independence and dependence in human causal reasoning … 14.pdf · · 2014-04-11independence and...

Cognitive Psychology 72 (2014) 54–107

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Cognitive Psychology

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Independence and dependence in human causalreasoning

http://dx.doi.org/10.1016/j.cogpsych.2014.02.0020010-0285/� 2014 Elsevier Inc. All rights reserved.

⇑ Address: Dept. of Psychology, 6 Washington Place, New York, NY 10003, United States.E-mail address: [email protected]

Bob Rehder ⇑Department of Psychology, New York University, New York, NY 10003, United States

a r t i c l e i n f o a b s t r a c t

Article history:Accepted 11 February 2014

Keywords:Causal reasoningCausal inferenceCausal Markov conditionConditional independenceScreening off

Causal graphical models (CGMs) are a popular formalism used tomodel human causal reasoning and learning. The key property ofCGMs is the causal Markov condition, which stipulates patterns ofindependence and dependence among causally related variables.Five experiments found that while adult’s causal inferences exhib-ited aspects of veridical causal reasoning, they also exhibited asmall but tenacious tendency to violate the Markov condition. Theyalso failed to exhibit robust discounting in which the presence ofone cause as an explanation of an effect makes the presence ofanother less likely. Instead, subjects often reasoned ‘‘associatively,’’that is, assumed that the presence of one variable implied thepresence of other, causally related variables, even those that were(according to the Markov condition) conditionally independent.This tendency was unaffected by manipulations (e.g., responsedeadlines) known to influence fast and intuitive reasoningprocesses, suggesting that an associative response to a causal rea-soning question is sometimes the product of careful and deliberatethinking. That about 60% of the erroneous associative inferenceswere made by about a quarter of the subjects suggests the pres-ence of substantial individual differences in this tendency. Therewas also evidence that inferences were influenced by subjects’assumptions about factors that disable causal relations and theiruse of a conjunctive reasoning strategy. Theories that strive toprovide high fidelity accounts of human causal reasoning will needto relax the independence constraints imposed by CGMs.

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B. Rehder / Cognitive Psychology 72 (2014) 54–107 55

0. Introduction

People possess numerous beliefs about the causal structure of the world. They believe that sunrisesmake roosters crow, that smoking causes lung cancer, and that alcohol consumption leads to trafficaccidents. The value of such knowledge lies in allowing one to infer more about a situation that whatcan be directly observed. For example, one generates explanations by reasoning backward to ascertainthe causes of the event at hand. One also reasons forward to predict what might happen in the future.On the basis of, say, a friend’s inebriated state, we predict dire consequences if he were to drive and sohide his car keys.

A large number of studies have investigated how humans make causal inferences. One simple ques-tion is: When two variables, X and Y, are causally related, do people infer one from the other? Unsur-prisingly, research has confirmed that they do, as X is deemed more likely in the presence of Y and viceversa (Fernbach, Darlow, & Sloman, 2010; Meder, Hagmayer, & Waldmann, 2008, 2009; Rehder &Burnett, 2005; see Rottman & Hastie, 2013, for a review). But causal inferences quickly become morecomplicated if just one additional variable is introduced. For example, suppose that X and Y are relatedto one another not directly but rather through a third variable Z. Under these conditions, the questionof how one should draw an inference between X and Y will depend on the direction of the causal rela-tions that link them via Z. Three possibilities are shown in Fig. 1. First, X and Y might both be effects ofZ (Fig. 1A), a topology referred to as a common cause network. For example, a doctor might diagnose adisease (Z) on the basis of a particular symptom (X), and then also predict that the patient will soonexhibit another symptom of that disease (Y). Second, the variables might form a causal chain in whichX causes Z which causes Y (Fig. 1B). For example, politicians may (X) calculate that pandering toextremists will lead to their support (Z), which in turn will galvanize members of the opposing party(Y). Finally, Z might be caused by X or Y, forming a common effect network (Fig. 1C). A police detectivemight release an individual (Y) suspected of murder (Z) upon discovering the murder weapon inpossession of another suspect (X).

A formalism that specifies the permissible forms of causal inferences and that is generally acceptedas normative is known as causal graphical models, hereafter CGM (Glymour, 1998; Jordan, 1999; Koller& Friedman, 2009; Pearl, 1988, 2000; Spirtes et al., 2000). CGMs are types of Bayesian networks (or di-rected acyclic graphs) in which variables are represented as nodes and directed edges between thosevariables are interpreted as causal relations. Note that a CGM need not be complete in the sense thatvariables may have exogenous influences (i.e., hidden causes) that are not part of the model; however,these influences are constrained to be uncorrelated. This property, referred to as causal sufficiency(Spirtes, Glymour, and Scheines, 1993, 2000), in turn has important implications for the sorts of

(A)

(B)

Z YX

(C)

Z YX

Z YX

Fig. 1. Three causal networks that can be formed from three variables. (A) A common cause network. (B) A chain network. (C) Acommon effect network.

56 B. Rehder / Cognitive Psychology 72 (2014) 54–107

inferences that are allowable. Specifically, CGMs stipulate the causal Markov condition, that specifiesthe conditions under which variables are conditionally independent of one another (Hausman &Woodward, 1999; Pearl, 1988, 2000; Spirtes et al., 2000).

This research tests whether the causal inferences people make follow the prediction of CGMs, par-ticularly whether they honor the constraints imposed by the Markov condition. This question isimportant because Bayes nets have become popular for modeling cognitive processes in numerous do-mains. For example, CGMs have been used as psychological models of not only various forms of causalreasoning (Holyoak, Lee, & Lu, 2010; Kemp, Shafto, & Tenenbaum, 2012; Kemp & Tenenbaum, 2009;Lee & Holyoak, 2008; Oppenheimer, 2004; Rehder, 2009; Rehder & Burnett, 2005; Shafto, Kemp, Bon-awitz, Coley, & Tenebaum, 2008), but also causal learning (Cheng, 1997; Gopnik, Glymour, Sobel,Schultz, & Kushnir, 2004; Griffiths & Tenebaum, 2005, 2009; Lu, Yuille, Liljeholm, Cheng, & Holyoak,2008; Sobel, Tenenbaum, & Gopnik, 2004; Waldmann, Holyoak, & Fratianne, 1995), interventions (Slo-man & Lagnado, 2005; Waldmann & Hagmayer, 2005), decision making (Hagmayer & Sloman, 2009),and classification (Rehder, 2003; Rehder & Kim, 2009, 2010). Graphical models have also been used asmodels of non-causal structured knowledge, such as taxonomic hierarchies (Kemp & Tenenbaum,2009). However, in all these domains the inferential procedures than accompany Bayes nets and thatare taken as candidate models of psychological processes rely on the Markov condition for their jus-tification. Said differently: the Markov condition is at the heart of Bayes nets. Without it, any claimthat knowledge is represented as a Bayes nets amount to no more than the claim that it consists ofnodes connected with arrows. Thus, a demonstration that humans sharply violate the Markov condi-tion would have implications for the role that Bayes nets currently occupy in cognitive modeling.

This article has the following structure. I first describe how the Markov condition constrains causalinferences. I then review previous research that bears on the psychological question of whether hu-mans violate that condition. Five new experiments testing the Markov condition are then presented.To foreshadow the results, subjects’ causal inferences and accompanying model-based analyses willshow that human reasoners systematically violate this principle.

1. Implications of the causal Markov condition

For tractability, this articles limits itself to restricted instances of the common cause, chain, andcommon effect networks. First, whereas nothing prevents CGMs from including continuous and ordi-nal variables, this work only considers binary variables that are either present or absent. Second,whereas CGMs can include inhibitory causal relations (a cause tends to prevent an effect) and relationsthat involve multiple variables, here I treat only simple facilitory (or generative) relations betweenpairs of variables. Third, those causal relations have a single sense: The presence of the cause facilitatesthe presence of the effect but the absence of the cause exerts no influence. Fourth, for the commoneffect network I will assume that X and Y are independent causes of Z. Under these assumptions, Idemonstrate how CGMs constrain inferences for the three networks in Fig. 1.

1.1. Common cause networks

The Markov condition specifies the pattern of conditional independence that arises given knowl-edge of the state of a subset of variables in a network. Specifically, when that subset includes a vari-able’s direct parents, that variable is conditionally independent of each of its non-descendants.(Hausman & Woodward, 1999). This condition has a natural causal interpretation: Apart from itsdescendants, one has learned as much as possible about a variable once one knows the state of allof its direct causes. Because non-descendants only provide information about the variable throughthe parents, the variable is said to be screened off from those non-descendants by the parents.

Fig. 2 illustrate this principle with the common cause network in Fig. 1A by presenting the eightdistinct situations in which one may infer the state of Y as a function of the states of X and Z. InFig. 2 a ‘‘1’’ means a binary variable is ‘‘present,’’ ‘‘0’’ means that it’s absent, and ‘‘x’’ means that itsstate is unknown. Y is always unknown and is the variable being inferred (‘‘?’’). The state of Y’s parentcause Z is known to be present in situations A, B, and C, known to be absent in F, G, and H, and its stateis unknown in D and E. Situations also vary according to whether X is present, absent, or unknown.

Z=0 Y?X=0

(H)Z=0 Y?X=x

(G)

Z=x Y?X=0

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Z=x Y?X=1

(D)

(I)

(II)

(III)

(IV) Z=0 Y?X=1

(F)

Z=1 Y?X=x

(B)Z=1 Y?X=1

(A)Z=1 Y?X=0

(C)

Fig. 2. Equivalence classes for common cause inference situations. 1 = causally-related value for a variable; 0 = causallyunrelated value; x = unknown value. In every panel Y is unknown and is the variable being predicted. Classes separated by asingle dashed line (I and II) are distinct only if the causal relations are not deterministically necessary. Classes separated by adouble dashed line (III and IV) are distinct only if the causal relations are not deterministically sufficient.


Because the labels ‘‘X’’ and ‘‘Y’’ are interchangeable in the common cause network, the situations inFig. 2 include those in which one infers X rather than Y.

Fig. 2 is arranged into equivalence classes I, II, III, and IV in which situations in the same class pro-vide the same inferential support for Y. Classes I and IV illustrate the Markov condition. In class I, thestate of Y’s immediate parent Z is known (it is present) and so knowledge about the state of Y’s non-descendants (namely, X) provides no additional information about Y. Because Z thus screens off Y fromX, situation types A, B, and C provide equivalent support for Y. Because the known (absent) value of Zscreens Y off from X in situation types F, G, and H, they also provide equal support for Y.

Assuming generative causes, CGMs also predict that inferences in favor of the causally related valueof Y generally become weaker as one moves from class I to IV. Problems in class I in which Y’s imme-diate cause Z is present generally provide stronger support for Y than that provided by the single prob-lem in class II (D), in which the state of Z is unknown but X is present. However, this distinctiondepends on the strength of the causal relations. For example, when the link between X and Z is deter-ministically necessary (an effect is always accompanied by its cause because it has no other potentialcauses), then the presence of Z is certain in problem type D and thus the probability of Y is the same asin problem types A, B, and C. This possible collapse of classes I and II into a single class due to deter-ministic necessity is represented in Fig. 2 with a dashed line.

The single situation in class III (E), provides weaker support than D because the causally related va-lue of X is absent (suggesting that Z is absent, and thus so too is Y). Finally, the class of problems inwhich Z is known with certainty to be absent (F, G, and H) provides the weakest support for Y ofall. However, when the causal link between X and Z is deterministically sufficient (a cause is alwaysaccompanied by its effect), then the absence of Z in situation D can be inferred with certainty, and thus


the absence of Y is as certain as in problem types F, G, and H. This possible collapse of classes III and IVdue to deterministic sufficiency is represented in Fig. 2 with a double dashed line.

In the ensuing experiments, subjects are presented with pairs of the situations shown in Fig. 2 andasked to choose the one in which variable Y (or X) is more likely to be present. The situations con-trasted will be those required to assess whether reasoners honor the Markov condition. For the com-mon cause network, those pairs are A vs. B, B vs. C, F vs. G, and G vs. H. For example, because Y shouldbe equally likely in situations A and B, subjects should be no more likely to choose one situation overthe other.

1.2. Chain networks

The normative pattern of inferences when X, Y, and Z form a causal chain are presented in Fig. 3,which presents the different situations in which one can predict Y as a function of X and Z. The anal-ysis of the chain network is similar to that of the common cause network. Situations A, B, and C forman equivalence class because the known value of Z screens off Y from the non-descendant X (so thatinformation about X is irrelevant to predicting Y). Next, situation D should support weaker inferencesto Y than types A, B, or C, because the presence of X in D suggests but does not guarantee the presenceof Z (unless the X ? Z link is deterministically sufficient, as discussed above). Situation E is weakerstill, because the absence of X suggests the absence of Z and thus the absence of Y. But, unless theX ? Z link is deterministically necessary (i.e., there are no alternative causes of Z), E will be strongerthan situations F, G, and H in which the absence of Z is known with certainty. Finally, types F, G, and Hform an equivalence class because the value of Z screens off X from Y.

Z=0 Y?X=x

(G)

Z=1 Y?X=x

(B)

Z=x Y?X=0

(E)

Z=x Y?X=1

(D)

(I)

(II)

(III)

(IV) Z=0 Y?X=0

(H)

Z=1 Y?X=1

(A)

Z=0 Y?X=1

(F)

Z=1 Y?X=0

(C)

Fig. 3. Equivalence classes for chain inference situations. 1 = causally-related value for a variable; 0 = causally unrelated value;x = unknown value. In every panel Y is unknown and is the variable being predicted. Classes separated by a single dashed line(I and II) are distinct only if the causal relations are not deterministically necessary. Classes separated by a double dashed line(III and IV) are distinct only if the causal relations are not deterministically sufficient.


Whereas for a common cause network inferences to either X or Y are qualitatively equivalent, thisis not the case in a chain network, because X is the initial cause and Y is the terminal effect. Neverthe-less, an analysis in which X rather than Y is the to-be-predicted variable yields the same result (prob-lem A, B, and C form one equivalence class and F, G, and H another). Although differences betweenpredicting the initial cause (X) as compared to the final effect (Y) are not uninteresting, I will generallycollapse over this distinction in what follows.

1.3. Common effect networks

The common effect networks in Fig. 1C illustrates a second sort of constraint stipulated by CGMs.Whereas in common cause and chain networks knowledge of Z renders X and Y independent, it hasthe opposite effect in common effect networks: X and Y are independent in the absence of knowledgeof Z but become dependent when the state of Z is known. The nature of that dependency depends onhow Z is functionally related to it causes. Although in general any functional form is possible (e.g., Xand Y may be conjunctive causes of Z such that X and Y must both be present to produce Z, Y mightdisable the causal relation that links X and Z, etc.) as mentioned I focus on cases in which X and Yare independent, generative causes of Z. Under this assumption, Fig. 4 presents the equivalence classes

Z=1 Y?X=0

(C)

Z=x Y?X=1

(D)

Z=0 Y?X=0

(H)Z=0 Y?X=x

(G)Z=0 Y?X=1

(F)

Z=1 Y?X=x

(B)

Z=1 Y?X=1(A)

Z=x Y?X=0

(E)

(I)

(II)

(III)

(IV)

(V)

Fig. 4. Equivalence classes for common effect inference situations. 1 = causally-related value for a variable; 0 = causallyunrelated value; x = unknown value. In every panel Y is unknown and is the variable being predicted. Classes separated by adouble dashed line (III and IV) are distinct only if the causal relations are not deterministically sufficient. Note that theseequivalence classes hold for the case in which X and Y are independent causes of Z.


for a common effect network. Of course, the presence of the common effect Z in situations A, B, and Cresults in them providing stronger evidence in favor of the presence of a cause than the other types.But, among these three problems, the probability that a cause is present when the other cause isknown to be absent (situation C) is larger when than when its state is unknown (B) which in turnis larger than when its known to be present (A), a phenomenon referred to as discounting or explainingaway.

As mentioned, when the state of Z is unknown, X and Y are conditionally independent. For example,problem types E and D each provide equally strong inferences to Y because X, as an independent cause,provides no information about Y (and thus one’s predictions regarding Y should correspond to its baserate, i.e., the probability with which one predicts Y in the absence of any information about X or Z).Finally, problem types F, G, and H also form an equivalence class. This is the case because of the singlesense interpretation of the causal relations, that is, the presence of X (or Y) causes the presence of Zbut the absence of X (or Y) does not cause the absence of Z.

Again, differences between some equivalence classes depend on the parameterization of the causalrelations. When those relations are deterministically sufficient, class III collapses into IV. This is thecase because the presence of variable X in problem type A completely accounts for the presence ofZ. Thus, the probability of Y in A should correspond to its base rate, as in problem types E and D.Whether subjects adhere to these predictions will be assessed by asking them to judge the probabilityof Y in the following situation pairs: A vs. B, B vs. C, D vs. E, F vs. G, and G vs. H. They should favor B andC in the first two (reflecting discounting) and be at chance otherwise.

2. Apparent Violations of the Markov condition in Psychological Research

Given the prominent use of CGMs in models of cognition, it is unsurprising that a number of inves-tigators have asked whether adult human reasoners in fact honor the constraints imposed by the Mar-kov condition. I now review three recent studies that bear on this question.

2.1. Walsh and Sloman (2008)

Walsh and Sloman (2008, Experiment 1; also see Park & Sloman, 2013; Walsh & Sloman, 2004)asked subjects to reason about a number of real-world vignettes that involved three variables relatedby causal knowledge into a common cause network. For example, subjects were told that worryingcauses difficulty concentrating and that worrying also causes insomnia. They were then asked twoinference questions. First, they were asked whether an individual had difficulty concentrating giventhat he or she was worried (this corresponds to situation type B in Fig. 2). Next, they were askedwhether a different individual had difficulty concentrating given that he or she was worried but didnot have insomnia (situation type C). Because the state of the common cause Z (worrying) is givenin both questions, Y (difficulty concentrating) is screened off from the additional information providedabout X (insomnia) in the second question. In fact, however, probability ratings were much higher forthe first question than the second one.

Although this result provides prima facie evidence against the Markov condition, results from afollow-up experiment suggested that subjects were reasoning with knowledge in addition to thatemphasized by the experimenters. Specifically, the absence of one of the effects in the second questionled subjects to assume the presence of a shared disabler that not only explained why the effect X failedto occur but also led them to expect that it would prevent the presence of the other effect Y. For exam-ple, some subjects assumed that the absence of insomnia was due to the individual performing relax-ation exercises, which in turn would also help prevent difficulty concentrating.

This finding is important because inferences that violate the Markov condition for one CGM may nolonger do so if that CGM is elaborated to include hidden variables (i.e., variables that were not pro-vided as part of the cover story and not explicitly mentioned as part of the inference question). Theleft panel of Fig. 5A presents a common cause model elaborated to include the sort of hidden disabler(W) assumed by many of Walsh and Sloman’s subjects. In the panel, arcs between two causal linksrepresent interactive causes such that the causal influence of Z on X and Y depends on W; in particular,

(B)

YX Z

W

(C)

YX

W

Z

(A)

YX

W

Z YX

W

Z

YX

W

ZYX

W

ZYX

W

Z

YX Z

M1 M2

W

YX Z

W

Fig. 5. The causal networks in Fig. 1 elaborated to include hidden causal influences. (A) Common cause, chain, and commoneffect networks in which the causal relationships have a shared disabler, represented by W. The arcs represent interactive causalinfluences, in which the influence of one causal factor depends on the state of the other. For example, for the common causenetwork in the left panel, when disabler W is present it prevents, with some probability, the operation of the causal mechanismbetween Z and its causes X and Y. (B) Common cause, chain, and common effect networks elaborated with a shared mediator.(C) Common cause, chain, and common effect networks in which X, Y, and Z share a cause W. Specifically, W is a generativecause that generates the causally related senses of X, Y, and Z.


that influence is absent when W is present. Because in this network the state of one of Y’s direct par-ents (W) is not known, Y is no longer screened off from X by Z; that is, because X (insomnia) providesinformation about W (relaxation exercises), it thus also provides information about Y (difficulty con-centrating) even when the state of Z (worrying) is known. For this causal network, the Walsh and Slo-man findings no longer constitute violations of the Markov condition.1

More recent work (Park & Sloman, 2013) suggests that reasoners may also assume the presence of ashared disabler with chain networks, where the presence of W now disables the X?Y and Y?Z causallinks (middle panel of Fig. 5A). Later, I will present a fuller analysis of how causal inferences are influ-enced by the possible presence of a shared disabler for all three types of networks, including commoneffect networks (right panel of Fig. 5A). But for now, these findings illustrate how situations that mayappear to be counterexamples to the Markov condition may turn out not to be when the causal rela-tions are represented with greater fidelity. Of course, the study of Walsh and Sloman has revealedsome interesting and important facts about causal reasoning. That people will respond to a cause-present/effect-absent situation with an ad hoc elaborations of their causal model to include additional

1 Said differently, representing the subjects’ causal knowledge as a common cause network omitting a shared disabler violatesthe causal sufficiency constraint described earlier: Because W is a causal influence that is common to both X and Y, omitting itmeans that exogenous influences are not uncorrelated. This in turn invalidates the expectations of independence stipulated by theMarkov condition.


causal factors is a significant finding; so too is that they then use this elaborated model to reasonabout new individuals. But what this study does not do is provide decisive evidence against theMarkov condition.

2.2. Mayrhofer, Hagmayer, and Waldmann (2010)

In another test of the Markov condition, Mayrhofer et al. (2010, Experiment 1; also see Mayrhofer &Waldmann, 2013) instructed subjects on scenarios involving mind reading aliens. In all conditions, thethoughts of one alien (Gonz) could be transmitted to three others (Murks, Brxxx, and Zoohng) but thecover story provided to subjects was varied. In the sending condition, they were told that Gonz couldtransmit its thoughts into the heads of the other aliens. In the reading condition, the other aliens couldread the thoughts of Gonz. Mayrhofer et al. construed both scenarios as involving a common causenetwork (with Gonz as the common cause) and thus tested the Markov condition by asking subjectsto predict the thoughts of one of the ‘‘effect’’ aliens (e.g., Murks) given the thoughts of Gonz and theremaining effects (Brxxx and Zoohng). They found that the effects were not independent: Subjectspredicted that Murks was more likely to have the same thought as Gonz if Brxxx and Zoohng did also.Importantly, this effect was much stronger in the sending condition as compared to the receivingcondition.

Rather than interpreting this as a violation of the Markov condition however, Mayrhofer et al. notedthat subjects’ were unlikely to have thought of the situation as involving a simple common cause mod-el. In the sending condition, it is natural to assume that Gonz’s ability to send thoughts relied on acommon sending mechanism. This situation corresponds to the causal model in the left panel ofFig. 5B in which Gonz’s sending mechanism is represented by W. On this account, if, say, Brxxx doesnot share Gonz’s thought, a likely reason is the malfunctioning of Gonz’s sending mechanism, in whichcase Murks is also unlikely to share Gonz’s thought. That is, in the left panel of Fig. 5B, an effect (e.g. Y)is not screened off from another effect (X) by Z, because X provides information about W and thus Y.The much smaller violations of the Markov condition in the receiving condition may have been due tosubjects’ belief that the process of reading mostly depended on some property of the reader itself(thus, the fact that Brxxx had trouble reading Gonz’s thought provides no reason to think that Murkswould too).2

Responses to supposed counterexamples to the Markov condition in the philosophical literaturehave also appealed to shared mediators. A situation presented by Cartwright (1993) involves two fac-tories that both produce a chemical used to treat sewage but that operate on different days. Whereasthe process used by the first factory produces the chemical 100% of the time, the one used by the sec-ond sometimes fails to produce the chemical at all, yielding a terrible pollutant instead. Cartwrightrepresents this situations as a common cause Z (which factory produced the chemical) producingtwo effects, X (the sewage-treating chemical) and Y (the pollutant), and observes that X and Y arenot independent given Z (e.g., even if one knows that the second factory was is in operation today,the presence of the pollutant implies the absence of the useful chemical). In response, Hausmanand Woodward (1999) noted that the situation is more accurately represented by the network shownin Fig. 5B in which the causal influence of factory (Z) is mediated by process (W) that in turn deter-mines the probabilities of the chemical and the pollutant (X and Y). On this analysis, X and Y are onlyindependent conditioned on W, and thus Cartwright’s scenario fails to serve as a counterexample tothe Markov condition (also see Salmon, 1984, and Sober, 2007, for similar problems with similarsolutions).

These examples again illustrate how failing to include relevant causal factors can invalidate thepatterns of conditional independence that would otherwise be stipulated by the Markov condition.Of course, the results from Mayrhofer et al. are important insofar as they reveal how subjects’ constru-al of agency in a situation (which actor initiates an event) can influence their causal model and thus

2 Mayrhofer et al. themselves followed Walsh and Sloman by modeling these results as involving a shared disabler, one that wasstronger in the sending versus receiving condition. The results of their Experiment 2, which tested a chain structure, are discussedlater.


the inferences they draw. But those findings fail to shed light on whether such inferences in fact honorthe Markov condition.

2.3. Rehder and Burnett (2005)

Rehder and Burnett tested the Markov condition by instructing subjects on categories with fea-tures that were linked by causal relations. These categories were artificial in that they denotedentities that do not really exist. For example, subjects who learned Lake Victoria Shrimp were toldthat such shrimp have a number of typical or characteristic features (e.g., a slow flight response, anaccelerated sleep cycle, etc.). Subjects were then presented with individual category members withmissing features (i.e., stimulus dimensions whose values were unknown) and asked to predict one ofthose features. These experiments went beyond those of Walsh and Sloman (2008) and Mayrhoferet al. (2010) by testing all three of the causal networks shown in Fig. 1 (albeit with four variablesrather than three). They also tested a wider variety of materials. Subjects learned not only biologicalkinds like Lake Victoria Shrimp but also nonliving natural kinds, artifacts, and ‘‘blank’’ materials(in which the categories were of ‘‘some sort of object’’ and the features were the letters ‘‘A,’’ ‘‘B,’’etc.).

Rehder and Burnett found that subjects appeared to violate the Markov condition in their causalinferences. These violations occurred for all three causal network topologies and all types of materials.The pattern was the same in all conditions: Predictions were stronger to the extent that the item hadmore typical category features, even when those additional features were (according to the Markovcondition) conditionally independent of the to-be-predicted feature.

Nevertheless, just as in the previous studies, Rehder and Burnett accounted for their results byappealing to subjects’ use of additional knowledge. They proposed that reasoners assume that catego-ries possess underlying properties or mechanisms that produce or generate a category’s observableproperties, a situation represented in Fig. 5C in which W serves as the shared generative cause. Be-cause one can reason from X to Y (or vice versa) via W, X and Y are conditionally dependent even givenZ. The common cause W also explains the inferences Rehder and Burnett found in the a-causal controlcondition (not shown in Fig. 5C): Although not directly causally related to one another, features arenonindependent because they are all indirectly related via W.

One might ask where these beliefs about categories’ underlying mechanisms come from. They didnot originate from experience with the categories themselves in Rehder and Burnett’s experiments be-cause artificial categories like Lake Victoria Shrimp do not exist. It was also unlikely to have originatedfrom more general knowledge associated with biological kinds (e.g., essential properties that generate,or cause, perceptual features, Gelman, 2003; Medin & Ortony, 1989), because the results also obtainedwith nonbiological kinds and artifacts (and with blank materials in which the ontological domain wasunspecified). Accordingly, Rehder and Burnett concluded that people possess a domain generalassumption that categories’ typical features are brought about by hidden causal mechanisms, thatis, even without knowing what those mechanisms might be. For present purposes, the important pointis that the Markov condition was again rescued by assuming that subjects reasoned with knowledgebeyond that provided by the experimenters.

In summary, the preceding review reveals that apparent violations of the Markov condition can beexplained away by appealing to additional knowledge structures brought to bear on the causal infer-ence. Of course, that prior knowledge can influence reasoning is hardly surprising given the long his-tory of research showing how beliefs affect performance on supposedly formal (content free)reasoning problems. The belief bias effect refers to reasoners’ tendency to more readily accept the con-clusion of a syllogistic reasoning problem as valid if it is believed to be true (Evans, Barston, & Pollard,1983; and see Evans, Handley, & Bacon, 2009, for an analogous effect with conditional reasoning). Clo-ser to home, suppression effects arise when conditional statements (if p then q) are interpreted causallyand the reasoner can easily retrieve counterexamples to the rule that imply the presence of alternativecauses or disabling conditions (Byrne, 1989; Byrne, Espino, & Santamaria, 1999; Cummins, 1995;Cummins, Lubart, Alksnis, & Rist, 1991; De Neys, Schaeken, & d’Ydewalle, 2003a, 2003b; Evans,Handley, & Bacon, 2009; Frosch & Johnson-Laird, 2011; Goldvarg & Johnson-Laird, 2001; Markovits& Quinn, 2002; Quinn & Markovits, 1998, 2002; Verschueren, Schaeken, & d’Ydewalle, 2005). Just as

Table 1Variables in the domains of economics, meteorology, and sociology.

Variable Value 1 Value 2

EconomicsInterest rates Low HighTrade deficits Small LargeRetirement savings High Low

MeteorologyOzone level High LowAir pressure Low HighHumidity High Low

SociologyDegree of urbanization High LowInterest in religion Low HighSocio-economic mobility High Low

Table 2Example of causal relationships in the domain of economics that form a common cause network.

Causally relationship Causal mechanism

Low interest rates ? Smalltrade deficits

Low interest rates cause small trade deficits. The low cost of borrowing money leadsbusinesses to invest in the latest manufacturing technologies, and the resulting low-cost products are exported around the world

Low interest rates ? Highretirement savings

Low interest rates cause high retirement savings. Low interest rates stimulate economicgrowth, leading to greater prosperity overall, and allowing more money to be saved forretirement in particular


in these previous lines of research, reasoners’ prior beliefs complicate the assessment of whetherpeople honor the rules of formal reasoning, in this case the Markov condition.

3. Overview of experiments

The following experiments taught university undergraduates three binary variables and two cau-sal relations in the domains of economics, meteorology, or sociology. For example, the economicvariables were interest rates (which they were told could be low or high), trade deficits (small or large),and retirement savings (low or high). The binary variables in each of the three domains are shown inTable 1. Subjects were provided with no information about the base rates of variables (e.g., subjectsin the domain of economics were only told that ‘‘some’’ economies have low interest rates and that‘‘some’’ have high interest rates). The causal relations specified how the sense of one variable causedanother (e.g., low interest rates causes small trade deficits). Which senses of the variables were de-scribed as causally related was randomized over participants (e.g., some participants were told thatlow interest rates cause small trade deficits, others that low interest rates cause large trade deficits,still others that high interest rates cause small trade deficits, etc.). The causal relationships formedeither a common cause, chain, or common effect causal network and were accompanied by descrip-tions of the mechanisms by which one variable produces another. See Table 2 for examples of thecausal mechanisms in the domain of economics. Note that the descriptions of the causal mechanismsmade it clear that they are unrelated (e.g., the two causal mechanisms in Table 2 indicate that theprocesses by which interest rates affect trade deficits and retirement savings are independent).These descriptions thus work against not only the shared mediator interpretation of common causenetworks (Cartwright, 1993; Hausman & Woodward, 1999; Mayrhofer et al., 2010; Salmon, 1984;Sober, 2007), but also the analogous interpretations of chain and common effect networks(Fig. 5B). As another safeguard, later experiments will explicitly instruct participants that eachmechanism operates independently.


Subjects were then presented with pairs of concrete situations (e.g., two particular economies)with an unknown variable and asked to judge, on the basis of the states of other variables in the sit-uations, in which one that variable was more likely to be present.

On the face of it, these materials appear to minimize several of the issues that have made previoustests of the Markov condition inconclusive. Although university students are unlikely to have exten-sive prior knowledge about these domains, (reducing the probability that they will elaborate theircausal models with sorts of structures shown in Fig. 5), any such knowledge that exists will tend tobe eliminated by averaging over the three domains and the counterbalancing conditions that variedwhich variable senses were described as causally related.3 As an additional safeguard, an experimentwill further minimize the use of domain knowledge by testing of blank materials, that is, the variablesare given the generic labels ‘‘A,’’ ‘‘B,’’ and ‘‘C’’. Finally, that the materials are not categories and so provideno basis for assuming that only certain dimension values (the ‘‘typical’’ ones) are causally related meansthere is no reason to think that the variables are related by a shared generative cause (as assumed byRehder & Burnett, 2005).

But although these materials provide a first line of defense against the use of prior knowledge, it isstill possible to conceive of ways that subjects might elaborate their causal model. The processes in-volved in comprehending the causal relations are likely to trigger a search of memory for relatedknowledge and this search may be biased so as to turn up only knowledge relating the variable sensesinvolved in the experimental causal relations (Chapman & Johnson, 1999; Heit & Bott, 2000; Mussweiler& Strack, 1999; Quinn & Markovits, 1998). For instance, if told that low interest rates causes smalltrade deficits, I may more readily retrieve facts involving low interest rates and small trade deficitsthan ones involving high interest rates and large trade deficits, perhaps yielding the structure inFig. 5C. These elaborations could produce apparent violations of independence despite the randomi-zation of the materials because different knowledge structures would get retrieved in the differentrandomized conditions. The search of memory might also turn up commonalities between the causalrelations, yielding the mediated structures in Fig. 5B. Once the test phase of the experiment begins,subjects may elaborate their models in response to the scenarios they reason about, just as Walshand Sloman’s (2008) subjects apparently did for cause-present/effect-absent situations (Fig. 5A) (anal-ogously, reasoners might postulate hidden causes to explain cause-absent/effect-present situations,Hagmayer & Waldmann, 2007; Luhmann & Ahn, 2007, 2011; Rottman et al., 2011; Saxe, Tenebaum,& Carey, 2005). Finally, a skeptic might argue that such concerns are not fully ruled out even by blankmaterials, because subjects might reason by analogy to familiar domains or assume the presence ofknowledge structures that are abstract (i.e., lack any concrete representation of the causal processesinvolved).

Accordingly, later I will present a theoretical analysis of each of the alternative models in Fig. 5 toassess their potential as accounts of the causal inferences made in the following experiments. As men-tioned, not only will those inferences exhibit numerous violations of the reasoning norms stipulatedby CGMs, all but one of the alternative structures in Fig. 5 will be unable to account for subjects’ aggre-gate responses for all three causal networks and the model that remains will be unable to account forthe responses of large numbers of individuals.

4. Experiment 1

Each participant was taught the three causal networks in Fig. 1, one each in the domains of eco-nomics, meteorology, and sociology. A forced-choice procedure was used in which participants were

3 Consider, for example, subjects who are taught the common cause knowledge in Table 2, which can be representedschematically as X1 Z1 ! Y1, where superscripts denote the value on a dimension (1 or 2). If, in addition to the Table 2 links, thesubject population tends to believe that one of the effects causes the other (say, that small trade deficits causes high retirementsavings, i.e., X1 ! Y1), then they will appear to violate independence. For example, the effect Y (whose role is played by Y1) will bejudged as more probable in situation A in Fig. 2 (in which its cause X1 is present) than in situation B (in which the state of X1 isunknown). The X1 ! Y1 link will have the opposite effect in other conditions however. For subjects who are instead taughtX1 Z1 ! Y2, the presence of X1 in situation A will make Y (now played by Y2) less likely as compared to situation B (becauseX1 ! Y1). In this manner, aggregating the results over the different randomized conditions will tend to average out the effects ofprior knowledge.


presented with a pair of situations and asked to choose which was more likely to possess a particularvariable value. They could also select a third ‘‘equally likely’’ response indicating that neither wasmore likely than the other to have that value. The choice problems were those needed to assess thekey predictions of conditional independence and dependence made by the three causal networks: Avs. B, B vs. C, D vs. E, F vs. G, and G vs. H. In the common cause and chain conditions, subjects shouldchoose D over E but choose the equally likely alternative otherwise (Figs. 2 and 3). In the common ef-fect condition they should prefer B over A and C over B but choose equally likely otherwise (Fig. 4).These predictions are summarized in the left hand side of Fig. 6.

4.1. Method

4.1.1. MaterialsThe three binary variables in the domains of economics, meteorology, and sociology are shown in

Table 1. In each domain participants were taught two causal relationships forming either a commoncause, chain, or common effect network. Each causal link was described as the sense of one variable(e.g., low interest rates) causing another (e.g., small trade deficits), and was accompanied with a shortdescription of the mechanism responsible for the causal relationship (Table 2). The senses of the var-iable that were described as causally related was randomized for each participant. The complete list ofcausal relationships used to construct common cause, chain, and common effect networks in each do-main are presented in Appendix A.

4.1.2. DesignChoice problem (A vs. B, B vs. C, D vs. E, F vs. G, and G vs. H.) and causal network were manipulated

as within-subject variables. In addition, there were two between-subject counterbalancing factors.First, the order in which the three causal networks were presented was either ceh, hce, or ehc (c = com-mon cause, h = chain, e = common effect). Second, the order in which the three domains were pre-sented was either mes, sme, or esm (m = meteorology, e = economics, s = sociology). As a result, eachcausal network was instantiated in each of the three domains, and was learned as the first, second,or third network, an equal number of times.

4.1.3. ParticipantsSixty-three New York University undergraduates received course credit for participating in this

experiment. They were assigned in equal numbers to the two between-subject counterbalancingconditions.

4.1.4. ProcedureFor each domain, participants first studied several computer screens of information about the do-

main and then performed the inference test. The initial screens presented a cover story and a descrip-tion of the domain’s three variables and their two values. Subsequent screens presented the twocausal relationships and their associated causal mechanisms. Participants also observed a diagramdepicting the topology of the causal links (common cause, chain, or common effect). When ready,participants took a multiple-choice test that tested them on this knowledge. While taking the test,participants were free to return to the information screens they had studied; however, doing soobligated them to retake the test. The only way to pass the test and proceed to subsequentphases was to complete it without error and without returning to the initial information screensfor help.

The feature inference phase presented participants with the five types of choice problems. The twoexamples were presented one above the other and participants were asked which was more likely tohave a particular value for one of the unknown variables. For example, the list of variables for oneeconomy might be ‘‘Low interest rates,’’ ’’Small trade deficits,’’ and ‘‘???’’ (indicating that the valuefor the third variable, retirement savings, was unknown), those for the second economy might be‘‘Low interest rates,’’ ‘‘???,’’ and ‘‘???,’’ and participants would be asked which economy was morelikely to have high retirement savings. Possible responses were 1 for the first example, 2 for the second

Fig. 6. Qualitative predictions of the normative model (left hand side) and Experiment 1’s choice scores (right hand side).Proportions reflect preference for the first response alternative in each problem (e.g., ‘‘A’’ in A vs. B). Independent and dependentchoice problems are depicted with white and shaded bars, respectively. Error bars are 95% confidence intervals.


example, and 3 for ‘‘equally likely.’’ There were two versions of each of the five types of choice prob-lems, one in which the participant was asked to choose which example was more likely to have Y (asshown in Figs. 2–4), and the corresponding version in which they were asked to which was more likely


to have X. To average over any bias for choosing the top or bottom example, each of these 10 problemswas presented twice, with the order of the two examples reversed. The order of these 20 problems wasrandomized for each participant.

4.2. Results

To construct a single choice score that summarizes subjects’ responses, choices in favor of the firstalternative (e.g., A in A vs. B) were coded as 1, those in favor of the second (B) were coded as 0, and an‘‘equally likely’’ response was coded as .5. Initial analyses revealed that choice scores were unaffectedby either domain or the order in which the causal networks were presented. Accordingly, subjects’choices are presented in Table 3 and their choice score are presented on the right hand side ofFig. 6 collapsed over these factors.

Fig. 6 reveals that responses in the common cause and chain conditions were approximately equaland substantially different from those in the common effect condition. This observation was sup-ported by statistical analysis. A 3 � 5 ANOVA with causal network and choice problem type as factorsyielded an overall effect of choice problem type, F(4,248) = 40.6, MSE = .041, p < .0001 and an interac-tion between problem type and network, F(8,496) = 5.5, MSE = .025, p < .0001. However, whereas theinteraction between problem type and the contrast between the common cause and chain networkcombined vs. the common effect network was significant (p < .0001), the interaction between thecommon cause and chain network was not (p > .20). Accordingly, I discuss the common cause andchain conditions together and then the common effect condition.

4.2.1. Common cause and chain resultsOn one hand, the common cause and chain choice scores in Figs. 6A and B exhibit some of the prop-

erties of normative causal reasoning shown in Fig. 6. When asked whether situation D or E was morelikely to have the causally relevant value of Y (or X), most participants chose D (choice scores of .79and .90 in the common cause and chain conditions, respectively), consistent with the predictions ofthe normative model. Both these scores were significantly different than .50, t(62) = 9.20 and 17.33,ps < .0001. This result indicates that in both conditions participants were willing to engage in indirectinferences, that is, from X to Y or Y to X when the state of Z was unknown.

Table 3Results from Experiment 1. Normative choices are shown in bold italic.

Choice problem Causal network

Common cause Chain Common effect

A vs. BA .21 .34 .25Equally likely .70 .60 .63B .08 .06 .12

B vs. CB .30 .44 .29Equally likely .65 .49 .58C .04 .07 .13

D vs. ED .67 .85 .37Equally likely .24 .10 .56E .10 .05 .07

F vs. GF .18 .23 .20Equally likely .73 .68 .65G .08 .09 .15

G vs. HG .13 .19 .18Equally likely .80 .75 .74H .07 .06 .08


Unfortunately, participants failed to honor independence on the remaining problems in Fig. 6. Re-call that when the state of Z is known, the state of X (Y) should have no influence on the whether Y (X)is present. In fact, the average choice scores on these problems (A vs. B, B vs. C, F vs. G, and G vs. H) was.57 and .62 in the common cause and chain conditions, respectively, t(62) = 4.57 and 7.12, ps < .0001.That is, the presence of one variable made the presence of the other more likely even when those vari-ables were supposedly screened off from one another. Nevertheless, that these scores were lower thanthose for the D vs. E, problem indicated that subjects exhibited some sensitivity to the difference be-tween independent and dependent problems, t(62) = 6.99 and 11.45, in the common cause and chainconditions, respectively, ps < .0001.

4.2.2. Common effect resultsRecall that an important property of common effect networks is discounting in which the presence

of one cause of an effect makes another less likely. Discounting suggests that B should be preferred inthe A vs. B choice problem and that C should be preferred in the B vs. C problem. Fig. 6C shows thatsubjects instead exhibited the opposite pattern, preferring A in the first problem and B in the second;their average choice score of .57 was significantly greater than .50, t(62) = 4.07, p < .0001. On the inde-pendent problems (D vs. E, F vs. G, and G vs. H), the average choice scores (.57) were also significantlygreater than .50, t(62) = 4.22, p < .0001. Only the score for the F vs. G problem (.52) was not signifi-cantly greater than .50.

4.2.3. Individual differencesIt is important to assess whether Experiment 1’s group results were manifested consistently by

all participants or only arose as a result of averaging over individuals with different response pro-files. In fact, cluster analyses revealed two subgroups of participants with qualitatively different re-sponses. The responses of one cluster of 18 participants, shown in the left side of Fig. 7, werevirtually identical for all three causal networks. That is, 29% of the participants—labeled ‘‘associativereasoners’’ for reasons discussed below—showed no sensitivity to causal direction and usually chosethe alternative in which more causally related variables were present. Indeed, a 3 � 5 ANOVA ofthese subjects with causal network and choice problem type as factors yielded no effects of network,ps > .12. The other cluster of 45 participants—labeled ‘‘causal reasoners’’ in the right side of Fig. 7—instead demonstrated sensitivity to causal direction by generating different responses in the com-mon effect condition as compared to the common cause and chain conditions. They also committedmany fewer violations of the Markov condition: These individual chose the correct ‘‘equally likely’’response alternative on 78% of independent choice problems as compared to 41% for the associativereasoners. Nevertheless, when they did not respond correctly, even these individual were morelikely to choose the alternative in which more causally related variables were present. As aresult, their choice scores continued to be significantly above chance on a number of independentproblems (e.g., B vs. C in the common cause and chain conditions and D vs. E in the common effectcondition).

4.3. Discussion

Experiment 1 paints a mixed picture. When reasoning with a common cause or chain network, par-ticipants correctly inferred that the states of X and Y provided information about one another whenthe state of Z was unknown. But participants also committed numerous apparent violations of theMarkov condition in which supposedly independent variables influenced one another. And, ratherthan discounting when reasoning with a common effect structure, they were more likely to predictthe presence of a cause when another cause was already present. Recall that these results obtaineddespite the steps intended to minimize the impact of prior knowledge (e.g., randomizing which var-iable senses were described as causally related). Additional tests of the role of prior knowledge will bepresented starting with Experiment 3. For now, the purpose of Experiment 2 is to further explore the‘‘associative’’ pattern of inferences found in the first experiment.

Fig. 7. Results of Experiment 1 segregated into two participant groups, the ‘‘associative reasoners’’ (N = 18) and the ‘‘causalreasoners’’ (N = 45). Independent and dependent choice problems are depicted with white and shaded bars, respectively. Errorbars represent 95% confidence intervals.



5. Experiment 2

The manner in which inferences in Experiment 1 departed from the normative model—in everycase, choice scores were higher than predicted—provides insight into the nature of those errors. Thispattern is consistent with subjects sometimes adopting an associative reasoning strategy in which thepresence of one variable makes the presence of another more likely. For example, Fig. 8 presents a rep-resentation of causal knowledge in which variable senses that are causally related in Fig. 1 are insteadrelated via symmetrical ‘‘associative’’ links. Violations of the Markov condition will occur if people rea-son with this representation as if it’s a spreading activation network. To take the materials in Table 2as an example, the presence of both low interests rates and small trade deficits (e.g., X and Z in choiceproblem A) will spread more activation to high retirement savings (Y) than small trade deficits alone(e.g., in choice problem B). (Later I will formalize this associative reasoning model and demonstratehow it provides an account of the associative reasoners.) Of course, that subjects showed an overallsensitivity to causal direction (i.e., inferences in the common effect condition differed from those inthe common cause and chain conditions) means that associative reasoning is by itself unable to ac-count for the results of Experiment 1. Instead, the claim is that subjects’ otherwise correct causal infer-ences are distorted by an associative strategy. The substantial minority of participants showed nosensitivity to causal direction shown in Fig. 7 provides especially direct evidence for the contributionof associative reasoning.

This apparent mixing of normative and associative responses raises the possibility that the presentresults may be a result of two separate reasoning processes, as stipulated by the well-known dual pro-cess theories of reasoning that distinguish between associative and ‘‘analytical’’ reasoning. Althoughthis distinction has been characterized in different ways, associative reasoning is generally thoughtto be non-deliberative, operates in parallel, is similarity-based, and consumes few cognitive resources,whereas the analytical system is conscious, operates sequentially, is rule-based and effortful (seeDarlow & Sloman, 2010; Evans, 2008; Kahneman & Frederick, 2002; Osman, 2004; Sloman, 1996;Smith & DeCoster, 2000 for reviews; see Sternberg & McClelland, 2011, for an analogous view of learn-ing). While past research has not emphasized the role of multiple systems in causal reasoning(although see Crisp-Bright & Feeney, 2010; Evans, Handley, Neilens, & Over, 2008; Evans et al.,2009; Rehder, 2009; Verschueren, Schaeken, & d’Ydewalle, 2005), Experiment 1 raises the possibilitythat people can engage in normative causal reasoning but resort to fast, associative processes in somecircumstances.

Moreover, dual process accounts suggest potential explanations of why individuals differ in theirtendency to reason associatively. A common assumption is that the associative system renders a fast,intuitive response that then might be ‘‘corrected’’ by the analytic reasoning system (e.g., Evans, 2008;Gilbert, 1989). However, because it is relies heavily on working memory, the analytic system may beless operative in those with less cognitive capacity (Evans & Over, 1996; Feeney, 2007; Stanovich &West, 1998; Stanovich, 1999). The associative reasoners in Fig. 7 may be examples of these less capa-ble individuals. Alternatively, these subjects might be distinguished not by their cognitive capacity butrather an unwillingness to ‘‘think hard,’’ that is, to deploy effortful analytical processes (reflecting, per-haps, a ‘‘need for cognition;’’ Cacioppo & Petty, 1982). This latter interpretation is important because itsuggests a deflationary interpretation of the results of Experiment 1, namely, that errors only occur onartificial laboratory tasks in which reasoners have little invested. Violations of the Markov conditionmay be rare during real-world reasoning problems in which people have a stake in the outcome. In-deed, Bless and Schwarz (1999) provide evidence indicating that increased motivation can reduce er-rors by promoting more deliberative processing.

Experiment 2 investigated whether causal inferences would be influenced by variables knownto affect the contribution of fast, associative processes, namely, time pressure (e.g., Evans &

Z YX

Fig. 8. Associative network interpretation of the causal networks in Fig. 1.


Curtis-Holmes, 2005; Evans et al., 2009; Finucane et al., 2000; Roberts & Newton, 2001) and askingsubjects to justify their choices (Smith & Sloman, 1994). Half the subjects were assigned to ajustification condition that was designed to reduce associative reasoning in two ways. First, partici-pants talked aloud into a tape recorder while making their decision to justify their answer. Evidencethat talking aloud promotes more analytical, rule-based processing was provided by Smith andSloman (1994) who found that classification decisions were more sensitive to features that were re-lated to participants’ theoretical understanding of a category and less sensitive to overall similaritywhen participants talked aloud, presumably because talking aloud promotes a search for a verbal-izable rule with which to justify the decision. Second, the justification condition sought to decreasetime pressure by (a) having participants learn only one causal network (rather than three as inExperiment 1), (b) informing them that they would have plenty of time to answer the inferencequestions in the 1 h allotted for the experiment, and (c) by asking them to emphasize accuracy overspeed.

The other half of the participants were assigned to a deadline condition designed to promote asso-ciative reasoning by placing them under time pressure. This was accomplished by giving participants adeadline of 10 s to make their response. To implement this deadline, the screen that presented achoice problem included a counter that began at 10 and counted down to 0 once per second. Note thatthe 10 s deadline was intended to induce only mild time pressure because extreme pressure wouldsimply induce random responding. (This would result in choice scores of �0.5, a finding that couldbe erroneously interpreted as the deadline leading to fewer violations of independence.) In addition,this group learned and answered inference questions about three causal networks (as in Experiment1), a fact that may also contribute to mild time pressure.

Fewer violations of independence and greater discounting in the justification condition will be con-sistent with the view that an analytical reasoning component can correct fast, intuitive responses gen-erated on the basis of associative rather than causal relations. The absence of these results will suggestthat associative reasoning is a mindful, deliberate strategy (or that analytic processes are poor at rec-ognizing and correcting such errors; more about this later).

5.1. Method

5.1.1. ParticipantsNinety New York University undergraduates received course credit for participating. They were

assigned in equal numbers to the deadline or justification condition. Because deadline participantslearned three causal networks, the same two counterbalancing factors used in Experiment 1 thatrotated the three networks through three presentation orders and three domains were used in thatcondition. Because justification participants learned just one network, they were assigned in equalnumbers to one of the three networks and one of the three domains. As in Experiment 1, thesenses of the variable that were described as causally related were randomized for eachparticipant.

5.1.2. ProcedureThe procedure was similar to that of Experiment 1 with changes to implement the deadline and

justification conditions. In the deadline condition, each choice problem was presented with a counterthat started at 10 and counted backwards once per second. If no response was made within the 10 s, awarning message was displayed asking the participant to respond in the allotted time, after which thecomputer presented the next problem. In the justification condition, participants were asked to ‘‘thinkabout this question out loud so that we can record your thinking process’’ and to ‘‘speak out loud whyyou made the choice you made, that is, the justification for that choice.’’ These participants were alsotold that they would only learn one causal network, that they would have plenty of time to answer theinference questions, and were asked to emphasize accuracy over speed.

All participants were presented with the same 20 choice problems used in Experiment 1. Fourwarm up trials were presented beforehand to familiarize participants with the procedure (the warmup trials were excluded from the following analyses).


5.2. Results

An initial analysis found that participants took much longer to respond in the justification condi-tion (average of 22.6 s) as compared to the deadline condition (5.5 s), confirming the effectiveness ofthe deadline vs. justification manipulation. The deadline participants failed to respond within 10 s onfewer than 1% of the trials.

As was the case in Experiment 1, there was no effect of the domain and so choice proportions arepresented in Table 4 and choice scores in Fig. 9 collapsed over this factor. As in Experiment 1, the re-sults were qualitatively different in the common effect as compared to the common cause and chainconditions and thus those results are reported separately.

5.2.1. Common cause and chain resultsFig. 9 shows that participants continued to commit screening off errors on independent problem

types for which they should (according to the normative model) have no preference. Moreover, thistendency was not weaker for those participants who were required to provide justifications. Separate2 � 5 ANOVAs of the common cause or chain conditions revealed that the deadline vs. justificationmanipulation yielded neither main effects nor interactions with problem type, all Fs < 1. Collapsingover the deadline and justification conditions, an analysis of the independent problems revealed thattheir choice scores (.60 and .63 in the common cause and chain conditions, respectively), were signif-icantly greater than .50, t(62) = 5.78 and 6.20, ps < .0001, just as they were in Experiment 1. Neverthe-less, participants continued to distinguished between the independent and dependent problems,generating higher scores (averages of .87 and .86) for the latter, t(59) = 9.84 and 7.95 in the commoncause and chain conditions, respectively, ps < .0001.

5.2.2. Common effect resultsThe results for the common effect condition (Fig. 9C) reveal that whereas responses in the deadline

condition were very much like those in Experiment 1 (Fig. 6C), the justification manipulation pro-duced a modest increase in normative responding. Choice scores on dependent problems that should



Common Cause Chain Common effect

Deadline Justification Deadline Justification Deadline Justification

A vs. BA .22 .25 .33 .33 .28 .20Equally likely .71 .70 .58 .63 .56 .58B .07 .05 .08 .03 .16 .22

B vs. CB .39 .30 .48 .55 .33 .15Equally likely .57 .65 .44 .43 .52 .48C .04 .05 .08 .02 .15 .37

D vs. ED .78 .78 .78 .92 .45 .12Equally likely .19 .15 .09 .07 .52 .87E .03 .07 .13 .02 .03 .02

F vs. GF .26 .28 .33 .22 .22 .08Equally likely .68 .68 .57 .75 .61 .58G .06 .03 .10 .03 .16 .33

G vs. HG .13 .18 .22 .17 .22 .27Equally likely .83 .75 .66 .80 .69 .62H .04 .07 .12 .03 .09 .12

Fig. 9. Results from Experiment 2. Independent and dependent choice problems are depicted with white and shaded bars,respectively. Error bars represent 95% confidence intervals.


exhibit discounting (A vs. B and B vs. C) were lower in the justification vs. deadline conditions (.44 vs.57). Likewise, scores on the three independent problems for which reasoners should have no prefer-ence also decreased (from .60 to .50). As a result, a 2 � 5 ANOVA yielded a significant main effect of the


justification vs. deadline manipulation, F(1,58) = 6.09, MSE = 0.119, p < .05 (and a marginal interactionwith problem type, F(4,232) = 2.10, MSE = 0.038, p = .08). Nevertheless, responses in the justificationcondition were still far from normative, as the degree of discounting in that condition failed to reachedsignificance, t(59) = 1.46, p = .15 (a separate analysis of the one problem exhibiting discounting, B vs. C,also failed to reach significance, p = .09). Also note that although the pattern of responding for theindependent problems in the justification condition was unexpectedly complex (choice scores <.50for the F vs. G problem and >.50 for the other two), none of these scores differed significantly from.50, ps > .08. Nevertheless, we will see that this pattern recurs in future experiments and so I defer fur-ther discussion until then.

5.2.3. Analysis of verbal protocolsVerbal protocols were relatively uninformative about subjects’ reasoning strategies as in the vast

majority of trials subjects merely repeated the information given in the problem and then verbalizedtheir choice. Nevertheless, a few trends of theoretical interest emerged. First, participants sometimesmade reference to outside knowledge, that is, to knowledge in addition to the causal relations theywere taught in the experiment. For example, one subject reasoned that ‘‘high interest in religionmeans more focused. . .good work ethics,’’ and their inferences were based on how the other two vari-ables might be related to good work ethic. Another reasoned that ‘‘when there are low interest ratespeople spend money’’ and this inference affected their subsequent choices. Overall, participantsshowed signs of using outside knowledge on 12% of the trials. Not only is this rate fairly low, recallthat I have argued that the use of prior knowledge cannot explain the overall pattern of results in theseexperiments (because such effects will be averaged away due to the randomization of causal materi-als). Nevertheless, Experiment 3 will take further steps to assess the use of domain knowledge (bycomparing how subjects reason with concrete vs. abstract materials in which no ontological domainis specified).

Second, recall that situations C and F can be construed as providing inconsistent information (causeabsent and effect present or vice versa). In fact, subjects noted these potential inconsistencies on 10%of the B vs. C and F vs. G trials. For example, some participants described situations F and C as giving‘‘incorrect information’’ and having ‘‘more factors wrong’’ (cf. Walsh & Sloman, 2008). Conversely, insituations A and H a cause and effect are either both present or both absent, and subjects noted thisfact on 16% of the A vs. B and G vs. H choice problems. For example, in these situations variable X (or Y)was sometimes described as ‘‘backing up’’ Z and as providing ‘‘coinciding’’ information. Experiment 4Awill assess the potential role of these kinds of consistencies and inconsistencies by manipulating thewhether or not the causal links are described as probabilistic.

Finally, that participants sometimes had trouble remembering the causal relations is evidenced bythe fact that participants described the causal relations incorrectly (e.g., misstated which variablesenses were related or the direction of causality) or explicitly stated they could not remember thoserelations on 7% of the trials. This result is a surprise given that subjects passed a multiple-choice teston this knowledge immediately before taking the inference test. Conceivably, an incomplete represen-tation of the causal links might contribute to the observed errors (e.g., they may have treated the cau-sal links as symmetrical relations because they forgot the direction of causality). Later experimentsaddress this possibility.

5.3. Discussion

Experiment 2 asked whether the departures from the normative model found in Experiment 1 weredue to fast associative reasoning process invoked by less capable or careless individuals. The answer isthey, because participants who provided justifications, were asked to emphasize accuracy over speed,and made inferences for one rather than three causal networks continued to violate independence andfail to discount. Providing justifications yielded some improvements on common effect inferences asdiscounting increased, although not to a significant level. Apparently, for some people most of thetime, and most people some of the time, providing an associative response to a causal reasoning prob-lem is the product of a careful and deliberate strategy.


6. Experiment 3

Experiment 3 returns to the question of whether the violations of causal reasoning norms can beattributed to subjects’ use of domain knowledge. It does so by comparing a concrete condition thattested the same materials as in Experiments 1–3 with an abstract condition that used a blank domainin which variables are simply labeled ‘‘A,’’ ‘‘B,’’ and ‘‘C.’’ Recall that the Introduction raised the possi-bility that the search of memory that occurred as part of the comprehending the materials might turnup additional inter-variable causal relations (perhaps yielding the model in Fig. 5C) or commonalitiesbetween the causal relations (5B). The verbal protocols in Experiment 2 showing how a subject in-ferred good work ethic (a variable not included as part of the instructions) from a high interest in reli-gion (and then considered how a good work ethic might affect the other variables) might be anexample of the latter. And, subjects may have elaborated their models in response to the situationsthey reasoned about (e.g., positing a shared disabler to account for cause-present/effect-absent situa-tions, Fig. 5A). That subjects’ verbal protocols revealed that they occasionally noted cause-effect incon-sistencies provides some support for this possibility.

Experiment 3 assesses the use of these sorts of elaborations by testing a condition in which domainknowledge is unavailable. If domain knowledge contributed substantially to subjects’ errors in the ear-lier experiments, then those errors should be less common in the abstract condition.

As in the justification condition of Experiment 2, all participants in Experiments 3 provided spokenjustifications, were asked to emphasize accuracy over speed, and answered questions for just one cau-sal network. In addition, two changes to the experimental procedure were made. First, because theverbal protocols also revealed that participants showed some signs of forgetting the causal relation-ships, in this experiment participants were provided with a printed diagram of those relationshipsduring the inference test. The second change was to emphasize the independence of causal mecha-nisms just before the start of the inference test.

6.1. Method

6.1.1. ParticipantsNinety New York University undergraduates received course credit for participating. They were

randomly assigned in equal numbers to one of the three causal networks. In the concrete condition,they were also randomly assigned in equal numbers to one of the three domains and the senses ofthe variable that were described as causally related was randomized for each participant.

6.1.2. MaterialsThe concrete condition used the same materials as in Experiments 1 and 2. Subjects in the abstract

condition were told that they were ‘‘learning about a new domain of knowledge’’ in which there were‘‘three different variables’’ and that they would learn that ‘‘some variables are responsible for causingother variables.’’ The variables were named ‘‘A,’’ ‘‘B,’’ and ‘‘C,’’ each of which could take on either a‘‘low’’ or ‘‘high’’ value. A causal link consisted of one variable sense causing another (e.g., ‘‘When Ais low it causes B to be high.’’). Descriptions of causal mechanisms were not provided in this condition.

6.1.3. ProcedureThe procedure was identical to the justification condition of Experiment 2, with two exceptions.

First, subjects were given a diagram of their causal network during the inference test. Second, atthe start of the inference test all participants were given additional instructions emphasizing the inde-pendence of the causal mechanisms. For example, in the common cause condition they were told‘‘Remember that X is a direct result of Z, and Y is independently a direct result of Z’’ (where for X,Y, and Z the experimenter pointed to that value on the causal network diagram). Common effect par-ticipants were told ‘‘Remember that both X and Y can each bring about Z on its own. That is, it’s not thecase that both of these two have to be present for Z to be present. Rather, X can independently produceZ on its own, and Y can independently produce Z on its own as well.’’


6.2. Results

The concrete condition showed no effect of whether the causal networks were instantiated in thedomain of economics, meteorology, or sociology, and so choice proportions in Table 5 and choicescores in Fig. 10 are collapsed over this factor.

6.2.1. Common cause and chain resultsFig. 10 reveals that participants continued to violate the predictions of normative model on inde-

pendent problem types. Moreover, the frequency of those errors did not vary as a function of whetherthe domain was concrete or abstract. 2 � 5 ANOVAs in the common cause and chain conditionsyielded the expected main effect of problem type, F(4,112) = 9.27, MSE = 0.023 and F(4,112) = 23.77,MSE = 0.022, respectively, both ps < .0001, but no effect of the concrete/abstract manipulation noran interaction with problem type, all Fs < 1. Collapsing over the concrete and abstract conditions, ananalysis of the independent problem types revealed that their choice scores (averages of .59 and .65in the common cause and chain conditions, respectively), were significantly greater than .50,t(29) = 2.96 and 4.88, ps < .01.

6.2.2. Common effect resultsFig. 10C shows that reasoning with the common effect network was also mostly unaffected by

using abstract vs. concrete domains. A 2 � 5 ANOVA revealed no effects other than a main effect ofproblem type, F(4,112) = 4.25, MSE = 0.027, p < .01; the type by concrete/abstract interaction was mar-ginal, F(4,112) = 1.75, MSE = 0.027, p = .14. Collapsing over the concrete and abstract conditions,choice scores on dependent problems that should exhibit discounting (.53) did not differ significantlyfrom .50, t < 1; those on the D vs. E problem that should exhibit independence (.62) were significantlygreater than .50, t(29) = 3.20, p < .01. As seen in the justification condition of Experiment 2, the choice




Abstract Concrete Abstract Concrete Abstract Concrete

A vs. BA .17 .22 .25 .32 .05 .20Equally likely .82 .75 .75 .68 .92 .63B .02 .03 0 0 .03 .17

B vs. CB .42 .34 .37 .42 .17 .32Equally likely .50 .64 .63 .58 .70 .52C .08 .02 0 0 .13 .17

D vs. ED .57 .52 .87 .90 .22 .28Equally likely .37 .47 .12 .10 .77 .70E .07 .02 .02 0 .02 .02

F vs. GF .13 .10 .27 .38 .08 .18Equally likely .85 .82 .65 .62 .87 .38G .02 .08 .08 0 .05 .43

G vs. HG .17 .21 .27 .25 .12 .28Equally likely .77 .75 .73 .72 .83 .55H .07 .03 0 .03 .05 .17

Fig. 10. Results from Experiment 3. Independent and dependent choice problems are depicted with white and shaded bars,respectively. Error bars represent 95% confidence intervals.


score for the F vs. G problem in the concrete condition was less than 0.5, although not significantly so,t(29) = 1.43, p = .17. Again, I defer discussion of this result until Experiment 4.


6.3. Discussion

Experiment 3 asked to what extent subjects’ prior domain knowledge affected their causal infer-ences in the earlier experiments. In fact, for all three networks the number of independence violationswas no smaller in a blank domain than in the domains of economics, meteorology, and sociology.Reasoners were also no more likely to discount with the blank materials. These errors persisted eventhough all subjects (a) learned one causal network rather than three, (b) were asked to emphasizespeed over accuracy, (c) provided justifications for their choices, (d) could refer to a diagram of thecausal relations during the inference test, and (e) were explicitly told that the two causal linksoperated independently.

7. Assessing elaborated causal models

The results of Experiment 3 notwithstanding, it might be argued that even the use of blank mate-rials is insufficient to fully rule out the influence of domain knowledge. As mentioned, it is possiblethat some subjects elaborated their causal model on the basis of analogies with familiar domain orassumed abstract versions of the knowledge structures in Fig. 5 (e.g., assumed the presence of a disa-bler without any concrete notion of what that disabler might be). This latter possibility is related toplaceholder notions that assume that reasoners have only an abstract representation of causal pro-cesses (Ahn, Kalish, Medin, & Gelman, 1995; Medin & Ortony, 1989).

While it may be impossible to definitively rule out such elaborations, it is possible to conduct a the-oretical analysis asking to what extent the models in Fig. 5 account for subjects’ inferences. Exact pre-dictions depend on the models’ parameters (e.g., the base rates of the causes, the strength of the causalrelations, etc.), information that was not provided during the experiment. Thus, each model was as-sessed by instantiating it with 10,000 sets of randomly generated parameter values. Predictions forthe five choice problems tested in Experiments 1–3 were computed for each instantiation. For modelm instantiated with parameters hm, the probability that target variable t is present in situation si is de-noted p(t = 1|si; m, hm). I assume that reasoners represent probabilities as log odds and that decidingthat t is more likely to be present in situation s1 than s2 is made according to a softmax rule,

4 Carcorresp

choicemðt; s1; s2; hm; sÞ ¼expðlogitðpðt ¼ 1js1; m; hmÞÞ=sÞ

expðlogitðpðt ¼ 1js1; m; hmÞÞ=sÞ þ expðlogitðpðt ¼ 1js2; m; hmÞÞ=sÞð1Þ

where s is a ‘‘temperature’’ parameter that controls the extremity of the responses. Details of thesesimulations are presented in Appendix B along with a qualitative description of each model’spredictions. The predictions averaged over the 10,000 instantiations are presented in Fig. 11. Thereare two versions of the shared disabler account. The specific version assumes that the disabler rendersinoperative the two explicit causal links but not other potential background causes of the effect(s). Thegeneral version assumes that the disabler prevents all occurrences of the effect.4 The error bars inFig. 11 bracket 95% of the 10,000 predictions made for each problem. For every problem for every model,the predictions were all less than, all greater than, or all equal to .50, with the exception of the predic-tions of the shared generative cause model for the chain and common effect networks (hatched bars inFig. 11B and C).

7.1. Shared disablers

When a common cause networks includes a shared disabler (W in the left panel of Fig. 5A), the firstand second panels of Fig. 11A confirm the observation, first made by Walsh and Sloman (2008), that adependency is introduced between the effects such that one should prefer A in the A vs. B problem andB in the B vs. C problem. Z-absent problems (F vs. G and G vs. H) are predicted to be independent whenthe disabler is specific but dependent when it is general, favoring F in F vs. G and (just barely) G in G vs.

roll and Cheng (2009) referred to a general disabler as a ‘‘broad preventer.’’ However, a specific disabler does notond to their ‘‘narrow preventer,’’ which disables only a single causal relationship.

Fig. 11. Predictions for the three causal networks of Fig. 1 elaborated with the different forms prior knowledge shown in Fig. 5.First column: specific shared disablers; second column: general shared disablers; third column: shared mediators; fourthcolumn: shared generative causes. Hatched bars represent indeterminate predictions, that is, cases in which the alternative thatis favored depends on the parameters of the model. Error bars bracket 95% of the 10,000 predictions generated for each choiceproblem.


H. Thus, a general disabler (or a mixture of specific and general disablers) can account for subjects’common cause inferences. In contrast, shared disablers are unable to account for the results in thechain or common effect condition. As observed by Park and Sloman (2013), introduction of a specificshared disabler to a chain network (middle panel of Fig. 5A) results in the loss of independence when Zis present (first panel of Fig. 11B). But when Z is absent, problems are either independent (specificdisabler) or dependent (general disabler) in a direction opposite to that of subjects’ preferences. Final-ly, for a common effect network elaborated (right panel of Fig. 5A), discounting should still occur andindependence should obtain on D vs. E (first two panels of Fig. 11C), at odds with the empirical results.

Although shared disablers thus fail to provide a comprehensive account of Experiments 1–3, closerexamination of those results provides some support for their influence. The presence of a specific disa-bler entails larger choices scores on the Z-present (A vs. B, B vs. C) as compared to the Z-absent (F vs. G,G vs. H) problems in both the common cause and chain conditions, a pattern in fact observed in allthree experiments.5 Later I revisit this potential contribution of disablers.

5 Common cause conditions: .60 vs. .54, t(62) = 3.18, p < .01, in Experiment 1; .62 vs. .58, t(59) = 1.97, p = .054, in Experiment 2;.60 vs. .55, t(29) = 3.62, p < .01 in Experiment 3. Chain conditions: .66 vs. .57, t(62) = 4.53, p < .0001, in Experiment 1; .67 vs. .58,t(59) = 4.69, p < .0001 in Experiment 2; .67 vs. .63, t(29) = 1.74, p = .09 in Experiment 3. Nevertheless, for both networks in bothexperiments, choice scores on the Z-absent problems were still greater than .50 (Experiment 1: ts = 2.83 and 4.27, ps < .01;Experiment 2: ts = 3.72 and 4.09, ps < .001; Experiment 3: t = 1.82, p = .08, and t = 4.23, p < .001).


7.2. Shared mediators

Assuming that common cause links are mediated by a common factor (left panel of Fig. 5B) yieldsthe (apparent) independence violations (third column of Fig. 11A) expected by previous investigators(Cartwright, 1993; Hausman & Woodward, 1999; Salmon, 1984; Sober, 2007) and exhibited by thepresent subjects. Consistent with the claims of Park and Sloman (2013), if the variables that mediatethe causal links (M1 and M2 in Fig. 5B) have a shared disabler (W), apparent violations of independencewill obtain on the Z-present chain inferences (Fig. 11B). This account fails to account for the >.50choice scores on Z-absent problems, however. For a common effect network (Fig. 11C), the mediationhypothesis predicts the presence of discounting and independence on the D vs. E choice problems, atodds with the empirical results. Note that the common effect results also rule out some combination ofdisablers and meditators, as both predict discounting and the independence of X and Y when Z isunknown.

7.3. Shared generative cause

The predictions of the shared generative cause hypothesis (Fig. 5C), shown in the fourth column ofFig. 11, indicate that it fares better than the previous accounts. It correctly predicts the pattern of inde-pendence violations in the common cause condition. In the chain condition, it correctly predicts the Z-absent problems. Its predictions for the Z-present problems (A vs. B and B vs. C) are parameter depen-dent (choice scores can be greater or less than .50); nevertheless, this means there exists parametervalues that can reproduce the independence violations on those problems as well. In the common ef-fect, it correctly predicts the Z-absent problems and is the only model that correctly predicts thatalternative D should be preferred on the D vs. E choice problem. Its prediction for the discountingproblems (Fig. 11C) are also parameter dependent and so it is also the only model that can potentiallyaccount for the absence of discounting.

Nevertheless, the shared generative cause model faces two challenges. The first concerns its theo-retical plausibility. To yield the predictions in Fig. 11, recall that this model must stipulate that thevariable senses that were described to participants as causally related that are made more likely bythe shared cause. But why would subjects assume just this structure? Rehder and Burnett (2005) pos-ited a shared generative cause to explain violations of independence but, unlike the current materials,they tested categories whose features can reasonably be thought to be linked by common causalmechanisms. As mentioned, the memory search that occurred when reading the materials might bebiased toward facts involving the causally-related variables senses, but why would those facts not in-clude inhibitory causal relations in addition to generative ones? And, although it is likely that reason-ers recognized that an effect must have an alternative cause when confronted with a cause-absent/effect-present situation (Carroll & Cheng, 2010; Hagmayer & Waldmann, 2007; Luhmann & Ahn,2007, 2011; Rottman et al., 2011; Saxe et al., 2005), why would they think that this alternative alsocaused the other two variables in the model? The second challenge is that, because it is superimposedon the instructed causal model, a shared generative cause predicts that subjects should be sensitive tocausal direction (e.g., make different predictions for common cause and common effect networks), atodds with the large number of ‘‘associative reasoners’’ in Experiment 1.6 I will return to these argu-ments in the General Discussion.

In summary, none of the elaborated models in Fig. 5 provide a comprehensive account of subjects’inferences in Experiments 1–3. Of course, these results do not rule out the possibility that the modelsin Fig. 5 are contributing to subjects’ inferences and Experiment 4 and quantitative model fitting thatfollows will provide additional support for this possibility.

6 Said differently, the shared cause account can only reproduce the observed results assuming very different parameter valuesfor the three causal networks. In particular, the probability of the shared cause and the strength of its causal links must be muchlarger in the common effect condition in order to overcome the effects of discounting and produce choice scores >.50 on the A vs. Band B vs. C problems.


8. Experiments 4A and 4B

The goal of Experiment 4A was to further generalize the results of Experiments 1–3 by testing aprobabilistic condition in which the causal links were described as operating probabilistically. Recallthat Experiments 1–3 provided subjects with no information about the strength of the causal rela-tions. Given research that suggests that reasoners’ default assumption is that causal links are deter-ministically sufficient (the cause always produces the effect, Bullock, Gelman, & Baillargeon, 1982;Goldvarg & Johnson-Laird, 2001; Goodman, in press; Lu et al., 2008; Schulz & Sommerville, 2006),the absence of strength information likely invited a deterministic construal. One potential conse-quence of this interpretation is that cause-present/effect-absent situations became especially salient,amplifying reasoners’ tendency to attribute the effect’s absence to the work of disablers. Moreover,because some studies suggest that cause-present/effect-absent situations are sufficient to ‘‘refute’’ acausal relation (e.g., Frosch & Johnson-Laird, 2011), it might have led subjects to the more radical con-clusion that the causal information they were taught must be in error, perhaps prompting them toadopt an alternative (e.g., associative) representation of those relations.

To address this possibility, links in the probabilistic condition were described as probabilisticallysufficient by stating that each cause brought about its effect with probability 75%. Because researchalso suggests that reasoners might assume that causal links are deterministically necessary (the effecthas no other causes; Lombrozo, 2007; Lu et al., 2008), links were described as probabilistically neces-sary by stating that each effect occurred with probability 25% even when its explicit cause(s) were ab-sent. Inferences in the probabilistic condition were compared to those in a no-strength condition inwhich no information about the strength of the causal relationships was provided (as in the first threeexperiments).

I also report the results of a follow-up study, Experiment 4B, that assessed the hypothesis that sub-jects’ non-normative responses reflect a form of response bias. Conceivably, subjects were biasedagainst choosing the ‘‘equally likely’’ response in the earlier experiments because it was the normativechoice on 80% of the trials in the common cause and chain conditions and 60% of those in the commoneffect condition. To test this, Experiment 4B compared a probabilistic/unbalanced condition that was areplication of the probabilistic condition of Experiment 4A with a probabilistic/balanced condition thatwas identical except that subjects were presented with additional trials such that ‘‘equally likely’’ wascorrect on 50% of the trials. It turns out that no effect of the unbalanced/balanced manipulation wasobserved and so the results of both experiments will be reported jointly. As in Experiment 3, partic-ipants in Experiments 4A and 4B provided spoken justification, were asked to emphasize accuracyover speed, answered questions for just one causal network, were told that the causal mechanismsoperated independently, and were given a diagram of the causal relations.

8.1. Method

8.1.1. ParticipantsExperiments 4A and 4B tested separate groups of 90 New York University undergraduates. Partic-

ipants were randomly assigned in equal numbers to the no-strength or probabilistic condition (Exper-iment 4A) or the balanced or unbalanced conditions (4B). Within each experiment, subjects wereassigned in equal numbers to one of the three causal networks and to one of the three domains.The senses of the variable that were described as causally related were randomized.

8.1.2. MaterialsThe materials were the same as those in the concrete conditions of the earlier experiments with the

exception of the probabilistic conditions. To specify that the causal links were probabilistically suffi-cient, subjects were told, for example, ‘‘Whenever an economy has low interest rates, it will cause thateconomy to have a small trade deficits with probability 75%;’’ to specify that the links were probabi-listically necessary, they were told ‘‘Even when the known causes of small trade deficits are absent,small trade deficits appear in 25% of economies.’’ The multiple-choice test was expanded to include


causal strength questions. The diagram provided before the inference test included the strengthinformation.

8.1.3. ProcedureThe procedure was identical to the concrete condition of Experiment 3.

8.2. Results

The choice proportions for Experiments 4A and 4B are presented in Tables 6 and 7, respectively.There was again no effect of the domain in which the causal networks were instantiated. Moreover,2 � 5 ANOVAs of each causal network with balanced/unbalanced and problem type as factors yieldedno effects of this manipulation, all ps > .33. Accordingly, I collapse the results from Experiment 4B andthose of the probabilistic condition of Experiment 4A. The choice scores for the no-strength and prob-abilistic conditions are presented in Fig. 12.

8.2.1. Common cause resultsFig. 12A reveals that subjects’ common cause choice scores were >.50 on independent problem

types, even when the causal relations were described as probabilistic. A 2 � 5 ANOVA revealed aneffect of problem type, F(4,232) = 23.49, MSE = 0.026, p < .0001, but no effect of the causal strengthmanipulation, F(1,58) = 1.17, MSE = 0.032, p = .28; the interaction was nonsignificant, F(1,58) = 2.33,MSE = 0.012, p = .13. Choice scores for the independent problems in the no-strength and probabilisticconditions (.62 and .58) were significantly greater than .50, t(14) = 3.44, p < .01, and t(44) = 4.46,p < .0001, respectively. Just as in the previous experiments (Footnote 5), the difference between theZ-present vs. Z-absent problems predicted by a shared disabler (Fig. 11A) was observed here (.63vs. .58), t(58) = 2.24, p = .03.

8.2.2. Chain resultsFig. 12B shows that chain subjects’ choice scores were >.50 on the independent problem types and

were unaffected by the manipulation of causal strength. A 2 � 5 ANOVA revealed an effect of problem

Table 6Results from Experiment 4A. Normative choices are shown in bold italic.



Deterministic Probabilistic Deterministic Probabilistic Deterministic Probabilistic

A vs. BA .32 .22 .20 .40 .17 .23Equally likely .68 .77 .68 .55 .67 .37B 0 .02 .12 .05 .17 .40

B vs. CB .35 .25 .27 .35 .22 .33Equally likely .65 .68 .65 .62 .48 .48C 0 .07 .08 .03 .30 .18

D vs. ED .75 .80 .70 .67 .22 .28Equally likely .23 .18 .27 .32 .70 .63E .02 .02 .03 .02 .08 .08


G vs. HG .25 .23 .15 .37 .20 .22Equally likely .70 .75 .78 .57 .65 .57H .05 .02 .07 .07 .15 .22

Table 7Results from Experiment 4B. Normative choices are shown in bold italic.



Unbalanced Balanced Unbalanced Balanced Unbalanced Balanced

A vs. BA .23 .15 .32 .17 .27 .27Equally likely .73 .78 .65 .77 .37 .37B .03 .07 .03 .07 .37 .37

B vs. CB .28 .18 .30 .27 .32 .38Equally likely .68 .80 .68 .70 .35 .48C .03 .02 .02 .03 .33 .13

D vs. ED .65 .68 .78 .88 .32 .42Equally likely .30 .30 .22 .08 .52 .52E .05 .02 0 .03 .17 .07


G vs. HG .15 .10 .17 .25 .17 .17Equally likely .80 .90 .78 .73 .52 .57H .05 0 .05 .02 .32 .27


type, F(4,232) = 23.13, MSE = 0.029, p < .0001, no effect of the causal strength, F(1,58) = 1.78,MSE = 0.058, p = .19, and no interaction, F < 1. Choice scores for the independent problems in theno-strength and probabilistic conditions (.55 and .61) were greater than .50, t(14) = 2.34, p = .03,and t(44) = 5.68, ps < .0001, respectively. The Z-present/Z-absent difference was marginal (.60 vs..57), t(58) = 1.50, p = .14.

8.2.3. Common effect resultsReasoning with a common effect network (Fig. 12C) was also unaffected by the causal strength

manipulation. A 2 � 5 ANOVA revealed no effects other than an effect of problem type,F(4,232) = 8.89, MSE = 0.043, p < .0001; the type by strength interaction was not significant,F(4,232) = 1.59, MSE = 0.043, p = .18. Just as in Experiments 1–3, choice scores on dependent problemsthat should exhibit discounting (.49) did not differ significantly from .50, t < 1; those on the indepen-dent D vs. E problem (.60) were significantly greater than .50, t(58) = 3.69, p < .001. As seen in Exper-iments 2 and 3, choice scores on the F vs. G problem were less than 0.5 (.31 and .37 in the no-strengthand probabilistic conditions, respectively), significantly so in this experiment, t(14) = 2.88, p < .05 andt(44) = 2.80, p < .01. This result is discussed immediately below.

8.3. Discussion

Experiment 4 asked whether the earlier errors arose because the causal relations were assumed tobe deterministic. The answer is that they did not, because they persisted even why they were de-scribed as probabilistic relations.

A recurring trend in Experiments 2–4, one that reached significance in this experiment, was sub-jects’ preference for situation G in the F vs. G common effect problem, a result that is predicted bynone of the alternative models considered thus far, including associative reasoning. This finding is sug-gestive of another sort of influence on causal judgments. On this problem, subjects chose whether acause is more likely in a situation in which a common effect is absent (situation G in Fig. 4) vs. onein which the alternative cause is also present (situation F). Rather than computing a conditional

Fig. 12. Results from Experiments 4A and 4B. Independent and dependent choice problems are depicted with white and shadedbars, respectively. Error bars represent 95% confidence intervals.


probability, subjects may have estimated the probability of the presence of a target variable (t = 1)conjoined with the variable states stipulated in a situation (s). That is, rather than deciding on the ba-sis of the normative common effect model (NCE) with parameters h,


choiceNCE ðt; s1; s2; h; sÞ ¼ expðlogitðpðt ¼ 1js1; NCE; hÞÞ=sÞexpðlogitðpðt ¼ 1js1; NCE; hÞÞ=sÞ þ expðlogitðpðt ¼ 1js2; NCE; hÞÞ=sÞ

ð2Þ

they may have substituted the conjunct p(t = 1, si) for the conditional p(t = 1| si), yielding,

choiceNCE ðt; s1; s2; h; sÞ ¼ expðlogitðpðt ¼ 1; s1; NCE; hÞÞ=sÞexpðlogitðpðt ¼ 1; s1; NCE; hÞÞ=sÞ þ expðlogitðpðt ¼ 1; s2; NCE; hÞÞ=sÞ

ð3Þ

This conjunctive reasoning strategy accounts for common effect subjects’ preference for situation Grelative to F because adding the cause to F means that the absence of the common effect is (implau-sibly) paired with the presence of two causes whereas adding it to G means that the absent commoneffect is paired with only one cause. Quantitative model fitting presented below will demonstrate thatreasoning of this sort contributes to the common effect F vs. G comparison.

9. Individual differences in Experiments 2–4

An important finding from Experiment 1 was that 29% of the participants exhibited virtually nosensitivity to causal direction, producing associative inferences instead. To determine whether thisfinding obtains when reasoners provide justifications for their choices (and learn one causal networkrather than three and are asked to emphasize speed over accuracy), I performed a cluster analysis ofthe subjects from Experiments 2’s justification condition combined with those from Experiments 3and 4. 210 common cause and chain participants were included in one analysis and 105 common ef-fect participants were included in the other. Participants again clustered into two types, shown inFig. 13. The clusters shown in the left hand side consisted of 23, 24, and 26 participants in the commoncause, chain, and common effect conditions, respectively, representing 23% of the total. These associa-tive reasoners committed a large number of Markov violations (of the responses that reflected asso-ciative reasoning but violated independence, 63% were made by this quarter of the participants)and exhibited anti-discounting.

Unlike the associative reasoners of Experiment 1, those in Fig. 13 exhibited some sensitivity to cau-sal direction. A 3 � 5 ANOVA yielded a significant main effect of network type, F(2,70) = 8.16,MSE = 0.045, p < .001, reflecting the generally lower choice scores in the common effect condition(the network by problem type interaction was marginal, p = .19). Model fitting presented in the fol-lowing section will shed light on the source of this asymmetry, which suggests that providing justifi-cations (and emphasizing speed over accuracy, etc.) was somewhat effective in promoting sensitivityto causal structure.

The causal reasoners in the right hand side of Fig. 13 conformed more closely to the normativemodel. Notable is the significant discounting on the A vs. B and B vs. C problems in the common effectcondition. Of all the discounting responses made in Experiments 2–4, 95% were made by the causalreasoners. This is the first and only analysis in this article showing significant discounting and pro-vides additional evidence that requiring justifications improved reasoning relative to Experiment 1.Nevertheless, as in Experiment 1, even the causal reasoners continued to display small but significantnumbers of Markov violations on some problems (e.g., on A vs. B in the common cause condition, on Gvs. H in the chain condition, and on B vs. C in both conditions). And, that choice scores on the F vs. Gproblem in the common effect condition were less than .5 is suggestive of the conjunctive reasoningstrategy described earlier.

Finally, an analysis of RTs revealed that the associative reasoners in Fig. 13 did not respond signif-icantly more rapidly than the causal reasoners, 26.0 and 27.5 s, respectively, t < 1, a result that corrob-orates the findings in Experiment 2 indicating that the associative inferences were not simply due toquick and careless responding. Rather, it is a deliberate strategy adopted by a substantial minority ofreasoners, one that takes as long to execute as causal reasoning itself.

10. Modeling multiple influences on causal judgments

The central conclusion drawn from Experiments 1–4 is that strategies other than the normative onecontribute to people’s causal inferences. This claim is now further supported by a quantitative analysis

Fig. 13. 315 Participants from Experiments 2 (justification condition), 3, 4A, and 4B grouped into ‘‘associative’’ and ‘‘causal’’(N = 23 and 82 in the common cause conditions, 24 and 81 in the chain condition, 26 and 79 in the common effect condition).


showing that a mixture of strategies is sufficient to reproduce the inferences of the subjects who pro-vided justifications, shown in Fig. 14. To characterize each strategy, their predictions averaged over10,000 random parameter sets are presented in Fig. 15. The first column shows the patterns of inde-


pendence and dependence produced by the normative models. The predictions of the computationalinstantiation of the associative reasoning model defined in Appendix C (second column of Fig. 15) ex-hibit the expected choice scores >.50 on independent problems and the absence of discounting. Thepredictions of the shared specific disabler models defined in Appendix B (third column) can accountfor greater choices scores on Z-present vs. Z-absent problems in the common cause and chain condi-tions. The predictions of the conjunctive reasoning models defined by Equation 3 (fourth column)show that only that class of model that can account for subjects’ preference for G on the F vs. G com-mon effect problem.

The data in Fig. 14 were modeled by choosing plausible parameters for the four strategies them-selves (c = d = .50, m = md = .99, b = .20, a2 = 0, and a3 = 3; see Appendices B and C for parameter defi-nitions) and then using a search program to find the mixture that provides the optimal fit. The mixedmodel is defined as,

7 Thewhethewhereassociatsharedreasoni

choiceMixedNetðt; s1; s2; c;m; b; a2; a3; d;md; sÞ ¼ wNchoiceNNet ðt; s1; s2; c;m; b; sÞ

þwAchoiceAssocðt; s1; s2; a2; a3; sÞþwDchoiceSSDNet ðt; s1; s2; c;m; b; d;md; sÞþwJchoiceConjunctNet

ðt; s1; s2; c;m; b; sÞ

where wN, wA, wD, and wJ control the relative contribution of the normative (N), associative (A), specificshared disabler (SSD), and conjunctive (Conjunct) models, respectively; Net represents the conditionbeing fit (CC = common cause, CH = chain, CE = common effect). The weights were constrained tosum to 1 for each network but were allowed to vary freely over networks. The s parameter was com-mon across networks. To obtain an estimate of the variability on the weights, a bootstrap procedurewas performed in which subjects were resampled with replacement.

The weights that minimized squared error averaged over the 100 resamples are shown in the leftbars in each panel of Fig. 16 (the average value of s was 1.95). The choice scores generated by this fitare shown superimposed on the data in Fig. 14, which shows that the model reproduces the qualita-tive phenomena for each network. Three aspects of the weights are worth noting. First, associative rea-soning contributes to the inferences of all three causal networks (wAs = .258, .494, and .183),reproducing the violations of independence and the absence of discounting.7 Second, shared disablerscontribute to the common cause inferences (wD = .293), accounting for the larger choice scores on Z-pres-ent vs. Z-absent problems. Third, conjunctive reasoning contributes to the common effect inferences(wJ = .357), accounting for the preference for G in the F vs. G problem. These results corroborate the claimthat human causal reasoning can be characterized a mixture of reasoning strategies.

It is illuminating to also examine the weights when the model is fit separately to the two groups inFig. 13. As expected, the role of normative reasoning was larger for the causal reasoners (right bars inFig. 16) for all three networks (average wN = .588). Nevertheless, the common cause subjects were alsoinfluenced by shared disablers, the common effect subjects by conjunctive reasoning, and the chainsubjects by associative and conjunctive reasoning. Even reasoners who largely grasp the logic of causalreasoning sometimes fall prey to these alternative strategies.

Conversely, the contribution of associative reasoning was larger for the associative group (middlebars in Fig. 16; average wA = .666). Nevertheless, that these individuals showed some sensitivity to theinstructed causal structure is indicated by the significant contribution of disablers to the commoncause subjects, conjunctive reasoning to the common effect subjects, and normative reasoning forall three networks. But while perhaps not the only influence, these weights bolster the claim that asso-ciative reasoning dominates the inferences of a large minority of causal reasoners.

contribution of each reasoning component was tested by setting its associated w = 0, refitting the model, and then askingr a poorer fit obtained according to a measure that corrects for the number of parameters p: RMSE = SQRT (AVG_SSE/(n � p))AVG_SSE = sum of squared error averaged over the 100 re-samples and n = number of data points fit (15). Eliminatingive reasoning (by setting each network’s associated wA = 0) resulted in a poorer fit for all three networks. Eliminatingdisablers (by settings each wD to 0) resulted in a poorer fit for the common cause network only. Eliminating conjunctiveng (by setting wJ = 0) resulted in a significantly poorer fit for the common effect network only.

Fig. 14. 315 Participants from Experiments 2 (justification condition), 3, 4A, and 4B. Independent and dependent choiceproblems are shown as white and shaded bars, respectively. Error bars represent 95% confidence intervals. Model fits (red roundplot symbols) are superimposed on the empirical results. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)


Fig. 15. Predictions for the four models contributing to causal inferences in Experiments 1–4. First column: the normativemodels; second column: an associative reasoning model; third column: the specific shared disabler models; fourth column: theconjunctive reasoning models. Error bars encompass 95% of the 10,000 predictions generated for each choice problem.


11. General discussion

This research asked whether adult human reasoners honor the patterns of conditional indepen-dence and dependence stipulated by causal graphical models. The answer is that they frequently donot. Of course, it is important to emphasize that subjects’ inferences were more often right thanwrong. They readily inferred a cause when its effect was present and vice versa and chose the correct‘‘equally likely’’ response on independent choice problems most of the time (68%). But on the rest ofthe independent problems they exhibited a systematic bias in which they chose the alternative withmore causally related variables 74% of the time. And, when asked to infer a cause given a commoneffect, subjects only discounted (i.e., chose the alternative in which the alternative cause was absent)24% of the time. That is, people exhibit a small but tenacious tendency to emit associative responses tocausal reasoning problems. There was also evidence that inferences were influenced by shareddisablers and a conjunctive reasoning strategy.

In the following sections I first review the evidence concerning whether these errors are due tosubjects’ domain knowledge. I then review other evidence for the use of alternative reasoning strate-gies during causal reasoning. I close with a discussion of future research and the implications theseresults have for the use of causal graphical models in cognitive modeling.

11.1. Causal inferences and prior knowledge

As discussed, there have been many past demonstrations of apparent violations of the Markovcondition in both the philosophical and psychological literatures. But in each instance the Markov

Fig. 16. Average model parameters bootstrapped over the 315 subjects from Experiments 2–4 who provided justifications. Ineach panel, bars on the left are the weights estimated from the responses of all 315 subjects, bars in the middle are those for the73 associative reasoners, and bars on the right are those for the 242 causal reasoners. Error bars are the standard errorsassociated with the 100 bootstrap samples.



condition has been defended by assuming that people reasoned with knowledge in addition to thatstipulated by the investigators. Indeed, in every previous presentation of the current research (e.g.,talks at conferences) at least one member of the audience has argued that the violations must bedue to the variables being related in ways in addition to the causal links provided as part of the exper-iment. Therefore, it is important to review the arguments for and against this possibility.

The Introduction noted several prima facie arguments against the use of domain knowledge inthese experiments. The materials were counterbalanced (thus averaging out any effect of prior knowl-edge that may exist), were not categories (eliminating any domain general assumption that typicalvalues were causally related, because there were no typical values), and sometimes specified no do-main at all (the blank materials in Experiment 3). But as mentioned, it is possible that subjects elab-orated their causal model as part of comprehending the materials (or in response to the situationsabout which they reasoned) and even used knowledge with the blank materials (by reasoning by anal-ogy with familiar domains or positing abstract knowledge structures). For these reasons, I also con-ducted a theoretical analysis of the models in Fig. 5. This analysis revealed that neither the shareddisabler nor the shared mediator model was able to provide a complete account of subjects’ infer-ences, particularly those for chain and common effect structures.

The model that posits a shared generative cause fared better as an account of subjects aggregateresponses, but faces two challenges, one theoretical and one empirical. It is first important to clarifythat this model does not merely posit that reasoners believe that the variables are (somehow) related.Although is likely that people believe that variables in complex domains like economics, meteorology,and sociology are intricately related (even without knowing what those relations might be), thisexpectation is of no use in making predictions. Imagine a person who believes that there are causalrelations that link economic variables like interest rates, trade deficits, and retirement savings andwho then learns that Venezuela has a small trade deficit. Should that person predict that Venezuelahas high or low retirement savings? Because the link between trade deficits and retirement savingsmay take many forms (small trade deficits may cause either high or low retirement savings, therelationship may be inhibitory rather than generative, etc.), the mere expectation that such links existprovide no information about how the probability of one variable should change given another.

Instead, to account for the present results the shared generative cause model must have a partic-ular functional form, namely, the Ws in Fig. 5C must be generative causes of the variables senses thatare involved in the other causal relations. The challenge then is to explain what led subjects to assumejust this structure. A couple of possibilities were considered. The search of memory initiated by com-prehending the materials may be biased toward information involving the causally related variablesenses (so that, e.g., low interest rates ? small trade deficits yields other facts about low interest ratesand small trade deficits but not high interest rates and large trade deficits). But although the resultmight be a shared cause structure (or even a model elaborated with additional direct inter-variablelinks), this alone is insufficient. One must also assume that those retrieved structures are generativerather than preventative.

The second possibility is to suppose that subjects postulated the presence of the shared generativecause when confronted with cause-absent/effect-present situations. If instructed that low interestrates causes small trade deficits (and high retirement savings) and then confronted with a high interestrate economy that nevertheless has small trade deficits, one can conclude that there must be someother cause of small trade deficits. But this is insufficient. One must also assume that that hiddencause is not only also a cause of the other variable in the model (retirement savings) but that it is agenerative cause of the variable sense that is causally related to interest rates (high retirementsavings).

In the absence of independent theoretical justification, both of these possibilities strike this authoras hopelessly post hoc. But readers who take them seriously should note the second difficulty, which isthat a model elaborated with a shared generative cause should still exhibit sensitivity to causal direc-tion, and of course that is what the associative reasoners in Experiment 1 failed to do. More generally,the existence of so many subjects whose common cause and common effect inferences were indistin-guishable casts doubt on virtually all explanations in terms of additional causal knowledge, because allpredict some residual sensitivity to causal direction.


In summary, there appears to be no comprehensive explanation of the reasoning errors in terms ofthe models in Fig. 5.8 The claim is not that such structures do not sometimes influence causal inferencesof course. As mentioned, domain knowledge is known to infiltrate formal reasoning problems (produc-ing, e.g., the belief bias and suppressions effects reviewed earlier) and so there is no doubt that causalreasoning scenarios, both in everyday reasoning and prior studies, are often embellished with knowledgebeyond the stated facts (shared disablers for common cause structures; see below). But this cannot bethe whole story. Above and beyond these embellishments, reasoners have a tendency to convert a causalinference into an associative one, as now discussed.

11.2. Associative reasoning as a heuristic for causal inferences

Associationist thinking has been found in other studies of causal reasoning. Rehder (2011, 2013)found independence violations consistent with associative reasoning when testing how people reasonwith conjunctive causes (in which, e.g., two causes are necessary for an effect). Rehder (2009) testedhow new properties are generalized to categories on the basis of causal relations and found, as here,that a minority of reasoners treated those relations as a symmetrical associative link (also see Rehder,2006a). Burnett (2004) found that the degree to which reasoners violated the Markov condition variedas a function of proximity in a network, consistent with the spreading activation view of inference de-scribed earlier.9 Evans et al. (2008) presented subjects with everyday causal conditionals of the form if pthen q (e.g., ‘‘if more people use protective sun cream then cases of skin cancer will be reduced’’) and thenasked to what extent each of four statements corresponding to pq, p�q; �pq; and �p�q supported, contradicted,or was irrelevant to the conditional (the ‘‘truth table task’’). The responses of a substantial minority ofparticipants were consistent with a symmetrical (i.e., biconditional) interpretation of the conditionalin which q implies p as much as p implies q (also see Newstead et al., 1997). Associative reasoning ap-pears to be a widespread response to requests to draw causal inferences.

Dual process accounts of reasoning readily explain such inferences by attributing them to the out-put of a fast, intuitive associative system before it is corrected by an analytic reasoning component.However, this simple story is undermined by both the failure of manipulations known to increase ana-lytic responding (e.g., instructions to justify answers) to reduce the rate of Markov violations and thefact that associative reasoners did not respond more quickly. Apparently, even careful and deliberatethinking sometimes produces an associative response to a causal reasoning problem.

These findings demand a theoretical account of how human reasoners’ can possess a tendency toreason associatively while also being responsible for the examples noted earlier of causal-based rea-soning in the domains of learning, decisions making, analogy, and categorization. One interpretation isthat people possess veridical causal reasoning abilities but sometimes fail to deploy them. Dual pro-cess theories may again be consulted to gain insight into the factors that determine when such pro-cesses are invoked. Kahneman and Frederick (2002) argue that the analytic reasoning systemmonitors responses of the associative system and intervenes when there is reason to suspect an error.The criterion it uses may often be lax, however. For example, adults often respond incorrectly to thewell-known question ‘‘A bat and ball cost $1.10 in total. The bat costs $1 more than the ball. Howmuch does the ball cost?’’ even though they are presumably able to carry out the elementary arithme-tic operations needed to compute the correct answer.

8 Other explanations of independence violations that rely on abstract, domain general knowledge structures have been offered.Buchanan, Tenenbaum, and Sobel (2010) proposed that people assume that causal links usually involve more than a singlegenerative process in which a cause produces an effect. Instead, they assume the link is constituted by one or more intermediatevariables that may themselves have disablers, enablers, and additional effects. The causal inferences predicted by this model(CERP) are a result of averaging over the many representations produced by a generative process that randomly generatesintermediate variables. But although Buchanan et al. claim that CERP reproduces the independence violations found here and inprevious studies (i.e., Mayrhofer et al., 2008; Rehder & Burnett, 2005; Walsh & Sloman, 2008), this is accomplished by simplyrunning CERP on a single causal link, after which all generated casual models that do not match the instructed model areeliminated. But this means that the CERP’s explanation of independence violations lies as much in this editing process as itsassumptions about how causal links are represented. It is also unclear how CERP could explain the reasoning errors found withchain and common effect networks.

9 For example, given the causal network W ? X ? Y ? Z and told the state of X, reasoners were more likely to infer the presenceof W when Y or Z was also present (violating the Markov condition) but this effect was larger for Y than Z.


Associative responses to causal reasoning questions may be especially unlikely to alert the analyticsystem because of the relative ease with which those inferences are presumably generated. For exam-ple, the fluency of processing can serve as a cue that affects a wide variety of judgments, includingfamiliarity (Monin, 2003), frequency (Tversky & Kahneman, 1973), typicality (Oppenheimer & Frank,2008), affect (Reber, Winkielman, & Schwarz, 1998) and even intelligence (Oppenheimer, 2005) andfame (Jacoby & Dallas, 1981). In the domain of reasoning more specifically, Thompson, Prowse Turner,and Pennycook (2011) have proposed that initial inferences generated by the associative system areaccompanied by a feeling of rightness (FOR), a claim supported by their finding that lower FOR judg-ments predicted longer rethinking times and a greater chance of answer change (also see Alter,Oppenheimer, Epley, & Eyre, 2009). On this account, the fluency or FOR that accompanies associativeinferences lulls the analytic system into acquiescence.

Another reason that associative inferences might seem acceptable is that they are sometimes usedas a defensible approximation to causal ones. Because everyday reasoning often involves many causalfactors (and because the required computations grow with the size of the causal network), peoplemight rely on (approximately correct) associative inferences instead. And, in real life variables are of-ten believed to be related in causal cycles (e.g., Kim & Ahn, 2002; Sloman, Love, & Ahn, 1998). Althoughthere are extensions to CGMs that address cycles (Murphy, 2002; Rehder & Martin, 2011; also see Kim,Luhmann, Pierce, & Ryan, 2009), a simple solution may be to reason associatively instead. That peoplelearn from experience that associative inferences are often useful may contribute to their feeling ofrightness even in circumstances that neither tax cognitive resources nor contain cycles (e.g., the smallcausal networks about which subjects reasoned in these experiments).

In summary, I take it as a given that individuals who, like the present subjects, have navigated lifesuccessfully enough to be admitted into a major university have reasoning abilities above the merefollowing of associations. Actions taken to achieve important life goals are unlikely to have the desiredconsequences if they are uninformed by cause/effect relations (Hagmayer & Sloman, 2009). But whatmany may lack is the metacognitive awareness that causal and associative inferences do not alwaysproduce the same answer. This ignorance leads us to readily accept associative answers that are easyto compute and which experience has taught are often correct.

Another question concerns the source of the individual differences in the use of associative reason-ing. Recall that about 60% of the associative inferences that violated independence were made byabout a quarter of the subjects. Because Experiment 2 suggested that this result is not due to differ-ences in subjects’ motivation (Bless & Schwarz, 1999), it may reflect differences in their cognitivecapacity. Indeed, Evans et al. (2008) found that biconditional responses to the truth table task werethe modal pattern of responding for subjects classified as lower ability via the AH4 intelligence test(also see Evans, Handley, Neilens, & Over, 2007). These results suggest research in whichmeasures of cognitive capacity are correlated with people’s tendency to violate independence andnot discount. The metacognitive monitoring abilities required to detect erroneous associative re-sponses may not map one-to-one onto an analytic reasoning system, however. Thompson et al.(2011) have proposed that the system responsible for monitoring can be dissociated from analyticreasoning per se, implying that the best predictors of independence violations may be scales that as-sess individuals’ willingness to engage the analytic system (see, e.g., Stanovich & West, 2007) ratherthan cognitive capacity per se.

11.3. Other influences on causal reasoning

Two other influences on subjects’ inferences were identified. Common cause inferences were con-sistent with reasoners sometimes assuming that the two links shared a disabler, as evidenced by thelarger violations of independence on Z-present vs. Z-absent choice problems and model fitting thatyielded a better fit when a shared disabler was included. Direct evidence for this conclusion was pro-vided by Park and Sloman (2013), who manipulated whether the verbally-described links of a com-mon cause structure were viewed as consisting of the ‘‘same’’ or ‘‘different’’ mechanisms and foundlarger Markov violations for the former, implying that the malfunctioning of one link can generalizeto the other. Nevertheless, independence violations obtained even in their different mechanism con-ditions, as predicted by an associative reasoning view. Although these effects were not consistently


significant (they were in their Experiment 1 but not 2 and 3), the apparent discrepancy with the re-sults reported here is resolved by noting Experiment 1–4’s use of a more sensitive forced choiceprocedure.

Evidence for use of disablers in a chain network is more equivocal. On one hand, Park and Slomanfound that a different kind of mechanism manipulation (whether or not sliders were the same color)had an analogous effect on Z-present chain inferences. The current experiments as well found the dif-ferences in the Z-present and Z-absent choice problems that are diagnostic of a chain structure with ashared disabler. On the other hand, Mayrhofer et al.’s (2010, Experiment 2) mechanism manipulation(whether alien mind readers were described as senders or receivers) failed to yield Z-present/absentdifferences when tested on a four element causal chain and the mixed model tested here did not yielda significantly better fit when it included a shared disabler, suggesting that reasoners may be lesslikely to posit shared disablers for chain as compared to common cause structures. There was no signof the influence of shared disablers on the common effect inferences.

A third non-normative influence on common effect inferences was a conjunctive reasoning strategyin which subjects apparently judged the probability of variable t in situation s as p(t, s) rather thanp(t|s). Research on the truth table task provides additional support for this strategy. Evans, Handley,and Over (2003) found that a substantial minority of subjects judged that the causal conditional if pthen q was supported by pq, but not p�q; �pq, and �p�q, consistent with a conjunctive interpretation (alsosee Oberauer & Wilhelm, 2003). This interpretation may be invoked more frequently by lower abilityindividuals (Evans et al., 2007; Oberauer, Geiger, Fisher, & Weidenfeld, 2007). However, while a con-junctive strategy is not uncommon when the materials are abstract, it is rare when they are concrete(Evans et al., 2008; Over, Hadjichristidis, Evans, Handley, & Sloman, 2007), in contrast to Experiment 4that found its use in the concrete domains of economics, meteorology, and sociology. But those causalrelations were also unfamiliar, raising the possibility that conjunctive reasoning might be promotedby unfamiliar rather than abstract materials. Future research should identify other conditions thatcontrol when this strategy is deployed. Open questions include, for example, why its use was espe-cially prominent for the common effect inferences and in the experiments in which reasoners pro-vided justifications.

11.4. Limitations and directions for future research

A number of other factors might affect people’s tendency to treat a causal inference as an associa-tive one. Although the domain variables tested here were described as binary, some subjects may haveinterpreted them as continuous (e.g., interest rates, trade deficits, and retirement savings in the domainof economics), and perhaps continuous variables are especially likely to elicit associative reasoning.10

Such reasoning may also be less likely with ‘‘additive’’ binary variables (Gati & Tversky, 1984), that areeither present or absent (vs. the ‘‘substitutive’’ ones tested here that were, e.g., ‘‘high’’ or ‘‘low’’). Substi-tutive variables might have invited an interpretation of the causal links as having a dual sense (e.g., thatlow interest rates ? small trade deficits implies high interest rates ? large trade deficits), exacerbatingthe Markov violations of some subjects. Consistent with this possibility, Mayrhofer et al. (2010) foundlarger violations when the variables were substitutive (an alien could be thinking ‘‘POR’’ or ‘‘TUS’’) ratherthan additive (‘‘POR’’ or nothing). Relatedly, Markov violations may be affected by whether the variablesrepresent temporally distinct and mechanical ‘‘events’’ rather than states. For example, Park and Slo-man’s (2013) slider experiment did not yield violations on Z-absent problems, consistent with the ideathat an ‘‘off’’ variable does not make other variables less likely to be ‘‘on.’’ More generally, the use ofmechanical events may have contributed to the success of Bayes net accounts of learning in the ‘‘blicketdetector’’ line of studies that rely on conditional and unconditional independence (e.g., Gopnik, Glymour,Sobel, Schulz, & Kushnir, 2004; Sobel et al., 2004).

Second, one might ask whether the errors were exacerbated by the forced-choice procedure inwhich subjects chose which of two scenarios were more likely to have a variable, a judgment that

10 For example, when instructed that low interest rates ? small trade deficits, some subjects might have reasoned that the lowinterest rates of an economy would go lower still when also told that its trade deficits were small, and these extra low interest ratesaffected retirement savings.


can be construed as requiring two causal inferences (one in each scenario). However, earlier experi-ments in this line of research in which subjects simply rated the likely presence of a variable in a sce-nario found that those ratings generally reflected the choices made in Experiments 1–4 (Rehder,2006b). These results suggest that the errors are unlikely to be an artifact of the forced choiceprocedure.

Third, the current subjects were asked to reason about observed scenarios but other cover storiesmight promote more causal reasoning. One might ask subjects to imagine intervening on one variable(e.g., an effect in a common cause structure) and to judge the probability of another (the other effect)before and after the intervention. The logic of CGMs dictates that the intervention renders the effectsindependent (Pearl, 2000). However, Waldmann and Hagmayer (2005) have shown that while reason-ers are sensitive to the difference between interventions and observations (e.g., the other effect isjudged more likely when the first effect is observed present rather than set to be present), full condi-tional independence does not obtain (e.g., the other effect is judged more likely when the first effect isset to be present vs. absent) (also see Hagmayer & Sloman, 2009; Sloman & Lagnado, 2005). Furtherafield, that people interpret their own actions as diagnostic of desirable properties (tolerance to coldwater is diagnostic of a healthy heart) even when the action is only taken to license the inference(Quattrone & Tversky, 1984; also see Sloman, Fernbach, & Hagmayer, 2010) suggests that interven-tions are no sure path to veridical causal inferences.

Finally, causal relations learned from observed data may be less susceptible to independence vio-lations. One extension of these experiments would be to instruct subjects on causal relations but alsopresent data that manifests the patterns of conditional independence implied by the relations. Or, onecould present data, ask subjects what causal relations they induced, and then present inferences thatassess the Markov condition for those learned relations. Past studies paint a mixed picture regardingthis possibility, however. On one hand, Von Sydow, Hagmayer, Meder, and Waldmann (2010) foundthat learners correctly treated causes of a common effect structure as uncorrelated after observingdata (also see Hagmayer & Waldmann, 2000; Perales, Catena, & Madonado, 2004 for related results).On the other, the violations of common cause and chain independence in Park and Sloman’s (2013)third experiment occurred after subjects observed data (see Fernbach & Sloman, 2009; Hagmayer &Waldmann, 2007; Luhmann & Ahn, 2007; Rottman & Hastie, 2013; Steyvers, Tenenbaum, Wagenmak-ers, & Blum, 2003; for more examples). Learning data may also not guarantee adherence to the Markovcondition.

11.5. Implications for cognitive modeling

It is important to consider the implications these results have for the use of causal graphical modelsin cognitive modeling generally. The inferential procedures that accompany CGMs all rely on the Mar-kov condition for their justification and so the finding that adults often violate independence raisesquestions about the appropriateness of CGMs as a framework for understanding the many cognitivephenomena to which they have been applied. How researchers should respond to this concern de-pends on their goals.

First, those wishing to establish the existence of causal representations in a domain are advised toavoid tests of independence inspired by the Markov condition, which, due to the virtual impossibilityof fully suppressing prior knowledge and associative reasoning, are highly likely to yield negative re-sults. A better approach is test for the specific patterns of reasoning asymmetries that are character-istic of causal knowledge. For example, whereas most empirical tests of independence have failed,many of the comparisons of, say, common cause and common effect networks have yielded asymme-tries in the expected direction (e.g., Rehder, 2003; Rehder & Hastie, 2001; Waldmann & Holyoak, 1992;Waldmann et al., 1995).

Second, researchers who are not especially interested in independence violations but who wish tomodel causal reasoning in a domain might treat those violations as a sort of ‘‘nuisance parameter’’ byincluding mechanisms that partial out their influence. One approach is to blend together the predic-tions of a CGM and an associative reasoning network, as done in the mixed model above. Dependingon context, it may be reasonable to assume that people are reasoning with one of the structure typesin Fig. 5. For example, inferences in the Rehder (2013) study described above that compared how


people reason with independent and conjunctive causes were fit to those networks embellished with ashared generative causes, which reproduced not only the differences between independent and con-junctive causes but also the independence violations. A shared generative cause model was appropri-ate in that study because the inferences were between features of categories that might be especiallylikely to be viewed as related by shared mechanisms. In addition to the models in Fig. 5, anotherapproach is to assume the presence of additional generative causal relations (one in each direction)between each pair of causally related variables (see Friedman, Murphy, & Russell, 1998; Richardson,1996; Spirtes, 1993, for techniques to deal with the cycles that result). Which approach is appropriatewill vary with context.

Most importantly, the present results should stem the development of new formalisms thatexplicitly model peoples’ associationist reasoning tendencies. Some possibilities suggest themselves.First, because they can represent non-causal interactions among variables not captured by CGMs,undirected graphical models known as Markov random fields may have the needed flexibility (seeDanks, 2007; Koller & Friedman, 2009; Smyth, 1997, and Appendix C for an implementation of asso-ciative reasoning as a Markov random field; also see Lauritzen & Richardson, 2002, for descriptionsof chain graphs that combine CGMs and Markov random fields). Note, however, that merely convert-ing CGMs into a Markov random field by removing arrowheads is insufficient, not only because rea-soning asymmetries are eliminated entirely but because Markov random fields stipulateindependence constraints that are analogous to those of CGMs (e.g., interpreted literally as a Markovrandom field, the graph of Fig. 8 stipulates that X and Y are independent given Z). Second, one mightcontemplate abandoning graph-based formalisms. After all, a central motivation for such represen-tations is that one can ‘‘read off’’ independence relations without regard to the exact functionalrelationships between variables (i.e., how the graph is ‘‘parameterized’’), a property that has littlevalue for a psychological theory if people systematically violate independence. Yet, this alternativeseems undesirable as graphs support other key operations, such as counterfactual reasoning(Hiddleston, 2005; Pearl, 2000; Rips, 2010) and anticipating the impact of interventions (see litera-ture cited earlier). Finally, independence violations may be modeled by considering the psychologicalprocesses by which a joint distribution (and hence inferences) is derived from a graph. The networkwith which people reason is likely to be constructed piecemeal and on the fly, and our lab has foundthat certain algorithmic shortcuts that expand an existing joint to accommodate a new networkvariable can yield independence violations.

In summary, graph based representations have the potential to make lasting contributions to thepsychology of reasoning because of the structured, generative, and probabilistic representations theyinspire. They will also serve the field by fulfilling the function of any normative model, which includesproviding approximate, ‘‘back of the envelope’’ predictions and identifying those places wherethe development of new theory is needed. However, that the independence relations they stipulateare not surviving experimental scrutiny means they will fail to reproduce some important qualitativeaspects of human reasoning. Future research should be oriented toward the development of newmodels that incorporate what are now two well-established facts: (a) that people are sophisticatedcausal reasoners and (b) they violate the patterns of independence stipulated by causal graphicalmodels.

11.6. Conclusion

Five experiments established that adults’ tendency to reason associatively resulted in them oftenviolating independence in causal inferences. That the rate of these violations was unaffected bymanipulations known to affect fast and intuitive reasoning processes suggests that an associative re-sponse to a causal reasoning question is sometimes the product of careful and deliberate thinking.That about 60% of the associative inferences were made by about a quarter of the subjects suggeststhat such reasoning may be influenced by individual difference variables in monitoring. Theories thatstrive to provide high fidelity accounts of human causal reasoning will need to relax the independenceconstraints imposed by graphical models.


Acknowledgments

I thank David Danks, Juhwa Park, Ben Rottman, and Michael Waldmann for their comments on pre-vious versions of this manuscript.

Appendix A

A.1. Causal relationships

The complete list of causal relationship in the domains of economics, meteorology, and sociologyare presented in Tables A1–A3, respectively.

Table A1Causal relationships for the domain of economics.

Version Causal link

Interest Rates ? Tradedeficits

Interest Rates ? Ret. savings Trade deficits ? Ret. savings

Value 1 ? Value 1 Low interest rates causesmall trade deficits. The lowcost of borrowing moneyleads businesses to invest inthe latest manufacturingtechnologies, and theresulting low-cost productsare exported around theworld

Low interest rates cause highretirement savings. Lowinterest rates stimulateeconomic growth, leading togreater prosperity overall,and allowing more money tobe saved for retirement inparticular

Small trade deficits causehigh retirement savings.When the economy is good,people can cover their basicexpenses and so have enoughmoney left over to contributeto their retirement accounts

Value 1 ? Value 2 Low interest rates causeslarge trade deficits. Becausemoney is cheap forconsumers to borrow (e.g., oncredit cards) demand for ‘‘bigticket’’ consumer goods ishigh and large commercialretailers increase theirimports from foreigncountries

Low interest rates cause lowretirement savings. The goodeconomic times produced bythe low interest rates leads togreater confidence and lessworry about the future, sopeople are less concernedabout retirement

Small trade deficits cause lowretirement savings. When theeconomy is good, people areoptimistic and so spendrather than save

Value 2 ? Value 1 High interest rates causesmall trade deficits. Becauseconsumers spend less inorder to avoid high-interestcredit card debt, largecommercial retailers importfewer goods

High interest rates cause highretirement savings. The highinterest rates result in highyields on government bonds,which are especiallyattractive for retirementsavings because they are sucha safe investment

Large trade deficits causehigh retirement savings.People become nervous whentheir economy is no longercompetitive enough in theworld economy to exportproducts, and begin savingfor retirement as a result

Value 2 ? Value 2 High interest rates causelarge trade deficits. The highinterest rates leads to afavorable exchange ratebetween the local currencyand foreign currencies, andconsumers buy manyimported goods because thefavorable exchange ratemakes them cheap

High interest rates cause lowretirement savings. Becauseso many people are makinglarge monthly interestpayments on credit card debt,they have no money left tosave for retirement

Large trade deficits cause lowretirement savings. The lossof local manufacturing jobsmeans that there are peopleout of work, andcontributions to retirementaccounts decreases

Table A2Causal relationships for the domain of meteorology.

Version Causal link

Ozone ? Air Pressure Ozone ? Humidity Air Pressure ? Humidity

Value 1 ? Value 1 A high amount of ozonecauses low air pressure.Ozone, being an allotrope ofoxygen (O3), combines withmore atoms. These densercollections of atoms sinkfrom open regions, creatingless pressure

A high amount of ozonecauses high humidity. Ozoneattracts extra oxygen atomsfrom water molecules,creating a concentration ofwater vapor in that region

Low air pressure causes highhumidity. When pressuredoes not force water vapor tobreak into oxygen andhydrogen atoms, water vaporremains in abundance

Value 1 ? Value 2 A high amount of ozonecauses high air pressure.With more molecules presentin the atmosphere, morepressure is exerted

A high amount of ozonecauses low humidity. Ozoneaccepts extra oxygen atoms,decreasing the amount ofoxygen available to formwater molecules. With fewerwater molecules, there islower humidity

Low air pressure causes lowhumidity. Low air pressurepoorly facilitatescondensation; as a result,there are less watermolecules in the air

Value 2 ? Value 1 A low amount of ozonecauses low air pressure. Thelower number of ozonemolecules means that airmolecules are less dense,which results in lower airpressure

A low amount of ozonecauses high humidity. Theoxygen atoms that wouldnormally be part of ozonemolecules are free tocombine with hydrogenatoms instead, creating watermolecules

High air pressure causes highhumidity. The higherpressure means that thecomponents of watermolecules (hydrogen andoxygen) tend to notdissociate from one another.Because there are more watermolecules, humidity is higher

Value 2 ? Value 2 A low amount of ozonecauses high air pressure.When the amount of ozone(O3) is low, there are moreoxygen (O2) atoms present inthe atmosphere, resulting inhigher air pressure

A low amount of ozonecauses low humidity. The lowamount of ozone allows alarge number of ultra-violet(UV) rays to enter theatmosphere, and the UV raysbreak up water molecules,resulting in low humidity

High air pressure causes lowhumidity. When air pressureis high, water vaporcondenses into liquid water(rain), and the atmosphere isleft with little moisture


Appendix B

B.1. Predictions of elaborated causal models

The sections below present the predictions for the elaborated models shown in Fig. 5 for the fivechoice problems presented in this study. These qualitative arguments were confirmed by computersimulations that instantiated each of the models with 10,000 randomly generated parameter setsand deriving the predictions each instantiation makes for each choice problem. The temperatureparameter s was set to 1. As described in the main text, there is both specific and general versionof the shared disabler hypothesis. The parameters for the common cause (CC), chain (CH), and com-mon effect (CE) versions of the normative (N), specific shared disabler (SSD), general shared disabler(SSD), shared mediator (SM), and shared generative cause (SGC) models are presented in Table B1. Eachof the elaborated models inherits parameters from the normative (N) model, which are: c, the proba-bility of the cause(s); m, the power of the explicit causal links; and b, the strength of alternative causesof the effect(s). The parameters that are specific to each model are defined in the sections below.

To sample from a more realistic parameter space, values for the parameters that represented thestrength of alternative causes not shown in the model (b and bw) were uniformly sampled from therange [0, .25]; the remaining parameters were sampled from [0, 1]. For each instantiation of eachmodel, choices were computed according to Equation 1. For the chain networks, all the models make

Table A3Causal relationships for the domain of sociology (SE Mobility = Socio-economic mobility).

Version Causal Link

Urbanization ? Religion Urbanization ? SE Mobility Religion ? SE Mobility

Value 1 ? Value 1 A high degree of urbanizationcauses low interest inreligion. Big cities tend tofoster materialism ratherthan spiritualism

A high degree of urbanizationcauses high socio-economicmobility. Big cities providemany opportunities forfinancial and socialimprovement

Low interest in religioncauses high socio-economicmobility. Without therestraint of religious-basedmorality the impulse towardgreed dominates and peopletend to accumulate materialwealth

Value 1 ? Value 2 A high degree of urbanizationcauses high interest inreligion. People are exposedto a large number of differentkinds of religions in cities andusually become interested inone of them as a result

A high degree of urbanizationcauses low socio-economicmobility. In big cities manypeople are competing for thesame high-status jobs andoccupations

Low interest in religioncauses low socio-economicmobility. Many religionsreinforce a strong work ethic;without this motivation,workers become complacentat their jobs

Value 2 ? Value 1 A low degree of urbanizationcauses low interest inreligion. The lack of culturaldiversity in these rural areaslimits access to and thelearning of diverse religiousconcepts. Without this,religion becomes‘‘background noise’’ inpeople’s daily lives

A low degree of urbanizationcauses high socio-economicmobility. People in rural areasare rarely career oriented,and so take time off fromworking and switchfrequently between different‘‘temp’’ jobs

High interest in religioncauses high socio-economicmobility. Religion fosterscommunal care, and those ofthe same religion tend tosupport each other with jobs,financial favors, and so on

Value 2 ? Value 2 A low degree of urbanizationcauses high interest inreligion. Poor rural societieslook to religion as a source ofhope in the future and ameans to escape theirproblems

A low degree of urbanizationcauses low socio-economicmobility. The low density ofpeople prevents the dynamiceconomic expansion neededfor people to get ahead

High interest in religioncauses low socio-economicmobility. The spiritualisminduced by religion works toreduce the desire for materialwealth


the same qualitative predictions regardless of whether the predicted variable is X (the root cause) or Y(the terminal effect) and so the predictions presented in Fig. 11B are the average of these twoinferences.

B.2. Shared disablers

In the shared disabler models in Fig. 5A, W disables the operation of other causal relations. WhenW is a specific disabler and is present, it disables the other two causal links in the model. When it is ageneral disabler and is present, it disables those two links plus any hidden causes of the effects (thuspreventing all occurrences of the effects). Besides c, m, and b (defined above), there were two addi-tional parameters: d, the probability of W; and md, the probability that W, when present, disablesthe other causal links (specific version) or prevents the occurrence of the effects (general version).

Common cause network (Fig. 5A, left panel). When Z is present (situations A, B, and C in Fig. 2), X andY are no longer screened off because from the presence of X (or Y) one can infer the likely absence ofthe disabler W and thus the likely presence of Y (or X). Analogously, from the absence of X (or Y) onecan infer the likely presence of W and thus the likely absence of Y (or X). Thus, reasoners should preferchoice A in the A vs. B problem and B in the B vs. C problem. The predictions for situations in which Z isknown to be absent (situations F, G, and H in Fig. 2) depend on whether the disabler is specific orgeneral. First consider a specific disabler. In situation F, X must have been brought about by somefactor other than Z, and so its presence reveals nothing about the state of W. In situation H, the

Table B1Parameters for the normative models (Fig. 1) and their elaborations (Fig. 5).


Normative Model: NCC Model: NCH Model: NCE

hNCC ¼ fc;m; bg hNCH ¼ fc;m; bg hNCE ¼ fc;m; bg

Shared disabler (Specific) Model: SSDCC Model: SSDCH Model: SSDCE

hSSDCC¼ fc;m; b;d;mdg hSSDCH

¼ fc;m; b;d;mdg hSSDCE¼ fc;m; b; d;mdg

Shared disabler (General) Model: GSDCC Model: GSDCH Model: GSDCE

hGSDCC¼ fc;m; b; d;mdg hGSDCH

¼ fc;m; b; d;mdg hGSDCE¼ fc;m; b;d;mdg

Shared mediator Model: SMCC Model: SMCH Model: SMCE

hSMCC¼ fc;m!w;mw!; bw; bg hSMCH

¼ fc;mm; b; d;mdg hSMCE¼ fc;m!w;mw!; bw; bg

Shared generative cause Model: SGCCC Model: SGCCH Model: SGCCE

hSGCCC¼ fc;m; b; d;mwg hSGCCH

¼ fc;m; b;w;mwg hSGCCE¼ fc;m; b;w;mwg

B.Rehder/Cognitive

Psychology72

(2014)54–

107101


absence of X is fully explained by the absence of Z, again telling us nothing about W (which only mod-erates the influence of Z on X but not X’s other potential causes). Thus, independence still obtains be-tween X and Y in the F vs. G and G vs. H choice problems. These predictions are reflected in thesimulation results in the first panel of Fig. 11A. When the disabler is general, the presence of X in sit-uation F tells us that the disabler W is likely to be absent (and thus Y is likely to be present) whereasthe absence of X in situation H tells us that W is likely to be present (and thus Y is likely to be absent).In other words, a general disabler predicts dependence between X and Y when Z is absent such that Fshould be preferred in the F vs. G choice problem and G should be preferred in G vs. H. These predic-tions are shown in the second panel of Fig. 11A.

Chain network (Fig. 5A, middle panel). First consider the case when the disabler is specific. When Z ispresent, the state of X provides information about the state of specific disabler W (it is less likely to bepresent if X is) and thus Y. When Z is absent, the WZ ? Y interactive link is already disabled, and soany additional information about the state of W provided by the state of X is irrelevant to Y (i.e., X andY are independent). These predictions are confirmed by the simulation results shown in first panel ofFig. 11B. Different predictions arise for a general disabler (second panel of Fig. 11B). When Z is present,W is either absent (or its inhibitory causal mechanism did not operate). Thus, the state of X revealsnothing further about the state of W and so Y; X and Y are thus independent. When Z is absent, thepresence (absence) of X makes the presence of W more (less) likely and thus the presence of Y less(more) likely. Thus, a general disabler predicts a preference for alternative G in the F vs. G problemand H in the G vs. H problem.11 Note that Mayrhofer et al. (2010) also assumed the presence of disablersin a chain network, but in their model the disablers were not shared.

Common effect network (Fig. 5A, right panel). The shared disabler account makes the same predic-tions regardless of whether the disabler is specific or general. First, reasoners should discount because,when Z is present, W is either absent (or its inhibitory causal mechanism did not operate), in whichcase the network reverts to the common effect network in Fig. 1C. Second, when Z is absent, W is morelikely to be present. When X is also present then W becomes even more likely, and thus Y is exoner-ated from responsibility for the absence of Z; thus, Y is also more likely. When X is absent then W be-comes less likely and so Y becomes more responsible for the absence of Z; thus Y also becomes lesslikely. That is, independence should no longer obtain between X and Y in the F vs. G and G vs. H choiceproblems. Finally, when Z is unknown, nothing can be inferred about W and so X and Y remain inde-pendent. These predictions are summarized in the first two panels of Fig. 11C.

B.3. Shared mediator hypothesis

In the common cause and common effect versions of the shared mediator models (Fig. 5B), the cau-sal links are mediated by W. Besides c and b, the additional parameters were: m?W, the power of thecausal links into W; mW?, the power of the causal links out of W; and bW, the strength of the alter-native causes of W. For the chain network, X ? Z and Z ? Y are mediated by M1 and M2, respectively,which in turn have a shared disabler W (Park & Sloman, 2013). The additional parameters were: mm,the power of the causal links into and out of M1 and M2; d, the probability of W; and md, the proba-bility that W, when present, disables the X ? M1 and Z ? M2 causal links.

The Introduction noted how a common cause model elaborated with a shared mediator results inthe effects becoming independent when Z is known, a conjecture confirmed by the simulation resultspresented in the third panel of Fig. 11A. For a chain network when Z is present, the presence (absence)of X raises (lowers) the probability of M1, which lowers (raises) the probability of W, which raises(lowers) the probability of M1, which raises (lowers) the probability of Y; thus, X and Y are dependent.When Z is absent, the WZ ? M2 interactive link is already disabled, so any information that X provides

11 The same pattern holds for chain inferences in the opposite direction, that is, when predicting X from Y and Z. For a specificdisabler, when Z is present, the presence of Y makes W less likely, which means that it is more likely that X is the cause of Z (asopposed to some extraneous factor). But when Z is absent, nothing further can be inferred about W (and thus X) from Y, i.e., thestate of Y is irrelevant to the state of X. For a general disabler, when Z is present, either W is absent or its inhibitory mechanism didnot operate and so X is independent of W (and thus also Y). When Z is absent, the presence (absence) of Y implies the absence(presence) of W and thus that X is more (less) responsible for the absence of Z, i.e., it is less (more) likely to be present.


about W is irrelevant to Y. That is, independence should still obtain on the F vs. G and G vs. H choiceproblems. These predictions are presented in the third panel of Fig. 11B.

For a mediated common effect network, X and Y remain independent when Z is unknown and dis-counting still obtains when Z is present. When Z is absent, the probability that X and Y are both absentbecomes relatively improbable, resulting in a positive correlations between them; thus F is preferredin F vs. G and G is preferred in G vs. H (third panel of Fig. 11C).

B.4. Shared generative cause

In the shared generative cause models (Fig. 5C), X, Y and Z are each linked to the generative causeW. Besides c, m, and b, the additional parameters were: w, the probability that W is present; and mw,the power of the causal links from W to X, Y, and Z. The fourth panel of Fig. 11A indicates that a com-mon cause network elaborated with a shared generative cause results in each variable becoming morelikely to the extent that other variables are present in the model. Applied to a common effect model,the introduction of W results in X and Y no long being independent when Z is absent; thus F is pre-ferred in F vs. G and G is preferred in G vs. H. When Z is present (the discounting trials A vs. B andB vs. C), it results in either discounting or anti-discounting depending on the parameters of the model(fourth panel of Fig. 11C). Choice scores on these problems will tend to be >.50 when the causal linksfrom W are weak and those to Z are strong and <.50 when the links from W are strong and those to Zare weak. Finally, a shared generative cause for a chain network results in dependence between Z andY when Z is absent. Predictions when Z is present are indeterminate (<, =, or >.50), because X, W, and Zform a common effect subnetwork. Thus, X and W will be negatively correlated when W?X is rela-tively weak and X?Z is relatively strong and positively correlated when W?X is relatively strongand X?Z is relatively weak (fourth panel of Fig. 11B).

Appendix C

C.1. Modeling association reasoning

To model associative reasoning, the associative network in Fig. 8 is represented as a Markov ran-dom field with the following factors,

eXY ¼

X ¼ 0 Y ¼ 0 a2

X ¼ 0 Y ¼ 1 0X ¼ 1 Y ¼ 0 0X ¼ 1 Y ¼ 1 a2

26664

37775; eYZ ¼

Y ¼ 0 Z ¼ 0 a2

Y ¼ 0 Z ¼ 1 0Y ¼ 1 Z ¼ 0 0Y ¼ 1 Z ¼ 1 a2

26664

37775; eXYZ ¼

X ¼ 0 Y ¼ 0 Z ¼ 0 a2

X ¼ 0 Y ¼ 0 Z ¼ 1 0X ¼ 0 Y ¼ 1 Z ¼ 0 0X ¼ 0 Y ¼ 1 Z ¼ 1 0X ¼ 1 Y ¼ 0 Z ¼ 0 0X ¼ 1 Y ¼ 0 Z ¼ 1 0X ¼ 1 Y ¼ 1 Z ¼ 0 0X ¼ 1 Y ¼ 1 Z ¼ 1 a2

266666666666664

377777777777775

where a2 and a3 are nonnegative real numbers. The joint distribution is formed by computing,

pðX ¼ x;Y ¼ y; Z ¼ zÞ ¼ expð�ð2XYðx; yÞ þ 2YZðy; zÞ þ 2XYZðx; y; zÞÞÞ

and then normalizing the result (Koller & Friedman, 2009). a2 and a3 capture the pairwise andthree-way degrees of association, respectively, between variables X, Y, and Z. In particular, a3 > 0 im-plies that X and Y are conditionally dependent regardless of the state of Z.

Predictions of this associative reasoning model for the five choice problems presented in Experi-ments 1–4 were computed in a manner analogous to Eq. (1), namely,

choiceassocðt;s1;s2;a2;a3Þ ¼expðlogitðpassocðt¼ 1js1;a2;a3ÞÞ=sÞ

expðlogitðpassocðt¼ 1js1;a2;a3ÞÞ=sÞþ expðlogitðpassocðt¼ 1js2;a2;a3ÞÞ=sÞ


The predictions shown in Fig. 15 were generated from 10,000 random draws of a2 and a3 from therange [0, 2].

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