ORIGINAL RESEARCH published: 24 January 2019 doi: 10.3389/fpsyg.2019.00024 Persistence of Causal Illusions After Extensive Training Itxaso Barberia 1* , Miguel A. Vadillo 2 and Javier Rodríguez-Ferreiro 1,3 1 Departament de Cognició, Desenvolupament y Psicologia de la Educació, Universitat de Barcelona, Barcelona, Spain, 2 Departamento de Psicología Básica, Universidad Autónoma de Madrid, Madrid, Spain, 3 Institut de Neurociències, Universitat de Barcelona, Barcelona, Spain We carried out an experiment using a conventional causal learning task but extending the number of learning trials participants were exposed to. Participants in the standard training group were exposed to 48 learning trials before being asked about the potential causal relationship under examination, whereas for participants in the long training group the length of training was extended to 288 trials. In both groups, the event acting as the potential cause had zero correlation with the occurrence of the outcome, but both the outcome density and the cause density were high, therefore providing a breeding ground for the emergence of a causal illusion. In contradiction to the predictions of associative models such the Rescorla-Wagner model, we found moderate evidence against the hypothesis that extending the learning phase alters the causal illusion. However, assessing causal impressions recurrently did weaken participants’ causal Edited by: illusions. Tom Beckers, KU Leuven, Belgium Keywords: causal illusion, illusion of causality, contingency learning, causal learning, extensive training, Rescorla-Wagner model Reviewed by: Jessica Lee, University of New South Wales, Australia INTRODUCTION Kosuke Sawa, Senshu University, Japan Several studies have found that, when certain conditions are met, people are easily led to develop erroneous causal beliefs. In the canonical experiment, volunteers are initially informed that their *Correspondence: Itxaso Barberia aim is to explore the extent to which a causal relationship exists. For example, they may be asked to itsasobarberia@ub.edu find out the extent to which a fictitious drug (a candidate cause) is effective in producing recovery from a fictitious disease (the outcome). Then, participants observe several patients with the disease, Specialty section: presented sequentially on a trial-by-trial basis. For each patient, they are told if the patient took This article was submitted to the drug or not (i.e., if the candidate cause is present or not), and whether the patient recovered Cognition, from the disease or not (if the outcome occurred or not). After seeing all patients, they have to a section of the journal rate the effectiveness of the drug in a numerical scale, usually ranging from 0 to 100. In reality, the Frontiers in Psychology experimenter has set up the task for the probability of recovery to be the same both for patients Received: 21 September 2018 taking the drug and for patients not taking it. Therefore, the normative answer in these conditions Accepted: 07 January 2019 should be that the drug is ineffective, because taking it does not increase the chances of recovery. Published: 24 January 2019 However, under some circumstances, people tend to perceive that the drug is effective, showing Citation: what could be called a causal illusion or an illusion of causality (see Matute et al., 2015, for a review). Barberia I, Vadillo MA and Previous research on this subject reveals that two factors play an essential role in the Rodríguez-Ferreiro J (2019) Persistence of Causal Illusions After development of causal illusions. In our previous scenario, even if the chances of recovery are Extensive Training. the same among patients taking the drug and patients not taking it, the drug has greater Front. Psychol. 10:24. chances of being perceived as effective if the recovery is, in general, very frequent (high outcome doi: 10.3389/fpsyg.2019.00024 density) and if there is a high proportion of patients taking the drug (high cause density) Frontiers in Psychology | www.frontiersin.org 1 January 2019 | Volume 10 | Article 24 Barberia et al. Persistent Causal Illusions (e.g., Alloy and Abramson, 1979; Allan and Jenkins, 1983; Allan strength of this association is given by equation: et al., 2005; Hannah and Beneteau, 2009; Blanco et al., 2013). X Table 1 represents an example with both a high outcome density 1Vn = α · β · (λ − Vn−1 ) (1) and a high cause density, illustrating a situation that should encourage strong causal illusions. where 1Vn is the change in the associative strength on trial n, α, The relevance of this phenomenon is partly based on the fact and β are two learning rate parameters dependent on the saliences that many erroneous beliefs in our daily life might appear just of the two events to be associated, λ denotes whether the outcome P in the same manner as causal illusions emerge in these simple (i.e., the recovery) was observed or not in that trial, and Vn−1 computer tasks in the lab (Blanco et al., 2015; Matute et al., 2015; is the sum of the associative strengths previously accumulated by Griffiths et al., 2018). This would be the case, for example, for all the cues present in that trial. Of note, although the associative pseudomedicines or “miracle products” with no proven capacity strength of a cue (i.e., taking the drug) only changes when that cue to produce improvement. In these cases, some people develop the is present, the model assumes that in addition to the target cue conviction of the effectiveness of a treatment, even when there there is also a constant contextual cue, present in all trials, that is no real covariation between using the pseudomedicine or the competes with the target cue to get associated with the outcome. miracle product and the improvement (Matute et al., 2011). In With this simple implementation, the model predicts that after fact, as previously noted by Blanco et al. (2014), pseudomedicines a representative sequence of trials, the associative strength of the seem to proliferate in contexts that are ideal for the rise of causal target cue will approach a value close to the objective correlation illusions: first, they are usually applied to health problems (e.g., between taking the drug and recovery (Chapman and Robbins, back pain, headache) that tend to show spontaneous recovery 1990; Wasserman et al., 1993), as measured by an index called (high outcome density) and, since they are frequently advertised 1P (Allan, 1980), which can be computed as: as lacking side effects, they tend to be used recurrently (high cause 1P = P(recovery|drug) − P(recovery| ∼ drug) (2) density). As far as misbeliefs such as these can strongly impact relevant daily life decisions, understanding the mechanisms by In that sense, the Rescorla-Wagner model can be seen as which they are formed, the circumstances in which they most a “rational” theory of causal learning (Shanks, 1995), because probably proliferate, and the expected evolution of these over once sufficient information has been experienced, the strength time seems essential. of the association between causes and effects will approach the Associative learning models, such as the error-correction objective degree of correlation between them1 . However, before algorithm proposed by Rescorla and Wagner (1972) are able reaching the learning asymptote, the model also predicts that to predict the appearance of causal illusions (e.g., Shanks, the associative strengths will be biased by factors that, from a 1995; Musca et al., 2008; Vadillo et al., 2016; Vadillo and rational point of view, should play no role at all. Specifically, the Barberia, 2018). The Rescorla-Wagner model assumes that model predicts that the development of the associations will be information about the relationship between different events positively biased by the overall probabilities of the two events to in the environment is encoded in the form of associations be associated (i.e., outcome density and cause density). whose strengths are dynamically updated as new evidence As an example, Figure 1 shows the results of four simulations is encountered. In a causal learning situation like the one of the Rescorla-Wagner model. The simulation represented by summarized in Table 1, the model assumes that whenever we black circles refers to the condition summarized in Table 1. encounter a patient who takes the drug and recovers from the As can be seen, the associative strength between taking the disease, the link between the mental representations of these two drug and recovering from the disease is strongly biased in the events (drug and recovery) are strengthened, while encountering initial trials and slowly declines as more information is gathered. a patient who takes the drug but does not experience recovery With sufficient training, the associative strength would eventually weakens the association between both events. The change in the reach an asymptotic value of zero. The comparison of this condition with the remaining three conditions shows that the size TABLE 1 | A contingency table showing an example of a situation prone to of the initial bias depends heavily on the marginal probabilities of generating the illusory perception of the drug being effective. the cause and the effect. The condition denoted by red triangles is identical to the one denoted by black circles, except that in Patients recovered Patients not recovered the latter the density of the cause is P(drug) = 0.75 while in Patients taking the drug 27 (a) 9 (b) the former the density of the cause is P(drug) = 0.50. Similarly, Patients not taking the drug 9 (c) 3 (d) 1 It is worth noting that, for the asymptotic predictions of the Rescorla-Wagner Out of 48 patients (i.e., trials) 27 took the drug and recovered from the disease model to converge with 1P, it is necessary to assume a “restricted” version of (cell a), 9 took the drug but did not recover (cell b), 9 recovered without taking the the model, where β is assumed to have the same value both in trials in which drug (cell c), and 3 did not take the drug and did not recover (cell d). Therefore, the the outcome occurs and in trials in which it does not occur. An “unrestricted” probability of recovery is 0.75 both among patients that took the drug [27 out of version of the model, allowing β to vary depending on the presence or absence of 36; a/(a+b)] and among patients that did not take the drug [9 out of 12; c/(c+d)]. the outcome, is able to predict deviations from 1P on causal ratings at asymptote Also, the outcome density is high (36 out of 48 patients recovered from the disease, (e.g., Wasserman et al., 1993; Buehner and Cheng, 1997; Lober and Shanks, 2000; irrespective of the intake of the drug), and the cause density is high (36 out of 48 Buehner et al., 2003). However, this is only the case for non-zero (positive and patients took the drug, irrespective of them experiencing subsequent recovery or negative) correlation values, whereas unrestricting the model does not alter the not). Although the drug does not increase the chances of recovery, a strong causal predictions for situations in which there is an absence of contingency (i.e., 1P = 0), illusion is expected in this situation. as it is the case in settings exploring causal illusions. Frontiers in Psychology | www.frontiersin.org 2 January 2019 | Volume 10 | Article 24 Barberia et al. Persistent Causal Illusions shortest cases (e.g., Barberia et al., 2013; Blanco et al., 2015; Griffiths et al., 2018) to about 100 trials in the longest cases (e.g., Matute et al., 2011; Yarritu et al., 2014). For this study, we decided to greatly extend the amount of training the participants were exposed to and explore if this manipulation diminished the intensity of the illusion developed. With this goal in mind, all participants were exposed to a causal learning task typically used in the research on causal illusions (Matute et al., 2015). The scenario involved discovering if a fictitious drug was effective in making patients overcome the crises produced by a fictitious disease. Half of the participants observed 48 patients (Standard Training group or “Standard” group) and then evaluated the effectiveness of the drug, whereas for the other half (Long Training group or “Long” group) the length of training was multiplied by six (48 × 6) and thus, participants in this condition observed a total of 288 patients FIGURE 1 | Four simulations of the Rescorla-Wagner model. The figure before evaluating the drug. If causal illusions are a product of legend summarizes the number of a, b, c, and d trials (see Table 1) included in each simulation. Learning rate parameters αcue , αcontext , and β were set to incomplete learning, causal ratings given after 48 trials (Standard 0.4, 0.2, and 0.6, respectively. The value of λ was set up to 1 for trials in group) should be higher than those given after 288 trials (Long which the outcome was present and to 0 for trials in which the outcome was group). absent. The figure shows the average results of 2,000 iterations with random We note that studies from a related area of research, illusory trial orders. correlation in stereotype formation (Hamilton and Gifford, 1976), could be relevant for drawing informed hypotheses regarding the results of our present study. Much like in the the condition denoted by green diamonds is identical to the case of causal illusion experiments, illusory correlation studies condition denoted by black circles, except that in the latter the rely on contingency detection tasks, in which volunteers tend density of the effect is P(recovery) = 0.75, while in the former to overestimate the degree of covariation between two given the density of the effect is P(recovery) = 0.50. Finally, in the variables. Importantly, Murphy et al. (2011) demonstrated that last condition, represented by blue inverted triangles, both the the illusory correlation effect might be a preasymptotic or density of the cause and the density of the effect are 0.50. The incomplete learning effect. In their experiments, the effect simulations show that increasing any of these probabilities makes emerged early in training but disappeared after more extended a substantial difference in the predictions of the model, even training. However, another paper published that same year though the degree of relationship between the cause and the effect (Kutzner et al., 2011) found a persistent illusory correlation effect is zero in all cases. even after a longer training than that implemented by Murphy Determining the extent to which the causal illusion is pre- et al. (2011). We consider that one important difference between asymptotic or permanent is crucial, both because of theoretical the procedures of both studies was that, whereas Murphy et al. and applied reasons. At a theoretical level, the question of interest (2011) asked participants to give their evaluations every few trials, is whether associative models such as Rescorla and Wagner’s Kutzner et al. (2011) only asked for explicit evaluations at the end can account for the evolution of causal misbeliefs after extended of training. exposure to the candidate cause and the outcome. That is, are This observation led us to explore the effects of recurrent these erroneous beliefs subject to normative adjustment following ratings of causal relationship during causal illusion formation a more extended exposure to them? In applied terms this issue as a secondary goal of our research. With this goal in mind, is also relevant as it could help us envisage if misbeliefs in the participants in what we have called the Standard group daily life, such as those related to pseudomedicine and miracle were, in fact, further trained after observing the first 48 patients products, can be expected to vanish as more information is and giving their first evaluation of the drug. Specifically, they gathered or if, once implemented, these beliefs remain relatively continued observing patients and evaluating the effectiveness of stable unless other counteractive, e.g., informative or educational, the drug after every 48 trials, until reaching the same amount of initiatives are introduced. In this sense, it is worth noting that training as those participants in the Long group. Note that this previous evidence in the domain of anti-vaccination and political subsequent training does not influence the main measure taken misinformation suggests that, once erroneous beliefs are accepted in the Standard group, which happened after the first 48 patients as valid, they tend to be quite resistant to any attempt to refute were observed. Inspired by the differences detected between them (e.g., Nyhan and Reifler, 2010; Lewandowsky et al., 2012; previous results of illusory correlation studies (Kutzner et al., Nyhan et al., 2014). 2011; Murphy et al., 2011), we hypothesized that, whereas no In the present study, we adapted the standard laboratory task differences would appear between the first causal ratings obtained used in research on causal illusions to an extended learning in the two groups, regardless of training length, repeatedly asking situation. Most of the previous experiments exploring causal participants to evaluate the effects of the drug might decrease the illusions employed training sessions of about 40 trials in the causal illusion effect. Frontiers in Psychology | www.frontiersin.org 3 January 2019 | Volume 10 | Article 24 Barberia et al. Persistent Causal Illusions MATERIALS AND METHODS trial, the same suffering emoticon appeared in the left side of the screen, together with an image of a pill on the right side of the Participants screen, if the trial corresponded to a patient that had taken the A total of 150 Psychology undergraduate students participated drug, or with the same pill crossed out, if the trial corresponded in the experiment in return for course credits, and they were to a patient that did not take the drug. A text in the upper side randomly assigned to the Standard group or Long group (75 of the screen indicated “This patient suffers from a crisis and participants per group, 62 female, mean age = 21.28, SD = 7.06 in receives Batatrim” or “This patient suffers from a crisis and does the Standard group and 63 female, mean age = 19.80, SD = 3.26 not receive anything,” respectively. An additional text in the lower in the Long group). Sample size was determined following part of the screen asked participants “Do you think the patient the criteria that participants would be added sequentially until would overcome the crisis?” Participants could answer by clicking reaching a Bayes factor (BF) higher than 10 in favor or against on one of two (yes/no) buttons. After the participants had made the alternative hypothesis, i.e., that participants in the Standard their prediction, the image of the suffering emoticon and all the group would show a stronger causal illusion than participants texts disappeared from the screen for 1 s. After this period, the in the Long group, as measured by the causal question (see emoticon reappeared on the screen, either with the same suffering section “Materials and Methods”). If none of these values were expression or with a happy expression, and with the “The patient reached after 150 participants (75 per group) the experiment did not overcome the crisis” or “The patient overcame the crisis!” would be stopped (see e.g., Wagenmakers et al., 2015, for text, respectively. One second later, an additional button that a similar sampling plan). In the end, the BF did not reach allowed participants to see the next patient was activated. any of the target boundaries and the final sample was of 150 We assessed the causal illusion by means of three questions: participants. Although this stopping rule was based on a Bayesian the causal question and two conditional probability questions. approach, we report both Bayesian and frequentist statistics for For the causal question, which was our main dependent variable all analyses. In the few occasions when the results of both types (see e.g., Blanco et al., 2011; Barberia et al., 2013, for similar of analyses seem to lead to different conclusions, we highlight questions), participants were asked “To what extent do you think these discrepancies. Participants gave written informed consent that Batatrim is effective to overcome the crises produced by the before participating in the experiment. The study protocol was Lindsay Syndrome?” and they could answer in a scale from 0 (not approved by the ethics committee of the University of Barcelona effective at all) to 100 (totally effective). Given that the normative (Institutional Review Board IRB00003099). calculation of covariation between cause and outcome depends on the direct comparison between cause-present and cause- Procedure absent outcome probabilities, we also introduced two conditional The experiment was programmed using Xojo2 and all the probability questions in order to gain insight on the process that instructions were provided in Spanish. Participants were exposed might underlie the participants’ responses to the causal question. to a standard causal learning task. Specifically, they were asked For these two conditional probability questions, participants were to imagine that they were specialists in a strange and dangerous asked to estimate the number of patients that would overcome the disease called Lindsay Syndrome (e.g., Blanco et al., 2011, crisis among 100 new patients taking the drug [P(recovery|drug) 2013; Matute et al., 2011). They were further told that the question], and among 100 new patients not taking the drug crises produced by this disease might be overcome with a new [P(recovery|∼drug) question]: “Imagine 100 NEW PATIENTS experimental drug (Batatrim) whose effectiveness was still to be that are suffering a crisis produced by the Lindsay Syndrome and determined. In order to evaluate the effectiveness of this drug, TAKE BATATRIM [DO NOT TAKE ANYTHING]. How many participants were told that they would see the medical records do you think they will overcome the crisis?” The order in which of several patients, presented one by one. Each patient would be the questions were answered was balanced across participants: suffering from a crisis and some of them would receive the drug about half of the participants of each condition answered the whereas others would not receive anything. Whether the patient causal question first and then the two questions referring to the had taken the drug or not, participants would have to predict conditional probabilities (also balanced in order), whereas for if the patient would overcome the crisis. The prediction would the other half this order was reversed, resulting in four possible be followed by a feedback indicating if the patient overcame orders. the crisis or not. Participants were further informed that, since In the case of the participants in the Standard group, after each they would see many patients, they were free to take short block of 48 patients, participants answered the causal question rests whenever they needed. (They were informed that they and conditional probability questions and they were told that they were not allowed to perform any other activity, such as using would continue observing more patients. They were requested their cellphones, during these periods.) The instructions ended to answer the three questions (causal question and conditional informing the participants that their goal was to find out if the probability questions) again after every 48 trials and, therefore, drug was effective and that, after observing a good number of they experienced a total of six blocks of ratings until reaching the patients, they would be asked some questions. same amount of training as the participants in the Long group. After the initial instructions, participants were exposed to a For the Long group these questions were answered only after all sequence of 288 patients, presented one after the other. On each 288 patients had been observed. Participants in both groups were exposed to six blocks 2 https://www.xojo.com/ of randomly presented 48 trials, of which 27 corresponded Frontiers in Psychology | www.frontiersin.org 4 January 2019 | Volume 10 | Article 24 Barberia et al. Persistent Causal Illusions to a trials, 9 to b trials, 9 to c trials, and 3 to d trials (see the alternative hypothesis that ratings would be different in Table 1). Every 48 trials the program would re-randomize both groups (modeled as a two-sided Cauchy distribution with the sequence. For the Standard group, this re-randomization r = 0.707) yielded a BF01 = 5.43, suggesting moderate evidence coincided with the moment in which the causal and conditional in favor of the null hypothesis. Frequentist statistics yielded no probability questions were answered after each block of trials. significant differences between both groups, t(148) = −0.317, Note that the situation was set up so that the contingency p = 0.752, d = −0.052. between the drug and recovery was zero, because the In the case of the estimated conditional probabilities, visual probability of recovery among patients that took the drug [i.e., inspection of Figure 2 (panel B) indicates that the probability P(recovery|drug) = a/(a+b) = 27/(27+9) = 0.75] was the same of recovery was assumed to be higher among those patients as the probability of recovery among those patient that did not who took the drug than among those patients who did not take take any drug [i.e., P(recovery|∼drug) = c/(c+d) = 9/(9+3) the drug, which would be consistent with participants giving = 0.75]. Moreover, both the outcome density causal ratings higher than zero in the causal question. Also, [P(recovery) = (a+c)/(a+b+c+d) = (27+9)/(27+9+9+3) = 0.75] Figure 2 suggests that the ratings of both probabilities seemed and the cause density [P(drug) = (a+b)/(a+b+c+d) = (27+9)/(27 to be somewhat higher in the Long than in the Standard group. +9+9+3) = 0.75] were high, therefore emulating the conditions A 2 [question: P(recovery|drug) vs. P(recovery|∼drug)] × 2 under which the causal illusions are known to reach strong [group: Standard vs. Long] Bayesian ANOVA confirmed these levels (Blanco et al., 2013; Matute et al., 2015, see simulations in impressions. The model including only the main effects of Figure 1). question and group outperformed all other models, with a BF = 2.70 over the model including just question, a BF = 5.58 over the model including both main effects and their interaction, and RESULTS a BF = 1.99 × 1013 over the model including just the main effect of group. The frequentist ANOVA yielded significant main effects The dataset for this study can be found at: https://osf.io/35ytz/ of question and group, F(1,148) = 68.690, p < 0.001, ηp 2 = 0.317, All the analyses reported in this section were conducted and F(1,148) = 7.266, p = 0.008, ηp 2 = 0.047, respectively, but the with JASP, https://jasp-stats.org/ (JASP Team, 2018; see also interaction failed to reach significance, F < 1. Wagenmakers et al., 2018). Following Wagenmakers et al. (2018, A potential criticism to these results raised by a reviewer see their Table 1) we interpret BFs between 1 and 3 as anecdotal is that perhaps participants had already reached the learning evidence, BFs between 3 and 10 as moderate evidence, and BFs asymptote after just 48 trials, in which case associative models >10 as strong-to-extreme evidence. Figure 2 shows the first of learning predict no further change with additional training. causal and conditional probabilities’ ratings given by participants (Note, however, that the asymptotic level predicted by the of each group, that is, the ratings given after 48 trials in the Rescorla-Wagner model is an associative strength of zero.) To Standard group and the ratings given after all 288 trials for explore this possibility, we analyzed the trial-by-trial predictions the Long group. As can be seen, mean causal ratings (our made by participants. Specifically, we calculated the proportion main dependent variable, see panel A) were similar in both of outcome predictions (i.e., “yes” responses) in cause-present groups. A Bayesian t-test contrasting the null hypothesis against and cause-absent trials for the initial 48 trials in the Standard FIGURE 2 | Mean causal ratings (A) and conditional probability ratings (B) after 48 trials in the Standard group and after all 288 trials in the Long group. Error bars denote 95% confidence intervals. Frontiers in Psychology | www.frontiersin.org 5 January 2019 | Volume 10 | Article 24 Barberia et al. Persistent Causal Illusions group and for the last 48 trials in the Long group. The descriptive equivalent frequentist ANOVA, F(5,370) = 3.641, p = 0.003, statistics show that outcome predictions tended to be higher for ηp 2 = 0.047. A direct (two-sided) comparison of blocks 1 the Long group, both in cause-present (Standard: Mean = 0.82, and 6 shows that causal judgments declined from the first SD = 0.12; Long: Mean = 0.86, SD = 0.12) and cause-absent to the last block of testing, t(74) = 2.720, p = 0.008, trials (Standard: Mean = 0.50, SD = 0.23; Long: Mean = 0.73, d = 0.314, BF10 = 3.84. Regarding estimated conditional SD = 0.22). These impressions were confirmed by a 2 [type probabilities, Figure 3 suggests that there was an increase of trial: cause-present vs. cause-absent] × 2 [group: Standard over time in the estimation of the P(recovery|∼drug) but vs. Long] Bayesian ANOVA. The model including both main not in the estimation of the P(recovery|drug). A Bayesian effects and their interaction achieved the best performance, with repeated measures ANOVA with the factors question [i.e., a BF = 9317.55 over the model including just the two main P(recovery|drug) vs. P(recovery|∼drug)] and block (1–6) showed effects and BFs > 4.11 × 109 over the rest of models. The that the model including only question outperformed the rest frequentist ANOVA yielded significant main effects of type of of models, with a BF = 2.34 over the model including question trial, F(1,148) = 113.39, p < 0.001, ηp 2 = 0.434, and group, and block, a BF = 19.3 over the model including both main F(1,148) = 42.47, p < 0.001, ηp 2 = 0.223, and a significant effects and their interaction, and a BF = 3.60 × 1016 over type of trial × group interaction, F(1,148) = 21.01, p < 0.001, the model including just block. Interestingly, the frequentist ηp 2 = 0.124. The analysis of the interaction showed that outcome ANOVA returned significant effects not only for question, predictions significantly differed between groups in the case of F(1,74) = 21.372, p < 0.001, ηp 2 = 0.224, but also for block, the cause-absent trials, t(148) = −6.255, p < 0.001, d = −1.021, F(5,370) = 3.948, p = 0.002, ηp 2 = 0.051, and for the interaction as BF10 = 2.39 × 106 , and only marginally differed between well, F(5,370) = 3.047, p = 0.010, ηp 2 = 0.040. Further separate groups for the cause-present trials, t(148) = −1.862, p = 0.065, ANOVAs for each of the questions revealed no effect of block d = −0.304, BF01 = 1.17. in the case of the P(recovery|drug), BF01 = 72.63, F < 1, but a As noted in the introduction, a complementary goal of our significant block effect for the P(recovery|∼drug), BF10 = 355.50, study was to explore if the fact that the participants were F(5,370) = 5.826, p < 0.001, ηp 2 = 0.073. requested to answer the causal and conditional probability Another way of exploring the effect of repeated testing questions several times during training would affect their is to compare ratings at the end of the experiment in estimations. Figure 3 shows the evolution of causal and both experimental conditions. At that time, both groups of conditional probabilities’ estimations over time in the Standard participants have experienced the same number of trials, but group. A Bayesian repeated measures ANOVA on causal ratings participants in the Standard group have already been tested yielded a BF10 = 3.91 in favor of the model including the factor five times. The descriptive statistics show that at that time block over the null model. This result was confirmed by an causal ratings in the Standard group were numerically lower (Mean = 43.77, SD = 31.08) than causal ratings in the Long group (Mean = 55.31, SD = 25.46). The Bayesian analysis yielded a BF10 = 2.905 in favor of the hypothesis that causal ratings were different in both groups over the null hypothesis. A frequentist t-test confirmed that the comparison was statistically significant, t(148) = −2.486, p = 0.014, d = −0.406. A similar comparison of conditional probability ratings revealed that, overall, estimations of P(recovery|drug) tended to be higher (Standard: Mean = 68.73, SD = 17.36; Long: Mean = 72.07, SD = 11.81) than estimations of P(recovery|∼drug) (Standard: Mean = 61.76, SD = 19.32; Long: Mean = 57.91, SD = 15.33). These ratings were analyzed through a 2 [question: P(recovery|drug) vs. P(recovery|∼drug)] × 2 [group: Standard vs. Long] Bayesian ANOVA. The model including only question outperformed all other models, with a BF = 3.18 over the model including both main effects and their interaction, a BF = 5.89 over the model including both main effects, and a BF = 1.29 × 108 over the model including just group. Note, however, that the frequentist ANOVA yielded both a significant main effect of question, F(1,148) = 44.038, p < 0.001, ηp 2 = 0.229, and a significant question × group interaction, F(1,148) = 5.093, p = 0.025, ηp 2 = 0.033, but not a main effect of group, F < 1. The analysis of this interaction showed that estimations of P(recovery|drug) and P(recovery|∼drug) differed both in the Standard group, t(74) = 2.716, p = 0.008, d = 0.314, BF10 = 3.80, and FIGURE 3 | Mean causal and conditional probability ratings after each block of 48 trials in the Standard group. Error bars denote 95% confidence intervals. in the Long group, t(74) = 7.518, p < 0.001, d = 0.868, BF10 = 9.22 × 107 . Frontiers in Psychology | www.frontiersin.org 6 January 2019 | Volume 10 | Article 24 Barberia et al. Persistent Causal Illusions We also conducted a parallel set of frequentist analyses relative frequencies of the four cells in the contingency table including the order in which the causal question and conditional (EI rule) remain unaltered along training, the predictions of probability estimations were answered by the participants. These these models would be equal for any amount of trials, which new analyses showed that the order of presentation of the means that these models do not anticipate changes in mean questions did not significantly impact our dependent variables causal impressions after 48 trials versus 288 trials in our (see Supplementary Materials). experiment. Regarding previous results on causal illusions literature pointing out in the same direction as the present study, Blanco DISCUSSION et al. (2011) carried out two studies using a causal learning task with a cover story similar to the one employed in our study. The main goal of the present study was to investigate the course Their Experiment 1 involved 50 trials and their Experiment of causal illusions after an extensive training with the candidate 2 was composed of 100 trials. When they performed a cross- cause and its potential connection with the outcome. To reach experiments comparison they found that, if anything, causal this aim, we compared causal impressions originated after a illusions were stronger in their second experiment. However, conventional length of training (48 trials, Standard group) with their procedure involved an active contingency learning task, those emerged after a markedly extended training (288 trials, i.e., participants could decide in every trial if they wanted Long group). Our results indicated that lengthening the amount the candidate cause to be present or not, that is, if they of training did not produce a significant decrease in the intensity wanted to administer the drug or not. Therefore, participants of causal illusions. Specifically, there was no change neither in the could decide the cause density to which they were exposed. causal ratings given by the participants nor in the difference in Since, as discussed in the Introduction, it has been previously their estimation of the two conditional probabilities. shown that higher cause density produces stronger causal This result is at odds with associative models such as illusions, and Blanco et al. (2011) participants tended to (at Rescorla-Wagner model, which predict that biased causal least marginally) increase the cause density they were exposed impressions at zero contingency conditions would be pre- to over blocks of trials, this cause density could have affected asymptotic and should therefore adjust to the normative absence the evolution of causal impressions over time. To this respect, of contingency as training moves forward. However, other the present experiment shows a sustained causal illusion models could accommodate the present results, such as some in a passive task in which the cause density is externally rule-based models. These models postulate that people encode controlled by the experimenter and kept constant over all the frequencies of the different events (cells a, b, c, and d in training. Table 1) or conditional probabilities of the outcome in the This finding has practical implications, because it suggests presence and absence of the candidate cause (see Equation 2) that causal misbeliefs, once acquired, might be quite resistant and apply some algebraic combination of them to form their to change, even in light of persistent covariation information causal impressions (see Perales and Shanks, 2007; Perales et al., that contradicts such beliefs. As we have already noted, causal 2017, for reviews). For example, the weighted 1P model (e.g., illusions are widely present in our everyday life, and they Lober and Shanks, 2000) assumes that people compute the influence some relevant quotidian decisions like those related two conditional probabilities involved in the calculation of 1P to health. Pseudomedicines tend to be repeatedly applied to but assign different weights to the probability of the outcome health conditions associated to high spontaneous recovery rates given the potential cause [P(recovery|drug) in Equation 2] and (Blanco et al., 2014). These high cause density and high outcome to the probability of the outcome when the potential cause is density situations are ideal for the appearance of causal illusions. absent [P(recovery|∼drug) in Equation 2]. Typically, the best According to our results, we cannot expect the perceived fit is found if the probability of the outcome is given more effectiveness of these pseudomedicines or miracle products to weight when the potential cause is present than when it is spontaneously diminish, even after extended zero-contingency absent (see Lober and Shanks, 2000; Perales and Shanks, 2007). experiences with them. This finding points out the necessity to As another example, the Evidence Integration (EI) rule (Perales develop and apply specific interventions (Barberia et al., 2013, and Shanks, 2007) proposes that causal ratings result from the 2018) aimed to prevent the appearance of causal illusions or comparison between confirmatory (cells a and d in Table 1) to reduce their influence. As mentioned in the Introduction, and disconfirmatory (cells b and c in Table 1) information, this result dovetails with previous research showing that, once and allows the four cells of the contingency table to be given accepted as valid, erroneous ideas about topics as diverse as health different weights (see Perales and Shanks, 2007, p. 583, for or politics tend to be resistant to change (e.g., Nyhan and Reifler, more details about the proposed rule). Perales and Shanks 2010; Lewandowsky et al., 2012; Nyhan et al., 2014). (2007) found in their meta-analysis that the best fit was for Superficially, our results might seem contradictory with those weights (w) ordering corresponding to wa > wb > wc > wd . of previous studies where causal ratings (López et al., 1999; Interestingly, both of these alternative models would anticipate see also Shanks, 1995) or illusory correlations (Murphy et al., a biased (non-zero) causal impression for the participants in our 2011) did decline over training. It is worth noting that we too experiment, if the typical weighting is applied to the conditional found a decrease in causal illusions, but only after repeated probabilities or cell frequencies. Crucially, as long as the two testing. Taken together, our results and previous evidence suggest relevant conditional probabilities (weighted 1P model) or the that the decline might be driven by repeated testing and not Frontiers in Psychology | www.frontiersin.org 7 January 2019 | Volume 10 | Article 24 Barberia et al. Persistent Causal Illusions extensive training. This result implies that the fact of asking disappear just because persistent empirical evidence against participants to repeatedly evaluate the potential cause may make them is gathered. Moreover, our analyses indicate that, the a difference in the way they solve the task. We hypothesize that, fact of recurrently asking people to evaluate the potential by asking participants to estimate both the P(recovery|drug) and causal relationship and to estimate the conditional probabilities P(recovery|∼drug) in our experiment, we could have brought involved in the contingency calculation, decreases the intensity their attention to the comparison of these two conditional of the illusion. Further studies should investigate how exactly the probabilities when evaluating the effectiveness of the drug. In introduction of recurrent ratings during contingency detection fact, previous debiasing interventions that have proven effective tasks could be applied to real-life situations in order to positively in decreasing causal illusions (Barberia et al., 2013) have focused influence participants’ decisions regarding the existence of causal on instructing participants on the importance of considering not relationships. only the P(recovery|drug) or the drug-recovery coincidences, but also the P(recovery|∼drug), which might be typically given less importance. Note that this idea might be implemented on the AUTHOR CONTRIBUTIONS previously discussed rule-based models as changes in the weights given to the different pieces of evidence. It could also be the case IB, MV, and JR-F contributed to the conception and design of that participants start the task with some prior assumptions about the study. IB programmed the computer task. MV performed both the P(recovery|drug) and P(recovery|∼drug), expecting the statistical analysis. IB and MV wrote the first draft of the the first to be high and the second to be low. This could manuscript. All authors contributed to manuscript revision, read, be a consequence of the specific cover story employed in this and approved the submitted version. experiment, involving a drug and the recovery from a disease. Alternatively, it could be associated to a more general default tendency to a priori assume that the cause under examination FUNDING produces the outcome, and to ignore the potential influence of other unknown generative causes on the same outcome. When This study was supported by the grant PSI2016-75776-R participants face the task, the prior about the P(recovery|drug) (AEI/FEDER, UE) from Agencia Estatal de Investigación of the would reasonably fit the actual data they encounter (a lot of a Spanish Government and the European Regional Development trials and much fewer b trials), whereas the P(recovery|∼drug) Fund to IB. MV was supported by grant 2016-T1/SOC-1395 would show a stronger discrepancy with participants’ prior. from Comunidad de Madrid (Programa de Atracción de Talento This might require special attention to be driven to trials in Investigador). which recovery happens without the drug in order to this prior to be modified and recurrently asking participant to estimate the expected percentage of recovery among patients not taking ACKNOWLEDGMENTS the drug might precisely act in this direction. It is also a possibility that, by asking participants to estimate these two We thank Yesua Chavarrías Pascual and Jordi González Guerra conditional probabilities, we untangled the inherent ambiguity for running the experiments. We also thank Yesua Chavarrías of the causal question (Buehner et al., 2003; Baker et al., Pascual for testing the computer task. 2005), making participants focus on those probabilities when evaluating the effectiveness of the drug, as predicted by the 1P model. SUPPLEMENTARY MATERIAL To conclude, the present study suggests that causal illusions are not the consequence of incomplete learning, because they The Supplementary Material for this article can be found show resistance to extensive training. This result implies that online at: https://www.frontiersin.org/articles/10.3389/fpsyg. erroneous causal beliefs of daily life are not expected to 2019.00024/full#supplementary-material REFERENCES Baker, A. G., Murphy, R., Mehta, R., and Baetu, I. (2005). “Mental models of causation: a comparative view,” in New Directions IN Human Associative Allan, L. G. (1980). A note on measurement of contingency between two binary Learning, ed. A. J. Wills (Mahwah, NJ: Lawrence Erlbaum Associates), 11–40. variables in judgment tasks. Bull. Psychon. Soc. 15, 147–149. doi: 10.3758/ Barberia, I., Blanco, F., Cubillas, C. P., and Matute, H. (2013). Implementation BF03334492 and assessment of an intervention to debias adolescents against causal illusions. Allan, L. G., and Jenkins, H. M. (1983). The effect of representations of binary PLoS One 8:e71303. doi: 10.1371/journal.pone.0071303 variables on judgment of influence. Learn. Motiv. 14, 381–405. doi: 10.1016/ Barberia, I., Tubau, E., Matute, H., and Rodríguez-Ferreiro, J. (2018). A short 0023-9690(83)90024-3 educational intervention diminishes causal illusions and specific paranormal Allan, L. G., Siegel, S., and Tangen, J. M. (2005). A signal detection analysis of beliefs in undergraduates. PLoS One 13:e0191907. doi: 10.1371/journal.pone. contingency data. Learn. Behav. 33, 250–263. doi: 10.3758/BF03196067 0191907 Alloy, L. B., and Abramson, L. Y. (1979). Judgment of contingency in depressed and Blanco, F., Barberia, I., and Matute, H. (2014). The lack of side effects of an nondepressed students: sadder but wiser? J. Exp. Psychol. Gen. 108, 441–485. ineffective treatment facilitates the development of a belief in its effectiveness. doi: 10.1037/0096-3445.108.4.441 PLoS One 9:e84084. doi: 10.1371/journal.pone.0084084 Frontiers in Psychology | www.frontiersin.org 8 January 2019 | Volume 10 | Article 24 Barberia et al. Persistent Causal Illusions Blanco, F., Barberia, I., and Matute, H. (2015). Individuals who believe in the Musca, S. C., Vadillo, M. A., Blanco, F., and Matute, H. (2008). “Associative and paranormal expose themselves to biased information and develop more causal connectionist accounts of biased contingency detection in humans,” in From illusions than nonbelievers in the laboratory. PLoS One 10:e0131378. doi: 10. Associations to Rules: Connectionist Models of Behavior and Cognition, eds 1371/journal.pone.0131378 R. M. French and E. Thomas (London: World Scientific), 16–27. doi: 10.1142/ Blanco, F., Matute, H., and Vadillo, M. A. (2011). Making the uncontrollable seem 9789812797322_0002 controllable: the role of action in the illusion of control. Q. J. Exp. Psychol. 64, Nyhan, B., and Reifler, J. (2010). When corrections fail: the persistence of political 1290–1304. doi: 10.1080/17470218.2011.552727 misperceptions. Polit. Behav. 32, 303–330. doi: 10.1007/s11109-010-9112-2 Blanco, F., Matute, H., and Vadillo, M. A. (2013). Interactive effects of the Nyhan, B., Reifler, J., Sean, R., and Freed, G. L. (2014). Effective messages in vaccine probability of the cue and the probability of the outcome on the overestimation promotion: a randomized trial. Pediatrics 133, e835–e842. doi: 10.1542/peds. of null contingency. Learn. Behav. 41, 333–340. doi: 10.3758/s13420-013-0 2013-2365 108-8 Perales, J. C., Catena, A., Cándido, A., and Maldonado, A. (2017). “Rules of Buehner, M. J., and Cheng, P. W. (1997). “Causal induction: the power PC theory causal judgment: mapping statistical information onto causal beliefs,” in Oxford versus the rescorla-Wagner model,” in Proceedings of the Nineteenth Annual Handbook of Causal Reasoning, ed. M. R. Waldmann (New York, NY: Oxford Conference of the Cognitive Science Society, eds M. G. Shafto and P. Langley University Press), 29–51. doi: 10.1093/oxfordhb/9780199399550.013.6 (Mahwah, NJ: Lawrence Erlbaum), 55–60. Perales, J. C., and Shanks, D. R. (2007). Models of covariation-based causal Buehner, M. J., Cheng, P. W., and Clifford, D. (2003). From covariation to judgment: a review and synthesis. Psychon. Bull. Rev. 14, 577–596. doi: 10.3758/ causation: a test of the assumption of causal power. J. Exp. Psychol. Learn. Mem. BF03196807 Cogn. 29, 1119–1140. doi: 10.1037/0278-7393.29.6.1119 Rescorla, R. A., and Wagner, A. R. (1972). “A theory of pavlovian conditioning: Chapman, G. B., and Robbins, S. J. (1990). Cue interaction in human contingency variations in the effectiveness of reinforcement and nonreinforcement,” in judgment. Mem. Cognit. 18, 537–545. doi: 10.3758/BF03198486 Classical Conditioning II: Current Research and Theory, eds A. H. Black and Griffiths, O., Shehabi, N., Murphy, R. A., and Le Pelley, M. E. (2018). Superstition W. F. Prokasy (New York, NY: Appleton-Century Crofts), 64–99. predicts perception of illusory control. Br. J. Psychol. doi: 10.1111/bjop.12344 Shanks, D. R. (1995). Is human learning rational? Q. J. Exp. Psychol. A 48, 257–279. [Epub ahead of print]. doi: 10.1080/14640749508401390 Hamilton, D. L., and Gifford, R. K. (1976). Illusory correlation in interpersonal Vadillo, M. A., and Barberia, I. (2018). A comparator-hypothesis account of biased perception: a cognitive basis of stereotypic judgments. J. Exp. Soc. Psychol. 12, contingency detection. Behav. Processes 154, 45–51. doi: 10.1016/j.beproc.2018. 392–407. doi: 10.1016/S0022-1031(76)80006-6 02.009 Hannah, S. D., and Beneteau, J. L. (2009). Just tell me what to do: bringing Vadillo, M. A., Blanco, F., Yarritu, I., and Matute, H. (2016). Single-and dual- back experimenter control in active contingency tasks with the command- process models of biased contingency detection. Exp. Psychol. 63, 3–19. performance procedure and finding cue density effects along the way. Can. J. doi: 10.1027/1618-3169/a000309 Exp. Psychol. 63, 59–73. doi: 10.1037/a0013403 Wagenmakers, E. J., Beek, T. F., Rotteveel, M., Gierholz, A., Matzke, D., JASP Team (2018). JASP (Version 0.9.01) Computer Software. Available at: https: Steingroever, H., et al. (2015). Turning the hands of time again: a purely //jasp-stats.org/ confirmatory replication study and a Bayesian analysis. Front. Psychol. 6:494. Kutzner, F., Vogel, T., Freytag, P., and Fiedler, K. (2011). A robust classic: illusory doi: 10.3389/fpsyg.2015.00494 correlations are maintained under extended operant learning. Exp. Psychol. 58, Wagenmakers, E. J., Love, J., Marsman, M., Jamil, T., Ly, A., Verhagen, J., et al. 443–453. doi: 10.1027/1618-3169/a000112 (2018). Bayesian inference for psychology part ii: example applications with Lewandowsky, S., Ecker, U. K. H., Seifert, C. M., Schwarz, N., and JASP. Psychon. Bull. Rev. 25, 58–76. doi: 10.3758/s13423-017-1323-7 Cook, J. (2012). Misinformation and its correction: continued influence Wasserman, E. A., Elek, S. M., Chatlosh, D. L., and Baker, A. G. (1993). and successful debiasing. Psychol. Sci. Public Interest 13, 106–131. Rating causal relations: role of probability in judgments of response-outcome doi: 10.1177/1529100612451018 contingency. J. Exp. Psychol. Learn. Mem. Cogn. 19, 174–188. doi: 10.1037/ Lober, K., and Shanks, D. R. (2000). Is causal induction based on causal power? 0278-7393.19.1.174 critique of cheng (1997). Psychol. Rev. 107, 195–212. doi: 10.1037/0033-295X. Yarritu, I., Matute, H., and Vadillo, M. A. (2014). Illusion of control: the role of 107.1.195 personal involvement. Exp. Psychol. 61, 38–47. doi: 10.1027/1618-3169/a000225 López, F. J., Almaraz, J., Fernández, P., and Shanks, D. (1999). Adquisición progresiva del conocimiento sobre relaciones predictivas: curvas de aprendizaje Conflict of Interest Statement: The authors declare that the research was en juicios de contingencia. Psicothema 11, 337–349. conducted in the absence of any commercial or financial relationships that could Matute, H., Blanco, F., Yarritu, I., Díaz-Lago, M., Vadillo, M. A., and Barberia, I. be construed as a potential conflict of interest. (2015). Illusions of causality: how they bias our everyday thinking and how they could be reduced. Front. Psychol. 6:888. doi: 10.3389/fpsyg.2015.00888 Copyright © 2019 Barberia, Vadillo and Rodríguez-Ferreiro. This is an open-access Matute, H., Yarritu, I., and Vadillo, M. A. (2011). Illusions of causality at the heart article distributed under the terms of the Creative Commons Attribution License of pseudoscience. Br. J. Psychol. 102, 392–405. doi: 10.1348/000712610X532210 (CC BY). The use, distribution or reproduction in other forums is permitted, provided Murphy, R. A., Schmeer, S., Vallée-Tourangeau, F., Mondragón, E., and Hilton, D. the original author(s) and the copyright owner(s) are credited and that the original (2011). Making the illusory correlation effect appear and then disappear: publication in this journal is cited, in accordance with accepted academic practice. the effects of increased learning. Q. J. Exp. Psychol. 64, 24–40. doi: 10.1080/ No use, distribution or reproduction is permitted which does not comply with these 17470218.2010.493615 terms. Frontiers in Psychology | www.frontiersin.org 9 January 2019 | Volume 10 | Article 24