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Two Heads Are Better Than One, but How Much?: Evidence That People’s Use of Causal Integration Rules Does not Always Conform to Normative Standards

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  • Universidad Loyola Andalucía - Human Neuroscience Lab

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Many theories of causal learning and causal induction differ in their assumptions about how people combine the causal impact of several causes presented in compound. Some theories propose that when several causes are present, their joint causal impact is equal to the linear sum of the individual impact of each cause. However, some recent theories propose that the causal impact of several causes needs to be combined by means of a noisy-OR integration rule. In other words, the probability of the effect given several causes would be equal to the sum of the probability of the effect given each cause in isolation minus the overlap between those probabilities. In the present series of experiments, participants were given information about the causal impact of several causes and then they were asked what compounds of those causes they would prefer to use if they wanted to produce the effect. The results of these experiments suggest that participants actually use a variety of strategies, including not only the linear and the noisy-OR integration rules, but also averaging the impact of several causes.
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Research Article
Two Heads Are Better
Than One, but How Much?
Evidence That Peoples Use of Causal Integration Rules
Does not Always Conform to Normative Standards
Miguel A. Vadillo,
1
Nerea Ortega-Castro,
2
Itxaso Barberia,
2
and A. G. Baker
3
1
University College London, UK,
2
Universidad de Deusto, Bilbao, Spain,
3
McGill University, Montréal, Canada
Abstract. Many theories of causal learning and causal induction differ in their assumptions about how people combine the causal impact of
several causes presented in compound. Some theories propose that when several causes are present, their joint causal impact is equal to the
linear sum of the individual impact of each cause. However, some recent theories propose that the causal impact of several causes needs to be
combined by means of a noisy-OR integration rule. In other words, the probability of the effect given several causes would be equal to the sum
of the probability of the effect given each cause in isolation minus the overlap between those probabilities. In the present series of experiments,
participants were given information about the causal impact of several causes and then they were asked what compounds of those causes they
would prefer to use if they wanted to produce the effect. The results of these experiments suggest that participants actually use a variety of
strategies, including not only the linear and the noisy-OR integration rules, but also averaging the impact of several causes.
Keywords: causal reasoning, integration rules, summation
It is difficult to find any single behavior in our daily life that
does not require some sort of causal thinking. From the
instant we wake up and turn off the alarm clock to the
moment we brush our teeth before going to bed, all our
activities are influenced by causal beliefs. However, it is
still far from obvious how we acquire and use this vast
amount of knowledge. Establishing with some certainty that
a given event is the cause of another is a deeply complex
cognitive process. Imagine, for example, that after eating
a peach your throat suddenly feels irritated. You will prob-
ably suspect that the peach has provoked an allergic reac-
tion. But the simple contiguity of eating the peach and
the sore throat does not guarantee that there is an actual
causal relation between them. There are a myriad of events
that also took place just before your reaction and any of
them could also be responsible for it (e.g., other foods
you have eaten, the smoke from a cigarette, or, who knows,
radiation from the sun, to name just a few).
Ideally, if you could discard the influence of all the
other causal factors by eating a peach in a context free from
all the other potentially relevant events, you would be much
more confident that there is a true causal connection
between the peach and the allergic reaction. Unfortunately,
in most instances, such an ideal context simply does not
exist. Moreover, it is often impossible to know what alter-
native causal factors need be taken into account. Fortu-
nately, there is an indirect way to test the causal role of
the peach. Although you often cannot eliminate all the other
potential causes of the allergy, you can try to keep them
constant while you test the causal impact of eating the
peach. If, other things being equal, the probability of an
allergic reaction is higher when you eat peaches than when
youdonot,thenpeachesarelikelytomakeacausalcontri-
bution to your allergy. Most models of causal induction
compute the causal strength of a potential candidate cause
by subtracting the probability of the effect given the
absence of the candidate cause from the probability of the
effect given its presence. However, how this subtraction
is done depends on assumptions about how the causal
impact of multiple causes should be combined.
The simplest assumption about how several causes
jointly determine the probability of the effect is that the
impact of each cause should be integrated by simple linear
addition. If, for example, we know that cause A produces
an outcome 30% of the time and B produces the same out-
come 20% of the time, then the notion of linear addition
would argue that when both are present (and no other effec-
tive cause is present) the probability of the effect should be
50%. This is the law for combining mutually exclusive
probabilities. Following this reasoning, isolating the
role of a single cue from the effect of a larger collection
of causes is based on a simple linear subtraction.
Experimental Psychology 2014; Vol. 61(5):356–367
DOI: 10.1027/1618-3169/a000255 Hogrefe OpenMind License (http://dx.doi.org/10.1027/a000001)
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This assumption is made in many formal models of causal
learning (Allan, 1980; Cheng & Novick, 1992; Jenkins &
Ward, 1965; Rescorla & Wagner, 1972).
Although this mental operation is simple, it poses some
normative problems if the causes are independent. We illus-
trate the nonnormative nature of simple linear addition by
using a coin-tossing game as an example. Imagine a game
in which two coins (i.e., candidate causes, Coin 1 and Coin
2) are tossed. To win (i.e., for the outcome to occur) either
or both of the coins must come up heads. That is to say, a
head on either coin represents the necessary generative con-
dition to guarantee an effect on that trial. If both coins are
tossed together, either or both of them might come up heads
and hence meet the conditions for generating the desired
effect (winning). When both coins are tossed, there are four
possible permutations of heads and tails: both are heads; the
first is heads and the second tails; the first is tails and the
second heads; or both are tails. In the first three cases the
outcome occurs because each includes at least one head,
but in the final case the outcome does not happen – you
lose. When only Coin 1 is tossed, we either get heads
and win or we get tails and lose (we win on half of the
tosses). One can see from the example that if Coin 2 is
tossed along with Coin 1 we win on 3/4 of the tosses. If
we follow the logic of the linear integration rule, the contri-
bution of tossing Coin 2 to winning is the linear difference
between the probability of winning if Coin 1 is tossed (.50)
and the probability of winning when both are tossed (.75).
Thus by the linear integration rule, the power of Coin 2 is
.25. However we know that the actual likelihood of an out-
come given the tossing of Coin 2 is .50. Thus the linear
integration rule leads to problems of coherence when
extrapolating the results of a causal inference to a novel
causal context (Cheng, Novick, Liljeholm, & Ford, 2007;
Liljeholm & Cheng, 2007). If a coin is tossed first its effec-
tivenessis.50;ifitistossedseconditis.25.
In order to solve these and other problems, Cheng
(1997) suggested that, from a rational point of view, the
combined power of several causes should be computed
using a different integration rule. In our case it is the law
of compound independent probabilities. It is clear from
our example that when the second coin is tossed at least
part of its effectiveness is masked because sometimes both
it and the first coin meet the conditions for an outcome.
Cheng suggested that people can control for these com-
bined effective causes. Instead of using simple addition,
she argued that generative causal powers should be com-
bined by means of a noisy-OR integration rule. According
to the noisy-OR rule, the probability of the effect if two
potential causes, A and B, are present (and no other cause
is present) can be computed as
pejA&BðÞ¼qAþqBqAqB
ðÞ;ð1Þ
where q
A
and q
B
are the generative causal powers of
cause A and B, respectively. The power of a cause is sim-
ply the proportion of times it produces the outcome, in a
hypothetical context in which no other effective causes of
the same outcome exist (Cheng, 1997).
If the above assumptions are made it is not correct to
assume that the unique contribution of a candidate cause
to the probability of the effect may be found by subtracting
the effect when it is present from that when it is absent. For
example, if we calculated the causal power of Coin 2 by
subtracting the probability of winning when only Coin 1
is tossed from the probability of winning when both Coin
1 and Coin 2 are tossed, we would underestimate the real
contribution of Coin 2. Following this reasoning, Cheng
(1997) proposed an alternative computational analysis of
causal induction. According to her Power PC model, when
some assumptions are met, the generative power of a can-
didate cause is given by the rule:
q¼pðejcÞpðej:cÞ
1pðej:cÞ;ð2Þ
where p(e|c) represents the probability of the effect given
the cause and p(e|c) represents the probability of the
effect given the absence of the cause. Equation 2 can be
easily applied to our coin-tossing game. If the probability
of winning is .50 when we only toss Coin 1 and .75 when
we toss both coins, Equation 2 correctly yields a causal
power of .50 for Coin 2. Equation 2 isolates the p(e|c)
in a hypothetical situation where no alternative causes
of the effect exist. In our example, this would be the prob-
ability of getting a head in a situation in which only Coin
2 is tossed.
The idea that causal powers should be combined by
means of a noisy-OR integration rule is particularly clear
in the Power PC model, but it can be introduced in associa-
tive models as well (Danks, Griffiths, & Tenenbaum,
2003). Bayesian models of causal induction do not commit
to a specific integration rule, but most formal versions of
these models have favored the noisy-OR rule over the linear
one (Griffiths & Tenenbaum, 2005, 2009). Precise proba-
bilities aside, the logic of the noisy-OR rule is that the
effectiveness of two independent causes, when they are
combined, is somewhat more than the effectiveness of the
stronger of the two alone and somewhat less than their
sum. If people internalize this rule then one might expect
that when asked to think about combining independent
causes their estimates of the effectiveness of this combina-
tion should be higher than either of the individual causes
and lower than their sum.
During the last decade, many experiments have tried to
find out whether causal inferences spontaneously made by
people conform to Equation 2. Unfortunately, this literature
has failed to yield a clear and consistent pattern of results
(cf., Allan, 2003; Barberia, Baetu, Sansa, & Baker, in press;
Buehner, Cheng, & Clifford, 2003; Liljeholm & Cheng,
2007; Lober & Shanks, 2000; Perales & Shanks, 2003,
2007). Most of these experiments have relied on experi-
mental preparations in which the participants received
information about the probability of the effect in the pres-
ence of the candidate cause and the probability of the effect
in the absence of that cause. In these preparations, the par-
ticipants are expected to use this information to isolate the
contribution of the candidate cause from the contribution of
M. A. Vadillo et al.: Causal Integration Rules 357
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all the background causes that might also influence the
effect. Applied to our coin-tossing example, this is analo-
gous to showing people the probability of getting heads
when Coins 1 and 2 are tossed together and when Coin 1
is tossed by itself, and subsequently ask them to evaluate
the contribution of Coin 2 to winning.
Quite surprisingly, none of those experiments has pro-
vided participants with information about the influence of
several potential causes and asked them to combine that
information so that they can predict the probability of the
effect given a combination of causes. In our example, this
would be equivalent to showing people the probability of
getting heads when either Coin 1 or Coin 2 is tossed and
then to ask them to predict the probability of getting heads
if we toss both coins simultaneously. This would provide a
more direct test of whether people spontaneously use the
noisy-OR integration rule when directly asked to predict
the effects of several causes. As we have just shown, the
noisy-OR rule makes very specific predictions for these sit-
uations: When people combine the causal impact of several
events, estimates of the combined effect should logically
fall in between the value of the individual effects and their
sum. Some studies in the associative learning tradition have
addressed the issue of how people combine information
about several causes when predicting events (Collins &
Shanks, 2006; Glautier, Redhead, Thorwart, & Lachnit,
2010; Soto, Vogel, Castillo, & Wagner, 2009; Van Ossel-
aer, Janiszewski, & Cunda, 2004), but most have focused
on the predictions of several associative models and have
not used experimental designs that would contrast the pre-
dictions made on the basis of the noisy-OR rule. The pres-
ent series of experiments aimed at providing such a
contrast.
Experiments 1A and 1B
In Experiments 1A and 1B, the participants were first given
information about the impact of several causes of the same
outcome. Then, they were presented with different combi-
nations of these causes presented in compound and were
asked to report which combinations they expected to be
more effective. On the basis of the noisy-OR integration
rule, the participants were expected to be indifferent to
some choices but not to others. Table 1 depicts the informa-
tion given to the participants about the effectiveness of five
causes in each different experiment. Table 2 shows the pre-
dictions of the noisy-OR, linear summation, and an averag-
ing rule for the test compounds in each experiment.
Individual participants should prefer compounds with
higher values and be indifferent to compounds that have
the same value. Even though participants were always
forced to choose one of the compounds, and therefore this
indifference cannot be detected at the individual level, we
should observe that choices distribute similarly between
the two compounds at the group level. The table shows that
if the participants, in all Experiments, use the noisy-OR
they should prefer compound CE to AB and be indifferent
to the choice between DE and AB.
For example, in Experiment 1A, participants were told
that the probability of an effect was .40 in the presence
of either A or B, .80 in the presence of C, .64 in the pres-
ence of D, and .00 in the presence of E. The cover story in
these experiments was carefully chosen so that the partici-
pants could assume that the probability of the effect was .00
in the absence of any of these causes. On the basis of this
information, participants were asked which compounds
they would prefer in order to maximize the likelihood that
they would produce the effect. Would the participants pre-
fer a compound including A and B to one including C and
E? If participants use a simple nonnormative linear summa-
tion strategy to make this decision, they should be indiffer-
ent, because the linear addition of A and B (.40 + .40) and
the linear addition of C and E (.80 + .00) are exactly the
same. However, if participants use a noisy-OR integration
rule they should prefer the compound CE, whose ability
to produce the effect is [.80 + .00 (.80 ·.00)] = .80,
to the compound AB, whose ability to produce the effect
is [.40 + .40 (.40 ·.40)] = .64. The participants were
also asked to decide between the compound AB and the
compound DE (.64 + .00). In this case, if the participants
use a linear integration rule, they are expected to choose
AB. However, if they use a noisy-OR integration rule they
are expected to be indifferent to both compounds. The spe-
Table 1. Design summary of the experiments
Experiment
Cause 1A, 2, 4A, 5A 1B, 3A, 3B 4B, 5B
A .40 .60 .50
B .40 .65 .50
C .80 .95 1.00
D .64 .80 .75
E .00 .30 .00
Note. The numbers represent the probability of the outcome
given each of the causes on each of the experiments.
Table 2. Predictions made by several integration rules
Compound Cause 1 Cause 2 Noisy-OR Linear Average
Experiments 1A, 2, 4A, 5A
AB .40 .40 .64 .80 .40
CE .80 .00 .80 .80 .40
DE .64 .00 .64 .64 .32
Experiments 1B, 3A, 3B
AB .60 .65 .86 1.25 .625
CE .95 .30 .965 1.25 .625
DE .80 .30 .86 1.10 .55
Experiments 4B, 5B
AB .50 .50 .75 1.00 .50
CE 1.00 .00 1.00 1.00 .50
DE .75 .00 .75 .75 .375
358 M. A. Vadillo et al.: Causal Integration Rules
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cific predictions of the noisy-OR rule are represented in
Figure 1.
Experiment 1B used different probabilities than Exper-
iment 1A (see Table 1), but the pattern of choices predicted
by the noisy-OR rule is the same as that of Experiment 1A:
In all cases the participants were expected to prefer CE to
AB and to be indifferent to the AB-DE choice. The main
reason why each experiment used different probability sets
is to make sure that the results are not dependent on specific
details of the parameters used. For example, in Experiment
1A one of the causes, E, had a .00 effectiveness, which
might have induced participants to avoid any compound
that contained E. Similarly, another cause, D, has a .64
effectiveness, which may be a highly unconventional num-
ber. Therefore, in Experiment 1B we used only multiples of
five, which are more conventional and commonly reported
values, and we changed the probability associated with
cause E from .00 to .30. Experiment 1B also included com-
pounds whose linearly-summed values would yield proba-
bilities with values larger than 1.00. For instance, if cause
A produces the effect with a .60 probability and B does
so with a .65 probability, then if the participants are asked
to compute the probability of the effect given A and B and
they apply a linear integration rule, then they might expect
the effect to occur with a probability of 1.25, which is an
impossible probability. The goal was to investigate if this
modification would increase the odds of observing behavior
more consistent with the noisy-OR rule, because it avoids
these inconsistencies.
Method
Participants
All the participants included in Experiments 1A and 1B
were psychology students from the University of Deusto.
Twenty-six of them volunteered for Experiment 1A and
57 for Experiment 1B.
Procedure and Design
All the materials were presented on a single sheet of paper.
The Appendix shows an example of the materials used for
the participants in one of the conditions, translated from the
Spanish original. The participants were asked to imagine
that the cosmetics industry had just discovered substances
that could change eye color to pink and that customers were
interested in buying products that included these. Then,
they were given information about the efficacy of each of
the substances (Alpha, Beta, Gamma, Delta, and Omega).
For example, they were told that when substance Alpha
was injected, eyes turned pink 40% of the time. The sub-
stances played the role of causes A–E in Table 1. The
assignment of substance names, and hence probabilities
of an outcome, to causes was partially counterbalanced, fol-
lowing a latin-square design. The order in which the
substances appeared on the sheet of paper was also counter-
balanced. After receiving this information, participants
were told that different companies were selling products
containing several combinations of these substances and
they were asked to choose which product they would buy
if they wished to change their eye color to pink. Specifi-
cally, participants were asked whether they would prefer
a product containing substances A and B to one containing
C and E, and whether they would prefer a product contain-
ing A and B to another containing D and E. The order of
these two questions and the order in which both compounds
were presented in each question (AB first vs. CE/DE first)
were counterbalanced across participants. The materials
and procedure used in each of the three experiments were
exactly the same, except for the probabilities of changing
the eye color shown in Table 1.
Results and Discussion
The proportion of choices of Compound AB over Com-
pounds CE and DE from Experiments 1A and 1B (and
the other four Experiments) are shown in Figure 1. The fig-
ure also indicates the proportion of preferences for AB over
CE and DE predicted by the noisy-OR rule. Instances
where these proportions differ from chance (p= .50) by
the binomial test are marked with an asterisk. Quite clearly
these proportions do not conform to the predictions of the
noisy-OR rule. For both experiments, it was predicted that
people should prefer CE to AB. However, this pattern was
not found in either experiment. The proportion of partici-
pants choosing either AB or CE did not differ in Experi-
ment 1A (binomial test, p= .327) and, contrary to the
prediction of the noisy-OR rule, a reliably high proportion
of participants chose AB over CE in Experiment 1B. Sim-
ilarly, it was predicted that participants should have no par-
ticular preference when choosing between AB and DE.
However, they showed a clear preference for AB in both
experiments. These results are more consistent with the lin-
ear summation rule that predicts the participants should be
indifferent to AB or CE but prefer AB to DE.
Experiment 2
The previous experiments were conducted with psychology
students, who receive training in probability theory and sta-
tistics as part of their formal education. In spite of this they
did not seem to use the more normative noisy-OR rule.
Nonetheless, it is possible that the results are not represen-
tative of the performance that one would find in the general,
untrained population. To address this, we replicated Exper-
iment 1A with middle-school age students.
Method
Participants, Procedure, and Design
Thirty-eight school students participated in Experiment 2.
All the participants were 12 or 13 years old. The procedural
M. A. Vadillo et al.: Causal Integration Rules 359
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details and materials were exactly the same as in Experi-
ments 1A–1B. Experiment 2 used the probabilities of
Experiment 1A (see Table 1).
Results and Discussion
AscanbeseeninFigure1,theresultsofExperiment2
were very similar to those of Experiment 1A, which was
based in the same probability set. Again participants did
not prefer CE to AB. The modest preference for AB over
CE did not reach statistical significance (binomial test,
p= .143). However, participants preferred the compound
AB to the compound DE. So, again, these results are at
odds with the predictions of the noisy-OR integration rule
and are consistent with linear summation. This implies that
the results of the previous experiments cannot be attributed
to the peculiarities of psychology students.
Experiments 3A and 3B
In the previous experiments, the information was provided
to the participants as probabilities. However, people com-
monly fail to use information presented as probabilities or
percentages correctly (e.g., Gigerenzer & Hoffrage, 1995;
Sloman, Over, Slovak, & Stibel, 2003). Therefore, in
Experiment 3A we presented information in a frequency
format (e.g., we told participants that when cause A was
present the effect occurred in 80 out of 200 occasions,
instead of telling them that produced the effect 40% of
the time). In Experiment 3B we used pie charts to present
the probabilities in a graphical manner.
Additionally, we also wanted to ensure that our results
would generalize to other scenarios. Therefore we used a
new cover story for Experiments 3A and 3B. Testing the
generality of the previous results is not only important for
methodological reasons, but also to rule out alternative
explanations. Although the results of our previous experi-
ments are in principle inconsistent with the predictions of
the noisy-OR rule, they can be accommodated by assuming
that the participants believed the two elements might inter-
act in producing the effect. In other words, the presence of
one substance would alter the ability of the other to produce
the effect, so that the effect of the compound might be dif-
ferent than the sum of its elements. In the following exper-
iments we used different materials to ensure that the results
of Experiments 1–2 were not due to specific details of that
particular cover story that might suggest potential interac-
tions between causes.
Method
Participants, Procedure, and Design
Fifty-three undergraduate psychology students volunteered
to participate in Experiment 3A and 34 volunteered for
Experiment 3B. In both experiments the set of probabilities
from Experiment 1B was again used (see Table 1). The new
cover story was inspired by a short science-fiction story
(Chiang, 2002) that we assumed most participants would
be unfamiliar with. The participants were asked to imagine
that scientists have discovered a treatment that makes peo-
ple insensitive to physical beauty. Some people demand
this treatment because they want to be more sensitive to
other peoples inner beauty than to their physical appear-
ance. Participants were given information about the effec-
tiveness of five different proteins that could produce this
effect. In Experiment 3A this probabilistic information
was given in a frequency format (e.g., ‘‘120 out of 200 peo-
ple injected with protein Alfa stop perceiving physical
beauty’’). In Experiment 3B, the information was given
by means of pie charts. After seeing this information, the
participants were asked to say which compounds they
would choose, if they wanted to stop being influenced by
the physical beauty of other people.
Results and Discussion
The pattern of results, depicted in Figure 1, is very similar
to that of previous experiments. Contrary to the predictions
of the noisy-OR rule, rather than showing indifference, the
majority of participants in both experiments reliably pre-
ferred the compound AB to the compound DE. In Experi-
ment 3A a reliable majority preferred AB to CE, but in
Experiment 3B they showed no clear preference for either
compound at the group level (binomial test, p=.392).
These results strengthen the idea that the results observed
in Experiments 1 and 2 are not attributable to either pecu-
liarities of the cover story or to the precise way in which
causal information was presented to participants.
Experiments 4A and 4B
In all the previous experiments, the results disagreed sys-
tematically with the predictions of the noisy-OR integration
rule. If anything, they are more consistent with alternative
integration rules such as linear summation, according to
which the causal impact of several factors is equal to the
simple sum of the causal impact of the elements. The
results are also quite consistent with the averaging strategy,
shown in the right-most column of Table 2, according to
which participants might estimate that the causal impact
of a compound is the average of the causal impact of the
elements. Although the binary decision test we have used
through Experiments 1–3 was well suited for contrasting
the predictions of the noisy-OR integration rule, it might
not be the best method for an exploratory analysis of alter-
native integration rules that might be used by the partici-
pants. Therefore, in addition to asking participants to
choose between AB and CE and between AB and DE, in
Experiments 4A and 4B we also requested ratings of the
probability of the effect given AB, CE, and DE.
360 M. A. Vadillo et al.: Causal Integration Rules
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Additionally, in Experiment 4B we asked participants to
decide between AB and C and between AB and D. As can
be seen in Table 1, in Experiment 4B, element E lacks any
power to produce the effect (i.e., the probability of the out-
come in the presence of E is zero). Therefore, removing it
from any compound should have no impact on choices: The
participantsbehavior should be largely similar when decid-
ing between AB and CE and when deciding between AB
and C. However, if removing this component has an effect
on the pattern of choices, this might provide an interesting
insight into the strategy used by the participants. The linear
and the noisy-OR rules predict no effect of adding the neu-
tral compound E, but if a participant relies on an averaging
strategy then adding a noneffective cause (such as E) to a
relevant cause (such as C or D) should reduce the perceived
causal impact. In this case, preference for C and D should
be higher when they are presented in isolation than when
they are presented in compound with E.
Method
Participants and Apparatus
Forty-eight psychology students from McGill University
participated in Experiment 4A and 58 psychology students
from the University of Deusto took part in Experiment 4B.
Three participants in Experiment 4A and two in Experi-
ment 4B wrote open responses (e.g., ‘‘some value between
0and40’’) in the space provided for their probability rat-
ings. The data from these participants were removed from
the analyses.
Design and Procedure
The pink-eyes cover story used in Experiments 1 and 2 was
used in Experiments 4A and 4B. As shown in Table 1, the
participants in Experiment 4A were exposed to the set of
probabilities used in Experiments 1A and 2A. For Experi-
ment 4B, a new probability set was used and is summarized
in Table 1. These probabilities were chosen so that, as in the
previous experiments, preference of CE over AB could be
taken as an evidence for the noisy-OR integration rule. As
mentioned in the introduction to this set of experiments,
Experiment 4B also differed from the previous ones in that
participants were not only asked to choose between AB
and CE, on the one hand, and AB and DE, on the other,
but also between AB and each of the elements C and D.
As in the previous experiments, the order in which informa-
tion about the different causes was given and the order of the
choices were partially counterbalanced following a latin-
square procedure.
Unlike the previous experiments, after making their
choices, participants from both Experiments 4A and 4B
were also asked to rate the probability of the effect given
the combinations AB, CE, and DE by means of the follow-
ing question: ‘‘If you inject substances A and B into 100
people, how many will get the pink eyes?’’ These extra
questions were added to further explore which strategy
might be responsible for the pattern of choices observed.
The order in which these three judgments were requested
was also counterbalanced following a latin-square design.
Results and Discussion
The choices made by participants in Experiments 4A and
4B are shown in Figures 1 and 2, respectively. As can be
seen, the pattern of preference of AB over CE and DE in
Experiment 4A replicates the results of previous experi-
ments. Again a larger proportion of participants preferred
AB to DE but, in spite of a trend in favor of CE, there
was no reliable difference in the proportions choosing AB
or CE. The same pattern of results concerning the AB,
CE, and DE compounds was found in Experiment 4B.
However, a different pattern was observed when partici-
pants chose between AB and the elements C and D now
presented alone. Participants tended to choose C and D over
AB more when they were presented on their own than when
they were presented in compound with E. This unexpected
effect of removing cause E from the CE and DE com-
pounds is not consistent with the noisy-OR integration rule
but is also not consistent with the simpler, linear integration
rule. However, as discussed above, it is consistent with an
averaging strategy.
Participants also estimated the number of times out of
100 that each causal configuration would turn the eyes pink.
Figure 3 plots the number of participants who made various
estimates for configurations AB, CE, and DE. The descrip-
tive statistics (mean and standard error) for each condition
are shown in each graph. As explained in the Introduction,
on the basis of the noisy-OR integration rule, we would
expect participantsratings to fall somewhere in between
the individual effects of each of the causes and their sum.
However, only a small number of participants gave ratings
within this range. The frequency graphs for the AB com-
pound show that the most frequent responses correspond
either to the linear sum of the probability of the effect given
A and the probability of the effect given B or the arithmetic
mean of the two probabilities. Ratings for CE and DE are
less informative because the responses predicted by the lin-
ear and the noisy-OR rules are exactly the same. However, it
is still interesting to note that many participants in those con-
ditions provided responses equal to the arithmetic mean of
the probabilities of the elements. This pattern of judgments
further suggests that the reduction in preference for AB over
CE and DE caused by removing element E from the
compounds may be due to the factthat some participants rely
on an averaging strategy.
Experiments 5A and 5B
In Experiments 5A and 5B we implemented a number of
modifications that we expected would promote behavior
consistent with the noisy-OR rule. First, we used two cover
stories that precluded the perception of any potential
rational interaction between causes. In Experiment 5A, par-
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ticipants were asked to imagine that they were playing a
coin-tossing game similar to that described in the Introduc-
tion. They were given the probabilities of obtaining heads
in several biased coins and then were asked which combi-
nations of coins would provide the higher probabilities of
winning (i.e., getting a head in at least one of the coins).
In Experiment 5B, participants were alerted about several
independent potential dangers of living in a fictitious pla-
net. Then they were asked to choose among several routes
for a mission on that planet. Each route involved a different
combination of dangers. Most importantly, according to the
description that we gave to participants, each of these dan-
gerous events could produce the death by means of com-
pletely different mechanisms that should not interact with
one another. Therefore, participants should have no reason
to assume the causes might interact.
As in Experiments 4A and 4B, participants not only had
to choose between different combinations of causes, but
they also had to provide probability ratings of each of the
combinations. Furthermore, in Experiment 5B we presented
the information using a frequency format that has been pre-
viously used in studies that found support for the noisy-OR
integration rule (e.g., Liljeholm & Cheng, 2009). We also
used different probability sets in both experiments (see
Table 1).
Method
Participants and Apparatus
Forty-nine psychology students from McGill University
participated in Experiment 5A and 30 engineering and
medicine students from the University of the Basque Coun-
try took part in Experiment 5B. All of them performed the
experiment on paper and pencil questionnaires. One partic-
ipant in Experiment 5A failed to provide probability rating
so was removed from the final analyses.
Design and Procedure
Two new cover stories were used in Experiments 5A and
5B. In Experiment 5A participants were asked to imagine
Figure 2. Proportion of participants preferring the com-
pound AB over CE and over DE in Experiments 4B and 5B.
To make the comparison with Experiment 4B easier,
preferences were reverse-scored in Experiment 5B (see the
main text for further details). Asterisks are placed upon
values that are significantly different from .50 with a= .05.
Figure 1. Pattern of preferences predicted by the noisy-OR rule and actual proportion of participants preferring the
compound AB over CE and over DE in Experiments 1A–4A and Experiment 5A. The 0.50 axis represents indifference
between AB and the other options. Proportions above 0.50 represent a preference for AB over the other options.
Asterisks are placed upon values that are significantly different from .50 in a binomial test with a= .05.
362 M. A. Vadillo et al.: Causal Integration Rules
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that they were playing a coin-tossing game with three dif-
ferent biased coins. In this game, they would win if they
obtained at least one head. The three coins produced heads
40%, 80%, and 64% of the times, respectively. Then they
were asked to choose whether they would prefer to toss
twice the coin with the 40% chance of winning (and win
if they got at least a head in one of the tosses) or to toss just
once the coin with the 80% chance of winning. Similarly,
they had to choose whether they preferred tossing the
40% coin twice or tossing the 64% just once. After making
these decisions, participants were asked to rate the probabil-
ity of winning if they tossed the 40% coin twice.
In Experiment 5B we used a different science fiction
scenario. Participants were asked to imagine that they were
scientists in charge for colonizing an alien planet. They had
been ordered to install an atmosphere purification system
that would allow humans to live on the planet. They were
also told that the routes to the ideal candidate locations
for the purification system were dangerous. They were
given information about five specific problems that they
might encounter. They might: Approach a methane ocean,
which might cause death by burning; step onto mercury
quicksand, which might drown them; swallow alien lichen,
which might poison them; fall into a cave, causing death by
impact; and, finally, be exposed to electric storms, which
might cause death by electrocution. These dangers were
mentioned in random order and were randomly assigned
to play the role of causes A–E in Table 1. On a second sheet
they were given detailed information about the probability
that each of these events might cause death. Specifically,
for each event, they were told how many of the last 24
explorers who had faced those situations had died. This
information was shown by means of frequency charts in
which each explorer who had survived was represented as
a smiley face and each explorer who had died was repre-
sented as a skull. In these charts, all the smiley faces were
grouped together as were all the skulls (for a similar proce-
dure, see Liljeholm & Cheng, 2009). The final sheet of the
questionnaire asked whether the participants would prefer
to use a route where they would face dangers A and B to
one in which they would face dangers C and E. Similarly,
they were asked to choose between AB and DE, between
AB and C, and between AB and D. Unlike all the previous
experiments, because participants here had to avoid a neg-
ative outcome, they should choose the less effective combi-
nation of causes (instead of the more effective). Finally, as
in Experiments 4A and 4B, participants were asked to rate
the probability of dying if they faced situations AB, situa-
tions CE, and situations DE. As in previous experiments,
these questions were asked in a frequency format (e.g.,
‘‘Out of 100 new explorers taking a route that exposed them
to X and Y, how many would die?’’). We expected that
framing this test question in terms of frequencies would
promote more normative responses (Gigerenzer &
Hoffrage, 1995; Sloman et al., 2003).
Results and Discussion
In contrast to the rest of the results represented in Figure 1,
the pattern of choices made by participants in Experiment
5A was surprisingly consistent with the predictions of the
Figure 3. Frequencies of responses to the probability ratings requested to the participants in Experiments 4A, 4B, and
5B. The x-axis represents specific values of judgments given by participants and the y-axis the frequency of those values
in our sample.
M. A. Vadillo et al.: Causal Integration Rules 363
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noisy-OR rule: They preferred to toss a single coin with an
80% chance of winning over twice tossing a coin with a
40% chance of winning. They were, however, indifferent
to the choice between tossing the 40% coin twice or the
64% coin once. Their numerical ratings of the probability
of winning if they tossed the 40% coin twice are repre-
sented in Figure 4. These ratings are in stark contrast with
the pattern of decisions. Following the noisy-OR rule, we
would expect most ratings to fall within the 40–80 interval.
However, few participants gave responses within that
range. Instead, most of them responded either 40 or 80.
These numerical responses are consistent with the use of
an averaging or a linear summation rule, respectively.
The preference responses for Experiment 5B are
depicted in Figure 2. To facilitate comparison with the pre-
vious experiments (in which the outcome was a desired
event) the choices made by participants in Experiment 5B
(in which the outcome was not desired) were reverse-
scored. As can be seen in Figure 2, participants did not
think that the compound of two causes with a 50% efficacy
was more dangerous than the compound of a 75% effective
cause and a 0% effective cause. However, they thought the
combination of the 100% effective cause and the 0% effec-
tive cause was more dangerous than the combination of the
two 50% effective causes. This pattern of results is consis-
tent with the noisy-OR rule. Unlike in Experiment 4B (also
depicted in Figure 2) the pattern of choices was not influ-
enced by the presence or absence of the 0% effective cause,
suggesting that in this experiment participants did not aver-
age the effect of several causes. Figure 3 shows the ratings
of the probability of the effect given the combinations AB,
CE, and DE. The first of them provides the most valuable
insight into the strategies used by participants. As can be
seen, most participants thought that the probability of the
effect given two 50% effective causes was 75%. Again, this
is consistent with the noisy-OR integration rule. Note, how-
ever, that one third of participants gave ratings equal to
either 50 or 100. These are the values from the averaging
and linear integration rules, respectively. Moreover four
of the participants gave ratings that were outside the 50–
100 range. Therefore even in this experiment only a bare
majority of participants gave ratings consistent with the
noisy-OR rule.
General Discussion
The results of the present series of experiments are more
consistent with the idea that people use a diversity of inte-
gration rules than with the assumption that there must be a
single, monolithic rule (either the linear or the noisy-OR
one) that is systematically adopted in all cases. On the
one hand, the preference choices of participants in Experi-
ments 1A, 1B, 2, 3A, 3B, and 4A agreed with the predic-
tions of the linear rule (see Figure 1). On the other hand,
choices in Experiments 5A and 5B were consistent with
the predictions of the noisy-OR rule (see Figures 1 and
2). However the participantsratings of the probabilities
weaken this conclusion. The probability ratings provided
in Experiments 4A–4B and 5A–5B show a complex pattern
(see Figures 3 and 4). With only the exception of Experi-
ment 5B, most of the ratings suggested that participants
were using either a linear integration rule or, alternatively,
an averaging heuristic. Note that the averaging strategy can
account for all the preference choices that can be explained
by the linear rule, plus the negative effect of adding a 0%
effective cause to a compound found in Experiment 4B.
Furthermore, even in the experiments that showed the pat-
tern of choices predicted by the noisy-OR rule, the proba-
bility ratings of many participants still seemed to be
based on an averaging strategy.
The diversity of strategies used by participants in the
present series of experiments converges with the results
of previous studies on summation in causal learning and
reasoning. Although none of those experiments was specif-
ically designed to contrast the predictions of the noisy-OR
integration rule with those of the linear rule, their results
show that summation is highly unstable and that it can
change dramatically depending on minor procedural details.
For instance, Glautier et al. (2010) found that increasing the
similarity between two causes by means of adding a com-
mon feature reduces and even reverses summation. Simi-
larly, Van Osselaer et al. (2004) found that seemingly
minor alterations in the procedure for collecting judgments
or presenting the information could completely abolish
summation. Note that this reduction in the size of summa-
tion is what one would expect on the basis of an averaging
strategy: If participants average the causal power of each
factor, then the whole need not be better than either of its
parts.
Most interestingly, some experiments show that even
within a single experimental task and with the same mate-
rials, participants might sometimes use different integration
rules. Waldmann (2007) found that subtle manipulations
could influence whether participants combined causes using
an averaging or an additive (linear or noisy-OR) rule. How-
ever, when participants had to subtract instead of combine
causal powers, their responses tended to be consistent with
an additive rule most of the time, even when the same
materials and instructions were used. This is consistent with
the divergences that we found in some of our experiments
between the pattern of choices and the probability ratings.
For instance, in Experiment 5A, participantschoices were
consistent with the noisy-OR rule. But their probability rat-
Figure 4. Frequencies of responses to the probability
ratings requested to the participants in Experiment 5A.
364 M. A. Vadillo et al.: Causal Integration Rules
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ings were more consistent with either the averaging or the
linear rules.
The results of our experiments pose problems for mod-
els relying exclusively on either the linear rule or on the
noisy-OR rule. Those models can only accommodate our
results by assuming that, in spite of our efforts to prevent
it, participants assumed potential interactions between
causes. From this perspective, theories that rely on the lin-
ear integration rule could explain behavior that seems con-
sistent with the noisy-OR integration rule by assuming that
participants believed there were interactions. Alternatively,
the theories that rely on the noisy-OR rule might explain
behavior consistent with the linear rule by making exactly
the opposite assumption. Based on this divergent predic-
tion, an interesting idea for future research would be to test
whether participants are more likely to believe there are
interactions in scenarios that promote behavior consistent
with the linear rule or in scenarios that promote behavior
consistent with the noisy-OR rule. Given that we did not
ask whether participants assumed interactions between
causes, our data are not suited to test this prediction. How-
ever, it is interesting to note that we observed decisions and
judgments were more consistent with the noisy-OR rule in
the two experiments (5A and 5B) that most strongly
emphasized the independence or lack of interaction
between causes. In principle, this supports the predictions
of the noisy-OR rule. As explained above, Experiments
1–4 used cover stories in which drugs and chemical sub-
stances were used as potential causes of physiological
and psychological effects. Perhaps the participantsfamil-
iarity with the, sometimes complex, effects of drugs might
have encouraged them to perceive potential interactions
between the causes. This assumption, in turn, would be
responsible for any behavior departing noticeably from
the predictions of the noisy-OR rule. Although plausible,
this assumption does not account well for previous experi-
mental findings, given that experiments seemingly consis-
tent with the noisy-OR rule also used cover stories
involving chemicals and drugs that are fairly similar to
the materials used in our Experiments 1–4 (e.g., Liljeholm
& Cheng, 2007, 2009). In light of these discrepancies, we
suggest that future experiments contrasting these two inte-
gration rules include a direct measure of whether or not par-
ticipants believe there to be potential interactions.
Stressing the noninteractivity of causes might not be the
only factor improving the normativity of judgments and
choices in our last experiments. Quite interestingly, the only
experiment that yielded results perfectly consistent with the
predictions of the noisy-OR rule (Experiment 5B) not only
emphasized that each cause acted by means of different and
independent mechanisms, but also included an important
change in the way information was presented. Specifically,
while the other experiments presented the information by
means of probabilities, frequencies, or pie charts, in Exper-
iment 5B the information was presented in frequency charts
in which each instance of the effect was represented as an
individual item among other instances where the effect was
absent. Quite interestingly, this information format has been
used extensively in other experiments that found support
for rational models relying on the noisy-OR rule (e.g.,
Buehner et al., 2003; Liljeholm & Cheng, 2009; Novick
& Cheng, 2004). Given that the information format is
known to have an effect on causal judgments (Shanks,
1991; Valle-Tourangeau, Payton & Murphy, 2008; Ward
& Jenkins, 1965), this coincidence might deserve further
attention. It is possible that presenting the information in
numeric format, as is frequently done in many causal learn-
ing experiments, invites participants to combine the infor-
mation using simple algebraic operations that, despite
their intuitive appeal, yield nonnormative behavior. Unfor-
tunately, our Experiment 5B differed from previous exper-
iments in other procedural details, apart from the
presentation format, and therefore it remains unclear to
what extent the information format, rather than any of the
other differences, is responsible for these results. However,
in light of the prevalence of using this method of presenting
information in many of the experiments supporting the
noisy-OR rule, an in-depth exploration of the effects of
information formats on the normativity of causal judgment
and decision-making seems a promising idea for future
research.
In the Introduction we mentioned that some associative
models rely on the linear integration rule (Rescorla &
Wagner, 1972) while some others rely on the noisy-OR rule
(Danks et al., 2003). Interestingly, there is a third category
of models that, even without committing specifically to any
rule, make predictions consistent with an averaging strat-
egy. In contrast to so-called elemental models, configural
models (Kruschke, 1992; Pearce, 1987, 1994) assume that
participants do not encode causes individually, but treat
every combination of causes as a completely different
entity. Very interestingly, this idea implicitly implements
the assumption that causes interact, given that from this per-
spective the effect of a combination of causes need not be
the same as the sum of the effects of its elements. In con-
figural models, the extent to which participants generalize
what they know about a single cause to a novel combina-
tion of causes depends on the similarity between the ele-
ment and the compound. In the case of our experiments,
according to the rule proposed by Pearce (1987, 1994),
the similarity between A and AB would be .50, because
half of the elements in AB are present in A. For the same
reasons, the similarity between B and AB would also be
.50. If the associative strength of A is .40 and that of B is
.40 as well, as in many of our experiments, then the asso-
ciative strength that generalizes to the compound AB is
(.40 ·.50) + (.40 ·.50) = .40. In other words, these mod-
els predict that the associative strength of the compound
AB should be equal to the average of the associative
strengths of A and B. Therefore, the generalization rules
used by these models could potentially explain why many
participants showed a pattern of responses consistent with
an averaging rule.
Quite interestingly, just as we found that the behavior of
participants can be consistent with different integration
rules, research on associative learning shows that people
and other animals sometimes behave as predicted by ele-
mental models and sometimes behave as predicted by con-
figural models (for a review, see Melchers, Shanks, &
Lachnit, 2008). Therefore, instead of deciding which of
M. A. Vadillo et al.: Causal Integration Rules 365
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these models fits better to the data, researchers are now
trying to find out which conditions promote elemental or
configural learning and to develop formal models that
can incorporate both processing strategies (Schmajuk &
DiCarlo, 1992; Wagner, 2003). In light of the diversity of
strategies used by participants in our experiments and in
the previous reports on summation, we think that the area
of causal learning and reasoning would benefit from a sim-
ilar approach: Instead of deciding whether it is the linear or
the noisy-OR rules that underlies causal learning, it seems
more sensible to explore when and how participants behave
according to those or other rules and to develop models that
can incorporate a diversity of strategies.
Acknowledgments
MAV, NOC, and IB were supported by Grant IT363-10
from Departamento de Educacin, Universidades e Investi-
gacin of the Basque Government and Grants PSI2011-
26965 (NOC and MAV) and PSI2010-20424 (IB) from
Ministerio de Ciencia e Innovacin. NOC was also sup-
ported by fellowship BFI09.102 from the Basque Govern-
ment. AGB was supported by a Discovery Grant from the
National Sciences and Engineering Research Council
(NSERC) of Canada. We would like to thank Fernando,
Blanco, Pedro Cobos, Francisco Lpez, David Luque, Hel-
ena Matute, Joaqun Mors, Robin Murphy, Cristina Orgaz,
Carmelo Prez, Frdric Valle-Tourangeau, and Ion Yar-
ritu for their helpful comments on several drafts of this
article.
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Received November 18, 2012
Revision received October 21, 2013
Accepted October 22, 2013
Published online March 11, 2014
Miguel A. Vadillo
Division of Psychology and Language Sciences
University College London
26 Bedford Way
London WC1H 0AH
UK
Tel. +44 20 7679 5364
E-mail m.vadillo@ucl.ac.uk
Appendix
Please, answer the questions in the order they are written.
Once you have answered one question, please do not come
back to this question later to change your answer. We are
interested in your first answers.
Try to imagine the following situation. The cosmetics
industry has recently been revolutionized by the discovery
of a number of natural substances that can change the color
of the peoples eyes. Interestingly, the color most demanded
by consumers is not any of the normal ones (blue, green,
brown...), but pink. Unfortunately, there are just a few
known substances that produce this eye color.
For the time being, scientists have identified only five
substances that may cause the pink eyes. The results of
these studies are as follows:
If you inject substance Alpha, eye color turns pink 40%
of the time.
If you inject substance Beta, eye color turns pink 40% of
the time.
– If you inject substance Gamma, eye color turns pink
80% of the time.
If you inject substance Delta, eye color turns pink 64%
of the time.
If you inject substance Omega, eye color turns pink 0%
of the time.
There are several cosmetic companies that sell products
to change the color of the eyes. Most of these products are
based on different compounds of one or several of these
substances. Imagine that you also want to change the color
of your eyes and answer the following questions accord-
ingly:
1. If there is a product that contains a compound of the
Alpha and Beta substances, and another that contains a
compound of the Gamma and Omega substances, which
one would you choose? (Please circle your choice)
Alpha and Beta / Gamma and Omega
2. If there is a product that contains a compound of the
Alpha and Beta substances, and another that contains
a compound of the Delta and Omega substances, which
one would you choose? (Please circle your choice)
Alpha and Beta / Delta and Omega
M. A. Vadillo et al.: Causal Integration Rules 367
2014 Hogrefe Publishing. Distributed under the Experimental Psychology 2014; Vol. 61(5):356–367
Hogrefe OpenMind License (http://dx.doi.org/10.1027/a000001)
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