and neuropsychological tasks, administered in a quasirandomized order, in a single session that lasted approximately 2 h. Some participants were invited to participate in an extra session in a different day, in which psychophysiological or neuroimaging measures were recorded. There is thus some overlap between the current sample and the one in other studies of our research group: Megías et al. (2018) Participants were debriefed about study aims and signed an informed consent prior to their participation, and received a €10/hour compensation. In the case of patients, the compensation was paid via an authorized relative. The study was approved by the Ethic Committee of the University of Granada and complied with the Helsinki Declaration.

Instruments
Matrix Reasoning Task [WAIS-IV (Wechsler, 2008)] This instrument consists of 26 sequences of geometric figures, with each one following a unique organizational pattern, and a blank cell. Participants are asked to guess the underlying logic in the sequence, and to fill the blank cell with the option that best fits among the five possible alternatives. This is a standardized task that has excellent psychometric properties and is adapted for Spanish populations (Wechsler, 2012).

Berlin Numeracy Test [BNT (Cokely et al., 2012)]
This is a paper-and-pencil test in which participants are asked to answer 4 different questions on probability in ascending order of difficulty [e.g., Imagine we are throwing a five-sided die 50 times. On average, out of these 50 throws, how many times would this five-sided die show an odd number (1, 3 or 5)?]. A final score of numeracy skills is calculated as the sum of correct answers.
Gambling-Related Cognitions Scale [GRCS; Raylu and Oei, 2004;Spanish version: Del Prete et al., 2017)] This is a self-reported measure of gambling-related cognition based on Raylu and Oei s model. It consists of 23 items to be answered using a five-point Likert scale that assess five cognitive distortions: inability to stop gambling (e.g., My desire to gamble is so overpowering), gambling expectancies (e.g., Gambling makes things seem better), predictive control (e.g., Losses when gambling, are bound to be followed by a series of wins), illusion of control (e.g., I have specific rituals and behaviors that increase my chances of winning), and interpretative bias (e.g., Relating my winnings to my skill and ability makes me continue gambling). Given that individuals in the PGD group had been in therapy for some time (from 15 days to 6 months), these participants were specifically instructed to refer their answers to the GRCS items to the time when they initiated treatment [see also (Navas et al., 2017a)].
This scale has shown good psychometric properties (Del Prete et al., 2017). In the present study, internal consistency values (Cronbach's α) were 0.866, 0.914, 0.709, 0.826, and 0.920 for gambling expectancies, inability to stop, illusion of control, predictive control and interpretive bias, respectively, and 0.963 for the total scale.
South Oaks Gambling Screen [SOGS (Lesieur and Blume, 1987); Spanish Version (Echeburúa et al., 1994)] This instrument was used to assess disordered gambling symptoms' severity. The Spanish version has shown good psychometric properties. For this study, SOGS showed an excellent level of internal consistency (Cronbach's α = 0.929).

Statistical Analyses
Descriptive statistics are provided for age, education years, gender composition, WAIS-IV matrices scores, BNT scores, SOGS total severity scores, and the five dimensions of the GRCS questionnaire (gambling expectancies, inability to stop, control illusion, predictive control, and interpretative bias). For quantitative or quasi-quantitative variables, these descriptives include mean, standard deviation, and maximum and minimum values. These descriptives are complemented with Bayesian and frequentist tests to check for differences between participants showing non-problematic gambling involvement (NPG) and patients with gambling disorder (PGD). Scores in the five dimensions of the GRCS are submitted to a first multivariate analysis of covariance (MANCOVA), with group (sample: PGD, NPG) as a between-participant factor, and WAIS-IV matrices score as a continuous predictor. These are followed by GRCS dimension-by-dimension analyses of covariance (ANCOVA), with the same independent variables. The same analyses will be performed with BNT (instead of matrices) scores as continuous predictor.
Given the nature of the dependent variables involved, these analyses are likely to be affected by two limitations: (a) violation of homogeneity of covariance matrices and multivariate normality assumptions, and (b) the unsuitability of nullhypothesis significance testing (NHST) to provide evidence in favor of the null hypothesis. In view of that, non-parametric correlations (Kendall's τ) will be computed for correlations of each GRCS subscore with matrices and BNT scores. These correlations will be interpreted using bidirectional Bayes factors (BF 10 ) instead of NHST.
Regarding these statistical analyses, there are two important points that require further consideration. First, we did not use stratified sampling (or any other method to ensure populational representativity; see section "Limitations and Final Remarks"), but the sampling strategy and the inclusion/exclusion criteria were very similar for the two groups, and we did not force matching on psychological/cognitive variables [please see Perandrés-Gómez et al. (2020), for a discussion on the consequences of IQ non-matching in cross-sectional analyses of a sample largely overlapping with the present one]. Using convenience samples of gamblers with and without gambling problems is quite a standard practice in correlational research in the field (Barrada et al., 2019). Still, and in order to surpass the problems that this sample composition may cause, we ran analyses with the whole sample, while controlling for group (first part of the section "Main Analyses"), with the whole sample without controlling for group (Supplementary Materials), and with the two groups separately (second part of the section "Main Analyses"). As detailed below, results were robust across statistical approaches.
And second, please note that frequentist tests are aimed at checking for statistical significance of effects (i.e., whether the observed test statistic is extreme enough for the null hypothesis to be rejected), so null results can be explained as resulting from either the absence of an effect or the lack of power of the test. That implies that frequentist tests cannot distinguish between evidence of absence and absence of evidence (Altman and Bland, 1995). In the present study, however, we are as much interested on the possible inexistence of certain relations as we are in their existence. Bayesian tests expressed in the form of Bayes factors (BF 10 ) are aimed at comparing two models of the world, one in which the effect of interest is zero, and another one in which it is non-zero (with a given probability density distribution over the populational effect size). These two models representing the null and the alternative hypothesis are treated symmetrically, in such a way that BF 10 < 1 is interpreted as supporting the null, whereas BF 10 > 1 is interpreted as supporting the alternative. The arbitrary thresholds to consider evidence in favor of one or the other substantial enough vary across reference guidelines, so BFs will be interpreted here as strictly continuous measures of evidence (Dienes, 2014). For a discussion on equivalence tests and Bayes factors as tools to establish evidence for the null, see Lakens et al. (2020).
Data and reproducible analysis files are fully available in the OSF framework 1 . Table 2 shows group means, maximum, and minimum values, and standard deviations for age, education years, matrices, BNT, SOGS severity, and GRCS dimensions scores; proportions for gender; as well as Bayes factors and p-values for differences between groups in all variables. Detailed distributions for all these variables across groups are reported in the Supplementary Materials.

Preliminary Analyses
As expected, the two groups differed in SOGS and GRCS scores, and were closely matched in gender composition and mean age. Although education years was also controlled across phases of the project, the pooling of samples across phases made the difference between groups in this variable to get close to the significance threshold (p = 0.064), and to yield a virtually uninformative BF (BF 10 ≈1).
The two groups, however, differed in both Matrices and BNT scores. In other words, differences in reasoning abilities remained in spite of control of sociodemographic variables. Actually, a MANCOVA with BNT and matrices scores as dependent variables, group as independent variable, and sociodemographics (age, gender, and education years) as covariates yielded significant effects for both the multivariate effect (Wilks' λ = 0.910, p = 0.002), and the univariate effects [F (1, 130) = 8.109, p = 0.005; and F (1, 130) = 8.335, p = 0.005, for matrices and BNT scores, respectively]. In other words, despite sociodemographic matching, general reasoning scores remained associated with GD, which is in line with the abovementioned evidence of links between reasoning abilities and risk of being diagnosed with GD.

Main Analyses
The MANCOVA with group as between-participants factor, matrices score as continuous predictor, and GRCS subscores as dependent variables, yielded a significant effect for group, Wilks' λ = 0.378, F (5, 128) = 42.181, p = 0.001, but not for the matrices score, Wilks' λ = 0.991, F (5, 128) = 0.231, p = 0.948. Table 3 (left panel) shows the results of separate ANCOVAs for the five GRCS dimensions. In accordance with the global MANCOVA, all dependent variables showed significant effects of group, but not of matrices score.
Similarly, the MANCOVA with group as between-participants factor, BNT score as continuous covariate, and GRCS subscores as dependent variables yielded a significant effect for group, Wilks' λ = 0.374, F (5, 128) = 42.884, p < 0.001, but not for the BNT score, Wilks' λ = 0.977, F (5, 128) = 0.607, p = 0.695. Table 3 (right panel) shows the results of separate ANCOVAs for the five GRCS dimensions. In accordance with the global MANCOVA, all dependent variables showed significant effects of group, but not of BNT score 2 .
2 These MANCOVAs were re-run including either education years or age as covariates, and allowing both covariates to interact with either BNT or Matrices (i.e., estimating the possible dependence of the effect of reasoning abilities on age The Box's test [χ 2 (15) = 201, p < 0.001], and the Shapiro-Wilks' test [W = 0.875, p < 0.001], showed clear violations of the homogeneity of covariance matrices and multivariate normality assumptions, respectively. In view of that, we computed nonparametric correlations (Kendall's τ) between reasoning abilities and GRCS dimensions for the two groups separately, and interpreted the evidence portrayed by them using bidirectional Bayes factors (BF 10 ), computed with the default settings in JASP software (JASP Team, 2019). Figure 1 and Table 4 show the results of these analyses for the PGD and the NPG group, respectively.
As expected, in both groups, substantial correlations were found between the different subdimensions of GRCS. In the NPG group, the SOGS score correlated positively with all GRCS dimensions, with the strength of evidence for H 1 ranging from BF 10 = 2.36 to BF 10 > 100. Correlations between SOGS and GRCS were weaker in the PGD group, with only three BFs above 1, i.e., for inability to stop (BF 10 = 7.59), interpretative bias (BF 10 = 6.83), and predictive control (BF 10 = 1.93, anecdotal) 3 . BNT and matrices also correlated positively between them, and with education years, and negatively with age.
Most importantly, BFs for correlation coefficients between reasoning abilities (matrices and BNT) and GRCS scores mostly provided moderate (BF 10 < 0.33) evidence in favor of the null hypothesis. The only exceptions (i.e., BF 10 > 1) were the BF 10 = 15.04, Kendall's τ = 0.27 between BNT and gambling expectancies, the BF 10 = 3.41, Kendall's τ = 0.22 between BNT or education years). Results from those analyses are reported in the second section, Supplementary Table 1 and Supplementary Figure 2. The potential relevance of these results is detailed in the section "Discussion." 3 The different size of correlations in the two groups could be due to range restriction in the SOGS scale. As shown in Supplementary Figure 1, GRCS scores showed a large range of variation in the PGD group, but SOGS scores in that group were more tightly concentrated around the mean (M = 10.35, SD = 2.99), which results from the fact that PGD in our study were severe enough to have sought treatment. Still, results in this group are also consistent with the fact that inability to stop and interpretative biases are the best GRCS indicators of severity at pathological levels (Michalczuk et al., 2011;Del Prete et al., 2017). and inability to stop, and the BF 10 = 1.64 (anecdotal), Kendall's τ = 0.19 between BNT and interpretative bias, in the NPG group. In other words, there is some weak evidence of a direct link between BNT and some gambling-related cognitions (mainly excluding gambling biases) in the NPG group, with stronger cognitions in individuals with higher BNT scores. There were not any cases in which evidence supported an inverse relationship between reasoning abilities and gambling-related cognitions.

DISCUSSION
The goal of the present study was to explore the relationships between domain-general reasoning abilities and gamblingrelated cognitions in non-problematic gamblers (NPG) and patients with gambling disorder (PGD). Results from NHST (MANCOVAs on the association between BNT/Matrices and gambling-related cognitions, and subsequent dimension-bydimension ANCOVAs) did not yield any significant associations. This result holds regardless of whether group (PGD, NPG) was included in the model or not. Subsequent Bayesian analyses yielded consistent support for the null hypothesis, i.e., no association between BNT/Matrices and gamblingrelated cognitions, except for anecdotal-to-substantial support for positive associations in the NPG subsample between BNT, on the one side, and gambling expectancies, inability to stop, and interpretative bias, on the other. These results converge with the ones of some previous works. For instance, Perales et al. (2017) found gamblers with stronger biases to perform better in a causal learning task than those with weaker biases. This result was interpreted as originating in the fact that gambling-related cognitive distortions are significantly more intense in gamblers preferring skill-based games (i.e., sports betting, casino and card games) than in those preferring chance games (i.e., slots, bingo, or lottery) [see also (Myrseth et al., 2010;Navas et al., 2017b;Mallorquí-Bagué et al., 2019)]. Individuals preferring skill-based games are, on average, younger, better educated, and more sensitive to reward (Navas et al., 2017b), so that their distorted beliefs about gambling are unlikely to be originated in any general-domain reasoning disadvantage. Relatedly, Xue et al. (2012a) found students with higher cognitive abilities (intelligence and executive function) to be more prone to show the gambler's fallacy. And in Lambos and Delfabbro (2007) disordered gamblers were found to be more susceptible to cognitive biases than non-gamblers and nondisordered gamblers, but no significant differences were observed between the three groups for their knowledge of gambling odds [see also (Benhsain et al., 2004)].
This lack of substantial inverse relationships between domaingeneral reasoning abilities and gambling-related cognitions renders two theoretical puzzles unresolved. First, to describe the mechanisms responsible for bias generation and their activation during and between gambling sessions; and, second, accounting for the seemingly robust link between domain-general cognitive abilities and the risk developing gambling problems, without the mediation of gambling-related distorted cognitions.
With regard to the first question, a possible solution arises from the cognitive switching (Sévigny and Ladouceur, 2003) hypothesis. According to this hypothesis, individuals with disordered gambling "switch off " their rational beliefs during gambling, so that their behavior becomes governed by features of the game or the gambling device, and "switch them on" again when they finish. In other words, in-game behavior and cognitions remain impermeable to general-domain reasoning. The cognitive switching hypothesis is inspired by dual-process models of cognition, according to which two competing systems, the intuitive and the analytic, filter the information necessary to control action. The intuitive system is regarded as fast, efficient, and heuristic-based, whereas the analytic system is slower and more effortful, but also more rational (Armstrong et al., 2020). The term cognitive reflection has been coined to denote the degree to which an individual is more or less willing to invest the necessary cognitive resources to engage in analytic thinking [see (Stange et al., 2018), for a discussion of its potential link with gambling]. Importantly, being less prone to cognitive reflection, especially under certain environmental and affective circumstances, does not imply having poorer reasoning abilities, but somehow eschewing the effort to use them, especially when motivated to do so. In words of Armstrong et al. (2019), "gamblers are often unlikely or unwilling to reflect on the veracity of beliefs as they are often used to justify gambling behaviors" (p. 183) [see also (Emond and Marmurek, 2010;Armstrong et al., 2019;Cosenza et al., 2019)]. This mechanism reminds of the "tilt" phenomenon in poker (Barrault et al., 2014), and some recent studies using functional magnetic resonance imaging (fMRI) (Xue et al., 2011), and transcranial direct current stimulation (tDCS) (Xue et al., 2012b) also indirectly support it.
A second, non-exclusive possibility is that some gamblers do remain reflective during gambling episodes, but they invest their cognitive resources in trying to "outsmart" the gambling device, and to find causal patterns where there are not any. Indirect evidence supporting this mechanism comes from the abovementioned reports that, especially in some sociodemographic sectors, individuals with preserved -or even superior-cognitive skills are more vulnerable to certain gambling-related fallacies. To our knowledge, there is no direct evidence of this mechanism, although the deleterious effects of trying to outsmart random devices on judgment and decisionmaking are well known [see (Gaissmaier and Schooler, 2008)].
That connects with a third possibility, emerging from the putative interaction of domain-general reasoning skills with age and/or education. Actually, when matrices scores were allowed to interact with age and education years (see Supplementary Materials, second section), some non-significant trends suggested that, in younger and more educated individuals, matrices scores were positively associated with GRCS scores, whereas in older and less educated individuals the association was non-existing or in the opposite direction. It is definitely premature to make any inferences from these trends, but they open the possibility that in younger, more educated people, distorted gambling cognitions were fueled by domain-general reasoning skills, whereas in older, less educated gamblers, poorer reasoning skills were a risk factor for developing gamblingrelated biases. Additionally, this interaction would explain why some studies have found no associations whatsoever between reasoning skills and gambling-related biases, whereas others have found a direct link (Xue et al., 2012a;Perales et al., 2017).
In summary, low domain-general reasoning skills are not necessary to develop gambling-related distorted beliefs, which reinforces the idea that, at least in some gamblers, in-or aboutgame emotion-laden states (e.g., urges triggered by conditioned cues, or negative affect caused by losses) can take control over gambling-related cognition, and probably motivate the individual to stick to irrational cognitions. Such possibility is one of the main tenets of the GSM, according to which the main source of gambling-related cognitive distortions is motivated reasoning, that is, the individual's tendency to regulate affect by overestimating their degree of control or reinterpreting gambling outcomes in a more favorable, ego-protecting light (Navas et al., 2017bRuiz de Lara et al., 2019). Whether this motivated reasoning mechanism is specific to some gamblers (more educated, younger ones) or generalizes to a wider range of individuals remains an open question for future research.
The second puzzle, namely the moderate but seemingly robust relationship of intelligence and abstract reasoning with gambling problems without the mediation of gambling related cognitions, seems more difficult to address. In our sample, this link held for GD diagnosis across groups, but not for severity of gambling problems within groups, and its interpretation is limited by features of the design. This result resonates with the one from Rai et al. (2014), in which a link between IQ and gambling problems was also corroborated at the populational level, but no association was found between IQ and non-problematic gambling. Unfortunately, none of the possible explanations for this link has been explored in detail. Tentatively, the association between poorer reasoning abilities and a higher risk of developing gambling problems can be partially accounted for by the overlap between these abilities and aspects of executive function as selfcontrol and top-down regulation of impulses (Meldrum et al., 2017). A detailed review of the role of executive functions related to cognitive control in gambling problems, and its neurobiological correlates, can be found at Moccia et al. (2017).
Clinical implications of our results, and the abovementioned related ones, are far-reaching. Gambling-related cognitions are hard to restructure, and the efficacy of cognitive therapy, although well-established, remains modest (Petry et al., 2017). Furthermore, individuals with problematic gambling are normally reluctant to change their beliefs when faced with disconfirming evidence, and often counterargument it (Delfabbro et al., 2006). In a variety of domains, this sort of reluctance has been related to the fact that, when motivated to maintain a given belief, individuals perceive information disconfirming it as confronting or uncomfortable (Gilbert et al., 1990;Mezirow, 1990;Stange et al., 2018). In consequence, altering beliefs will not only require more (or more accurate) information, but an increased degree of metacognition about how motives to gamble and to regulate emotions derived from gambling (and its consequences) relate to one's beliefs (Wells, 2009;Lindberg et al., 2014b;Caselli and Spada, 2016).

Limitations and Final Remarks
Results of our study should also be understood considering at least five main limitations. First, we cannot establish causal associations between the variables examined, since this is a cross-sectional study. Second, since the majority of the participants are male, generalizability to the entire population of gamblers should not be taken for granted. Third, assessing psychological constructs using self-report questionnaires may not fully represent the cognitive processes involved, and social desirability effects are possible. Fourth, no power analysis was performed a priori to determine sample size. As noted earlier, participants in this study were the ones in a larger project who had been assessed with all the measurements of current interest. This problem is, however, partially palliated by the use of Bayes factors, that provide evidence in support of the null or the alternative hypothesis in a continuous fashion, so that no dichotomous decisions leading to type I or type II errors are made. And fifth, we did not use stratified sampling (or any other method to ensure populational representativity), which means that the sampling strategy and the inclusion/exclusion criteria were very similar for the two groups, and we did not force matching on psychological/cognitive variables. That implies that the proportion of PGD in our sample is much larger than in the general population, but there are no reasons to expect substantial alterations of the correlations between psychological variables. Given that there is an association between gambling problems, on the one hand, and both stronger gambling-related biases and lower reasoning skills, on the other, the overrepresentation of PGD could have artificially inflated correlations between the latter when group was not controlled for (supplementary analyses). Despite this risk of inflation, gambling-related cognitions and domain-general reasoning remained mostly disconnected.
On the side of strengths, although some previous studies had explored the relationship between reasoning abilities and gambling-related beliefs, to our knowledge, this is the first one simultaneously assessing two core constructs of domaingeneral reasoning directly relevant to gambling (abstract and probabilistic reasoning), and their relationship with different dimensions of gambling-related cognitions in individuals without problem gambling and patients with gambling disorder. Additionally, the inclusion of Bayesian analyses allows to symmetrically assess the evidential support in favor of the null or the alternative hypothesis. Our results evidence that probabilistic and abstract reasoning abilities are mostly unrelated to the intensity of distorted gambling-related beliefs, and are thus unlikely to protect gamblers from them. This pattern or results reinforces the idea that distorted cognitions do not originate in a general lack of understanding of probability or low fluid intelligence, but probably result from motivated reasoning.

DATA AVAILABILITY STATEMENT
The datasets presented in this study can be found in the OSF repository: https://osf.io/8ksxa/.

ETHICS STATEMENT
The studies involving human participants were reviewed and approved by Ethic Committee of the University of Granada. The patients/participants provided their written informed consent to participate in this study.

AUTHOR CONTRIBUTIONS
JP, JN, and IM: conceptualization and writing-review and editing. JP: formal analysis. JP and JN: methodology. IM and JN: writingoriginal draft. All authors contributed to the article and approved the submitted version.

FUNDING
This work has been supported by grants from the Spanish Government (Ministerio de Economía y Competitividad, Secretaría de Estado de Investigación, Desarrollo e Innovación; co-funded by the Fondo Europeo de Desarrollo Regional, FEDER, European Union), with reference numbers PSI2013-45055-P and PSI2017-85488-P. IM was funded by an individual grant, linked to the latter, from the Spanish Government (Ministerio de Ciencia, Innovación y Universidades) with reference number PRE2018-085150. Additionally, JN was supported by a Spanish Ministry of Science and Innovation post-doctoral contract (Juan de la Cierva-Formación: FJC2018-036047-I). The funding agencies are public and played no role in planning or executing this work, nor interfered with the collection and interpretation of data.