Cognitive Psychology 72 (2014) 54–107
Contents lists available at ScienceDirect
Cognitive Psychology
journal homepage: www.elsevier.com/locate/cogpsych
Independence and dependence in human causal
reasoning
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
Article history:
Accepted 11 February 2014
Keywords:
Causal reasoning
Causal inference
Causal Markov condition
Conditional independence
Screening off
a b s t r a c t
Causal graphical models (CGMs) are a popular formalism used to
model human causal reasoning and learning. The key property of
CGMs is the causal Markov condition, which stipulates patterns of
independence and dependence among causally related variables.
Five experiments found that while adult’s causal inferences exhibited aspects of veridical causal reasoning, they also exhibited a
small but tenacious tendency to violate the Markov condition. They
also failed to exhibit robust discounting in which the presence of
one cause as an explanation of an effect makes the presence of
another less likely. Instead, subjects often reasoned ‘‘associatively,’’
that is, assumed that the presence of one variable implied the
presence of other, causally related variables, even those that were
(according to the Markov condition) conditionally independent.
This tendency was unaffected by manipulations (e.g., response
deadlines) known to influence fast and intuitive reasoning
processes, suggesting that an associative response to a causal reasoning question is sometimes the product of careful and deliberate
thinking. That about 60% of the erroneous associative inferences
were made by about a quarter of the subjects suggests the presence of substantial individual differences in this tendency. There
was also evidence that inferences were influenced by subjects’
assumptions about factors that disable causal relations and their
use of a conjunctive reasoning strategy. Theories that strive to
provide high fidelity accounts of human causal reasoning will need
to relax the independence constraints imposed by CGMs.
Ó 2014 Elsevier Inc. All rights reserved.
⇑ Address: Dept. of Psychology, 6 Washington Place, New York, NY 10003, United States.
E-mail address: bob.rehder@nyu.edu
http://dx.doi.org/10.1016/j.cogpsych.2014.02.002
0010-0285/Ó 2014 Elsevier Inc. All rights reserved.
B. Rehder / Cognitive Psychology 72 (2014) 54–107
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0. Introduction
People possess numerous beliefs about the causal structure of the world. They believe that sunrises
make roosters crow, that smoking causes lung cancer, and that alcohol consumption leads to traffic
accidents. The value of such knowledge lies in allowing one to infer more about a situation that what
can be directly observed. For example, one generates explanations by reasoning backward to ascertain
the 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 so
hide his car keys.
A large number of studies have investigated how humans make causal inferences. One simple question is: When two variables, X and Y, are causally related, do people infer one from the other? Unsurprisingly, research has confirmed that they do, as X is deemed more likely in the presence of Y and vice
versa (Fernbach, Darlow, & Sloman, 2010; Meder, Hagmayer, & Waldmann, 2008, 2009; Rehder &
Burnett, 2005; see Rottman & Hastie, 2013, for a review). But causal inferences quickly become more
complicated if just one additional variable is introduced. For example, suppose that X and Y are related
to one another not directly but rather through a third variable Z. Under these conditions, the question
of how one should draw an inference between X and Y will depend on the direction of the causal relations that link them via Z. Three possibilities are shown in Fig. 1. First, X and Y might both be effects of
Z (Fig. 1A), a topology referred to as a common cause network. For example, a doctor might diagnose a
disease (Z) on the basis of a particular symptom (X), and then also predict that the patient will soon
exhibit another symptom of that disease (Y). Second, the variables might form a causal chain in which
X causes Z which causes Y (Fig. 1B). For example, politicians may (X) calculate that pandering to
extremists 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 detective
might release an individual (Y) suspected of murder (Z) upon discovering the murder weapon in
possession of another suspect (X).
A formalism that specifies the permissible forms of causal inferences and that is generally accepted
as 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 directed acyclic graphs) in which variables are represented as nodes and directed edges between those
variables are interpreted as causal relations. Note that a CGM need not be complete in the sense that
variables 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)
X
Z
Y
X
Z
Y
X
Z
Y
(B)
(C)
Fig. 1. Three causal networks that can be formed from three variables. (A) A common cause network. (B) A chain network. (C) A
common effect network.
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
inferences that are allowable. Specifically, CGMs stipulate the causal Markov condition, that specifies
the 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, particularly whether they honor the constraints imposed by the Markov condition. This question is
important because Bayes nets have become popular for modeling cognitive processes in numerous domains. For example, CGMs have been used as psychological models of not only various forms of causal
reasoning (Holyoak, Lee, & Lu, 2010; Kemp, Shafto, & Tenenbaum, 2012; Kemp & Tenenbaum, 2009;
Lee & Holyoak, 2008; Oppenheimer, 2004; Rehder, 2009; Rehder & Burnett, 2005; Shafto, Kemp, Bonawitz, 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 (Sloman & Lagnado, 2005; Waldmann & Hagmayer, 2005), decision making (Hagmayer & Sloman, 2009),
and classification (Rehder, 2003; Rehder & Kim, 2009, 2010). Graphical models have also been used as
models 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 that
are taken as candidate models of psychological processes rely on the Markov condition for their justification. Said differently: the Markov condition is at the heart of Bayes nets. Without it, any claim
that knowledge is represented as a Bayes nets amount to no more than the claim that it consists of
nodes connected with arrows. Thus, a demonstration that humans sharply violate the Markov condition 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 causal
inferences. I then review previous research that bears on the psychological question of whether humans 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 will
show 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, and
common effect networks. First, whereas nothing prevents CGMs from including continuous and ordinal 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 relations
that involve multiple variables, here I treat only simple facilitory (or generative) relations between
pairs of variables. Third, those causal relations have a single sense: The presence of the cause facilitates
the presence of the effect but the absence of the cause exerts no influence. Fourth, for the common
effect network I will assume that X and Y are independent causes of Z. Under these assumptions, I
demonstrate 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 knowledge of the state of a subset of variables in a network. Specifically, when that subset includes a variable’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 its
descendants, one has learned as much as possible about a variable once one knows the state of all
of its direct causes. Because non-descendants only provide information about the variable through
the 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 eight
distinct situations in which one may infer the state of Y as a function of the states of X and Z. In
Fig. 2 a ‘‘1’’ means a binary variable is ‘‘present,’’ ‘‘0’’ means that it’s absent, and ‘‘x’’ means that its
state is unknown. Y is always unknown and is the variable being inferred (‘‘?’’). The state of Y’s parent
cause Z is known to be present in situations A, B, and C, known to be absent in F, G, and H, and its state
is unknown in D and E. Situations also vary according to whether X is present, absent, or unknown.
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(I)
(A)
X=1
(B)
Z=1
Y?
Z=1
Y?
X=1
Z=x
Y?
X=0
Z=x
Y?
X=0
Z=1
Y?
Z=0
Y?
(D)
(II)
(E)
(III)
(IV)
(C)
X=x
(F)
X=1
(G)
Z=0
Y?
X=x
(H)
Z=0
Y?
X=0
Fig. 2. Equivalence classes for common cause 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.
Because the labels ‘‘X’’ and ‘‘Y’’ are interchangeable in the common cause network, the situations in
Fig. 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 provide the same inferential support for Y. Classes I and IV illustrate the Markov condition. In class I, the
state of Y’s immediate parent Z is known (it is present) and so knowledge about the state of Y’s nondescendants (namely, X) provides no additional information about Y. Because Z thus screens off Y from
X, situation types A, B, and C provide equivalent support for Y. Because the known (absent) value of Z
screens 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 value
of Y generally become weaker as one moves from class I to IV. Problems in class I in which Y’s immediate cause Z is present generally provide stronger support for Y than that provided by the single problem in class II (D), in which the state of Z is unknown but X is present. However, this distinction
depends on the strength of the causal relations. For example, when the link between X and Z is deterministically necessary (an effect is always accompanied by its cause because it has no other potential
causes), then the presence of Z is certain in problem type D and thus the probability of Y is the same as
in problem types A, B, and C. This possible collapse of classes I and II into a single class due to deterministic 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 value of X is absent (suggesting that Z is absent, and thus so too is Y). Finally, the class of problems in
which Z is known with certainty to be absent (F, G, and H) provides the weakest support for Y of
all. However, when the causal link between X and Z is deterministically sufficient (a cause is always
accompanied by its effect), then the absence of Z in situation D can be inferred with certainty, and thus
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the absence of Y is as certain as in problem types F, G, and H. This possible collapse of classes III and IV
due 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 and
asked to choose the one in which variable Y (or X) is more likely to be present. The situations contrasted will be those required to assess whether reasoners honor the Markov condition. For the common cause network, those pairs are A vs. B, B vs. C, F vs. G, and G vs. H. For example, because Y should
be equally likely in situations A and B, subjects should be no more likely to choose one situation over
the 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 analysis of the chain network is similar to that of the common cause network. Situations A, B, and C form
an equivalence class because the known value of Z screens off Y from the non-descendant X (so that
information about X is irrelevant to predicting Y). Next, situation D should support weaker inferences
to Y than types A, B, or C, because the presence of X in D suggests but does not guarantee the presence
of Z (unless the X ? Z link is deterministically sufficient, as discussed above). Situation E is weaker
still, because the absence of X suggests the absence of Z and thus the absence of Y. But, unless the
X ? Z link is deterministically necessary (i.e., there are no alternative causes of Z), E will be stronger
than situations F, G, and H in which the absence of Z is known with certainty. Finally, types F, G, and H
form an equivalence class because the value of Z screens off X from Y.
(I)
(A)
X=1
(B)
Z=1
Y?
(C)
Z=1
Y?
X=1
Z=x
Y?
X=0
Z=x
Y?
Z=0
Y?
X=0
Z=1
Y?
Z=0
Y?
(D)
(II)
(E)
(III)
(IV)
X=x
(F)
X=1
(G)
Z=0
Y?
X=x
(H)
X=0
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.
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Whereas for a common cause network inferences to either X or Y are qualitatively equivalent, this
is not the case in a chain network, because X is the initial cause and Y is the terminal effect. Nevertheless, an analysis in which X rather than Y is the to-be-predicted variable yields the same result (problem A, B, and C form one equivalence class and F, G, and H another). Although differences between
predicting the initial cause (X) as compared to the final effect (Y) are not uninteresting, I will generally
collapse 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 has
the opposite effect in common effect networks: X and Y are independent in the absence of knowledge
of Z but become dependent when the state of Z is known. The nature of that dependency depends on
how Z is functionally related to it causes. Although in general any functional form is possible (e.g., X
and Y may be conjunctive causes of Z such that X and Y must both be present to produce Z, Y might
disable the causal relation that links X and Z, etc.) as mentioned I focus on cases in which X and Y
are independent, generative causes of Z. Under this assumption, Fig. 4 presents the equivalence classes
(C)
(I)
Z=1
Y?
X=x
Z=1
Y?
X=1
Z=1
Y?
(B)
(II)
(A)
(III)
(D)
(IV)
(V)
X=0
(E)
X=1
Z=x
(F)
X=1
Y?
X=0
(G)
Z=0
Y?
X=x
Z=x
Y?
(H)
Z=0
Y?
X=0
Z=0
Y?
Fig. 4. Equivalence classes for common effect 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
double dashed line (III and IV) are distinct only if the causal relations are not deterministically sufficient. Note that these
equivalence classes hold for the case in which X and Y are independent causes of Z.
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for a common effect network. Of course, the presence of the common effect Z in situations A, B, and C
results 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 is
known to be absent (situation C) is larger when than when its state is unknown (B) which in turn
is larger than when its known to be present (A), a phenomenon referred to as discounting or explaining
away.
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 base
rate, 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 single
sense interpretation of the causal relations, that is, the presence of X (or Y) causes the presence of Z
but 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 causal
relations. When those relations are deterministically sufficient, class III collapses into IV. This is the
case because the presence of variable X in problem type A completely accounts for the presence of
Z. 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 probability
of 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 and
C 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 investigators have asked whether adult human reasoners in fact honor the constraints imposed by the Markov 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 related
by causal knowledge into a common cause network. For example, subjects were told that worrying
causes difficulty concentrating and that worrying also causes insomnia. They were then asked two
inference questions. First, they were asked whether an individual had difficulty concentrating given
that he or she was worried (this corresponds to situation type B in Fig. 2). Next, they were asked
whether a different individual had difficulty concentrating given that he or she was worried but did
not have insomnia (situation type C). Because the state of the common cause Z (worrying) is given
in both questions, Y (difficulty concentrating) is screened off from the additional information provided
about X (insomnia) in the second question. In fact, however, probability ratings were much higher for
the first question than the second one.
Although this result provides prima facie evidence against the Markov condition, results from a
follow-up experiment suggested that subjects were reasoning with knowledge in addition to that
emphasized by the experimenters. Specifically, the absence of one of the effects in the second question
led subjects to assume the presence of a shared disabler that not only explained why the effect X failed
to occur but also led them to expect that it would prevent the presence of the other effect Y. For example, some subjects assumed that the absence of insomnia was due to the individual performing relaxation 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 no
longer do so if that CGM is elaborated to include hidden variables (i.e., variables that were not provided as part of the cover story and not explicitly mentioned as part of the inference question). The
left 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 links
represent interactive causes such that the causal influence of Z on X and Y depends on W; in particular,
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
(A)
X
W
Z
W
Y
X
W
Z
(B)
Y
X
Z
Y
W
W
X
Z
(C)
W
X
Z
M1
Y
X
M2
Z
W
Y
X
W
Y
X
Z
Z
Y
W
Y
X
Z
Y
Fig. 5. The causal networks in Fig. 1 elaborated to include hidden causal influences. (A) Common cause, chain, and common
effect networks in which the causal relationships have a shared disabler, represented by W. The arcs represent interactive causal
influences, in which the influence of one causal factor depends on the state of the other. For example, for the common cause
network in the left panel, when disabler W is present it prevents, with some probability, the operation of the causal mechanism
between 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 generative
cause 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 parents (W) is not known, Y is no longer screened off from X by Z; that is, because X (insomnia) provides
information about W (relaxation exercises), it thus also provides information about Y (difficulty concentrating) even when the state of Z (worrying) is known. For this causal network, the Walsh and Sloman 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 a
shared disabler with chain networks, where the presence of W now disables the X?Y and Y?Z causal
links (middle panel of Fig. 5A). Later, I will present a fuller analysis of how causal inferences are influenced by the possible presence of a shared disabler for all three types of networks, including common
effect networks (right panel of Fig. 5A). But for now, these findings illustrate how situations that may
appear to be counterexamples to the Markov condition may turn out not to be when the causal relations are represented with greater fidelity. Of course, the study of Walsh and Sloman has revealed
some interesting and important facts about causal reasoning. That people will respond to a causepresent/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 violates
the causal sufficiency constraint described earlier: Because W is a causal influence that is common to both X and Y, omitting it
means that exogenous influences are not uncorrelated. This in turn invalidates the expectations of independence stipulated by the
Markov condition.
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causal factors is a significant finding; so too is that they then use this elaborated model to reason
about new individuals. But what this study does not do is provide decisive evidence against the
Markov 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, the
thoughts of one alien (Gonz) could be transmitted to three others (Murks, Brxxx, and Zoohng) but the
cover story provided to subjects was varied. In the sending condition, they were told that Gonz could
transmit its thoughts into the heads of the other aliens. In the reading condition, the other aliens could
read the thoughts of Gonz. Mayrhofer et al. construed both scenarios as involving a common cause
network (with Gonz as the common cause) and thus tested the Markov condition by asking subjects
to predict the thoughts of one of the ‘‘effect’’ aliens (e.g., Murks) given the thoughts of Gonz and the
remaining effects (Brxxx and Zoohng). They found that the effects were not independent: Subjects
predicted 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 receiving
condition.
Rather than interpreting this as a violation of the Markov condition however, Mayrhofer et al. noted
that subjects’ were unlikely to have thought of the situation as involving a simple common cause model. In the sending condition, it is natural to assume that Gonz’s ability to send thoughts relied on a
common sending mechanism. This situation corresponds to the causal model in the left panel of
Fig. 5B in which Gonz’s sending mechanism is represented by W. On this account, if, say, Brxxx does
not share Gonz’s thought, a likely reason is the malfunctioning of Gonz’s sending mechanism, in which
case 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 to
subjects’ 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 Murks
would too).2
Responses to supposed counterexamples to the Markov condition in the philosophical literature
have also appealed to shared mediators. A situation presented by Cartwright (1993) involves two factories that both produce a chemical used to treat sewage but that operate on different days. Whereas
the process used by the first factory produces the chemical 100% of the time, the one used by the second sometimes fails to produce the chemical at all, yielding a terrible pollutant instead. Cartwright
represents this situations as a common cause Z (which factory produced the chemical) producing
two effects, X (the sewage-treating chemical) and Y (the pollutant), and observes that X and Y are
not 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, Hausman
and Woodward (1999) noted that the situation is more accurately represented by the network shown
in Fig. 5B in which the causal influence of factory (Z) is mediated by process (W) that in turn determines the probabilities of the chemical and the pollutant (X and Y). On this analysis, X and Y are only
independent conditioned on W, and thus Cartwright’s scenario fails to serve as a counterexample to
the Markov condition (also see Salmon, 1984, and Sober, 2007, for similar problems with similar
solutions).
These examples again illustrate how failing to include relevant causal factors can invalidate the
patterns 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’ construal 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 was
stronger in the sending versus receiving condition. The results of their Experiment 2, which tested a chain structure, are discussed
later.
B. Rehder / Cognitive Psychology 72 (2014) 54–107
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the inferences they draw. But those findings fail to shed light on whether such inferences in fact honor
the Markov condition.
2.3. Rehder and Burnett (2005)
Rehder and Burnett tested the Markov condition by instructing subjects on categories with features that were linked by causal relations. These categories were artificial in that they denoted
entities that do not really exist. For example, subjects who learned Lake Victoria Shrimp were told
that such shrimp have a number of typical or characteristic features (e.g., a slow flight response, an
accelerated sleep cycle, etc.). Subjects were then presented with individual category members with
missing features (i.e., stimulus dimensions whose values were unknown) and asked to predict one of
those features. These experiments went beyond those of Walsh and Sloman (2008) and Mayrhofer
et al. (2010) by testing all three of the causal networks shown in Fig. 1 (albeit with four variables
rather than three). They also tested a wider variety of materials. Subjects learned not only biological
kinds 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 causal
inferences. 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 had
more typical category features, even when those additional features were (according to the Markov
condition) conditionally independent of the to-be-predicted feature.
Nevertheless, just as in the previous studies, Rehder and Burnett accounted for their results by
appealing to subjects’ use of additional knowledge. They proposed that reasoners assume that categories possess underlying properties or mechanisms that produce or generate a category’s observable
properties, a situation represented in Fig. 5C in which W serves as the shared generative cause. Because one can reason from X to Y (or vice versa) via W, X and Y are conditionally dependent even given
Z. The common cause W also explains the inferences Rehder and Burnett found in the a-causal control
condition (not shown in Fig. 5C): Although not directly causally related to one another, features are
nonindependent because they are all indirectly related via W.
One might ask where these beliefs about categories’ underlying mechanisms come from. They did
not originate from experience with the categories themselves in Rehder and Burnett’s experiments because artificial categories like Lake Victoria Shrimp do not exist. It was also unlikely to have originated
from 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 obtained
with nonbiological kinds and artifacts (and with blank materials in which the ontological domain was
unspecified). Accordingly, Rehder and Burnett concluded that people possess a domain general
assumption that categories’ typical features are brought about by hidden causal mechanisms, that
is, even without knowing what those mechanisms might be. For present purposes, the important point
is that the Markov condition was again rescued by assuming that subjects reasoned with knowledge
beyond that provided by the experimenters.
In summary, the preceding review reveals that apparent violations of the Markov condition can be
explained away by appealing to additional knowledge structures brought to bear on the causal inference. Of course, that prior knowledge can influence reasoning is hardly surprising given the long history 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 conclusion 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). Closer to home, suppression effects arise when conditional statements (if p then q) are interpreted causally
and the reasoner can easily retrieve counterexamples to the rule that imply the presence of alternative
causes 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
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Table 1
Variables in the domains of economics, meteorology, and sociology.
Variable
Value 1
Value 2
Economics
Interest rates
Trade deficits
Retirement savings
Low
Small
High
High
Large
Low
Meteorology
Ozone level
Air pressure
Humidity
High
Low
High
Low
High
Low
Sociology
Degree of urbanization
Interest in religion
Socio-economic mobility
High
Low
High
Low
High
Low
Table 2
Example of causal relationships in the domain of economics that form a common cause network.
Causally relationship
Causal mechanism
Low interest rates ? Small
trade deficits
Low interest rates cause small trade deficits. The low cost of borrowing money leads
businesses to invest in the latest manufacturing technologies, and the resulting lowcost products are exported around the world
Low interest rates cause high retirement savings. Low interest rates stimulate economic
growth, leading to greater prosperity overall, and allowing more money to be saved for
retirement in particular
Low interest rates ? High
retirement savings
in these previous lines of research, reasoners’ prior beliefs complicate the assessment of whether
people 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 causal relations in the domains of economics, meteorology, or sociology. For example, the economic
variables 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 in
Table 1. Subjects were provided with no information about the base rates of variables (e.g., subjects
in 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 caused
another (e.g., low interest rates causes small trade deficits). Which senses of the variables were described as causally related was randomized over participants (e.g., some participants were told that
low 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 formed
either a common cause, chain, or common effect causal network and were accompanied by descriptions of the mechanisms by which one variable produces another. See Table 2 for examples of the
causal mechanisms in the domain of economics. Note that the descriptions of the causal mechanisms
made it clear that they are unrelated (e.g., the two causal mechanisms in Table 2 indicate that the
processes 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 cause
networks (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 each
mechanism operates independently.
B. Rehder / Cognitive Psychology 72 (2014) 54–107
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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 situations, 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 previous
tests of the Markov condition inconclusive. Although university students are unlikely to have extensive prior knowledge about these domains, (reducing the probability that they will elaborate their
causal models with sorts of structures shown in Fig. 5), any such knowledge that exists will tend to
be eliminated by averaging over the three domains and the counterbalancing conditions that varied
which variable senses were described as causally related.3 As an additional safeguard, an experiment
will further minimize the use of domain knowledge by testing of blank materials, that is, the variables
are given the generic labels ‘‘A,’’ ‘‘B,’’ and ‘‘C’’. Finally, that the materials are not categories and so provide
no basis for assuming that only certain dimension values (the ‘‘typical’’ ones) are causally related means
there is no reason to think that the variables are related by a shared generative cause (as assumed by
Rehder & Burnett, 2005).
But although these materials provide a first line of defense against the use of prior knowledge, it is
still possible to conceive of ways that subjects might elaborate their causal model. The processes involved in comprehending the causal relations are likely to trigger a search of memory for related
knowledge and this search may be biased so as to turn up only knowledge relating the variable senses
involved 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 small
trade deficits, I may more readily retrieve facts involving low interest rates and small trade deficits
than ones involving high interest rates and large trade deficits, perhaps yielding the structure in
Fig. 5C. These elaborations could produce apparent violations of independence despite the randomization of the materials because different knowledge structures would get retrieved in the different
randomized conditions. The search of memory might also turn up commonalities between the causal
relations, 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 Walsh
and Sloman’s (2008) subjects apparently did for cause-present/effect-absent situations (Fig. 5A) (analogously, 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 blank
materials, because subjects might reason by analogy to familiar domains or assume the presence of
knowledge structures that are abstract (i.e., lack any concrete representation of the causal processes
involved).
Accordingly, later I will present a theoretical analysis of each of the alternative models in Fig. 5 to
assess their potential as accounts of the causal inferences made in the following experiments. As mentioned, not only will those inferences exhibit numerous violations of the reasoning norms stipulated
by CGMs, all but one of the alternative structures in Fig. 5 will be unable to account for subjects’ aggregate responses for all three causal networks and the model that remains will be unable to account for
the 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 economics, 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 represented
schematically as X1
Z1 ! Y1 , where superscripts denote the value on a dimension (1 or 2). If, in addition to the Table 2 links, the
subject population tends to believe that one of the effects causes the other (say, that small trade deficits causes high retirement
savings, i.e., X1 ! Y1 ), then they will appear to violate independence. For example, the effect Y (whose role is played by Y1) will be
judged 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 is
unknown). The X1 ! Y1 link will have the opposite effect in other conditions however. For subjects who are instead taught
X1
Z1 ! Y2 , the presence of X1 in situation A will make Y (now played by Y2) less likely as compared to situation B (because
X1 ! Y1 ). In this manner, aggregating the results over the different randomized conditions will tend to average out the effects of
prior knowledge.
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presented with a pair of situations and asked to choose which was more likely to possess a particular
variable value. They could also select a third ‘‘equally likely’’ response indicating that neither was
more likely than the other to have that value. The choice problems were those needed to assess the
key predictions of conditional independence and dependence made by the three causal networks: A
vs. B, B vs. C, D vs. E, F vs. G, and G vs. H. In the common cause and chain conditions, subjects should
choose D over E but choose the equally likely alternative otherwise (Figs. 2 and 3). In the common effect 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. Materials
The 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 common
cause, 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 short
description of the mechanism responsible for the causal relationship (Table 2). The senses of the variable that were described as causally related was randomized for each participant. The complete list of
causal relationships used to construct common cause, chain, and common effect networks in each domain are presented in Appendix A.
4.1.2. Design
Choice 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 = common cause, h = chain, e = common effect). Second, the order in which the three domains were presented was either mes, sme, or esm (m = meteorology, e = economics, s = sociology). As a result, each
causal 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. Participants
Sixty-three New York University undergraduates received course credit for participating in this
experiment. They were assigned in equal numbers to the two between-subject counterbalancing
conditions.
4.1.4. Procedure
For each domain, participants first studied several computer screens of information about the domain and then performed the inference test. The initial screens presented a cover story and a description of the domain’s three variables and their two values. Subsequent screens presented the two
causal relationships and their associated causal mechanisms. Participants also observed a diagram
depicting 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 so
obligated them to retake the test. The only way to pass the test and proceed to subsequent
phases was to complete it without error and without returning to the initial information screens
for help.
The feature inference phase presented participants with the five types of choice problems. The two
examples were presented one above the other and participants were asked which was more likely to
have a particular value for one of the unknown variables. For example, the list of variables for one
economy might be ‘‘Low interest rates,’’ ’’Small trade deficits,’’ and ‘‘???’’ (indicating that the value
for 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 more
likely to have high retirement savings. Possible responses were 1 for the first example, 2 for the second
B. Rehder / Cognitive Psychology 72 (2014) 54–107
67
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 dependent
choice 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 problems, one in which the participant was asked to choose which example was more likely to have Y (as
shown in Figs. 2–4), and the corresponding version in which they were asked to which was more likely
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to have X. To average over any bias for choosing the top or bottom example, each of these 10 problems
was presented twice, with the order of the two examples reversed. The order of these 20 problems was
randomized for each participant.
4.2. Results
To construct a single choice score that summarizes subjects’ responses, choices in favor of the first
alternative (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 unaffected
by 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 of
Fig. 6 collapsed over these factors.
Fig. 6 reveals that responses in the common cause and chain conditions were approximately equal
and substantially different from those in the common effect condition. This observation was supported by statistical analysis. A 3 5 ANOVA with causal network and choice problem type as factors
yielded an overall effect of choice problem type, F(4, 248) = 40.6, MSE = .041, p < .0001 and an interaction between problem type and network, F(8, 496) = 5.5, MSE = .025, p < .0001. However, whereas the
interaction between problem type and the contrast between the common cause and chain network
combined vs. the common effect network was significant (p < .0001), the interaction between the
common cause and chain network was not (p > .20). Accordingly, I discuss the common cause and
chain conditions together and then the common effect condition.
4.2.1. Common cause and chain results
On one hand, the common cause and chain choice scores in Figs. 6A and B exhibit some of the properties of normative causal reasoning shown in Fig. 6. When asked whether situation D or E was more
likely to have the causally relevant value of Y (or X), most participants chose D (choice scores of .79
and .90 in the common cause and chain conditions, respectively), consistent with the predictions of
the 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 indirect
inferences, that is, from X to Y or Y to X when the state of Z was unknown.
Table 3
Results from Experiment 1. Normative choices are shown in bold italic.
Choice problem
Causal network
Common cause
Chain
Common effect
A vs. B
A
Equally likely
B
.21
.70
.08
.34
.60
.06
.25
.63
.12
B vs. C
B
Equally likely
C
.30
.65
.04
.44
.49
.07
.29
.58
.13
D vs. E
D
Equally likely
E
.67
.24
.10
.85
.10
.05
.37
.56
.07
F vs. G
F
Equally likely
G
.18
.73
.08
.23
.68
.09
.20
.65
.15
G vs. H
G
Equally likely
H
.13
.80
.07
.19
.75
.06
.18
.74
.08
B. Rehder / Cognitive Psychology 72 (2014) 54–107
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Unfortunately, participants failed to honor independence on the remaining problems in Fig. 6. Recall 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 variables were supposedly screened off from one another. Nevertheless, that these scores were lower than
those for the D vs. E, problem indicated that subjects exhibited some sensitivity to the difference between independent and dependent problems, t(62) = 6.99 and 11.45, in the common cause and chain
conditions, respectively, ps < .0001.
4.2.2. Common effect results
Recall 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 in
the A vs. B choice problem and that C should be preferred in the B vs. C problem. Fig. 6C shows that
subjects 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 independent problems (D vs. E, F vs. G, and G vs. H), the average choice scores (.57) were also significantly
greater than .50, t(62) = 4.22, p < .0001. Only the score for the F vs. G problem (.52) was not significantly greater than .50.
4.2.3. Individual differences
It 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 profiles. In fact, cluster analyses revealed two subgroups of participants with qualitatively different responses. The responses of one cluster of 18 participants, shown in the left side of Fig. 7, were
virtually identical for all three causal networks. That is, 29% of the participants—labeled ‘‘associative
reasoners’’ for reasons discussed below—showed no sensitivity to causal direction and usually chose
the alternative in which more causally related variables were present. Indeed, a 3 5 ANOVA of
these 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 common effect condition as compared to the common cause and chain conditions. They also committed
many 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 associative
reasoners. Nevertheless, when they did not respond correctly, even these individual were more
likely to choose the alternative in which more causally related variables were present. As a
result, their choice scores continued to be significantly above chance on a number of independent
problems (e.g., B vs. C in the common cause and chain conditions and D vs. E in the common effect
condition).
4.3. Discussion
Experiment 1 paints a mixed picture. When reasoning with a common cause or chain network, participants correctly inferred that the states of X and Y provided information about one another when
the state of Z was unknown. But participants also committed numerous apparent violations of the
Markov condition in which supposedly independent variables influenced one another. And, rather
than discounting when reasoning with a common effect structure, they were more likely to predict
the presence of a cause when another cause was already present. Recall that these results obtained
despite the steps intended to minimize the impact of prior knowledge (e.g., randomizing which variable senses were described as causally related). Additional tests of the role of prior knowledge will be
presented 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.
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Fig. 7. Results of Experiment 1 segregated into two participant groups, the ‘‘associative reasoners’’ (N = 18) and the ‘‘causal
reasoners’’ (N = 45). Independent and dependent choice problems are depicted with white and shaded bars, respectively. Error
bars represent 95% confidence intervals.
B. Rehder / Cognitive Psychology 72 (2014) 54–107
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5. Experiment 2
The manner in which inferences in Experiment 1 departed from the normative model—in every
case, choice scores were higher than predicted—provides insight into the nature of those errors. This
pattern is consistent with subjects sometimes adopting an associative reasoning strategy in which the
presence of one variable makes the presence of another more likely. For example, Fig. 8 presents a representation of causal knowledge in which variable senses that are causally related in Fig. 1 are instead
related via symmetrical ‘‘associative’’ links. Violations of the Markov condition will occur if people reason with this representation as if it’s a spreading activation network. To take the materials in Table 2
as an example, the presence of both low interests rates and small trade deficits (e.g., X and Z in choice
problem 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 demonstrate
how it provides an account of the associative reasoners.) Of course, that subjects showed an overall
sensitivity to causal direction (i.e., inferences in the common effect condition differed from those in
the common cause and chain conditions) means that associative reasoning is by itself unable to account for the results of Experiment 1. Instead, the claim is that subjects’ otherwise correct causal inferences are distorted by an associative strategy. The substantial minority of participants showed no
sensitivity to causal direction shown in Fig. 7 provides especially direct evidence for the contribution
of associative reasoning.
This apparent mixing of normative and associative responses raises the possibility that the present
results may be a result of two separate reasoning processes, as stipulated by the well-known dual process theories of reasoning that distinguish between associative and ‘‘analytical’’ reasoning. Although
this distinction has been characterized in different ways, associative reasoning is generally thought
to 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 (see
Darlow & 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 learning). 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 possibility
that people can engage in normative causal reasoning but resort to fast, associative processes in some
circumstances.
Moreover, dual process accounts suggest potential explanations of why individuals differ in their
tendency 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 be
less 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 capable individuals. Alternatively, these subjects might be distinguished not by their cognitive capacity but
rather an unwillingness to ‘‘think hard,’’ that is, to deploy effortful analytical processes (reflecting, perhaps, a ‘‘need for cognition;’’ Cacioppo & Petty, 1982). This latter interpretation is important because it
suggests a deflationary interpretation of the results of Experiment 1, namely, that errors only occur on
artificial laboratory tasks in which reasoners have little invested. Violations of the Markov condition
may be rare during real-world reasoning problems in which people have a stake in the outcome. Indeed, Bless and Schwarz (1999) provide evidence indicating that increased motivation can reduce errors by promoting more deliberative processing.
Experiment 2 investigated whether causal inferences would be influenced by variables known
to affect the contribution of fast, associative processes, namely, time pressure (e.g., Evans &
X
Z
Y
Fig. 8. Associative network interpretation of the causal networks in Fig. 1.
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Curtis-Holmes, 2005; Evans et al., 2009; Finucane et al., 2000; Roberts & Newton, 2001) and asking
subjects to justify their choices (Smith & Sloman, 1994). Half the subjects were assigned to a
justification condition that was designed to reduce associative reasoning in two ways. First, participants talked aloud into a tape recorder while making their decision to justify their answer. Evidence
that talking aloud promotes more analytical, rule-based processing was provided by Smith and
Sloman (1994) who found that classification decisions were more sensitive to features that were related to participants’ theoretical understanding of a category and less sensitive to overall similarity
when participants talked aloud, presumably because talking aloud promotes a search for a verbalizable rule with which to justify the decision. Second, the justification condition sought to decrease
time pressure by (a) having participants learn only one causal network (rather than three as in
Experiment 1), (b) informing them that they would have plenty of time to answer the inference
questions in the 1 h allotted for the experiment, and (c) by asking them to emphasize accuracy over
speed.
The other half of the participants were assigned to a deadline condition designed to promote associative reasoning by placing them under time pressure. This was accomplished by giving participants a
deadline of 10 s to make their response. To implement this deadline, the screen that presented a
choice problem included a counter that began at 10 and counted down to 0 once per second. Note that
the 10 s deadline was intended to induce only mild time pressure because extreme pressure would
simply induce random responding. (This would result in choice scores of 0.5, a finding that could
be 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 Experiment
1), a fact that may also contribute to mild time pressure.
Fewer violations of independence and greater discounting in the justification condition will be consistent with the view that an analytical reasoning component can correct fast, intuitive responses generated on the basis of associative rather than causal relations. The absence of these results will suggest
that associative reasoning is a mindful, deliberate strategy (or that analytic processes are poor at recognizing and correcting such errors; more about this later).
5.1. Method
5.1.1. Participants
Ninety New York University undergraduates received course credit for participating. They were
assigned in equal numbers to the deadline or justification condition. Because deadline participants
learned three causal networks, the same two counterbalancing factors used in Experiment 1 that
rotated the three networks through three presentation orders and three domains were used in that
condition. Because justification participants learned just one network, they were assigned in equal
numbers to one of the three networks and one of the three domains. As in Experiment 1, the
senses of the variable that were described as causally related were randomized for each
participant.
5.1.2. Procedure
The 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 counter
that started at 10 and counted backwards once per second. If no response was made within the 10 s, a
warning message was displayed asking the participant to respond in the allotted time, after which the
computer presented the next problem. In the justification condition, participants were asked to ‘‘think
about this question out loud so that we can record your thinking process’’ and to ‘‘speak out loud why
you made the choice you made, that is, the justification for that choice.’’ These participants were also
told that they would only learn one causal network, that they would have plenty of time to answer the
inference questions, and were asked to emphasize accuracy over speed.
All participants were presented with the same 20 choice problems used in Experiment 1. Four
warm up trials were presented beforehand to familiarize participants with the procedure (the warm
up trials were excluded from the following analyses).
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5.2. Results
An initial analysis found that participants took much longer to respond in the justification condition (average of 22.6 s) as compared to the deadline condition (5.5 s), confirming the effectiveness of
the deadline vs. justification manipulation. The deadline participants failed to respond within 10 s on
fewer than 1% of the trials.
As was the case in Experiment 1, there was no effect of the domain and so choice proportions are
presented in Table 4 and choice scores in Fig. 9 collapsed over this factor. As in Experiment 1, the results were qualitatively different in the common effect as compared to the common cause and chain
conditions and thus those results are reported separately.
5.2.1. Common cause and chain results
Fig. 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, this
tendency was not weaker for those participants who were required to provide justifications. Separate
2 5 ANOVAs of the common cause or chain conditions revealed that the deadline vs. justification
manipulation yielded neither main effects nor interactions with problem type, all Fs < 1. Collapsing
over the deadline and justification conditions, an analysis of the independent problems revealed that
their choice scores (.60 and .63 in the common cause and chain conditions, respectively), were significantly greater than .50, t(62) = 5.78 and 6.20, ps < .0001, just as they were in Experiment 1. Nevertheless, 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 common
cause and chain conditions, respectively, ps < .0001.
5.2.2. Common effect results
The 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 produced a modest increase in normative responding. Choice scores on dependent problems that should
Table 4
Results from Experiment 2. Normative choices are shown in bold italic.
Choice problem
Causal network
Common Cause
Chain
Common effect
Deadline
Justification
Deadline
Justification
Deadline
Justification
A vs. B
A
Equally likely
B
.22
.71
.07
.25
.70
.05
.33
.58
.08
.33
.63
.03
.28
.56
.16
.20
.58
.22
B vs. C
B
Equally likely
C
.39
.57
.04
.30
.65
.05
.48
.44
.08
.55
.43
.02
.33
.52
.15
.15
.48
.37
D vs. E
D
Equally likely
E
.78
.19
.03
.78
.15
.07
.78
.09
.13
.92
.07
.02
.45
.52
.03
.12
.87
.02
F vs. G
F
Equally likely
G
.26
.68
.06
.28
.68
.03
.33
.57
.10
.22
.75
.03
.22
.61
.16
.08
.58
.33
G vs. H
G
Equally likely
H
.13
.83
.04
.18
.75
.07
.22
.66
.12
.17
.80
.03
.22
.69
.09
.27
.62
.12
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
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 preference also decreased (from .60 to .50). As a result, a 2 5 ANOVA yielded a significant main effect of the
B. Rehder / Cognitive Psychology 72 (2014) 54–107
75
justification vs. deadline manipulation, F(1, 58) = 6.09, MSE = 0.119, p < .05 (and a marginal interaction
with problem type, F(4, 232) = 2.10, MSE = 0.038, p = .08). Nevertheless, responses in the justification
condition were still far from normative, as the degree of discounting in that condition failed to reached
significance, 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 the
independent problems in the justification condition was unexpectedly complex (choice scores <.50
for 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 further discussion until then.
5.2.3. Analysis of verbal protocols
Verbal 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 verbalized
their choice. Nevertheless, a few trends of theoretical interest emerged. First, participants sometimes
made reference to outside knowledge, that is, to knowledge in addition to the causal relations they
were taught in the experiment. For example, one subject reasoned that ‘‘high interest in religion
means more focused. . .good work ethics,’’ and their inferences were based on how the other two variables might be related to good work ethic. Another reasoned that ‘‘when there are low interest rates
people spend money’’ and this inference affected their subsequent choices. Overall, participants
showed signs of using outside knowledge on 12% of the trials. Not only is this rate fairly low, recall
that I have argued that the use of prior knowledge cannot explain the overall pattern of results in these
experiments (because such effects will be averaged away due to the randomization of causal materials). Nevertheless, Experiment 3 will take further steps to assess the use of domain knowledge (by
comparing how subjects reason with concrete vs. abstract materials in which no ontological domain
is specified).
Second, recall that situations C and F can be construed as providing inconsistent information (cause
absent 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, in
situations A and H a cause and effect are either both present or both absent, and subjects noted this
fact 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 4A
will assess the potential role of these kinds of consistencies and inconsistencies by manipulating the
whether or not the causal links are described as probabilistic.
Finally, that participants sometimes had trouble remembering the causal relations is evidenced by
the fact that participants described the causal relations incorrectly (e.g., misstated which variable
senses were related or the direction of causality) or explicitly stated they could not remember those
relations on 7% of the trials. This result is a surprise given that subjects passed a multiple-choice test
on this knowledge immediately before taking the inference test. Conceivably, an incomplete representation of the causal links might contribute to the observed errors (e.g., they may have treated the causal links as symmetrical relations because they forgot the direction of causality). Later experiments
address this possibility.
5.3. Discussion
Experiment 2 asked whether the departures from the normative model found in Experiment 1 were
due to fast associative reasoning process invoked by less capable or careless individuals. The answer is
they, 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 and
fail to discount. Providing justifications yielded some improvements on common effect inferences as
discounting increased, although not to a significant level. Apparently, for some people most of the
time, and most people some of the time, providing an associative response to a causal reasoning problem is the product of a careful and deliberate strategy.
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
6. Experiment 3
Experiment 3 returns to the question of whether the violations of causal reasoning norms can be
attributed to subjects’ use of domain knowledge. It does so by comparing a concrete condition that
tested the same materials as in Experiments 1–3 with an abstract condition that used a blank domain
in which variables are simply labeled ‘‘A,’’ ‘‘B,’’ and ‘‘C.’’ Recall that the Introduction raised the possibility that the search of memory that occurred as part of the comprehending the materials might turn
up additional inter-variable causal relations (perhaps yielding the model in Fig. 5C) or commonalities
between the causal relations (5B). The verbal protocols in Experiment 2 showing how a subject inferred good work ethic (a variable not included as part of the instructions) from a high interest in religion (and then considered how a good work ethic might affect the other variables) might be an
example of the latter. And, subjects may have elaborated their models in response to the situations
they reasoned about (e.g., positing a shared disabler to account for cause-present/effect-absent situations, Fig. 5A). That subjects’ verbal protocols revealed that they occasionally noted cause-effect inconsistencies provides some support for this possibility.
Experiment 3 assesses the use of these sorts of elaborations by testing a condition in which domain
knowledge is unavailable. If domain knowledge contributed substantially to subjects’ errors in the earlier 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 spoken
justifications, were asked to emphasize accuracy over speed, and answered questions for just one causal network. In addition, two changes to the experimental procedure were made. First, because the
verbal protocols also revealed that participants showed some signs of forgetting the causal relationships, in this experiment participants were provided with a printed diagram of those relationships
during the inference test. The second change was to emphasize the independence of causal mechanisms just before the start of the inference test.
6.1. Method
6.1.1. Participants
Ninety 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 of
the variable that were described as causally related was randomized for each participant.
6.1.2. Materials
The 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 causing
other 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 A
is low it causes B to be high.’’). Descriptions of causal mechanisms were not provided in this condition.
6.1.3. Procedure
The 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, at
the start of the inference test all participants were given additional instructions emphasizing the independence 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 participants were told ‘‘Remember that both X and Y can each bring about Z on its own. That is, it’s not the
case that both of these two have to be present for Z to be present. Rather, X can independently produce
Z on its own, and Y can independently produce Z on its own as well.’’
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
6.2. Results
The concrete condition showed no effect of whether the causal networks were instantiated in the
domain of economics, meteorology, or sociology, and so choice proportions in Table 5 and choice
scores in Fig. 10 are collapsed over this factor.
6.2.1. Common cause and chain results
Fig. 10 reveals that participants continued to violate the predictions of normative model on independent problem types. Moreover, the frequency of those errors did not vary as a function of whether
the domain was concrete or abstract. 2 5 ANOVAs in the common cause and chain conditions
yielded 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 nor
an interaction with problem type, all Fs < 1. Collapsing over the concrete and abstract conditions, an
analysis of the independent problem types revealed that their choice scores (averages of .59 and .65
in 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 results
Fig. 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 of
problem type, F(4, 112) = 4.25, MSE = 0.027, p < .01; the type by concrete/abstract interaction was marginal, 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 significantly
from .50, t < 1; those on the D vs. E problem that should exhibit independence (.62) were significantly
greater than .50, t(29) = 3.20, p < .01. As seen in the justification condition of Experiment 2, the choice
Table 5
Results from Experiment 3. Normative choices are shown in bold italic.
Choice problem
Causal network
Common cause
Chain
Common effect
Abstract
Concrete
Abstract
Concrete
Abstract
Concrete
A vs. B
A
Equally likely
B
.17
.82
.02
.22
.75
.03
.25
.75
0
.32
.68
0
.05
.92
.03
.20
.63
.17
B vs. C
B
Equally likely
C
.42
.50
.08
.34
.64
.02
.37
.63
0
.42
.58
0
.17
.70
.13
.32
.52
.17
D vs. E
D
Equally likely
E
.57
.37
.07
.52
.47
.02
.87
.12
.02
.90
.10
0
.22
.77
.02
.28
.70
.02
F vs. G
F
Equally likely
G
.13
.85
.02
.10
.82
.08
.27
.65
.08
.38
.62
0
.08
.87
.05
.18
.38
.43
G vs. H
G
Equally likely
H
.17
.77
.07
.21
.75
.03
.27
.73
0
.25
.72
.03
.12
.83
.05
.28
.55
.17
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
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.
B. Rehder / Cognitive Psychology 72 (2014) 54–107
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6.3. Discussion
Experiment 3 asked to what extent subjects’ prior domain knowledge affected their causal inferences in the earlier experiments. In fact, for all three networks the number of independence violations
was 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 even
though all subjects (a) learned one causal network rather than three, (b) were asked to emphasize
speed over accuracy, (c) provided justifications for their choices, (d) could refer to a diagram of the
causal relations during the inference test, and (e) were explicitly told that the two causal links
operated independently.
7. Assessing elaborated causal models
The results of Experiment 3 notwithstanding, it might be argued that even the use of blank materials is insufficient to fully rule out the influence of domain knowledge. As mentioned, it is possible
that some subjects elaborated their causal model on the basis of analogies with familiar domain or
assumed abstract versions of the knowledge structures in Fig. 5 (e.g., assumed the presence of a disabler without any concrete notion of what that disabler might be). This latter possibility is related to
placeholder notions that assume that reasoners have only an abstract representation of causal processes (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 theoretical analysis asking to what extent the models in Fig. 5 account for subjects’ inferences. Exact predictions depend on the models’ parameters (e.g., the base rates of the causes, the strength of the causal
relations, etc.), information that was not provided during the experiment. Thus, each model was assessed by instantiating it with 10,000 sets of randomly generated parameter values. Predictions for
the five choice problems tested in Experiments 1–3 were computed for each instantiation. For model
m instantiated with parameters hm, the probability that target variable t is present in situation si is denoted p(t = 1|si; m, hm). I assume that reasoners represent probabilities as log odds and that deciding
that t is more likely to be present in situation s1 than s2 is made according to a softmax rule,
choicem ðt; s1 ; s2 ; hm ; sÞ ¼
expðlogitðpðt ¼ 1js1 ; m; hm ÞÞ=sÞ
expðlogitðpðt ¼ 1js1 ; m; hm ÞÞ=sÞ þ expðlogitðpðt ¼ 1js2 ; m; hm ÞÞ=sÞ
ð1Þ
where s is a ‘‘temperature’’ parameter that controls the extremity of the responses. Details of these
simulations are presented in Appendix B along with a qualitative description of each model’s
predictions. The predictions averaged over the 10,000 instantiations are presented in Fig. 11. There
are two versions of the shared disabler account. The specific version assumes that the disabler renders
inoperative the two explicit causal links but not other potential background causes of the effect(s). The
general version assumes that the disabler prevents all occurrences of the effect.4 The error bars in
Fig. 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 predictions of the shared generative cause model for the chain and common effect networks (hatched bars in
Fig. 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 first
and second panels of Fig. 11A confirm the observation, first made by Walsh and Sloman (2008), that a
dependency is introduced between the effects such that one should prefer A in the A vs. B problem and
B in the B vs. C problem. Z-absent problems (F vs. G and G vs. H) are predicted to be independent when
the disabler is specific but dependent when it is general, favoring F in F vs. G and (just barely) G in G vs.
4
Carroll and Cheng (2009) referred to a general disabler as a ‘‘broad preventer.’’ However, a specific disabler does not
correspond to their ‘‘narrow preventer,’’ which disables only a single causal relationship.
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
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; fourth
column: shared generative causes. Hatched bars represent indeterminate predictions, that is, cases in which the alternative that
is favored depends on the parameters of the model. Error bars bracket 95% of the 10,000 predictions generated for each choice
problem.
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 the
chain or common effect condition. As observed by Park and Sloman (2013), introduction of a specific
shared disabler to a chain network (middle panel of Fig. 5A) results in the loss of independence when Z
is present (first panel of Fig. 11B). But when Z is absent, problems are either independent (specific
disabler) or dependent (general disabler) in a direction opposite to that of subjects’ preferences. Finally, for a common effect network elaborated (right panel of Fig. 5A), discounting should still occur and
independence 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, closer
examination of those results provides some support for their influence. The presence of a specific disabler 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 all
three 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 both
experiments, 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).
B. Rehder / Cognitive Psychology 72 (2014) 54–107
81
7.2. Shared mediators
Assuming that common cause links are mediated by a common factor (left panel of Fig. 5B) yields
the (apparent) independence violations (third column of Fig. 11A) expected by previous investigators
(Cartwright, 1993; Hausman & Woodward, 1999; Salmon, 1984; Sober, 2007) and exhibited by the
present subjects. Consistent with the claims of Park and Sloman (2013), if the variables that mediate
the causal links (M1 and M2 in Fig. 5B) have a shared disabler (W), apparent violations of independence
will obtain on the Z-present chain inferences (Fig. 11B). This account fails to account for the >.50
choice scores on Z-absent problems, however. For a common effect network (Fig. 11C), the mediation
hypothesis predicts the presence of discounting and independence on the D vs. E choice problems, at
odds with the empirical results. Note that the common effect results also rule out some combination of
disablers and meditators, as both predict discounting and the independence of X and Y when Z is
unknown.
7.3. Shared generative cause
The predictions of the shared generative cause hypothesis (Fig. 5C), shown in the fourth column of
Fig. 11, indicate that it fares better than the previous accounts. It correctly predicts the pattern of independence violations in the common cause condition. In the chain condition, it correctly predicts the Zabsent problems. Its predictions for the Z-present problems (A vs. B and B vs. C) are parameter dependent (choice scores can be greater or less than .50); nevertheless, this means there exists parameter
values that can reproduce the independence violations on those problems as well. In the common effect, it correctly predicts the Z-absent problems and is the only model that correctly predicts that
alternative D should be preferred on the D vs. E choice problem. Its prediction for the discounting
problems (Fig. 11C) are also parameter dependent and so it is also the only model that can potentially
account for the absence of discounting.
Nevertheless, the shared generative cause model faces two challenges. The first concerns its theoretical plausibility. To yield the predictions in Fig. 11, recall that this model must stipulate that the
variable senses that were described to participants as causally related that are made more likely by
the shared cause. But why would subjects assume just this structure? Rehder and Burnett (2005) posited 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 causal
mechanisms. As mentioned, the memory search that occurred when reading the materials might be
biased toward facts involving the causally-related variables senses, but why would those facts not include inhibitory causal relations in addition to generative ones? And, although it is likely that reasoners 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 also
caused the other two variables in the model? The second challenge is that, because it is superimposed
on the instructed causal model, a shared generative cause predicts that subjects should be sensitive to
causal direction (e.g., make different predictions for common cause and common effect networks), at
odds with the large number of ‘‘associative reasoners’’ in Experiment 1.6 I will return to these arguments 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 models
in Fig. 5 are contributing to subjects’ inferences and Experiment 4 and quantitative model fitting that
follows will provide additional support for this possibility.
6
Said differently, the shared cause account can only reproduce the observed results assuming very different parameter values
for the three causal networks. In particular, the probability of the shared cause and the strength of its causal links must be much
larger in the common effect condition in order to overcome the effects of discounting and produce choice scores >.50 on the A vs. B
and B vs. C problems.
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
8. Experiments 4A and 4B
The goal of Experiment 4A was to further generalize the results of Experiments 1–3 by testing a
probabilistic condition in which the causal links were described as operating probabilistically. Recall
that Experiments 1–3 provided subjects with no information about the strength of the causal relations. Given research that suggests that reasoners’ default assumption is that causal links are deterministically 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 consequence 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’’ a
causal relation (e.g., Frosch & Johnson-Laird, 2011), it might have led subjects to the more radical conclusion that the causal information they were taught must be in error, perhaps prompting them to
adopt an alternative (e.g., associative) representation of those relations.
To address this possibility, links in the probabilistic condition were described as probabilistically
sufficient by stating that each cause brought about its effect with probability 75%. Because research
also suggests that reasoners might assume that causal links are deterministically necessary (the effect
has no other causes; Lombrozo, 2007; Lu et al., 2008), links were described as probabilistically necessary by stating that each effect occurred with probability 25% even when its explicit cause(s) were absent. Inferences in the probabilistic condition were compared to those in a no-strength condition in
which no information about the strength of the causal relationships was provided (as in the first three
experiments).
I also report the results of a follow-up study, Experiment 4B, that assessed the hypothesis that subjects’ non-normative responses reflect a form of response bias. Conceivably, subjects were biased
against choosing the ‘‘equally likely’’ response in the earlier experiments because it was the normative
choice on 80% of the trials in the common cause and chain conditions and 60% of those in the common
effect condition. To test this, Experiment 4B compared a probabilistic/unbalanced condition that was a
replication of the probabilistic condition of Experiment 4A with a probabilistic/balanced condition that
was identical except that subjects were presented with additional trials such that ‘‘equally likely’’ was
correct on 50% of the trials. It turns out that no effect of the unbalanced/balanced manipulation was
observed and so the results of both experiments will be reported jointly. As in Experiment 3, participants in Experiments 4A and 4B provided spoken justification, were asked to emphasize accuracy
over speed, answered questions for just one causal network, were told that the causal mechanisms
operated independently, and were given a diagram of the causal relations.
8.1. Method
8.1.1. Participants
Experiments 4A and 4B tested separate groups of 90 New York University undergraduates. Participants were randomly assigned in equal numbers to the no-strength or probabilistic condition (Experiment 4A) or the balanced or unbalanced conditions (4B). Within each experiment, subjects were
assigned 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. Materials
The 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 sufficient, subjects were told, for example, ‘‘Whenever an economy has low interest rates, it will cause that
economy to have a small trade deficits with probability 75%;’’ to specify that the links were probabilistically 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
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
causal strength questions. The diagram provided before the inference test included the strength
information.
8.1.3. Procedure
The 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 yielded
no effects of this manipulation, all ps > .33. Accordingly, I collapse the results from Experiment 4B and
those of the probabilistic condition of Experiment 4A. The choice scores for the no-strength and probabilistic conditions are presented in Fig. 12.
8.2.1. Common cause results
Fig. 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 an
effect of problem type, F(4, 232) = 23.49, MSE = 0.026, p < .0001, but no effect of the causal strength
manipulation, 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 probabilistic
conditions (.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 the
Z-present vs. Z-absent problems predicted by a shared disabler (Fig. 11A) was observed here (.63
vs. .58), t(58) = 2.24, p = .03.
8.2.2. Chain results
Fig. 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 6
Results from Experiment 4A. Normative choices are shown in bold italic.
Choice problem
Causal network
Common cause
Chain
Common effect
Deterministic
Probabilistic
Deterministic
Probabilistic
Deterministic
Probabilistic
A vs. B
A
Equally likely
B
.32
.68
0
.22
.77
.02
.20
.68
.12
.40
.55
.05
.17
.67
.17
.23
.37
.40
B vs. C
B
Equally likely
C
.35
.65
0
.25
.68
.07
.27
.65
.08
.35
.62
.03
.22
.48
.30
.33
.48
.18
D vs. E
D
Equally likely
E
.75
.23
.02
.80
.18
.02
.70
.27
.03
.67
.32
.02
.22
.70
.08
.28
.63
.08
F vs. G
F
Equally likely
G
.23
.65
.12
.18
.77
.05
.22
.73
.05
.25
.67
.08
.10
.42
.48
.08
.48
.43
G vs. H
G
Equally likely
H
.25
.70
.05
.23
.75
.02
.15
.78
.07
.37
.57
.07
.20
.65
.15
.22
.57
.22
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Table 7
Results from Experiment 4B. Normative choices are shown in bold italic.
Choice problem
Causal network
Common cause
Chain
Common effect
Unbalanced
Balanced
Unbalanced
Balanced
Unbalanced
Balanced
A vs. B
A
Equally likely
B
.23
.73
.03
.15
.78
.07
.32
.65
.03
.17
.77
.07
.27
.37
.37
.27
.37
.37
B vs. C
B
Equally likely
C
.28
.68
.03
.18
.80
.02
.30
.68
.02
.27
.70
.03
.32
.35
.33
.38
.48
.13
D vs. E
D
Equally likely
E
.65
.30
.05
.68
.30
.02
.78
.22
0
.88
.08
.03
.32
.52
.17
.42
.52
.07
F vs. G
F
Equally likely
G
.35
.58
.07
.10
.85
.05
.12
.85
.03
.15
.78
.07
.20
.45
.35
.25
.22
.53
G vs. H
G
Equally likely
H
.15
.80
.05
.10
.90
0
.17
.78
.05
.25
.73
.02
.17
.52
.32
.17
.57
.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 the
no-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 results
Reasoning 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 problems
that should exhibit discounting (.49) did not differ significantly from .50, t < 1; those on the independent D vs. E problem (.60) were significantly greater than .50, t(58) = 3.69, p < .001. As seen in Experiments 2 and 3, choice scores on the F vs. G problem were less than 0.5 (.31 and .37 in the no-strength
and probabilistic conditions, respectively), significantly so in this experiment, t(14) = 2.88, p < .05 and
t(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 to
be deterministic. The answer is that they did not, because they persisted even why they were described as probabilistic relations.
A recurring trend in Experiments 2–4, one that reached significance in this experiment, was subjects’ preference for situation G in the F vs. G common effect problem, a result that is predicted by
none of the alternative models considered thus far, including associative reasoning. This finding is suggestive of another sort of influence on causal judgments. On this problem, subjects chose whether a
cause is more likely in a situation in which a common effect is absent (situation G in Fig. 4) vs. one
in which the alternative cause is also present (situation F). Rather than computing a conditional
B. Rehder / Cognitive Psychology 72 (2014) 54–107
85
Fig. 12. Results from Experiments 4A and 4B. Independent and dependent choice problems are depicted with white and shaded
bars, 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 basis of the normative common effect model (NCE) with parameters h,
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
choiceNCE ðt; s1 ; s2 ; h; sÞ ¼
expðlogitðpðt ¼ 1js1 ; NCE ; hÞÞ=sÞ
expðlogitðpðt 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=sÞ
expðlogitðpðt 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 G
relative to F because adding the cause to F means that the absence of the common effect is (implausibly) paired with the presence of two causes whereas adding it to G means that the absent common
effect is paired with only one cause. Quantitative model fitting presented below will demonstrate that
reasoning 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 no
sensitivity to causal direction, producing associative inferences instead. To determine whether this
finding obtains when reasoners provide justifications for their choices (and learn one causal network
rather than three and are asked to emphasize speed over accuracy), I performed a cluster analysis of
the subjects from Experiments 2’s justification condition combined with those from Experiments 3
and 4. 210 common cause and chain participants were included in one analysis and 105 common effect participants were included in the other. Participants again clustered into two types, shown in
Fig. 13. The clusters shown in the left hand side consisted of 23, 24, and 26 participants in the common
cause, chain, and common effect conditions, respectively, representing 23% of the total. These associative reasoners committed a large number of Markov violations (of the responses that reflected associative 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 causal 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 following section will shed light on the source of this asymmetry, which suggests that providing justifications (and emphasizing speed over accuracy, etc.) was somewhat effective in promoting sensitivity
to causal structure.
The causal reasoners in the right hand side of Fig. 13 conformed more closely to the normative
model. Notable is the significant discounting on the A vs. B and B vs. C problems in the common effect
condition. Of all the discounting responses made in Experiments 2–4, 95% were made by the causal
reasoners. This is the first and only analysis in this article showing significant discounting and provides 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 significant
numbers of Markov violations on some problems (e.g., on A vs. B in the common cause condition, on G
vs. H in the chain condition, and on B vs. C in both conditions). And, that choice scores on the F vs. G
problem in the common effect condition were less than .5 is suggestive of the conjunctive reasoning
strategy described earlier.
Finally, an analysis of RTs revealed that the associative reasoners in Fig. 13 did not respond significantly more rapidly than the causal reasoners, 26.0 and 27.5 s, respectively, t < 1, a result that corroborates the findings in Experiment 2 indicating that the associative inferences were not simply due to
quick and careless responding. Rather, it is a deliberate strategy adopted by a substantial minority of
reasoners, 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 one
contribute to people’s causal inferences. This claim is now further supported by a quantitative analysis
B. Rehder / Cognitive Psychology 72 (2014) 54–107
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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 provided justifications, shown in Fig. 14. To characterize each strategy, their predictions averaged over
10,000 random parameter sets are presented in Fig. 15. The first column shows the patterns of inde-
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pendence and dependence produced by the normative models. The predictions of the computational
instantiation of the associative reasoning model defined in Appendix C (second column of Fig. 15) exhibit the expected choice scores >.50 on independent problems and the absence of discounting. The
predictions of the shared specific disabler models defined in Appendix B (third column) can account
for greater choices scores on Z-present vs. Z-absent problems in the common cause and chain conditions. 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 common effect problem.
The data in Fig. 14 were modeled by choosing plausible parameters for the four strategies themselves (c = d = .50, m = md = .99, b = .20, a2 = 0, and a3 = 3; see Appendices B and C for parameter definitions) and then using a search program to find the mixture that provides the optimal fit. The mixed
model is defined as,
choiceMixedNet ðt; s1 ; s2 ; c; m; b; a2 ; a3 ; d; md ; sÞ ¼ wN choiceNNet ðt; s1 ; s2 ; c; m; b; sÞ
þ wA choiceAssoc ðt; s1 ; s2 ; a2 ; a3 ; sÞ
þ wD choiceSSDNet ðt; s1 ; s2 ; c; m; b; d; md ; sÞ
þ wJ choiceConjunctNet ðt; s1 ; s2 ; c; m; b; sÞ
where wN, wA, wD, and wJ control the relative contribution of the normative (N), associative (A), specific
shared disabler (SSD), and conjunctive (Conjunct) models, respectively; Net represents the condition
being fit (CC = common cause, CH = chain, CE = common effect). The weights were constrained to
sum to 1 for each network but were allowed to vary freely over networks. The s parameter was common across networks. To obtain an estimate of the variability on the weights, a bootstrap procedure
was performed in which subjects were resampled with replacement.
The weights that minimized squared error averaged over the 100 resamples are shown in the left
bars in each panel of Fig. 16 (the average value of s was 1.95). The choice scores generated by this fit
are shown superimposed on the data in Fig. 14, which shows that the model reproduces the qualitative phenomena for each network. Three aspects of the weights are worth noting. First, associative reasoning 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 disablers
contribute to the common cause inferences (wD = .293), accounting for the larger choice scores on Z-present 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 claim
that 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 in
Fig. 13. As expected, the role of normative reasoning was larger for the causal reasoners (right bars in
Fig. 16) for all three networks (average wN = .588). Nevertheless, the common cause subjects were also
influenced by shared disablers, the common effect subjects by conjunctive reasoning, and the chain
subjects by associative and conjunctive reasoning. Even reasoners who largely grasp the logic of causal
reasoning sometimes fall prey to these alternative strategies.
Conversely, the contribution of associative reasoning was larger for the associative group (middle
bars in Fig. 16; average wA = .666). Nevertheless, that these individuals showed some sensitivity to the
instructed causal structure is indicated by the significant contribution of disablers to the common
cause subjects, conjunctive reasoning to the common effect subjects, and normative reasoning for
all three networks. But while perhaps not the only influence, these weights bolster the claim that associative reasoning dominates the inferences of a large minority of causal reasoners.
7
The contribution of each reasoning component was tested by setting its associated w = 0, refitting the model, and then asking
whether a poorer fit obtained according to a measure that corrects for the number of parameters p: RMSE = SQRT (AVG_SSE/(n p))
where AVG_SSE = sum of squared error averaged over the 100 re-samples and n = number of data points fit (15). Eliminating
associative reasoning (by setting each network’s associated wA = 0) resulted in a poorer fit for all three networks. Eliminating
shared disablers (by settings each wD to 0) resulted in a poorer fit for the common cause network only. Eliminating conjunctive
reasoning (by setting wJ = 0) resulted in a significantly poorer fit for the common effect network only.
B. Rehder / Cognitive Psychology 72 (2014) 54–107
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Fig. 14. 315 Participants from Experiments 2 (justification condition), 3, 4A, and 4B. Independent and dependent choice
problems are shown as white and shaded bars, respectively. Error bars represent 95% confidence intervals. Model fits (red round
plot symbols) are superimposed on the empirical results. (For interpretation of the references to color in this figure legend, the
reader is referred to the web version of this article.)
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
Fig. 15. Predictions for the four models contributing to causal inferences in Experiments 1–4. First column: the normative
models; second column: an associative reasoning model; third column: the specific shared disabler models; fourth column: the
conjunctive 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 independence and dependence stipulated by causal graphical models. The answer is that they frequently do
not. Of course, it is important to emphasize that subjects’ inferences were more often right than
wrong. 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 of
the independent problems they exhibited a systematic bias in which they chose the alternative with
more causally related variables 74% of the time. And, when asked to infer a cause given a common
effect, 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 to
causal reasoning problems. There was also evidence that inferences were influenced by shared
disablers and a conjunctive reasoning strategy.
In the following sections I first review the evidence concerning whether these errors are due to
subjects’ domain knowledge. I then review other evidence for the use of alternative reasoning strategies during causal reasoning. I close with a discussion of future research and the implications these
results 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 Markov
condition in both the philosophical and psychological literatures. But in each instance the Markov
B. Rehder / Cognitive Psychology 72 (2014) 54–107
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Fig. 16. Average model parameters bootstrapped over the 315 subjects from Experiments 2–4 who provided justifications. In
each panel, bars on the left are the weights estimated from the responses of all 315 subjects, bars in the middle are those for the
73 associative reasoners, and bars on the right are those for the 242 causal reasoners. Error bars are the standard errors
associated with the 100 bootstrap samples.
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condition has been defended by assuming that people reasoned with knowledge in addition to that
stipulated 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 be
due to the variables being related in ways in addition to the causal links provided as part of the experiment. 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 in
these experiments. The materials were counterbalanced (thus averaging out any effect of prior knowledge that may exist), were not categories (eliminating any domain general assumption that typical
values were causally related, because there were no typical values), and sometimes specified no domain at all (the blank materials in Experiment 3). But as mentioned, it is possible that subjects elaborated their causal model as part of comprehending the materials (or in response to the situations
about which they reasoned) and even used knowledge with the blank materials (by reasoning by analogy with familiar domains or positing abstract knowledge structures). For these reasons, I also conducted a theoretical analysis of the models in Fig. 5. This analysis revealed that neither the shared
disabler nor the shared mediator model was able to provide a complete account of subjects’ inferences, particularly those for chain and common effect structures.
The model that posits a shared generative cause fared better as an account of subjects aggregate
responses, but faces two challenges, one theoretical and one empirical. It is first important to clarify
that 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), this
expectation is of no use in making predictions. Imagine a person who believes that there are causal
relations that link economic variables like interest rates, trade deficits, and retirement savings and
who then learns that Venezuela has a small trade deficit. Should that person predict that Venezuela
has high or low retirement savings? Because the link between trade deficits and retirement savings
may take many forms (small trade deficits may cause either high or low retirement savings, the
relationship may be inhibitory rather than generative, etc.), the mere expectation that such links exist
provide 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 particular functional form, namely, the Ws in Fig. 5C must be generative causes of the variables senses that
are involved in the other causal relations. The challenge then is to explain what led subjects to assume
just this structure. A couple of possibilities were considered. The search of memory initiated by comprehending the materials may be biased toward information involving the causally related variable
senses (so that, e.g., low interest rates ? small trade deficits yields other facts about low interest rates
and small trade deficits but not high interest rates and large trade deficits). But although the result
might be a shared cause structure (or even a model elaborated with additional direct inter-variable
links), this alone is insufficient. One must also assume that those retrieved structures are generative
rather than preventative.
The second possibility is to suppose that subjects postulated the presence of the shared generative
cause when confronted with cause-absent/effect-present situations. If instructed that low interest
rates causes small trade deficits (and high retirement savings) and then confronted with a high interest
rate economy that nevertheless has small trade deficits, one can conclude that there must be some
other cause of small trade deficits. But this is insufficient. One must also assume that that hidden
cause is not only also a cause of the other variable in the model (retirement savings) but that it is a
generative cause of the variable sense that is causally related to interest rates (high retirement
savings).
In the absence of independent theoretical justification, both of these possibilities strike this author
as hopelessly post hoc. But readers who take them seriously should note the second difficulty, which is
that a model elaborated with a shared generative cause should still exhibit sensitivity to causal direction, 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 indistinguishable casts doubt on virtually all explanations in terms of additional causal knowledge, because all
predict some residual sensitivity to causal direction.
B. Rehder / Cognitive Psychology 72 (2014) 54–107
93
In summary, there appears to be no comprehensive explanation of the reasoning errors in terms of
the models in Fig. 5.8 The claim is not that such structures do not sometimes influence causal inferences
of course. As mentioned, domain knowledge is known to infiltrate formal reasoning problems (producing, e.g., the belief bias and suppressions effects reviewed earlier) and so there is no doubt that causal
reasoning scenarios, both in everyday reasoning and prior studies, are often embellished with knowledge
beyond the stated facts (shared disablers for common cause structures; see below). But this cannot be
the whole story. Above and beyond these embellishments, reasoners have a tendency to convert a causal
inference 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 reason
with conjunctive causes (in which, e.g., two causes are necessary for an effect). Rehder (2009) tested
how 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 varied
as a function of proximity in a network, consistent with the spreading activation view of inference described earlier.9 Evans et al. (2008) presented subjects with everyday causal conditionals of the form if p
then q (e.g., ‘‘if more people use protective sun cream then cases of skin cancer will be reduced’’) and then
; p
supported, contradicted,
q; and p
q
asked to what extent each of four statements corresponding to pq, pq
or was irrelevant to the conditional (the ‘‘truth table task’’). The responses of a substantial minority of
participants were consistent with a symmetrical (i.e., biconditional) interpretation of the conditional
in which q implies p as much as p implies q (also see Newstead et al., 1997). Associative reasoning appears 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 output 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 analytic responding (e.g., instructions to justify answers) to reduce the rate of Markov violations and the
fact that associative reasoners did not respond more quickly. Apparently, even careful and deliberate
thinking sometimes produces an associative response to a causal reasoning problem.
These findings demand a theoretical account of how human reasoners’ can possess a tendency to
reason associatively while also being responsible for the examples noted earlier of causal-based reasoning in the domains of learning, decisions making, analogy, and categorization. One interpretation is
that people possess veridical causal reasoning abilities but sometimes fail to deploy them. Dual process theories may again be consulted to gain insight into the factors that determine when such processes are invoked. Kahneman and Frederick (2002) argue that the analytic reasoning system
monitors 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 the
well-known question ‘‘A bat and ball cost $1.10 in total. The bat costs $1 more than the ball. How
much does the ball cost?’’ even though they are presumably able to carry out the elementary arithmetic 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 single
generative process in which a cause produces an effect. Instead, they assume the link is constituted by one or more intermediate
variables 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 generates
intermediate variables. But although Buchanan et al. claim that CERP reproduces the independence violations found here and in
previous studies (i.e., Mayrhofer et al., 2008; Rehder & Burnett, 2005; Walsh & Sloman, 2008), this is accomplished by simply
running CERP on a single causal link, after which all generated casual models that do not match the instructed model are
eliminated. But this means that the CERP’s explanation of independence violations lies as much in this editing process as its
assumptions about how causal links are represented. It is also unclear how CERP could explain the reasoning errors found with
chain 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 presence
of W when Y or Z was also present (violating the Markov condition) but this effect was larger for Y than Z.
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Associative responses to causal reasoning questions may be especially unlikely to alert the analytic
system because of the relative ease with which those inferences are presumably generated. For example, the fluency of processing can serve as a cue that affects a wide variety of judgments, including
familiarity (Monin, 2003), frequency (Tversky & Kahneman, 1973), typicality (Oppenheimer & Frank,
2008), affect (Reber, Winkielman, & Schwarz, 1998) and even intelligence (Oppenheimer, 2005) and
fame (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 are
accompanied by a feeling of rightness (FOR), a claim supported by their finding that lower FOR judgments 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 associative
inferences lulls the analytic system into acquiescence.
Another reason that associative inferences might seem acceptable is that they are sometimes used
as a defensible approximation to causal ones. Because everyday reasoning often involves many causal
factors (and because the required computations grow with the size of the causal network), people
might rely on (approximately correct) associative inferences instead. And, in real life variables are often believed to be related in causal cycles (e.g., Kim & Ahn, 2002; Sloman, Love, & Ahn, 1998). Although
there 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 people
learn from experience that associative inferences are often useful may contribute to their feeling of
rightness even in circumstances that neither tax cognitive resources nor contain cycles (e.g., the small
causal 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 life
successfully enough to be admitted into a major university have reasoning abilities above the mere
following of associations. Actions taken to achieve important life goals are unlikely to have the desired
consequences if they are uninformed by cause/effect relations (Hagmayer & Sloman, 2009). But what
many may lack is the metacognitive awareness that causal and associative inferences do not always
produce the same answer. This ignorance leads us to readily accept associative answers that are easy
to compute and which experience has taught are often correct.
Another question concerns the source of the individual differences in the use of associative reasoning. Recall that about 60% of the associative inferences that violated independence were made by
about a quarter of the subjects. Because Experiment 2 suggested that this result is not due to differences in subjects’ motivation (Bless & Schwarz, 1999), it may reflect differences in their cognitive
capacity. Indeed, Evans et al. (2008) found that biconditional responses to the truth table task were
the 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 which
measures of cognitive capacity are correlated with people’s tendency to violate independence and
not discount. The metacognitive monitoring abilities required to detect erroneous associative responses 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 analytic
reasoning per se, implying that the best predictors of independence violations may be scales that assess individuals’ willingness to engage the analytic system (see, e.g., Stanovich & West, 2007) rather
than cognitive capacity per se.
11.3. Other influences on causal reasoning
Two other influences on subjects’ inferences were identified. Common cause inferences were consistent with reasoners sometimes assuming that the two links shared a disabler, as evidenced by the
larger violations of independence on Z-present vs. Z-absent choice problems and model fitting that
yielded a better fit when a shared disabler was included. Direct evidence for this conclusion was provided by Park and Sloman (2013), who manipulated whether the verbally-described links of a common cause structure were viewed as consisting of the ‘‘same’’ or ‘‘different’’ mechanisms and found
larger Markov violations for the former, implying that the malfunctioning of one link can generalize
to the other. Nevertheless, independence violations obtained even in their different mechanism conditions, as predicted by an associative reasoning view. Although these effects were not consistently
B. Rehder / Cognitive Psychology 72 (2014) 54–107
95
significant (they were in their Experiment 1 but not 2 and 3), the apparent discrepancy with the results reported here is resolved by noting Experiment 1–4’s use of a more sensitive forced choice
procedure.
Evidence for use of disablers in a chain network is more equivocal. On one hand, Park and Sloman
found 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 differences in the Z-present and Z-absent choice problems that are diagnostic of a chain structure with a
shared 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/absent
differences when tested on a four element causal chain and the mixed model tested here did not yield
a significantly better fit when it included a shared disabler, suggesting that reasoners may be less
likely to posit shared disablers for chain as compared to common cause structures. There was no sign
of the influence of shared disablers on the common effect inferences.
A third non-normative influence on common effect inferences was a conjunctive reasoning strategy
in which subjects apparently judged the probability of variable t in situation s as p(t, s) rather than
p(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 p
; p
, consistent with a conjunctive interpretation (also
q, and p
q
then q was supported by pq, but not pq
see Oberauer & Wilhelm, 2003). This interpretation may be invoked more frequently by lower ability
individuals (Evans et al., 2007; Oberauer, Geiger, Fisher, & Weidenfeld, 2007). However, while a conjunctive 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 4
that found its use in the concrete domains of economics, meteorology, and sociology. But those causal
relations were also unfamiliar, raising the possibility that conjunctive reasoning might be promoted
by unfamiliar rather than abstract materials. Future research should identify other conditions that
control when this strategy is deployed. Open questions include, for example, why its use was especially prominent for the common effect inferences and in the experiments in which reasoners provided 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 associative one. Although the domain variables tested here were described as binary, some subjects may have
interpreted them as continuous (e.g., interest rates, trade deficits, and retirement savings in the domain
of 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 are
either present or absent (vs. the ‘‘substitutive’’ ones tested here that were, e.g., ‘‘high’’ or ‘‘low’’). Substitutive variables might have invited an interpretation of the causal links as having a dual sense (e.g., that
low interest rates ? small trade deficits implies high interest rates ? large trade deficits), exacerbating
the Markov violations of some subjects. Consistent with this possibility, Mayrhofer et al. (2010) found
larger violations when the variables were substitutive (an alien could be thinking ‘‘POR’’ or ‘‘TUS’’) rather
than additive (‘‘POR’’ or nothing). Relatedly, Markov violations may be affected by whether the variables
represent temporally distinct and mechanical ‘‘events’’ rather than states. For example, Park and Sloman’s (2013) slider experiment did not yield violations on Z-absent problems, consistent with the idea
that an ‘‘off’’ variable does not make other variables less likely to be ‘‘on.’’ More generally, the use of
mechanical events may have contributed to the success of Bayes net accounts of learning in the ‘‘blicket
detector’’ 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 in
which 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 low
interest rates of an economy would go lower still when also told that its trade deficits were small, and these extra low interest rates
affected retirement savings.
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
can be construed as requiring two causal inferences (one in each scenario). However, earlier experiments in this line of research in which subjects simply rated the likely presence of a variable in a scenario 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 choice
procedure.
Third, the current subjects were asked to reason about observed scenarios but other cover stories
might 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 effects
independent (Pearl, 2000). However, Waldmann and Hagmayer (2005) have shown that while reasoners are sensitive to the difference between interventions and observations (e.g., the other effect is
judged more likely when the first effect is observed present rather than set to be present), full conditional independence does not obtain (e.g., the other effect is judged more likely when the first effect is
set to be present vs. absent) (also see Hagmayer & Sloman, 2009; Sloman & Lagnado, 2005). Further
afield, that people interpret their own actions as diagnostic of desirable properties (tolerance to cold
water 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 interventions are no sure path to veridical causal inferences.
Finally, causal relations learned from observed data may be less susceptible to independence violations. One extension of these experiments would be to instruct subjects on causal relations but also
present data that manifests the patterns of conditional independence implied by the relations. Or, one
could present data, ask subjects what causal relations they induced, and then present inferences that
assess the Markov condition for those learned relations. Past studies paint a mixed picture regarding
this possibility, however. On one hand, Von Sydow, Hagmayer, Meder, and Waldmann (2010) found
that learners correctly treated causes of a common effect structure as uncorrelated after observing
data (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, Wagenmakers, & Blum, 2003; for more examples). Learning data may also not guarantee adherence to the Markov
condition.
11.5. Implications for cognitive modeling
It is important to consider the implications these results have for the use of causal graphical models
in cognitive modeling generally. The inferential procedures that accompany CGMs all rely on the Markov condition for their justification and so the finding that adults often violate independence raises
questions about the appropriateness of CGMs as a framework for understanding the many cognitive
phenomena to which they have been applied. How researchers should respond to this concern depends on their goals.
First, those wishing to establish the existence of causal representations in a domain are advised to
avoid tests of independence inspired by the Markov condition, which, due to the virtual impossibility
of fully suppressing prior knowledge and associative reasoning, are highly likely to yield negative results. A better approach is test for the specific patterns of reasoning asymmetries that are characteristic 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 asymmetries 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 to
model causal reasoning in a domain might treat those violations as a sort of ‘‘nuisance parameter’’ by
including mechanisms that partial out their influence. One approach is to blend together the predictions of a CGM and an associative reasoning network, as done in the mixed model above. Depending
on context, it may be reasonable to assume that people are reasoning with one of the structure types
in Fig. 5. For example, inferences in the Rehder (2013) study described above that compared how
B. Rehder / Cognitive Psychology 72 (2014) 54–107
97
people reason with independent and conjunctive causes were fit to those networks embellished with a
shared generative causes, which reproduced not only the differences between independent and conjunctive causes but also the independence violations. A shared generative cause model was appropriate in that study because the inferences were between features of categories that might be especially
likely to be viewed as related by shared mechanisms. In addition to the models in Fig. 5, another
approach 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 appropriate
will vary with context.
Most importantly, the present results should stem the development of new formalisms that
explicitly 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 (see
Danks, 2007; Koller & Friedman, 2009; Smyth, 1997, and Appendix C for an implementation of associative reasoning as a Markov random field; also see Lauritzen & Richardson, 2002, for descriptions
of chain graphs that combine CGMs and Markov random fields). Note, however, that merely converting CGMs into a Markov random field by removing arrowheads is insufficient, not only because reasoning asymmetries are eliminated entirely but because Markov random fields stipulate
independence constraints that are analogous to those of CGMs (e.g., interpreted literally as a Markov
random field, the graph of Fig. 8 stipulates that X and Y are independent given Z). Second, one might
contemplate abandoning graph-based formalisms. After all, a central motivation for such representations is that one can ‘‘read off’’ independence relations without regard to the exact functional
relationships between variables (i.e., how the graph is ‘‘parameterized’’), a property that has little
value for a psychological theory if people systematically violate independence. Yet, this alternative
seems undesirable as graphs support other key operations, such as counterfactual reasoning
(Hiddleston, 2005; Pearl, 2000; Rips, 2010) and anticipating the impact of interventions (see literature cited earlier). Finally, independence violations may be modeled by considering the psychological
processes by which a joint distribution (and hence inferences) is derived from a graph. The network
with which people reason is likely to be constructed piecemeal and on the fly, and our lab has found
that certain algorithmic shortcuts that expand an existing joint to accommodate a new network
variable can yield independence violations.
In summary, graph based representations have the potential to make lasting contributions to the
psychology of reasoning because of the structured, generative, and probabilistic representations they
inspire. They will also serve the field by fulfilling the function of any normative model, which includes
providing approximate, ‘‘back of the envelope’’ predictions and identifying those places where
the development of new theory is needed. However, that the independence relations they stipulate
are not surviving experimental scrutiny means they will fail to reproduce some important qualitative
aspects of human reasoning. Future research should be oriented toward the development of new
models that incorporate what are now two well-established facts: (a) that people are sophisticated
causal reasoners and (b) they violate the patterns of independence stipulated by causal graphical
models.
11.6. Conclusion
Five experiments established that adults’ tendency to reason associatively resulted in them often
violating independence in causal inferences. That the rate of these violations was unaffected by
manipulations known to affect fast and intuitive reasoning processes suggests that an associative response 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 suggests
that such reasoning may be influenced by individual difference variables in monitoring. Theories that
strive to provide high fidelity accounts of human causal reasoning will need to relax the independence
constraints imposed by graphical models.
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
Acknowledgments
I thank David Danks, Juhwa Park, Ben Rottman, and Michael Waldmann for their comments on previous versions of this manuscript.
Appendix A
A.1. Causal relationships
The complete list of causal relationship in the domains of economics, meteorology, and sociology
are presented in Tables A1–A3, respectively.
Table A1
Causal relationships for the domain of economics.
Version
Causal link
Interest Rates ? Trade
deficits
Interest Rates ? Ret. savings
Trade deficits ? Ret. savings
Value 1 ? Value 1
Low interest rates cause
small trade deficits. The low
cost of borrowing money
leads businesses to invest in
the latest manufacturing
technologies, and the
resulting low-cost products
are exported around the
world
Low interest rates cause high
retirement savings. Low
interest rates stimulate
economic growth, leading to
greater prosperity overall,
and allowing more money to
be saved for retirement in
particular
Small trade deficits cause
high retirement savings.
When the economy is good,
people can cover their basic
expenses and so have enough
money left over to contribute
to their retirement accounts
Value 1 ? Value 2
Low interest rates causes
large trade deficits. Because
money is cheap for
consumers to borrow (e.g., on
credit cards) demand for ‘‘big
ticket’’ consumer goods is
high and large commercial
retailers increase their
imports from foreign
countries
Low interest rates cause low
retirement savings. The good
economic times produced by
the low interest rates leads to
greater confidence and less
worry about the future, so
people are less concerned
about retirement
Small trade deficits cause low
retirement savings. When the
economy is good, people are
optimistic and so spend
rather than save
Value 2 ? Value 1
High interest rates cause
small trade deficits. Because
consumers spend less in
order to avoid high-interest
credit card debt, large
commercial retailers import
fewer goods
High interest rates cause high
retirement savings. The high
interest rates result in high
yields on government bonds,
which are especially
attractive for retirement
savings because they are such
a safe investment
Large trade deficits cause
high retirement savings.
People become nervous when
their economy is no longer
competitive enough in the
world economy to export
products, and begin saving
for retirement as a result
Value 2 ? Value 2
High interest rates cause
large trade deficits. The high
interest rates leads to a
favorable exchange rate
between the local currency
and foreign currencies, and
consumers buy many
imported goods because the
favorable exchange rate
makes them cheap
High interest rates cause low
retirement savings. Because
so many people are making
large monthly interest
payments on credit card debt,
they have no money left to
save for retirement
Large trade deficits cause low
retirement savings. The loss
of local manufacturing jobs
means that there are people
out of work, and
contributions to retirement
accounts decreases
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
Table A2
Causal 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 ozone
causes low air pressure.
Ozone, being an allotrope of
oxygen (O3), combines with
more atoms. These denser
collections of atoms sink
from open regions, creating
less pressure
A high amount of ozone
causes high humidity. Ozone
attracts extra oxygen atoms
from water molecules,
creating a concentration of
water vapor in that region
Low air pressure causes high
humidity. When pressure
does not force water vapor to
break into oxygen and
hydrogen atoms, water vapor
remains in abundance
Value 1 ? Value 2
A high amount of ozone
causes high air pressure.
With more molecules present
in the atmosphere, more
pressure is exerted
A high amount of ozone
causes low humidity. Ozone
accepts extra oxygen atoms,
decreasing the amount of
oxygen available to form
water molecules. With fewer
water molecules, there is
lower humidity
Low air pressure causes low
humidity. Low air pressure
poorly facilitates
condensation; as a result,
there are less water
molecules in the air
Value 2 ? Value 1
A low amount of ozone
causes low air pressure. The
lower number of ozone
molecules means that air
molecules are less dense,
which results in lower air
pressure
A low amount of ozone
causes high humidity. The
oxygen atoms that would
normally be part of ozone
molecules are free to
combine with hydrogen
atoms instead, creating water
molecules
High air pressure causes high
humidity. The higher
pressure means that the
components of water
molecules (hydrogen and
oxygen) tend to not
dissociate from one another.
Because there are more water
molecules, humidity is higher
Value 2 ? Value 2
A low amount of ozone
causes high air pressure.
When the amount of ozone
(O3) is low, there are more
oxygen (O2) atoms present in
the atmosphere, resulting in
higher air pressure
A low amount of ozone
causes low humidity. The low
amount of ozone allows a
large number of ultra-violet
(UV) rays to enter the
atmosphere, and the UV rays
break up water molecules,
resulting in low humidity
High air pressure causes low
humidity. When air pressure
is high, water vapor
condenses into liquid water
(rain), and the atmosphere is
left 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 five
choice problems presented in this study. These qualitative arguments were confirmed by computer
simulations that instantiated each of the models with 10,000 randomly generated parameter sets
and deriving the predictions each instantiation makes for each choice problem. The temperature
parameter s was set to 1. As described in the main text, there is both specific and general version
of the shared disabler hypothesis. The parameters for the common cause (CC), chain (CH), and common 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. Each
of the elaborated models inherits parameters from the normative (N) model, which are: c, the probability of the cause(s); m, the power of the explicit causal links; and b, the strength of alternative causes
of 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 the
strength of alternative causes not shown in the model (b and bw) were uniformly sampled from the
range [0, .25]; the remaining parameters were sampled from [0, 1]. For each instantiation of each
model, choices were computed according to Equation 1. For the chain networks, all the models make
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
Table A3
Causal 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 urbanization
causes low interest in
religion. Big cities tend to
foster materialism rather
than spiritualism
A high degree of urbanization
causes high socio-economic
mobility. Big cities provide
many opportunities for
financial and social
improvement
Low interest in religion
causes high socio-economic
mobility. Without the
restraint of religious-based
morality the impulse toward
greed dominates and people
tend to accumulate material
wealth
Value 1 ? Value 2
A high degree of urbanization
causes high interest in
religion. People are exposed
to a large number of different
kinds of religions in cities and
usually become interested in
one of them as a result
A high degree of urbanization
causes low socio-economic
mobility. In big cities many
people are competing for the
same high-status jobs and
occupations
Low interest in religion
causes low socio-economic
mobility. Many religions
reinforce a strong work ethic;
without this motivation,
workers become complacent
at their jobs
Value 2 ? Value 1
A low degree of urbanization
causes low interest in
religion. The lack of cultural
diversity in these rural areas
limits access to and the
learning of diverse religious
concepts. Without this,
religion becomes
‘‘background noise’’ in
people’s daily lives
A low degree of urbanization
causes high socio-economic
mobility. People in rural areas
are rarely career oriented,
and so take time off from
working and switch
frequently between different
‘‘temp’’ jobs
High interest in religion
causes high socio-economic
mobility. Religion fosters
communal care, and those of
the same religion tend to
support each other with jobs,
financial favors, and so on
Value 2 ? Value 2
A low degree of urbanization
causes high interest in
religion. Poor rural societies
look to religion as a source of
hope in the future and a
means to escape their
problems
A low degree of urbanization
causes low socio-economic
mobility. The low density of
people prevents the dynamic
economic expansion needed
for people to get ahead
High interest in religion
causes low socio-economic
mobility. The spiritualism
induced by religion works to
reduce the desire for material
wealth
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 two
inferences.
B.2. Shared disablers
In the shared disabler models in Fig. 5A, W disables the operation of other causal relations. When
W is a specific disabler and is present, it disables the other two causal links in the model. When it is a
general disabler and is present, it disables those two links plus any hidden causes of the effects (thus
preventing all occurrences of the effects). Besides c, m, and b (defined above), there were two additional parameters: d, the probability of W; and md, the probability that W, when present, disables
the 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 and
Y are no longer screened off because from the presence of X (or Y) one can infer the likely absence of
the disabler W and thus the likely presence of Y (or X). Analogously, from the absence of X (or Y) one
can infer the likely presence of W and thus the likely absence of Y (or X). Thus, reasoners should prefer
choice A in the A vs. B problem and B in the B vs. C problem. The predictions for situations in which Z is
known to be absent (situations F, G, and H in Fig. 2) depend on whether the disabler is specific or
general. First consider a specific disabler. In situation F, X must have been brought about by some
factor other than Z, and so its presence reveals nothing about the state of W. In situation H, the
Common cause
Chain
Common effect
Normative
Model: N CC
hNCC ¼ fc; m; bg
Model: N CH
hNCH ¼ fc; m; bg
Model: N CE
hNCE ¼ fc; m; bg
Shared disabler (Specific)
Model: SSDCC
hSSDCC ¼ fc; m; b; d; md g
Model: SSDCH
hSSDCH ¼ fc; m; b; d; md g
Model: SSDCE
hSSDCE ¼ fc; m; b; d; md g
Shared disabler (General)
Model: GSDCC
hGSDCC ¼ fc; m; b; d; md g
Model: GSDCH
hGSDCH ¼ fc; m; b; d; md g
Model: GSDCE
hGSDCE ¼ fc; m; b; d; md g
Shared mediator
Model: SMCC
hSMCC ¼ fc; m!w ; mw! ; bw ; bg
Model: SMCH
hSMCH ¼ fc; mm ; b; d; md g
Model: SMCE
hSMCE ¼ fc; m!w ; mw! ; bw ; bg
Shared generative cause
Model: SGC CC
hSGC CC ¼ fc; m; b; d; mw g
Model: SGC CH
hSGC CH ¼ fc; m; b; w; mw g
Model: SGC CE
hSGC CE ¼ fc; m; b; w; mw g
B. Rehder / Cognitive Psychology 72 (2014) 54–107
Table B1
Parameters for the normative models (Fig. 1) and their elaborations (Fig. 5).
101
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
absence of X is fully explained by the absence of Z, again telling us nothing about W (which only moderates the influence of Z on X but not X’s other potential causes). Thus, independence still obtains between X and Y in the F vs. G and G vs. H choice problems. These predictions are reflected in the
simulation results in the first panel of Fig. 11A. When the disabler is general, the presence of X in situation F tells us that the disabler W is likely to be absent (and thus Y is likely to be present) whereas
the 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 F
should be preferred in the F vs. G choice problem and G should be preferred in G vs. H. These predictions 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 is
present, the state of X provides information about the state of specific disabler W (it is less likely to be
present if X is) and thus Y. When Z is absent, the WZ ? Y interactive link is already disabled, and so
any additional information about the state of W provided by the state of X is irrelevant to Y (i.e., X and
Y are independent). These predictions are confirmed by the simulation results shown in first panel of
Fig. 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 reveals
nothing further about the state of W and so Y; X and Y are thus independent. When Z is absent, the
presence (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 problem
and H in the G vs. H problem.11 Note that Mayrhofer et al. (2010) also assumed the presence of disablers
in 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 predictions 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 which
case the network reverts to the common effect network in Fig. 1C. Second, when Z is absent, W is more
likely to be present. When X is also present then W becomes even more likely, and thus Y is exonerated from responsibility for the absence of Z; thus, Y is also more likely. When X is absent then W becomes less likely and so Y becomes more responsible for the absence of Z; thus Y also becomes less
likely. That is, independence should no longer obtain between X and Y in the F vs. G and G vs. H choice
problems. Finally, when Z is unknown, nothing can be inferred about W and so X and Y remain independent. 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 causal links are mediated by W. Besides c and b, the additional parameters were: m?W, the power of the
causal links into W; mW?, the power of the causal links out of W; and bW, the strength of the alternative 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 probability 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 in
the effects becoming independent when Z is known, a conjecture confirmed by the simulation results
presented 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 specific
disabler, 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 (as
opposed to some extraneous factor). But when Z is absent, nothing further can be inferred about W (and thus X) from Y, i.e., the
state 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 did
not 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.
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B. Rehder / Cognitive Psychology 72 (2014) 54–107
about W is irrelevant to Y. That is, independence should still obtain on the F vs. G and G vs. H choice
problems. 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 discounting still obtains when Z is present. When Z is absent, the probability that X and Y are both absent
becomes relatively improbable, resulting in a positive correlations between them; thus F is preferred
in 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 cause
W. 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 common cause network elaborated with a shared generative cause results in each variable becoming more
likely 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 preferred in F vs. G and G is preferred in G vs. H. When Z is present (the discounting trials A vs. B and
B 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 links
from W are weak and those to Z are strong and <.50 when the links from W are strong and those to Z
are weak. Finally, a shared generative cause for a chain network results in dependence between Z and
Y when Z is absent. Predictions when Z is present are indeterminate (<, =, or >.50), because X, W, and Z
form a common effect subnetwork. Thus, X and W will be negatively correlated when W?X is relatively weak and X?Z is relatively strong and positively correlated when W?X is relatively strong
and 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 random field with the following factors,
2
2
eXY
X¼0
6X ¼ 0
6
¼6
4X ¼ 1
X¼1
Y ¼ 0 a2
3
2
Y ¼1 0 7
7
7;
Y ¼0 0 5
Y ¼ 1 a2
eYZ
Y
6Y
6
¼6
4Y
Y
¼ 0 Z ¼ 0 a2
3
¼0 Z¼1 0 7
7
7;
¼1 Z¼0 0 5
¼ 1 Z ¼ 1 a2
eXYZ
X¼0
6X ¼ 0
6
6
6X ¼ 0
6
6X ¼ 0
6
¼6
6X ¼ 1
6
6X ¼ 1
6
6
4X ¼ 1
Y ¼ 0 Z ¼ 0 a2
3
Y ¼0 Z¼1 0 7
7
7
Y ¼1 Z¼0 0 7
7
Y ¼1 Z¼1 0 7
7
7
Y ¼0 Z¼0 0 7
7
Y ¼0 Z¼1 0 7
7
7
Y ¼1 Z¼0 0 5
X ¼ 1 Y ¼ 1 Z ¼ 1 a2
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 expðð2XY ðx; yÞ þ 2YZ ðy; zÞ þ 2XYZ ðx; y; zÞÞÞ
and then normalizing the result (Koller & Friedman, 2009). a2 and a3 capture the pairwise and
three-way degrees of association, respectively, between variables X, Y, and Z. In particular, a3 > 0 implies 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 Experiments 1–4 were computed in a manner analogous to Eq. (1), namely,
choiceassoc ðt; s1 ; s2 ; a2 ; a3 Þ ¼
expðlogitðpassoc ðt ¼ 1js1 ; a2 ; a3 ÞÞ=sÞ
expðlogitðpassoc ðt ¼ 1js1 ; a2 ; a3 ÞÞ=sÞ þ expðlogitðpassoc ðt ¼ 1js2 ; a2 ; a3 ÞÞ=sÞ
104
B. Rehder / Cognitive Psychology 72 (2014) 54–107
The predictions shown in Fig. 15 were generated from 10,000 random draws of a2 and a3 from the
range [0, 2].
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