Abstract
In two experiments we tested the prediction derived from Tversky and Kahneman's (1983) work on the causal conjunction fallacy that the strength of the causal connection between constituent events directly affects the magnitude of the causal conjunction fallacy. We also explored whether any effects of perceived causal strength were due to graded output from heuristic Type 1 reasoning processes or the result of analytic Type 2 reasoning processes. As predicted, Experiment 1 demonstrated that fallacy rates were higher for strongly than for weakly related conjunctions. Weakly related conjunctions in turn attracted higher rates of fallacious responding than did unrelated conjunctions. Experiment 2 showed that a concurrent memory load increased rates of fallacious responding for strongly related but not for weakly related conjunctions. We interpret these results as showing that manipulations of the strength of the perceived causal relationship between the conjuncts result in graded output from heuristic reasoning process and that additional mental resources are required to suppress strong heuristic output.
In everyday life, people often make judgements based on their beliefs about the strength of the causal relationships between events. For example, people may think it more likely that petrol prices will rise because of an imminent conflict in the Middle East than because of a similar dispute in a Western country. Underlying these likelihood judgements is the belief that some events are more likely to cause an outcome than others. At the same time, people's likelihood estimations should also be constrained by the laws of probability. However, when estimating probabilities, people frequently fail to comply with logical constraints (Gilovich, Griffin, & Kahneman, 2002; Tversky & Kahneman, 1983), usually because there is a more intuitively appealing, but normatively fallacious, answer (Kahneman & Frederick, 2002). Tasks that produce the most robust reasoning errors often activate background knowledge that seems responsible for biasing the reasoning process. In perhaps the best known demonstration of this sort, Tversky and Kahneman (1983) established that knowledge about social categories and causal schemas can lead people to commit the famous conjunction fallacy. Here we focus on the causal case, and the prediction, based on Tversky and Kahneman's (1983) early work, that the rate of conjunctive errors should be proportional to the strength and saliency of a causal connection between constituent events. Somewhat surprisingly, given widespread current interest in causal cognition (e.g., see Gopnik et al., 2004; Luhmann & Ahn, 2005; Rehder & Hastie, 2001; Sloman, 2005), there has been relatively little work on causal conjunction fallacies. Studies that have employed the causal paradigm have not delivered consistent findings, and consequently there are still no robust experimental data to corroborate the claim about the effect of causal strength. Addressing this issue was the first goal of this study.
Our second goal was to consider the processes by which causal knowledge may exert its effects on the reasoning process. There is increasing evidence that causal knowledge affects inferences across many apparently disparate reasoning domains, such as categorization, associative learning and deductive as well as inductive reasoning (e.g., Glymour & Cheng, 1999; Gopnik et al., 2004; Griffiths & Tenenbaum, 2005; Sloman & Lagnado, 2005; Steyvers, Tenenbaum, Wagenmakers, & Blum, 2003; Tenenbaum & Griffiths, 2001, 2003; Waldmann, 2001). However, most approaches to causal cognition are largely computational or descriptive. That is, many researchers have focused on describing people's inferences or on comparing their inferences to some principle of causal reasoning. Instead, we wish to focus on the processes that mediate the effects of causal knowledge on probabilistic inferences. We do this with reference to the dual process theory of thinking (Epstein, Pacini, Denes-Raj, & Heier, 1996; Evans, 2003, 2006; Evans & Over, 1996; Kahneman & Frederick, 2002; Sloman, 1996; Stanovich & West, 1998b).
According to dual process accounts, our heuristic reasoning processes deliver effortless and automatic default responses. In contrast, slow and effortful analytical reasoning processes produce responses that are in line with normative standards. Evidence for such accounts comes from tasks where heuristic processes cue one type of response and analytic processes another. The conjunction fallacy is often interpreted as providing evidence to support dual process claims (see De Neys, 2006a; Sloman, 1996; Stanovich, 1999), because reasoners are said to respond in line with the output of the heuristic rather than the analytic system. To a large extent, causal beliefs have been ignored by dual process researchers (for important exceptions, see De Neys, Schaeken, & d'Ydewalle, 2005; Over, Hadjichristidis, Evans, Handley, & Sloman, 2007; Verscheuren, Schaeken, & d'Ydewalle, 2005), and consequently we know little about the processes underlying the effects of our causal beliefs on our probabilistic judgements. With the current experiments, we aimed to begin to address this gap in our understanding.
Background knowledge and the conjunction fallacy
The most widely known version of the conjunction fallacy is Tversky and Kahneman's (1983) classic Linda problem. People read the following personality description:
Linda is 31 years old, single, outspoken, and very bright. She majored in Philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
Participants then rank statements according to the probability that they apply to Linda, including “Linda is active in the feminist movement” (T), “Linda is a bank teller” (F), and “Linda is a bank teller and active in the feminist movement” (F&T). Up to 86% of participants believe the conjunction F&T to be more probable than the single constituent F. Given that the probability of a conjunction is the product of the probabilities of its constituent events, the probability of the conjunction can never exceed the probability of a single constituent. Thus, although the modal answer given on the Linda problem is intuitively appealing, it violates a fundamental law of probability theory.
The conjunction fallacy is robust even when the instructions are disambiguated (Tentori, Bonini, & Osherson, 2004), and participants have received statistical training (Agnoli & Krantz, 1989), ruling out explanations based on misinterpretation of the task (Cosmides & Tooby, 1996; Gigerenzer, 1996, 1998; Hertwig & Gigerenzer, 1999), or the application of pragmatic rather than mathematical principles (Dulany & Hilton, 1991; Fiedler, 1988; Politzer & Noveck, 1991). Similarly, when two likely events are conjoined, and people give numerical estimates rather than making a forced choice, the conjunction fallacy rate is reduced but by no means eliminated (Wedell & Moro, 2008).
Clearly then, other factors must be at work, leading to the persistence of the fallacy even under such stringent conditions. Tversky and Kahneman (1983) appeal to a representativeness heuristic, a natural assessment of the degree to which an actual outcome, event, or person matches a relevant mental schema. This implicitly suggests that background knowledge about social categories has a crucial influence on the appeal of a heuristic response, which at times is not reconcilable with the correct statistical derivation.
As mentioned above, causal knowledge and schemas also provide a rich and important source of background information (Aijzen, 1977). Tversky and Kahneman (1983) pointed to the role of causal knowledge in probability estimation. They reported a conjunction fallacy that arises because of a causal connection between the two constituent events. Known as the A → B paradigm, an unlikely event B is made more plausible by adding a causal event A. A further contention is that causal strength biases probability judgements, such that events with a strong causal link should attract more fallacious responding than do weakly related events. Unfortunately, they do not present any experimental data to corroborate the claim about causal strength.
Since then, only a handful of studies have included some form of manipulation of causality and/or causal strength, and their findings are contradictory. Thüring and Jungermann (1990) failed to detect an effect of presence versus absence of a causal link, which stands in direct contrast to Tversky and Kahneman's (1983) results. Similarly, Fisk and Pidgeon (1998) used a slightly modified version of the original causal conjunction fallacy. Although they found that conditional dependence between conjunctive events increased the occurrence of the fallacy, their regression analysis failed to find a correlation between susceptibility to the fallacy and the strength of the conditional relationship assessed in a pretest. Fabre, Caverni, and Jungermann (1995) on the other hand showed that conjunctions in which the causal link was said to be “frequent” attracted higher rates of fallacious responding than conjunctions with “possible” causal connections.
Although not explicitly addressing this issue, a more recent study by Sides, Osherson, Bonini, and Viale (2002) does point to the modulating effect of causal strength. Their findings suggest that the perceived strength of a causal link is an important determinant of degree of susceptibility to the fallacy. Participants were asked to place a monetary bet on one of two statements, such as:
“Smoking will decrease by 15%” (A)
“The government will increase tax by 1$ per pack and smoking will decrease by 15%” (B)
Sides et al. (2002) mention that the scenarios in their experiment attracted very different rates of fallacious responding, varying from 19% to 51%. Unfortunately they leave these differences unexplained, and it remains unclear what features of the problems account for the different rates of the fallacy. However, the items attracting the highest fallacy rates appear to share a very close causal connection between the two single constituents. It is plausible that the variability in susceptibility to the fallacy observed by Sides et al. (2002) is a function of causal strength. Thus, in order to demonstrate a relationship between perceptions of causal strength and susceptibility to the conjunction fallacy, in this study we adopted Sides et al.'s paradigm and explicitly manipulated the causal strength between conjunctive events.
Dual process accounts of reasoning
Perhaps the most commonly accepted account of the conjunction fallacy is provided by dual process theories of thinking (Evans, 2003, 2006, 2007; Evans & Over, 1996; Sloman, 1996; Stanovich & West, 1998a). Dual process frameworks of reasoning capture divisions between heuristic thinking that often produces erroneous output and effortful analytic thinking that is more likely to produce normatively sound responses. At their most rudimentary level, such frameworks hold that the mind consists of two separate and competing processes, each with contrasting characteristics. Heuristic processes are thought to be fast, automatic, associative, heavily context bound, and cognitively undemanding. Inferences are based on similarity and contiguity. In contrast, analytic processes are held to be slow and cognitively demanding with restrictions set by working-memory capacity. Although there is some disagreement about whether analytic reasoning must necessarily be decontextualized (Evans, 2006; Feeney, Shafto, & Dunning, 2007; Morsanyi & Handley, 2008; Verschueren et al., 2005), analytic inferences can often only be generated by teasing apart content and structure and applying formal rules to the latter.
To bestow an adaptive advantage and to facilitate cognitive economy, the heuristic response should at least approximate the answer resulting from more effortful reasoning. Thus, the two processes usually deliver congruent output. Under rare circumstances, the existence of two processes will become evident when they cue sharply opposing answers. Response-incongruent problems such as Wason's selection task (Wason, 1966), syllogistic reasoning under belief bias (for reviews, see Evans, 2003; Osman, 2004), and the conjunction fallacy highlight how the heuristic response can conflict with the logical answer based on a decontextualized representation of the problem. Thus, if people rely on the output from the initial process based on the contextualized problem representation, they are likely to give a normatively fallacious response. The precise relationship between the two processes remains controversial (see Evans, 2007). Whatever the exact nature of the relationship, whenever the heuristic system cues an erroneous response, a normative response can only be delivered if the analytical system ultimately controls the output. However, this operation is both time consuming and cognitively demanding.
The fact that analytic reasoning is said to come at a cognitive cost, with restrictions imposed by cognitive ability and working-memory resources, as well as the time available to complete the task, has enabled researchers to dissociate the two systems. For example, Stanovich and West (1998a) report an association between IQ and people's ability to resist the Linda conjunction fallacy. This is interpreted as showing that people high in analytic processes are able to overcome the default heuristic output in order to give the normative response. Other studies show that imposing a load on working memory increases rates of fallacious responding in the classic Linda problem (De Neys, 2006a), as well as on other deductive inference tasks (Gilhooly, Logie, Wetherick, & Wynn, 1993). Decreasing available response time also compromises analytic reasoning processes and leads to an increase in belief bias in syllogistic reasoning tasks (Evans & Curtis-Holmes, 2005). Hence, under conditions of time pressure and memory load people are less able to recruit effortful analytical mechanisms in order to correct the prepotent heuristic response, supporting the contention that the incorrect response is based on a less thorough analysis and is delivered without much effort.
Dual process theorists argue that heuristic processes are associative (e.g., Sloman, 1996). One very important characteristic of outputs from associative processes is that they vary in strength. Thus, the output of heuristic processes should vary in strength, and the ability of analytic processes to control such outputs should, in turn, vary with the strength of the heuristic process. This is a very important, but (to the best of our knowledge) as yet untested, prediction of dual process theories: On tasks where the analytic and heuristic systems cue different responses, the proportion of analytic responses should be in inverse proportion to the strength of the belief driving the heuristic response. Our interest in causal conjunction fallacies stems from our aim to test this prediction. If we can show that people commit the conjunction fallacy in inverse proportion to the perceived strength of the causal link between the conjuncts, then we can claim to have confirmed the prediction, derived from dual process theory, about the relationship between heuristic and analytic responses.
Causal beliefs and dual processes
One potential problem for the dual process account of the conjunction fallacy outlined above is that it is built on the assumption that the effects of causal knowledge on the output of our reasoning processes are driven by heuristic processes. That is, people commit the fallacy when their analytic processes fail to control the output of their heuristic processes. However, this assumption may not be warranted, and indeed some causal reasoning must be achieved via analytic processes. Consider an engineer engaging in the complex task of attempting to diagnose the cause of a fault in a power plant. Her reasoning is likely to be slow, effortful, and analytic. In the literature there have been recent studies of complex causal reasoning, such as predicting the outcome of an action or intervention (Sloman & Hagmayer, 2006), evaluating predictive and diagnostic relations amongst variables (Fenker, Waldmann, & Holyoak, 2005), and explaining why an event happened (Keil, 2006). In all of these cases, analytic processes are likely to be required in order for good reasoning to occur. On the other hand, other approaches to causality emphasize continuities between human and animal cognition (e.g., Shanks & Dickinson, 1987), suggesting that some causal reasoning may be associative in nature (see also Rogers & McClelland, 2004).
From a dual processes perspective, some causal reasoning is likely to be achieved by heuristic processes only whereas some causal tasks will require the involvement of heuristic and analytic processes. Consistent with the second of these possibilities, Evans (2006) has argued that analytic reasoning need not be decontextualized, and Morsanyi and Handley (2008) have recently shown that in children, contextualized reasoning is associated with cognitive ability. Other evidence that some causal reasoning draws on analytic processes comes from studies by Waldmann and Hagmayer (2001) who showed that estimations of causal strength are affected by processing constraints and cognitive load. Fugelsang and Thompson (2003) showed that whereas prior beliefs about both covariation and causal mechanisms may influence the estimation of causal strength in an automatic fashion, assimilating additional covariation information into causal strength estimations may involve a more deliberate, analytical process. Finally, Verscheuren et al. (2005) have demonstrated that the effects of causal knowledge on reasoning are mediated by heuristic processes whereas reasoning about causal relations is achieved by analytic processes (see also De Neys et al., 2005).
The foregoing suggests that there might be an alternative explanation for any result showing that an increase in the strength of the causal relationship between the conjuncts leads to a decrease in resistance to the conjunction fallacy. Causal conjunction fallacies may occur not because people's limited analytic capabilities cannot control the output from their heuristic processes, but because people commit effortful analytic processes to thinking about the causal relationship between the conjuncts. Such an analytical process might take the form of an effortful mental simulation (see Kahneman & Tversky, 1982), in which people suppose the cause and use their analytic processes to evaluate the feasibility or likelihood of the outcome (for more on the simulation heuristic in reasoning see Over et al., 2007). In a similar vein, people may simply construct a situation model (for a review, see Zwaan & Radvansky, 1998) containing the conjoined cause and effect. Because such a model might be more coherent when the conjuncts are strongly related, mental models of strongly related conjunctions may result in a failure to effortfully search for counterexamples (for a discussion of counterexample generation, see Johnson-Laird & Byrne, 1993) in which alternative events may cause the outcome (De Neys et al., 2005; Verschueren et al., 2005). As a consequence, fallacy rates would be higher for strongly related than for weakly related conjunctions.
One way of distinguishing between these competing explanations is to examine how variations in causal strength interact with manipulations designed to compromise analytic processes. If the output from heuristic processes varies with causal strength, then we might expect manipulations designed to interfere with analytic processing to have greater effects on problems where the heuristic output is likely to be stronger. In cases where the heuristic response is very strong, the effects of, for example, a concurrent memory load may be particularly large. However when the heuristic response is weaker, a concurrent memory load may have much less effect on responding. In contrast, if the output from the heuristic system does not vary with causal strength because the effect of causal strength is based on effortful and contextualized analytic processes, then we might expect to see a decreased rate of fallacious responding when analytical processes are compromised. This would occur regardless of causal strength because in all cases a concurrent memory load manipulation, for example, would mean that fewer mental resources could be dedicated to the effortful evaluation of the causal relationship.
In summary, the causal conjunction fallacy provides us with the opportunity to test how heuristic and analytic processes interact during probabilistic causal thinking. However, even if we succeed in finding that strength of causal relationship between the constituent events predicts rates of fallacious responding, before we can claim that the effects of causal strength are mediated by a heuristic rather than an analytical process, and fallacies arise because analytical processes fail to control the output of heuristic processes, we must test whether factors expected to affect analytic processes interact with causal strength in determining probabilistic reasoning.
Overview of the experiments
In Experiment 1 we used materials based on those used by Sides et al. (2002) to test our prediction derived from Tversky and Kahneman (1983) work that susceptibility to the conjunction fallacy should be proportional to the perceived strength of a causal link between the constituent events. We hoped that this would remediate the lack of conclusive experimental evidence to support their claim. In Experiment 2 we combined manipulations of causal strength with a secondary task paradigm. A secondary task burden should compromise analytical processing and force people to rely more heavily on the initial heuristic output. Thus, if the effects of causal strength in the causal conjunction fallacy are mediated by heuristic processes then we should find bigger effects of the secondary task when the causal relationships between constituent events are strong, and thus output from heuristic processes is compelling, than when the causal relationships are weak, and the output from heuristic processes is less compelling. Alternatively, if the effect of causal strength is mediated by a contextualized, but analytic, process there should be no interaction between causal strength and manipulation of mental resources, as people dedicate fewer resources to reconstructing or evaluating the causal link regardless of its strength.
Experiment 1
Experiment 1 explores how explicitly manipulating the strength of the causal relation between the conjuncts affects susceptibility to the conjunction fallacy. In addition to weakly and strongly related conjunctions we included control conjunctions whose constituents we expected to be unrelated. Based on Tversky and Kahneman's (1983) work, we predict a main effect of causal strength, with higher rates of susceptibility to the conjunction fallacy for conjuncts sharing a strong causal link than for conjuncts sharing a weak causal link. Weakly related conjuncts should in turn attract higher rates of fallacious responding than unrelated conjuncts.
Method
Design
The experiment used a repeated measures design. The independent variable was causal strength, which had three levels: unrelated, weakly related, and strongly related. The dependent variable was the proportion of trials on which participants were susceptible to the conjunction fallacy.
Materials
The scenarios in the current experiments were very loosely based on the original stimuli used by Sides et al. (2002). Each scenario consisted of a chain of three causally related events, A, B, and C, and an unrelated event, U. A full set of materials is reproduced in Appendix A, and an example is given in Figure 1. A and B were both potential causes of an outcome C. However, the causal link between Event A and Outcome C was much weaker and less immediate than the strong causal link between Event B and Outcome C. This resulted in three problems for each scenario, one in which there was a strong causal connection (Event B and Outcome C), one in which there was a weak causal link (Event A and Outcome C), and finally another where there was no causal link between the conjuncts (Event U and Outcome C). An example of a strongly related conjunction would be (C) “Pepsi will launch a large-scale advertising campaign” and (B ^ C) “Pepsi will launch a large-scale advertising campaign and sales of Pepsi will overtake those of Coca-Cola”. For weakly related conjunctions, (C) remained the same but the conjunction (A ^ C) was changed to “A new CEO will take over as leading director of Pepsi and sales of Pepsi will overtake those of Coca-Cola”. The unrelated conjunction was (U ^ C) “A new CEO will take over as leading director of Tesco and sales of Pepsi will overtake those of Coca-Cola”.
Schematic diagram of the causal structure of the scenarios.
Each pair of descriptions was presented on a single sheet of paper side by side, and their position was counterbalanced, so that each appeared equally often on the left and on the right side. Beneath these descriptions were two blank probability boxes. Participants were requested to separately rate how likely they thought it was that each event on the page would occur at some point in the future or in the time frame explicitly stated in the event. Ratings were given in a percentage format, ranging from 0% (will definitely not happen) to 100% (must happen). Nine distractor scenarios were randomly interspersed amongst the experimental scenarios. The three kinds of distractor items consisted either of two conjunctive events in which none of the conjuncts were identical—for example, A new drug that increases life expectancy for HIV patients will come onto the market and research to find a cure for HIV will intensify (W ^ X)/New cheaper treatments for Hepatitis C will be developed and sales for the new drug will meet record numbers by next year (Y ^ Z)—a conjunctive and unrelated single event—for example, the Government will reverse their classification of cannabis as a soft drug and drug offences will decrease by 10% (X ^ Y)/Cocaine use will decrease by 5% over the next two years (Z)—or two single events (e.g., A severe avalanche will cause devastation in several regions of the Swiss Alps this winter (X)/More than 10 climbers will die in an attempt to climb Mount Everest next year (Y)). Table 1 sets out the logical form of the conjunction and distractor scenarios.
Logical form for the four types of event pairs used in Experiments 1 and 2
The content for which participants rated the strongly related, weakly related, or unrelated conjunctions was counterbalanced across participants in a Latin-square-type design, resulting in three groups. Thus, although causal strength was a within-subjects manipulation, participants only ever engaged once with each particular scenario. The order of the scenarios was randomly determined for each of the three groups.
A preliminary experiment not reported in full detail here confirmed higher rates of fallacious responding for strongly related conjunctions than for weakly related conjunctions. Selection of scenarios for this earlier experiment was based on an extensive pretest in which we verified the manipulation of causal strength. A total of 20 students from Durham University were given sets of two sentences, one pertaining to either the weak (Event A) or strong (Event B) cause and the second referring to the outcome (Event C). Participants rated how strong the potential causal link between the events was on a scale from 1 (unrelated) to 9 (very strong causal relationship). Separate independent samples t tests with causal strength as the between-subjects variable showed that weakly related events received significantly lower strength ratings (mean strength rating = 4.5) than strongly related events (mean strength rating = 6.1).
To verify the results of the earlier pretest and to ensure that participants perceived a much weaker causal relationship between the unrelated and weakly related conjuncts in the main experiment, we rechecked our causal strength manipulation in a posttest causal strength rating task akin to the pretest for the earlier experiment. Thus, once they had completed the reasoning task, participants rated the causal link between the conjuncts from each of the experimental scenarios that they had attempted. These materials were presented in two different orders.
Participants
A total of 51 participants from the University of Durham took part in this experiment. There were 33 females and 18 males with a mean age of 24.6 years (SD = 8.4 years).
Procedure
Participants were given oral instructions and randomly received one of the three booklet versions. Once they had read the instructions they worked through the booklet at their own speed. Following this, they completed a posttest measuring causal strength. Similar to the pretest, the posttest required participants to rate the strength of the causal connection between each pair of conjuncts that participants had seen in the main experiment.
Results
Rates of susceptibility to the fallacy
The main focus of our analysis was on the normative validity of people's responses. Thus, for each participant we calculated the proportion of strong, weak, and unrelated trials in which they rated the conjunction higher than the single event.
Proportions of trials on which participants were susceptible to the fallacy were analysed with a 3 (group) by 3 (causal strength) mixed-design analysis of variance (ANOVA), with group as the between-subjects and causal strength as the within-subjects variables.
Neither the main effect of group, F(2, 48) = 1.14, p = .33, nor its interaction with causal strength, F(3.43, 82.41) = 1.27, p = .29, was significant (in this and all future cases where the sphericity assumption is violated, we have reported degrees of freedom corrected using the Greenhouse–Geisser procedure). Hence, this variable is not further commented upon. As predicted, there was a main effect of causal strength, F(1.72, 82.4) = 11.46, p < .001, effect size f = 0.49. Table 2 shows the means across the three levels of causal strength. Bonferroni post hoc tests corrected for multiple pairwise comparisons confirmed that susceptibility to the fallacy was significantly higher for strongly related conjunctions than for both the weakly related conjunctions (p < .05) and unrelated conjunctions (p < .001). Similarly, weakly related conjunctions attracted more fallacious responding than did unrelated conjunctions (p < .05). The fallacy rates for each individual scenario can be found in Appendix B.
Conjunction fallacy susceptibility proportions and standard errors from Experiment 1
To generalize these results across items, the mean unrelated, weak, and strong conjunction fallacy proportions for each of the nine items were calculated. Thus, for this item analysis, the conjunction fallacy susceptibility proportions were averaged across subjects rather than items. A repeated measures ANOVA with strength as the independent variable confirmed that there was a main effect of causal strength, F(2, 16) = 10.79, p = .001, effect size f = 1.16. Although due to the small number of items not all pairwise comparisons were statistically significant (ps between .002 and .23), the effect sizes were substantial for all comparisons (Cohen's d between 0.5 and 1.82).
Correlations between fallacy rates and posttest causal strength ratings
For each scenario, three posttest mean causal strength ratings were calculated. This reflected the mean perceived causal link between the strongly related, weakly related, and unrelated conjuncts. A repeated measures ANOVA collapsed across group with causal strength as the repeated measures independent variable confirmed the manipulation of causal strength, F(1.79, 89.23) = 62.48, p = .0005. Bonferroni-adjusted pairwise comparisons revealed that conjunctions constructed to be strongly related (mean = 6.4, SE = 0.18) were rated significantly higher than both weakly related (mean = 5.4, SE = 0.21; p < .001) and unrelated conjunctions (mean = 3, SE = 0.24; p < .001). The mean strength rating for weakly related events was significantly higher than that for the unrelated events (p = .001).
We examined the degree of association between posttest ratings and the number of participants who displayed the conjunction fallacy on each of the 27 unique conjunctive problems (one strongly related, weakly related, and unrelated conjunction for each of the nine scenarios). The correlation was highly significant, r(27) = .67, p < .001. Thus, 45% of the variation in the number of participants who succumbed to the fallacy on each problem was accounted for by the perceived strength of the causal relationship between the constituent events.
Discussion
The results of this experiment confirm Tversky and Kahneman's (1983) assertion that causal strength increases biased reasoning about conjunctive events. Adding a plausible cause to a single event thus increased its perceived likelihood, often resulting in a conjunction fallacy. As predicted, the fallacy rate was higher for strongly related conjunctions than for weakly conjunctions, which in turn produced higher fallacy rates than unrelated conjunctions. In Experiment 2 we compared strongly to weakly related conjunctions in order to examine whether the effects of causal strength are due to graded output from a heuristic process, or whether they are the result of a contextualized, effortful analysis of the causal relationship between events, which in turn affects evaluations about the likelihood of the events occurring.
Experiment 2
To explore which process is modulated by variations in causal strength and how heuristic and analytic processes interact in determining whether reasoners display susceptibility to the causal conjunction fallacy, Experiment 2 combined a secondary task manipulation, typically used to interfere with analytical processing (see De Neys, 2006a, 2006b), with a manipulation of causal strength. Based on the previous experiments, we should observe a main effect of causal strength, with higher conjunction fallacy susceptibility rates for the strong causal conjunctions than for the weak causal conjunctions.
Conflicting predictions may be made about the main effect of load and its interaction with causal strength. If causal conjunction fallacies are observed because analytic processes often fail to control the output of heuristic processes then we should find higher rates of susceptibility to the fallacy under conditions of load, where analytic processes are compromised. Similar findings have been reported by De Neys (2006a), who showed an increase in susceptibility to the fallacy on the Linda problem when people had to simultaneously memorize a random dot matrix pattern. Furthermore, if the effects of causal strength occur because stronger heuristic responses are harder for analytic processes to control, then the causal strength and load factors should interact. As to do so requires fewer mental resources, participants may succeed in overcoming a relatively weak heuristic response caused by a weak causal relationship between the conjuncts, even under conditions of load. In contrast, as more mental effort is needed to inhibit the stronger heuristic response that is cued by strongly related conjuncts, participants may be more likely to fail in inhibiting such a response under conditions of memory load.
A different set of predictions may be derived from the view that the effect of causal strength is the result of an effortful reasoning process. First, when participants are under conditions of load we might actually observe a decrease in the rate at which they succumb to the conjunction fallacy. That is, if the causal conjunction fallacy occurs because people assess the probability of the conjunction by using the analytical system to reconstruct or evaluate the causal relationship between the conjuncts, then they may be less able to do so under conditions of memory load.
Method
Design
The experiment used a 2 (causal strength: strong vs. weak) × 2 (memory load: loaded vs. unloaded) mixed design, with causal strength as the within-subjects and memory load as the between-subjects variables. As in Experiment 1, the dependent measure was the proportion of trials on which participants were susceptible to the conjunction fallacy.
Materials
The main materials, distractor items, and counterbalancing procedure were identical to those employed in Experiment 1. However, one of the scenarios that attracted low rates of the fallacy in Experiment 1 and did not show a difference for weakly and strongly related conjunctions was not used in this experiment.
The scenarios were presented on a laptop using PowerPoint, with a pair of descriptions from each scenario presented on an individual page side by side. In the loaded condition, De Neys's (2006b) secondary task was adapted so that each scenario was preceded by a 4 × 4 dot matrix, which contained four randomly placed light-blue dots.
Participants
A total of 40 participants (20 per condition) were recruited at Durham University Queen's Campus. There were 12 males and 28 females with a mean age of 26.7 years.
Procedure
After receiving instructions, people completed an example distractor problem not used in the main experiment to familiarize them with the format of the experiment and how to attach separate probability estimates to the two events. The experiment was self-paced. In the unloaded condition, people rated the probability of the conjunction and the single event on a scale from 0 (will definitely not happen) to 100 (must happen) on a separate score sheet and proceeded to the next scenario in their own time. In the loaded condition, participants were instructed to memorize the dot matrix that was presented before each scenario for 1,000 ms. After rating the probability of the events, they also recalled the position of the dots in an empty matrix on a separate score sheet. Following the main experiment, participants in both conditions completed a posttest similar to that used in Experiment 1.
Results
Conjunction fallacy rates
Figure 2 shows the overall rates of fallacious responding across the experiment. As may be seen in Appendix B, in all but one of the scenarios, strongly related conjunctions attracted higher rates of fallacious responding than did weakly related conjunctions. A 2 (group) × 2 (memory load) × 2 (causal strength) mixed-design ANOVA, with group and condition as the between-subjects and causal strength as the within-subjects variables, was carried out on the fallacy susceptibility proportions. The analysis revealed no effects of group. However, the main effect of causal strength was highly significant, F(1, 36) = 24.89, p < .001, effect size f = 0.82, with strong causal conjunctions attracting a higher rate of fallacious responding than weakly related conjunctions.
Rates of conjunction fallacy (and standard errors) from Experiment 2 broken down by causal strength and load.
The main effect of memory load did not quite reach statistical significance, F(1, 36) = 3.56, p < .07, effect size f = 0.25. However, it was qualified by a significant interaction between memory load and causal strength, F(1, 36) = 4.08, p < .05, effect size f = 0.34. Thus, post hoc pairwise tests were carried out using a Bonferroni correction for multiple comparisons. These showed that the difference between rates of the fallacy for strong and weak conjunctions was significant both in the loaded condition (p < .001, with a very large effect size of d = 1.29), and in the unloaded condition (p < .05, with a medium effect size of d = 0.51, see Cohen, 1988). Furthermore, post hoc comparisons showed that memory load only had an effect on the strongly related arguments (p < .02, effect size d = 0.85), such that the fallacy rate was higher for participants who were burdened by a secondary memory load. In contrast, rates of susceptibility to the fallacy for the weakly related problems were unaffected by a secondary task (p = .64, effect size d = 0.17).
For the item analysis, the mean loaded and unloaded weak and loaded and unloaded strong conjunction fallacy proportions were calculated for each of the eight items. A repeated measures ANOVA was carried out with causal strength (weak vs. strong) and memory load (loaded vs. unloaded) as the repeated measures variables. The main effect of causal strength was significant, F(1, 7) = 14.55, p < .01, effect size f = 1.44, as was the main effect of memory load, F(1, 7) = 18.05, p < .005, effect size f = 1.61. The interaction was only marginally significant, F(1, 7) = 3.86, p = .09, effect size f = 0.74. This was probably due to the small number of items, and, as indicated by the effect size, the interaction was substantial. Post hoc Bonferroni comparisons confirmed that as in the main analysis, there was a significant difference between the loaded and unloaded conditions for strongly related conjunctions (p = .011) but no effect of memory load on the fallacy susceptibility proportion for weakly related conjunctions (p = .83).
Correlations between conjunction fallacy rates and posttest ratings of causal strength
For each participant, two posttest mean strength ratings were calculated, one for the strongly related conjunctions and one for the weakly related conjunctions. A paired-samples t test collapsed across group and condition with causal strength as the repeated measures independent variable confirmed the manipulation of causal strength. Conjunctions constructed to be strongly related received strength ratings of 6.23 (SD = 1.16), whereas weakly related conjunctions received an average causal strength rating of 3.71 (SD = 1.81), t(39) = 7.23, p < .001.
The number of participants susceptible to the conjunction fallacy for each of the 16 unique conjunctive problems (one weak and one strong conjunction problem for each of the eight scenarios) was significantly correlated with the corresponding posttest causal strength rating, r(16) = .58, p < .02. This suggests that the magnitude of the conjunction fallacy is indeed proportional to the perceived strength of the causal relationship between the conjuncts
Secondary task analysis
In a dual task paradigm dissociable effects of the secondary task can reflect strategic trade-offs between primary and secondary tasks (see Hegarty, Shah, & Miyake, 2000). To guard against the possibility that the dissociable effect of memory load reflected such a strategic trade-off, we calculated the number of dots correctly recalled separately for weakly and strongly related conjunctions. A paired-sample t test with causal strength as the within-subjects variable showed that there was no significant difference in the mean number of dots correctly recalled, t(19) = 0.45, p = .66. Participants recalled a mean of 1.7 dots (SD = 0.82) for trials in which the conjunctions were strongly related and 1.64 dots (SD = 0.76) for weakly related conjunctions. This suggests that people were consistent in how they allocated their mental resources to the primary and secondary tasks across the weakly and strongly related conjunction problems.
Discussion
As predicted, causal strength had a large effect on rates of susceptibility to the causal conjunction fallacy. Although the effect of memory load did not quite reach significance, its interaction with causal strength was significant. Thus, the secondary task increased fallacious responding for strongly related events, but had no effect on the fallacy rate for weakly related scenarios. This suggests that strongly related conjunctions cue a much stronger heuristic output than do the weak conjunctions. and the strength of this output determines the ease with which the analytical system can inhibit the heuristic response. Under memory load conditions, most participants may not have enough cognitive resources to inhibit the compelling heuristic output that is activated by the strong causal link. Accordingly, we observe higher rates of the fallacy under conditions of memory load. When the causal link is weaker, the heuristic response is less compelling, and participants can overcome it, even under conditions of memory load. If the causal conjunction fallacy was the result of effortful reconstruction and evaluation of the causal relationship between the two constituent events, we should have observed a decrease in the fallacy rate under conditions of memory load. Given that this was not the case, we can be more confident that variations in causal strength lead to variations in the output from the heuristic reasoning process.
Note here that the results of Experiment 1 also rule out a possible alternative explanation. In that experiment we demonstrated that our weak materials produced significantly higher rates of the fallacy than did control materials where there was little or no perceived causal link. Without this demonstration it might have been possible to argue that we were comparing cases where there is a strong causal relationship to cases where there is no such relationship. This argument is defeated by the results of Experiment 1.
General Discussion
The two experiments in this study suggest that causal knowledge has a crucial influence on probabilistic reasoning. As proposed by Tversky and Kahneman (1983), the perceived strength of the causal connection between conjunctive events influenced the magnitude of the conjunction fallacy. Thus, Experiment 1 showed that conjunctions with stronger causal links attracted more biased responses than weakly associated conjunctions. Conjunctions with a weak causal link in turn attracted higher rates of fallacious responses than did unrelated conjunctions. Experiment 2 showed that limiting available mental resources increased the fallacy rate for strongly related conjunctions but not for weakly related conjunctions. The results strongly suggest that the initial heuristic output can be graded and depends upon the strength of the belief driving this initial response. The interaction also indicates that the effects of experimental manipulations designed to compromise analytic reasoning processes are dependent on the strength of the output from the heuristic system.
Causal conjunction fallacies and dual process accounts
How do the current findings fit with the assumptions made by dual process theorists? With the exception of De Neys et al.'s (2005) and Verschueren et al.'s (2005) accounts of causal conditional reasoning, there are few studies that have tried to characterize the reasoning mechanisms that may underlie the output from a heuristic and analytical reasoning process. To date, the emphasis has largely been on distinguishing effortless, heuristic responses from cognitively demanding, analytical answers (e.g., Kokis, Macpherson, Toplak, West, & Stanovich, 2002) that vie for control during the reasoning process. One way of pursuing this has been to manipulate the availability of effortful Type 2 reasoning processes. Examples include limiting response times (e.g., Evans & Curtis-Holmes, 2005) and using secondary task manipulations (e.g., De Neys, 2006a, 2006b). Within the individual differences approach, researchers have investigated plausible correlates of more effortful and normatively correct reasoning, such as cognitive resources and IQ (see Stanovich & West, 1998b). However, as Evans (2006) notes, it is unclear how relevant information is represented and processed by the heuristic system. The experiments reported here explore whether the effect of causal strength was due to graded output from a Type 1 automatic and heuristic reasoning process or the result of a Type 2 effortful and contextualized reasoning process evaluating the probability of the outcome based on the strength of the causal link between constituent events.
Interestingly the results of both experiments suggest that causal strength leads to a graded output from the heuristic process, favouring the explanation that in manipulating causal strength we influenced the strength of the initial heuristic output. Although the strength of this output appears to vary in proportion to the perceived casual link, the secondary task does not appear to reduce people's ability to inhibit heuristic responses of varying strengths in a linear manner. When the heuristic response is relatively weak, on the majority of trials people are still able to inhibit the heuristic response. On the other hand, a strong heuristic signal coupled with a secondary task results in a failure to inhibit the heuristic response on almost two thirds of trials. The reasoning process thus appears to be determined by subtle interactions between effortless heuristic processes and effortful analytical processes. This depends both on the robustness of the graded initial heuristic output and on the amount of cognitive resources available for effortful processing.
Our results are relevant to recent claims about the relationship between reasoning processes. For example, Evans (2006) has claimed that both heuristic and analytic processes are involved in determining responses on all reasoning tasks. However, people follow a principle of cognitive economy (the satisficing principle) so that they will tend to accept a response based on heuristic processing whenever possible. The analytical process will only intervene if there is good reason to discard the initial response or to consider alternative representations one at a time (e.g., Evans, Venn, & Feeney, 2002; Mynatt, Doherty, & Dragon, 1993). It is conceivable that this satisficing bias increases in proportion to the perceived causal strength between constituent events and hence the appeal of the heuristic answer. The stronger the link, the more robust the initial output and the more effort and resources are required by the analytical system to intervene and correct the final output.
Evans (2007) has identified three separate accounts of how heuristic and analytic processes combine to determine response output on reasoning problems. Our claim that the strength of the satisficing bias might be related to causal strength accords naturally with his default-interventionist model of conflict resolution, whereby the success of the effortful interventionist process is determined by the appeal of the default heuristic response. However, our results can also be accommodated by a parallel-competitive model, in which both processes are in simultaneous operation, generating two opposing outputs and vying for control over the final response: Strong heuristic output is most likely to win any competition for control of output, whereas analytic processes burdened by a secondary task are least likely to succeed in determining the response. It is harder to see how the current results could be explained by an alternative preemptive conflict resolution process, in which a decision about whether to engage a heuristic or analytical response precedes the reasoning process. Given the linear relationship between causal strength and susceptibility to the conjunction fallacy, it may be hard to pinpoint a qualitative characteristic that would prompt either an analytical or a heuristic response from the outset.
There has also been recent debate about the point at which the analytical system intervenes in the reasoning process. Behavioural research on base-rate neglect suggests that people detect a conflict between a response based on the base rates and a response based on the individuating information, but nonetheless fail to inhibit the appealing heuristic response in favour of the correct probabilistic answer (De Neys & Glumicic, 2008). Recent brain imaging data (De Neys, Vartanian, & Goel, 2008) show that when heuristic and analytical processes cue contrasting answers, regardless of which response is given, there is activation in the anterior cingulate cortex (ACC), an area involved in conflict detection. However, when people refrain from giving the heuristic response, there is additional activation in the right lateral prefrontal cortex (RLPFC), a region involved in response inhibition (Aron, Robbins, & Poldrack, 2004). This suggests that at least for their paradigm, increased fallacious responding is a result of inhibition rather than from a failure to detect a conflict. It is not entirely clear that this analysis can be extended to the conjunction fallacy as it is likely that some participants do not have access to the appropriate reasoning principle. Nevertheless, future research might investigate whether there is a relationship between RLPFC activation and the strength of the causal link when people manage to resist the causal conjunction fallacy. This would underscore our suggestion that stronger causal links between conjuncts make it increasingly more effortful to inhibit the heuristic response.
Although the current study is the first dual process specification signifying the crucial impact that causal strength has on judgements about probability, the importance of causal strength has been observed in other related domains. For example, in the domain of causal conditional reasoning Quinn and Markovits (1998) have shown that people draw more invalid inferences for strongly associated premises than for weakly associated premises. Analogous to the current study, causal strength may have affected the appeal of the initial invalid response. This in turn could have made it more difficult to retrieve disabling counterexamples for invalid premises, a process shown to be more effortful and time consuming (Verschueren et al., 2005).
Similarly, assessing the causal strength of a variable is influenced by the complexity of the abstract, underlying causal structure and is constrained by processing capacity. Thus, causal impact assessments can be biased if processing effort is increased, and the causal relevance of a variable is not obvious (Waldmann & Hagmayer, 2001). In this view, causal strength influences the heuristic response because of the way experiential, causal structures are organized in semantic memory and how this affects the ease of construction and consequent appeal of the initial problem representation.
In the current experiments, causal information was activated fairly automatically when the events were closely related. However, this does not rule out the possibility that causal reasoning will be effortful when task requirements make the underlying causal structure most relevant. The normative structure of the current task is not conducive to examining cases in which causal knowledge must be effortfully retrieved. In everyday situations, the reliance on knowledge about causal relations can effectively reduce complexity and aids the establishment of simple heuristic reasoning strategies (Garcia-Retamero, Wallin, & Dieckmann, 2007). However, if the subsequent use of such causal representations conflicts with the normative structure of a reasoning task, people are likely to manifest biases such as the causal conjunction fallacy or base-rate neglect (Krynski & Tenenbaum, 2007). In terms of actual mental processes, it remains for future research within the dual process theory paradigm to explore how the analytical system may appraise the relevance of causal or other background information when there is no competing alternative normative response, further elucidating the influence that background knowledge has on the processing effort of various mental operations.
CONCLUSION
This research provides strong experimental evidence illustrating how causal knowledge interacts with the mental processes underlying probabilistic judgements. In particular, it confirmed our assertion derived from Tversky and Kahneman's (1983) work on causal conjunction fallacies that the causal conjunction fallacy is a function of the strength of the causal connection between constituent events. The most parsimonious dual process account of these findings is that causal strength results in graded output from the heuristic process. Our novel experimental manipulation suggests that causal background knowledge has a crucial modulating role on the initial problem representation. This subsequently interacts with more effortful processes to produce a final response.
Footnotes
Appendix A
Appendix B
Proportion of participants susceptible to the fallacy on each scenario, broken down by strength, in Experiments 1 and 2
| Experiment |
|||||
|---|---|---|---|---|---|
| 1 |
2 |
||||
| Strong | Weak | Unrelated | Strong | Weak | |
| World Wide Web usage | .41 | .12 | .24 | .35 | .15 |
| Housing market | .65 | .24 | .24 | .75 | .40 |
| Adolescent smoking | .47 | .47 | .24 | .55 | .35 |
| Pepsi sales | .59 | .35 | .35 | .55 | .45 |
| Tsunami hits Far East | .59 | .24 | .12 | .15 | .25 |
| Smoking decrease | .53 | .41 | .24 | .70 | .45 |
| Psychology applications | .47 | .29 | .41 | .35 | .20 |
| Popularity of collie breed | .71 | .59 | .24 | .70 | .30 |
| Durham ranked in Top 20 | .29 | .35 | .12 | — | — |
| Overall | .52 | .34 | .24 | .51 | .32 |