imagine that you were asked to evaluate whether a new medicine produces an allergic 
 reaction as a side effect .
to accomplish this task , you are shown the individual 
 medical records of a number of patients where you can find out whether each patient 
 took the medicine and whether he / she suffered an allergic reaction .
how should you 
 assess the relation between taking the medicine and suffering the allergy ?
as shown 
 in figure 1a , to make this 
 judgment you would need four pieces of information that can be summarized in a 
 2    2 contingency table .
you would need to know how many 
 patients took the medicine and suffered an allergy ( cell a ) , how 
 many patients took the medicine and did not suffer an allergy ( cell 
 b ) , how many patients did not take the medicine but suffered an 
 allergy nevertheless ( cell c ) , and , finally , how many patients did 
 not take the medicine and did not suffer an allergy ( cell d ) .
based on this information , you could compute some measure of contingency and estimate 
 whether or not that level of contingency is substantially different from zero . 
 although there are alternative ways to measure contingency , the 
 p rule is usually considered a valid normative index 
 ( allan , 1980 ; cheng & novick , 1992 ; jenkins & ward , 1965 ) . 
 according to this rule ,
if you want to assess the degree of contingency between a 
 cue ( e.g. , taking a medicine ) and an outcome ( e.g. , an allergy ) , then you need to 
 compute(1)where p(o|c ) is the probability of the outcome given 
 the cue and p(o|~c ) is the probability of the outcome in the 
 absence of the cue . as shown in figure 
 1a , these two probabilities can be easily computed from the information 
 contained in a contingency table .
positive values of p 
 indicate that the probability of the outcome is higher when the cue is present than 
 when it is absent ( see , e.g. , figure 
 1b ) .
in contrast , negative values indicate that the probability of the 
 outcome is reduced when the cue is present ( e.g. , figure 1c ) . finally , if the probability of the outcome is the 
 same in the presence as in the absence of the cue , the value of 
 p is always 0 ( e.g. , figure 1d ) . in these latter cases , there is no contingency 
 between cue and outcome .
when laypeople are asked to estimate the contingency between two events , do their 
 judgments agree with this normative rule ? as we will see , the answer is both 
  yes  and  no .  to study how people detect contingency , 
 researchers typically rely on a very simple task that has become a standard 
 procedure in contingency - learning research . during the task
, participants are 
 exposed to a series of trials in which a cue and an outcome may be either present or 
 absent , and they are instructed to discover the relationship between both . as in our 
 previous example , the cue can be a fictitious medicine taken by some patients and 
 the outcome can be an allergic reaction . on each trial ,
participants are first shown 
 information about whether a patient took the drug on a specific day and they are 
 asked to predict whether or not they think that this patient will develop an 
 allergic reaction .
after entering a yes / no response , they receive feedback and they 
 proceed to the next trial . once they have seen the whole sequence of trials in 
 random order , the participants are asked to rate their perceived strength of the 
 relationship between the medicine and the allergic reaction .
the usual result is that participants judgments tend to covary with the 
 objective drug - allergy contingency as measured by p ( e.g. , 
 lpez et al . , 1998 ; 
 shanks & dickinson , 1987 ; 
 wasserman , 1990 ) .
therefore , 
 to some extent participants seem to be able to track the actual cue - outcome 
 contingency .
for instance , participants judgments tend to be biased by the marginal 
 probability of the outcome , defined as the proportion of trials in which the outcome 
 is present , that is , 
 p(outcome )  =  ( a  +  c)/(a  +  b  +  c  +  d ) . 

figure 1e depicts an example 
 where there is no contingency between cue and outcome , but the outcome tends to 
 appear very frequently . in situations like this , participants tend to overestimate 
 contingency ( allan & jenkins , 
 1983 ; allan et al . , 
 2005 ;
buehner , cheng , & 
 clifford , 2003 ; lpez 
 et al . , 1998 ; musca , vadillo , 
 blanco , & matute , 2010 ; wasserman et al . , 1996 ) .
similarly , other things being equal , 
 participants judgments tend to covary with the marginal probability of the 
 cue , defined as the proportion of trials in which the cue is present ; that is , 
 p(cue )  =  ( a  +  b)/(a  +  b  +  c  +  d ) . 

figure 1f represents an 
 example where the probability of the cue is high , but there is no contingency 
 between cue and outcome .
again , participants tend to overestimate contingency in 
 situations like this ( allan & jenkins , 
 1983 ; matute et al . , 
 2011
; perales et al . , 
 2005 ; vadillo , musca , blanco , 
 & matute , 2011 ; wasserman et al . , 1996 ) .
the biasing effects of the probability of the 
 outcome and the probability of the cue are typically known as 
 outcome- and cue - density biases . as the astute 
 reader might guess , the most problematic situation is that in which both the 
 probability of the outcome and the probability of the cue are large ( figure 1 g ) .
participants seem to 
 find particularly difficult to detect the lack of contingency in these cases ( blanco , matute , & vadillo , 
 2013 ) .
it is interesting to note that biases akin to these have also been found in the 
 social psychology literature on illusory correlations in stereotype formation ( hamilton & gifford , 1976 ; kutzner & fiedler , 2015 ; murphy , schmeer , valle - tourangeau , 
 mondragn , & hilton , 2011 ) . in these experiments , participants 
 are shown information about the personality traits of members of two social groups . 
 across trials ,
participants see more information about one of the groups than about 
 the other and they also see more people with positive traits than people with 
 negative traits .
most importantly , the proportion of positive and negative traits is 
 identical in both social groups .
therefore , there is no correlation between 
 membership to the majority or the minority group and the quality ( positive vs. 
 negative ) of personality traits . as can be seen , if one assumes that social groups 
 play the role of cues and that positive and negative traits play the role of 
 outcomes , this situation is identical to the one represented in figure 1 g .
although there is no correlation 
 between groups and traits , when participants are asked to rate the traits of both 
 groups , they systematically tend to judge the majority group more favorably than the 
 minority group . in other words , despite the absence of a real correlation , 
 participants tend to associate the majority group with the most frequent ( positive ) 
 personality traits and the minority group with the least frequent ( negative ) 
 personality traits .
the interesting point , for our current purposes , is that we can 
 interpret illusory correlations as a combination of cue- and outcome - density biases , 
 which means that this effect may be explained as a contingency - learning 
 phenomenon .
traditionally , demonstrations of cue / outcome - density biases and illusory correlations 
 have been explained in terms of simple associative processes analogous to those 
 assumed to account for classical and instrumental conditioning ( e.g. , alloy & abramson , 1979 ; lpez et al . , 1998 ; matute , 1996 ; murphy et al . , 2011 ; shanks , 1995 ; sherman et al . , 2009 ;
van rooy , van overwalle , vanhoomissen , labiouse , & french , 
 2003 ) .
the associative learning rule proposed by rescorla and wagner ( 1972 ) provides the simplest 
 example of this family of models . according to the rescorla - wagner model , when a cue 
 is followed by an outcome
, an association or link is formed between the 
 representations of both stimuli . after each pairing , the strength of the association 
 is assumed to increase or decrease according to the formula:(2)where vc - o is the increase in the 
 strength of the cue - outcome association after that trial ,  
 and  are learning rate parameters depending on the salience 
 of the cue and the salience of the outcome , respectively ,  
 is a dummy variable coding whether the outcome was present or absent in that trial , 
 and vtotal is the sum of the associative strengths of all the potential 
 cues of the outcome present in that trial .
in addition to the target cue , a 
 contextual cue is assumed to remain present in all trials .
the association of the 
 contextual cue with the outcome is also updated according to equation 2 . to illustrate how this simple model accounts for cue- and outcome - density biases , in 
 figure 2 we show the 
 predictions of the model when given as input the six contingencies depicted in figures 1b1 g .
the top panel shows the 
 predictions of the model when the contingency is positive ( 1b ) , negative ( 1c ) , or 
 zero ( 1d ) .
as can be seen , eventually the strength of the cue - outcome association 
 tends to converge to the true contingency , as defined by p . 
 by the end of training
, the model learns a positive association when the cue - outcome 
 contingency is positive and a negative association when the cue - outcome contingency 
 is negative . when the cue - outcome contingency is exactly zero , the associative 
 strength of the cue also tends to move toward this value .
therefore , the model does 
 a good job at explaining why people are good at detecting contingencies ( see chapman & robbins , 1990 ; danks , 2003 ; wasserman , elek , chatlosh , & baker , 1993 ) . 

however , the model also predicts some systematic deviations from the true 
 contingency . in the four conditions where the contingency is zero , depicted in the 
 bottom panel ,
these overestimations are larger when the outcome ( 1e ) or the 
 cue ( 1f ) is very frequent , and even larger when both of them are very frequent ( 1 g )
. 
 therefore , the model also provides a nice explanation for cue- and outcome - density 
 biases ( matute , vadillo , blanco , & 
 musca , 2007 ; shanks , 
 1995 ; vadillo & luque , 
 2013 ) .
regardless of the merits and limits of associative models ( mitchell , de houwer , & lovibond , 2009 ; shanks , 2010 ) , for our present 
 purposes , their most important feature is that , according to them , the same 
 mechanism explains ( 1 ) why people are sensitive to contingency and ( 2 ) why their 
 judgments are also biased under some conditions .
a single process accounts for 
 accurate and biased contingency detection . as we will discuss below ,
this is the key 
 feature of single - process models that distinguishes them from their dual - process 
 counterparts .
it is interesting to note that this property is also shared by other early models of 
 biased contingency detection that do not rely on associative learning algorithms . 

for example , instance - based models assume that each cue - outcome trial is stored in a 
 separate memory trace in long - term memory ( fiedler , 1996 , 2000 ; meiser & hewstone , 
 2006 ; smith , 1991 ) . 

parts of these memory traces may be lost during the encoding process . in a situation 
 like
the one represented in figure 
 1 g , the loss of information has very little impact on the encoding of 
 cell a events , because there are many redundant memory traces 
 representing the same type of event .
however , information loss can have a severe 
 impact on the encoding of cells c and d 
 events because there are fewer traces representing them . as a result
, the 
 information encoded in memory contains more ( or better ) information about frequent 
 events ( cells a ) than about infrequent events ( cells 
 c and d ) .
therefore , information loss explains 
 why participants tend to perceive a positive contingency whenever type 
 a events are more frequent than other events in the contingency 
 table .
most importantly , according to these models we do not need to invoke 
 different mechanisms to explain the cases in which participants are sensitive to the 
 actual contingency and the instances in which their judgments are biased .
accurate 
 and biased contingency detection are supposed to arise from the same operating 
 mechanisms .
therefore , from our point of view , instance - based theories also belong 
 to the category of single - process models . for our present purposes ,
propositional models can be considered yet another case of 
 single - process models . in a thought - provoking series of papers , de houwer and 
 colleagues ( de houwer , 2009 , 
 2014 ; mitchell et al . ,
2009 ) have suggested that all 
 instances of human contingency learning might depend exclusively on the formation 
 and truth evaluation of propositions .
in contrast to simple associations , 
 propositions do not just represent that events in the environment are related to 
 each other : they also qualify how they are related ( lagnado , waldmann , hagmayer , & sloman , 2007 ) . 

however , the difference between them can not be represented in terms of 
 a simple association .
although propositional models do not necessarily exclude the 
 contribution of associative processes ( see moors , 2014 ) , the representational power of propositions allows 
 these models to explain aspects of learning that fall beyond the scope of simple 
 associative models ( de houwer , beckers , 
 & glautier , 2002 ; gast 
 & de houwer , 2012 ; zanon , de houwer , & gast , 2012 ) .
these ideas have not been 
 formalized in a mathematical model , but nothing in their current formulation 
 suggests that separate mechanisms would be needed to account for accurate and biased 
 contingency detection . for our present purposes , the idea that all learning depends 
 on the evaluation of propositions represents yet another example of a single - process 
 model .
during the last decade , some researchers have abandoned these explanations in favor 
 of more complex dual - process theories ( allan et al . , 2005 , 2007 ; perales et al . , 
 2005 ; ratliff & nosek , 
 2010 ) .
although differing in the detail , the core idea of these proposals 
 is that they call upon one mechanism to explain how people ( correctly ) track 
 contingencies and a different mechanism to explain why their judgments are sometimes 
 biased by cue and outcome density .
this proposal is based on the results of several 
 experiments showing what appear to be systematic dissociations between different 
 dependent measures .
cue / outcome density biases and illusory correlations are typically assessed with a 
 numerical causal or contingency rating that participants provide at the end of the 
 experiment . as explained ,
these judgments show sensitivity to both actual 
 contingency and to the biasing effects of cue and outcome density . however , 
 according to these authors ( allan et al . , 
 2005 , 2007 ; perales et al . , 2005 ; ratliff & nosek , 2010 ) ,
other 
 dependent measures seem to be sensitive only to the actual contingency , showing no 
 trace of cue- or outcome - density biases .
these alternative measures are assumed to 
 be relatively uninfluenced by higher - order reasoning processes , or at least , less 
 influenced by them than the numerical judgments typically used as dependent 
 variables . from this point of view , it follows naturally that there must be a very 
 basic learning mechanism that explains how people accurately track contingencies and 
 whose output can be directly observed in these dependent variables . in contrast , 
 judgments are affected both by contingency and by cue / outcome density biases . 
 because measures that address directly the learning mechanism do not seem to be 
 sensitive to biases , these must be operating through a different mechanism that 
 influences judgments but not the original encoding of information .
this is the 
 reason why these models incorporate different processes to account for accurate and 
 biased contingency detection .
a schematic representation of the role of learning and 
 judgment processes of biased contingency detection is offered in figure 3 .
therefore , in dual - process models two different and successively operating mechanisms 
 are invoked to explain why people are sensitive to contingency but they are also 
 biased by the marginal probabilities of the cue and the outcome .
the first mechanism 
 would include basic encoding and retrieval processes that are highly sensitive to 
 the objective contingency .
the information gathered by this mechanism would then 
 feed forward to other mechanisms involved in judgment and decision - making processes . 

these findings are certainly challenging for the theories of contingency detection 
 discussed in the previous section , which in the absence of additional assumptions 
 would typically anticipate similar effects in all dependent measures of contingency 
 learning .
note , however , that dissociations are not a perfect basis to draw 
 inferences about the presence of one or multiple systems .
borrowing an example from 
 chater ( 2003 ) , following the 
 logic of dissociations one might conclude that human beings must have different 
 digestive systems , because some people are allergic to prawns , while others are 
 allergic to peanuts .
tiny differences in the way a single mechanism tackles similar 
 problems might create the illusion that several mechanisms are involved in an 
 operation that is actually best described in terms of a single - system process . in spite of these concerns about the interpretation of dissociations , in the 
 following sections we do not question this logic , but the reliability of the 
 findings that support dual - process models of contingency learning .
the evidence for 
 these models stems from three papers published during the last decade ( allan et al . , 2005 ; perales et al .
although their theoretical conclusions are quite consistent , the empirical findings 
 reported in each of them are noticeably different . in the following sections we 
 review each of them in turn and discuss their merits and shortcomings . to overview 
 our criticisms , we argue that some of these findings seem to be poorly replicable or 
 generalizable , while others are based on possibly faulty dependent measures .
the first piece of evidence suggesting that these biases are not observed in all 
 dependent variables comes from an interesting experiment conducted by allan et al .
two groups of 
 participants were instructed to discover the effect of a series of fictitious 
 chemicals ( playing the role of the cue ) on the survival of a sample of bacteria in a 
 petri dish ( playing the role of the outcome ) . in each trial , participants saw 
 whether or not the chemical was present in a sample and they were asked to predict 
 whether or not the bacteria would survive . immediately after entering their 
 responses ,
they were informed of the outcome of the trial ( i.e. , whether the 
 bacteria survived ) and they proceeded to the next trial . at the end of training , 

participants were asked to rate to what extent the chemicals had a positive or a 
 negative impact on the survival of bacteria , using a numerical scale from 
 100 ( negative impact ) to 100 ( positive impact ) .
for one 
 of them , the contingency was moderately positive 
 ( p  =  .467 ) , while for the other one the 
 contingency was always null ( p  =  .000 ) .
each 
 participant was asked to complete three of these contingency - detection problems , all 
 of them with the same overall contingency , but with different probabilities of the 
 outcome .
therefore , the experiment relies on a 2    3 factorial 
 design with contingency as a between - groups manipulation and probability of the 
 outcome as a within - participant factor . as the reader might expect , the first finding of allan et al .
( 2005 ) was that the numerical ratings that 
 participants provided at the end of each problem were influenced both by contingency 
 and outcome - probability .
that is to say , participants were able , in general , to 
 track the objective degree of contingency between each of the chemicals and the 
 survival of bacteria ; however , their ratings were also biased by the probability of 
 the outcome .
this is a replication of the well - known outcome - density bias discussed 
 in previous sections .
( 2005 ) found that other dependent measures seemed to be unaffected by 
 outcome density , although they were sensitive to the overall cue - outcome 
 contingency .
specifically , allan et al . used the discrete yes / no predictions made by 
 participants in every trial to compute an alternative measure of their sensitivity 
 to contingency .
if the participant believes that there is a statistical connection 
 between the chemicals and the survival of bacteria , then he or she should predict 
 the survival ( i.e. , respond 
yes  to the question of whether the 
 bacteria would survive in the current trial ) more frequently when the chemicals are 
 present than when they are not .
following this reasoning , it would be possible to 
 measure the extent to which a participant believes that there is a relationship 
 between the cue and the outcome using the formula:(3 ) 
  note that this index is based on the same logic that underlies the computation of 
 p in equation 1 , only that the real occurrence of the 
 outcome is replaced by the outcome predictions made by the participant ( see collins & shanks , 2002 ) . 

therefore , ppred does not measure the objective 
 contingency between cue and outcome , but it aims at measuring the subjective 
 contingency that the participant perceives , as revealed by the trial - by - trial 
 predictions made during training . when using ppred as their dependent variable , 
 allan et al .
( 2005 ) found that 
 this measure was sensitive to cue - outcome contingency .
however , it was absolutely 
 unaffected by manipulations of the probability of the outcome .
this result led them 
 to conclude that there must be a stage of processing in which the contingency 
 between cue and outcome or the target conditional probabilities have already been 
 encoded but the outcome - density bias is still absent .
then , 
 ppred would provide an insight into this 
 basic mechanism that encodes contingency in a format that is not yet influenced by 
 the outcome - density bias .
given that the numerical judgments are influenced by both 
 contingency and outcome - probability , this must mean that judgments are affected not 
 only by the original encoding of information ( sensitive to contingency but free from 
 bias ) but also by processes that take place after encoding ( see figure 3 ) .
in other words , the outcome - density 
 bias is not due to learning or encoding , but to more sophisticated processes related 
 to judgment and decision making .
( 2005 ) explored only the outcome - density bias , they 
 suggested that a similar approach might explain the complementary cue - density bias 
 as well .
if they proved to be reliable , 
 they would pose insurmountable problems for single - process models that aim to 
 explain cue- and outcome - density biases as learning effects .
but , how strong is this 
 evidence ? to answer this question , we decided to reanalyze data from our own 
 laboratory using the strategy followed by allan et al .
specifically , we reanalyzed 
 data from nine experimental conditions exploring the cue - density bias ( originally 
 published in blanco et al . , 2013 ; 
 matute et al . , 2011 ;
vadillo et al . , 2011 ; yarritu & matute , 2015 ; yarritu , matute , & vadillo , 
 2014 ) and three experimental conditions exploring the outcome - density 
 bias ( originally published in musca et 
 al . , 2010 ; vadillo , miller , 
 & matute , 2005 ) .
all these experiments were conducted using the 
 standard experimental paradigm outlined above . in the original reports of those 
 experiments
we only analyzed the judgments that participants reported at the end of 
 the training phase .
however , we also collected trial - by - trial predictive responses 
 to maintain participants attention and to make sure that they were following 
 the experiment .
this allows us 
 to compare the size of the bias observed in the 
 ppred scores with the size of the bias that 
 we observed in judgments .
cue- and 
 outcome - density biases should have an effect on judgments , but not on 
 ppred . in the following analyses , we included data from 848 participants tested in 12 
 conditions included in the articles mentioned in the previous paragraph.figure 4 plots the effect size ( cohen s 
 d ) of density biases on the 
 ppred index against the effect size of the 
 same manipulation on judgments collected at the end of the experiment . as can be 
 seen , overall , most experiments found clear evidence for density biases in 
 ppred .
a random - effects meta - analysis 
 yielded a statistically significant effect of size 
 d  =  0.49 , 95% ci [ 0.08 , 0.90 ] , 
 z  =  2.37 , p  =  .018 . 

therefore , overall our results do not replicate the original findings of allan et al .
( 2005 ) : cue and outcome 
 density do have an effect on ppred scores . 

however , the confidence intervals plotted in figure 4 show that the effect of density bias did not reach 
 statistical significance in some experimental conditions . in two cases ,
. a closer look at the data shows that in the rare occasions when 
 biases were not observed in ppred , they also 
 tended to be absent or smaller than usual in judgments .
a meta - regression confirmed 
 that the effect size of biases on ppred was 
 moderated by the effect size of biases on judgments , 
 q(1 )  =  5.54 , p  =  .019 . 

these analyses suggest that cue- and outcome - density biases can be observed in 
 ppred and that , when they are absent , it is 
 not due to a dissociation between judgments and 
 ppred , but to any other factor that affects 
 both measures . to show that these results are robust
, we also analyzed the individual data of the 
 participants tested in all the experimental conditions shown in figure 4 . across all conditions , the 
 cue-/outcome - density manipulation had an effect on 
 ppred , 
 t(846 )  =  6.20 , 
 p  <  .001 , d  =  0.43 . 

this result also held when data from experiments exploring cue- and outcome - density 
 biases were analyzed separately : t(659 )  =  4.97 , 
 p  <  .001 , 
 d  =  0.37 , and 

t(185 )  =  4.01 , p < .001 , 
 d  =  0.59 , respectively . not surprisingly , the two 
 dependent measures , judgments and ppred , were 
 significantly correlated , r  =  .27 , 
 p  <  .001 , and this correlation remained 
 significant when data from cue- and outcome - density biases were analyzed 
 independently : r  =  .28 , 
 p  <  .001 , and 
 r  =  .24 , p  <  .001 , 
 respectively .
overall the data shown in figure 4 and these additional analyses are inconsistent with the 
 hypothesis that trial - by - trial predictions are unbiased by the probability of the 
 cue / outcome or that radically different results are observed with judgments and 
 trial - by - trial predictions .
thus , there is no need to postulate different mechanisms 
 to account for judgments and for trial - by - trial predictions .
this being said , figure 4 also 
 reveals that some of our studies failed to find a significant effect of the 
 cue / outcome density manipulation on ppred . to 
 some extent
, this feature of our results can be considered a replication of allan et al .
but as seen in 
 figure 4 and in the previous 
 analyses , this can hardly be considered strong evidence for a dissociation between 
 judgments and ppred .
the fact that sometimes 
 ppred fails to yield significant results may 
 be due to its reduced reliability compared to judgments .
there are many situations 
 where the use of unreliable or insensitive measures can produce patterns of results 
 that look like dissociations but do not require a dual - process account ( shanks & st .
consistent with this interpretation , there are good reasons why 
 ppred might be an imperfect index of 
 contingency learning , as we discuss in subsequent sections .
( 2005 , experiment 
 1 ) found a dissociation strikingly similar to the one reported by allan et al .
et al . , two groups of participants were exposed to several 
 contingency - learning problems where they had to learn the relationship between the 
 activation of a fictitious minefield and the explosion of enemy tanks . in each 
 trial , participants were first presented with information about whether the 
 minefield was active and were asked to predict by means of a yes / no response whether 
 or not they thought that the tanks would explode in that trial . after entering their 
 predictions , they were given feedback and they proceeded to the next trial . for one 
 of the groups ,
the contingency between the minefield and the explosions was always 
 positive , p  =  .50 , while the contingency was 
 always zero for all the problems presented to the other group of participants . 
 within participants ,
the probability of the cue was manipulated with two levels , 
 high ( .75 ) and low ( .25 ) . at the end of each problem
, participants were asked to 
 rate the strength of the causal relation between the activation of the minefield and 
 the explosion of tanks .
consistent with previous reports , perales et al . found that 
 these numerical judgments were sensitive not only to the contingency manipulation , 
 but also to the cue - density manipulation .
that is to say , for a specific level of 
 contingency , judgments tended to vary with the probability of the cue , replicating 
 the well - known cue - density bias .
perales et 
 al . found that other dependent measures , also computed from participants 
 trial - by - trial responses , were sensitive only to contingency and were largely immune 
 to the cue - density bias . unlike allan et al .
( 2005 , experiment 1 ) 
 did not convert trial - by - trial predictions to 
 ppred but , instead , they computed two 
 alternative measures inspired in signal detection theory ( sdt ) analyses .
one of 
 them , the criterion for responding , was a measure of participants overall 
 tendency to predict that the outcome will occur .
the second one , 
 d , was the discriminability index of sdt analyses and aimed 
 at measuring participants ability to discriminate when the outcome was more 
 likely to appear and when it was less likely to appear . for reasons that will become 
 obvious later ,
only this second measure is relevant for the present discussion . to 
 compute the d index , perales et al .
registered the 
  hit rate  of each participant ( i.e. , the proportion of trials in 
 which they correctly predicted that the outcome would occur , among all trials where 
 the outcome was present ) and their  false alarm rate 
( i.e. , the 
 proportion of trials in which they incorrectly predicted the outcome , among all 
 trials where the outcome was absent ) . based on these two measures , 
 d can be easily computed as(4)where z is the inverse of the normal cumulative 
 density function .
crucially , this equation ignores completely whether participants 
 made their predictions on cue - present or on cue - absent trials .
the only important 
 thing is whether they correctly predicted the outcome when it was going to happen 
 and whether they incorrectly predicted the outcome when it was not going to happen . 
 in other words , the d index measures to
what extent 
 participants are good at discriminating when the outcome will be presented and when 
 it will not .
the rationale for using this index as a measure of participants 
 sensitivity to contingency is that , in principle , if participants have learned the 
 correct cue - outcome contingency , they should be able to make more accurate 
 predictions , and this should yield a higher d. the key finding of perales et al . 

( 2005 , experiment 1 ) was that participants 
 d scores turned out to be sensitive just to the contingency 
 manipulation , but not to the cue - density manipulation .
this parallels allan et al.s ( 2005 ) finding 
 that ppred was affected by manipulations of 
 contingency , but not by manipulations of the probability of the outcome .
taken 
 collectively , both experiments converged on the same idea : there are some dependent 
 measures that reflect that participants have learned the cue - outcome contingency , 
 but which nevertheless show no trace of cue- or outcome - density bias .
this stands in 
 stark contrast with the patterns of results found in numerical judgments , which are 
 sensitive to both contingency and density biases .
perales et al . discussed several dual - process accounts that could explain these 
 dissociations . although differing in the detail , all of them dovetailed with the 
 idea that there is a basic encoding mechanism that tracks cue - outcome contingency in 
 a format free from any cue- or outcome - density bias .
the density 
 biases observed in judgments must then be attributed to other mechanisms that 
 intervene in later stages of processing .
a first striking feature of the results is 
 that the manipulation of the probability of the cue did in fact seem to have an 
 effect on d , although this effect was only marginally 
 significant ( p  =  .09 , 
 p  =  .067 ) .
the 
 authors argued that this effect was  far too small to account for the 
 significant effect of cause - density [ i.e. , cue - density ] on judgments . 
however , this argumentation is only valid if one assumes that the validity , 
 reliability , and sensitivity of d as a dependent measure 
 are comparable to those of judgments . if d turned out to be 
 less sensitive , then the smaller effect size of the cue - density effect found of 
 d would be very poor evidence for a dissociation . are there any reasons to suspect that d is not a sensitive 
 measure for cue- or outcome - density biases ?
. the problems of 
 d as a measure of learning are particularly obvious in 
 the two null - contingency conditions of perales et al .
the 
 probability of the outcome was always .50 , no matter whether the cue was present or 
 not .
in this situation , there was nothing participants could do to predict the 
 outcome successfully .
the outcome and its absence were equally likely and the cue 
 did not offer any information to make the outcome more predictable .
furthermore , 
 this was the case in both the high cue - density and the low cue - density conditions : 
 the probability of the cue was higher in one condition than in the other , but this 
 did not change the fact that the outcome was equally unpredictable in both 
 conditions . therefore , it is not surprising that participants in both conditions got 
 d values indistinguishable from zero . note that this 
 does not mean that participants had the same perception of contingency in those two 
 conditions .
it only means that in those particular conditions , nothing participants 
 learn can help them make better outcome predictions as measured by 
 d. but crucially , if all participants produce a 
 d of zero regardless of their predictions , then their 
 score can not be used as a measure of their perception of contingency .
a computer simulation provides a simple means to illustrate the problems of 
 d as a measure of contingency learning . in the 
 following simulation we computed d for a large number of 
 simulated participants exposed to the same contingencies used in perales
the labels pc - hd , pc - ld , nc - hd , and nc - ld shown in figure 5 refer to the four 
 experimental conditions tested by perales et al .
these labels denote whether 
 contingency was positive ( pc ) or null ( nc ) and whether the density of the cue was 
 high ( hd ) or low ( ld ) . for each experimental condition
, we computed how many correct 
 and incorrect responses a simulated participant would get .
then , we used this 
 information to compute the d of that simulated participant 
 using equation 4 . 
  for each condition , we simulated 1,000 participants with nine different response 
 distributions .
the rationale for simulating a wide variety of response strategies is 
 that , if d is a valid measure of learning , this index 
 should adopt different values when participants behave differently .
imagine that one 
 participant learns that there is positive contingency between the cue and the 
 outcome .
yes  very frequently when asked 
 whether the outcome will follow the cue and he / she should say  no  
 very frequently when asked whether the outcome will appear in a trial in which the 
 cue was absent .
now imagine a second participant who learns that there is no 
 contingency between cue and outcome .
this participant will be just as likely to 
 predict the outcome in cue - present and in cue - absent trials .
if 
 d is a good measure of contingency learning , these two 
 participants should get different d scores .
in contrast , if 
 participants who act on the basis of different beliefs about the cue - outcome 
 contingency obtain the same d score , this would imply that 
 d is not a valid measure of contingency detection .
to 
 minimize the impact of sampling error on the exact number of hits and false alarms , 
 we run 1,000 simulations for each combination of experimental condition and response 
 distribution .
each of the nine series of data shown in figure 5 refers to a different response distribution ( e.g. , 25/75 , 
 50/50 ,  ) . the first number ( 25 , 50 , or 75 ) refers to the probability of 
 predicting the outcome in the presence of the cue and the second number ( also 25 , 
 50 , or 75 ) refers to the probability of predicting the outcome in the absence of the 
 cue .
for instance , a simulated participant with response strategy 75/25 would 
 predict the outcome with probability .75 if the cue was present and with probability 
 .25 if the cue was absent ( which is consistent with the belief in a positive 
 contingency ) . similarly , a simulated participant with response strategy 75/75 would 
 predict the outcome with probability .75 regardless of whether the cue is present or 
 absent ( which is consistent with the belief in a null contingency ) .
consistent with our previous discussion , figure 5 shows that all simulated participants got almost identical 
 d scores in the null - contingency conditions ( right - most 
 half of the figure ) . in other words ,
the d scores obtained 
 by those participants reveal absolutely nothing about their pattern of performance . 
 in these null - contingency conditions ,
a participant who acted as if there were a 
 positive cue - outcome contingency ( e.g. , 75/25 ) would receive virtually the same 
 score as a participant who acted as if contingency were negative ( e.g. , 25/75 ) . 

similarly , a participant who was very prone to predicting the outcome ( e.g. , 75/75 ) 
 and a participant who was very reluctant to predict the outcome ( e.g. , 25/25 ) would 
 obtain similar d scores .
( 2005 , experiment 1 ) in the nc conditions tell 
 us nothing about the participants performance , let alone about their beliefs 
 regarding the cue - outcome contingency or their outcome expectancies . in contrast
, the left - hand side of figure 
 5 shows that d can be a sensitive measure of the 
 perception of contingency in the positive contingency ( pc ) conditions tested by 
 perales et al . in these conditions , different patterns of responding do give rise to 
 different d scores , suggesting that 
 d can reveal something about participants beliefs 
 or about their response strategies . therefore , it is only in these conditions with 
 positive contingencies that one can expect to measure differences in performance 
 with d. it is interesting to note that visual inspection of 
 the data reported by perales et al . 

( 2005 ) confirms that the trend towards a cue - density bias was stronger in 
 the pc condition than in the nc condition .
when one takes into account that one half 
 of the experiment ( the two nc conditions ) is affected by a methodological artifact , 
 it becomes less surprising that the cue - density manipulation only had a marginally 
 significant effect on d.
this being said , we note that in the pc conditions the cue - density effect observed in 
 d still looks relatively small , compared to the large 
 effect found in participants judgments .
going back to the reanalysis of our 
 own studies presented in our previous section , we also found there that , in some 
 occasions , cue / outcome density manipulations seemed to have a stronger effect on 
 judgments than on ppred . based on this evidence
, 
 it appears that , in general , all the dependent measures computed from trial - by - trial 
 predictions ( either ppred or 
 d ) are less sensitive than numerical judgments .
is there 
 any reason why these dependent measures should be less reliable ? as we will show 
 below , we suspect that the data reported by allan et al .
imagine that two participants , a and b , have been exposed to exactly the same 
 sequence of trials and that , as a result , they end up having the same beliefs about 
 the relationship between a cue and an outcome .
for instance , imagine that both of 
 them have learned that the probability of the outcome given the cue is .75 and that 
 the probability of the outcome given the absence of the cue is .25 .
this means that 
 both of them would believe , implicitly or explicitly , that there is a moderate 
 positive contingency between cue and outcome ( i.e. , that the outcome is more likely 
 to appear in the presence of the cue than in its absence ) .
now , let us assume that 
 both participants are presented again with a series of trials where the cue is 
 present or absent and they are asked to predict whether or not the outcome will be 
 presented in each trial .
participant a might consider that , because the probability 
 of the outcome given the cue is .75 , he should predict the outcome in roughly 75% of 
 the trials where the cue is present . and ,
similarly , because the probability of the 
 outcome given the absence of the cue is .25 , he predicts the outcome in 
 approximately 25% of the trials where the cue is absent .
the behavior of this 
 participant would show what researchers call  probability matching  

( nies , 1962 ; shanks , tunney , & mccarthy , 
 2002 ; tversky & edwards , 
 1966 ) , that is , his predictions would match the probabilities seen ( or 
 perceived ) in the environment .
participant b has a 
 different goal in mind : he wants to be right as many times as possible .
if the 
 outcome appears 75% of the times when the cue is present , then predicting the 
 outcome on 75% of the trials is not a perfect strategy .
if he did that , on average , 
 he would be right on 56.25% of the trials ( i.e. , .75    .75 ) . in 
 contrast ,
if he always predicts the outcome when the cue is present , he will be 
 correct 75% of the times ( i.e. , 1.00    .75 ) .
if he wants to 
 maximize the number of correct outcome predictions , this is a much more rational 
 strategy .
following the same logic , if the probability of the outcome in the absence 
 of the cue is .25 , it makes sense to always predict the absence of the outcome . 

our point is that if a 
 participant wants to maximize the number of correct predictions , he will predict the 
 outcome whenever he thinks that its probability is higher than .50 and he will 
 predict the absence of the outcome in any other case .
not surprisingly , research 
 with human and nonhuman animals shows that maximization is a typical response 
 strategy in many situations ( e.g. , unturbe & coromias , 2007 ) .
interestingly , although both participants , a and b , base their responses on the same 
  beliefs , their behavior is radically different because they pursue 
 different goals .
this has important implications for our review of the results 
 reported by allan et al .
if 
 trial - by - trial predictions do not only depend on the perceived contingency but also 
 on response strategies like probability matching or maximization , then any dependent 
 variable computed from them ( like ppred or 
 d ) will be necessarily noisy and unreliable .
for 
 instance , if we computed ppred for participant a 
 in our previous example , that would yield an approximate value of .50 , which 
 reflects quite well his beliefs about the contingency between cue and outcome . 

however , if we computed ppred for participant b , 
 this would yield a value of 1.00 , which is a gross overestimation of his true 
 beliefs .
these two participants would also receive different 
 d scores . in the case of perales et al . 

( 2005 , experiment 1 ) , there is clear evidence of probability 
 maximization .
the data reported in the appendix of perales et al . confirm that 
 participants in the pc group predicted the outcome in around 9293% of 
 cue - present trials and in 1213% of cue - absent trials . in the case of allan et al .
( 2005 ) there is no 
 obvious evidence for maximization in their noncontingent condition , but there 
 appears to be such a trend in the contingent condition .
their figure 5 suggests that although the actual 
 probability of the outcome given the cue varied from .567 to .900 , participants 
 predicted the outcome in 7090% of cue - present trials .
similarly , although 
 the probability of the outcome in the absence of the cue ranged from .100 to .667 , 
 participants predicted the outcome in 1030% of cue - absent trials .
this 
 pattern is perhaps less extreme than the one found in perales et al .
( 2005 ) , but it does nevertheless 
 suggest that many of their participants must have used a maximization strategy . in 
 either case , this strategy makes ppred and 
 d less sensitive to any manipulation .
the lack of 
 sensitivity might explain why they failed to find any effect of cue and outcome 
 density on these dependent variables .
it is interesting to note that participants are more likely to become 
  maximizers  in relatively long experiments , which provide more 
 opportunities to develop optimal response strategies ( shanks et al . , 2002 )
cue and outcome 
 density were manipulated within - participants . to accomplish this , all participants 
 had to complete the contingency learning several times .
in contrast , in our 
 experiments all the manipulations were conducted between groups , reducing 
 substantially the length of the experiment and , consequently , the opportunities to 
 develop sophisticated response strategies like maximization .
dissociations between judgments and trial - by - trial predictions are not the only piece 
 of evidence in favor of dual - process models of biased contingency detection .
this 
 approach received convergent support from a recent study by ratliff and nosek ( 2010 ) that found a similar 
 dissociation between different measures of illusory correlations in stereotype 
 formation , suggesting that two or more processes might also be involved in this 
 effect .
as explained in previous sections , most experiments on illusory correlations in 
 stereotype formation rely on a fairly standard procedure ( hamilton & gifford , 1976 ) .
participants are 
 presented with positive and negative traits of members of two different social 
 groups on a trial - by - trial basis . crucially , there are more members of one group 
 ( majority ) than of the other ( minority ) and , regardless of group , most of the 
 members show positive traits . although the proportion of positive and negative 
 traits is identical for the majority and the minority groups , people tend to make 
 more positive evaluations of the majority group when asked to judge both groups at 
 the end of training .
illusory correlations in stereotypes and cue / outcome density 
 biases have been explored in quite different literatures , but both effects are 
 clearly related and can be explained by the same or very similar models ( murphy et al . , 2011 ; sherman et al . , 2009 ; van rooy et al . , 2003 ) .
illusory correlations are typically assessed by means of numerical ratings ( similar 
 to judgments in contingency - detection experiments ) or by asking participants to 
 recall which positive or negative traits were observed in the majority or the 
 minority group
. however , ratliff and 
 nosek ( 2010 ) wondered whether the same illusory - correlation effects would 
 be found in an alternative test that is supposed to provide a cleaner measure of the 
 underlying attitudes of participants : the implicit association test ( iat ; greenwald , mcghee , & schwartz , 
 1998 ) .
unlike traditional questionnaires , the iat is a reaction - time test 
 that is traditionally assumed to measure implicit attitudes with little interference 
 from higher - order cognitive processes ( de houwer , teige - mocigmba , spruyt , & moors , 2009 ; gawronski , lebel , & peters , 
 2007 ; nosek , hawkins , & 
 frazier , 2011 ) .
following this idea , if illusory correlations require the 
 operation of reasoning or inferential processes , then they should not be observed in 
 the iat .
in contrast , if only very elemental associative processes are responsible 
 for illusory correlations , then the iat should be able to detect them . 

ratliff and nosek ( 2010 ) found 
 the standard illusory - correlation effect in the responses to the explicit 
 questionnaire .
however , there was no hint of the effect in the iat scores in any of 
 their two experiments .
most importantly , the absence of effects can not be attributed 
 to the lack of validity of the iat : the iat was sensitive to the valence of the 
 majority and the minority groups when there was a correlation between membership to 
 one of them and the positive or negative personality traits .
nor can they be 
 attributed to a lack of statistical power , given that the results were replicated in 
 an online experiment with almost 900 participants . 

ratliff and nosek ( 2010 ) 
 interpreted these results in terms of a dual - process model ( gawronski & bodenhausen , 2006 ) surprisingly 
 similar to that invoked by allan et al . 

according to them , the iat 
 would be only sensitive to the original encoding of associations in memory . from 
 this point of view
, the fact that performance in the iat was unaffected by the 
 illusory - correlation manipulation indicates that participants in their experiment 
 correctly learned that there was no correlation between belonging to one group or 
 the other and having positive or negative personality traits .
therefore , the 
 illusory - correlation effect observed in explicit judgments must have been due to 
 additional higher - order cognitive processes that took place on a later stage , and 
 not to the original learning mechanism responsible for the initial encoding of the 
 information ( see figure 3 ) . as in 
 the case of the results reported by allan et al . and perales et al .
, these results 
 pose problems for any single - process model that assumes that illusory correlations 
 are the product of the same mechanisms responsible for the detection of contingency . 

if a single process were responsible for both sensitivity to contingency and for 
 density biases , why should an iat be sensitive to one of these manipulations 
 ( contingency ) but not to the other ( density ) ? before drawing any conclusion , it is convenient to review all the available evidence 
 regarding this dissociation . until recently ,
the study conducted by ratliff and nosek ( 2010 ) was the 
 only experimental work that had tried to detect illusory correlations with the iat . 

however , the latest attempt to replicate this result using a similar methodology has 
 failed to find any dissociation between explicit measures and the iat .
using a very 
 similar procedure to ratliff and nosek , carraro , negri , castelli , and pastore ( 2014 ) did find an illusory 
 correlation on the iat , showing that the original dissociation was either not 
 reliable or , more likely , not generalizable to similar but not identical conditions . 
 in a similar vein ,
a recent experiment conducted in our laboratory ( vadillo , de houwer , de schryver , 
 ortega - castro , & matute , 2013 ) found an outcome - density effect using 
 the iat .
the divergences with ratliff and nosek are less surprising in this case , 
 because vadillo et al . used a radically different design and procedure .
however , 
 this discrepancy converges to the idea that the failure of the iat to detect 
 illusory correlations might not be a generalizable result .
to better illustrate the results found with the iat , figure 6 depicts a forest plot with the divergent results of 
 these studies .
as can 
 be seen , the only firm conclusion that can be drawn on the basis of this evidence is 
 that the results are strikingly variable .
in fact , the meta - analysis of these 
 studies yielded an unusually large heterogeneity , 
 q(4 )  =  134.18 , 
 p  <  .001 .
( 2014 ) reported in 
 their general discussion yielded results notably distant from those of their main 
 study , although both of them were statistically significant .
this variability 
 suggests that whether or not illusory correlations are observed in the iat probably 
 depends on a number of moderators that we still ignore . as shown in figure 6 , the 95% confidence 
 interval of the random - effects model includes zero .
an advocate of dual - process 
 models might claim that this null result of the meta - analysis supports the claim 
 that illusory correlations are not found in implicit measures like the iat .
however , 
 the confidence interval does not just include zero : it extends over a large number 
 of positive effect sizes . on the basis of the collective evidence
, any value from 
 0.13 to a massive 1.16 could be an accurate estimate of the average 
 cohen s d. we doubt that this evidence is clear or robust 
 enough to abandon single - process models of illusory correlations , which offer a 
 simple and parsimonious explanation for a large body of data ( allan , 1993 ; fiedler , 2000 ; lpez et al .
even more so , if we keep in mind that the only converging 
 evidence from allan et al .
in the previous sections we have reviewed the studies that have found dissociations 
 in cue- and outcome - density effects across dependent variables ( allan et al . , 2005 ; perales et al .
a 
 common result of these experiments is that there are some dependent measures that 
 only show sensitivity to contingency ( e.g. , 
 ppred , d , or iat 
 scores ) , while other dependent measures ( e.g. , contingency judgments ) show 
 sensitivity to both contingency and cue / outcome density . on the basis of this 
 evidence
, it has been suggested that two separate mechanisms are needed to explain 
 ( 1 ) why people are able to learn the correct cue - outcome contingencies and ( 2 ) why 
 their judgments are influenced not only by contingency but also by the overall 
 probabilities of the cue and the outcome .
however , on closer inspection , it appears 
 that this evidence might not be reliable enough to justify this theoretical 
 interpretation .
a review of the available evidence from our laboratory and from 
 other research groups shows that some of these results do not seem to be replicable 
 or do not generalize easily to similar experimental settings . in our experiments , 
 cue- and outcome - density manipulations seem to have a significant impact on all 
 dependent measures .
similarly , our simulations show that some of the dissociations 
 previously reported in the literature might be due to simple methodological 
 artifacts resulting from aspects of the design and the computation of the dependent 
 variable .
given the lack of strong evidence in favor of these dual - process accounts , we think 
 that it is premature to abandon the idea that sensitivity to contingency and density 
 biases are both attributable to the operation of a single mechanism .
as mentioned in 
 the introduction , this idea is an essential feature of many associative models that 
 were originally invoked to account for cue- and outcome - density effects ( lpez et al . , 1998 ; shanks , 1995 ;
it is also a central feature 
 of alternative models of biased contingency detection , such as instance - based models 
 ( fiedler , 1996 ; meiser & hewstone , 2006 ; smith , 1991 ) . beyond the specific 
 details of these models
, their common feature is that they all share the assumption 
 that the same mechanism that is responsible for detecting and encoding the 
 relationship between cues and outcomes is also responsible for density biases . in 
 other words
, there is no level of representation in the cognitive system where 
 contingency information is represented in a format that is free from cue- or 
 outcome - density biases .
note that , although we favor single - process models of biased contingency detection , 
 we do not ignore the fact that different processes might contribute to each of the 
 dependent variables used in this kind of research .
we do not question the idea that 
 there are manipulations that might affect one dependent variable without affecting 
 others .
in fact , part of our own research has been directed at showing that 
 judgments of contingency can vary considerably depending on seemingly minor 
 procedural details like the wording of the test question ( matute , vegas , & de marez , 2002 ; vadillo & matute , 2007 ; vadillo et al . , 2005 ; see also 
 crocker , 1982 ; de houwer et al .
( 2005 ) relies on the idea that participants sometimes 
 adopt response strategies that might mask their true perception of contingency .
what 
 we are suggesting here is that there are no strong reasons to assume that there are 
 two different levels of representation of contingency information , one that closely 
 mirrors objective contingencies and one where that information is biased by factors 
 like the probability of the cue or the probability of the outcome . in the absence of 
 stronger evidence
, it appears more parsimonious to assume that a single 
 representation is learned during contingency - learning experiments and that this 
 representation is biased by cue and outcome density at the encoding level .
dual - process models of biased contingency detection share some ideas with other 
 dichotomist models of cognition ( evans 
 & over , 1996 ; kahneman , 
 2011 ; osman , 2004 ; 
 sloman , 1996 ; stanovich & west , 2000 ) .
there 
 is a crucial difference , though , between the dual models reviewed in this article 
 and other dual - process models .
traditionally , dual models have tended to explain 
 biases and cognitive illusions attributing them to very simple cognitive mechanisms 
 that operate effortlessly and in a relatively automatic manner , usually related to 
 simple encoding or retrieval processes .
more complex cognitive mechanisms , usually 
 strategic processes related to judgment and decision making , have been invoked to 
 explain why people are sometimes able to overcome the harmful effect of these biases 
 and intuitive reactions ( e.g. , kahneman , 
 2011 ) .
interestingly , the dual - process models of biased contingency 
 detection that we review here make the opposite interpretation : correct contingency 
 detection is attributed to the operation of basic encoding and retrieval processes , 
 while biased judgments are attributed to more sophisticated judgment and 
 decision - making processes .
we might even say that these models present a benign view 
 of biases in contingency detection : although people might show a bias in their 
 judgments , deep inside their cognitive system there is some level of representation 
 where information is represented accurately ( a similar perspective can be found in 
 de neys , 2012 ) .
the question of whether biases in contingency detection are due to basic encoding and 
 retrieval processes or whether they reflect the operation of judgment and 
 decision - making processes is not only important from a theoretical point of view . as 
 mentioned above ,
it has been suggested that these biases might contribute to the 
 development of superstitious and pseudoscientific thinking ( gilovich , 1991 ; lindeman & svedholm , 2012 ; matute et al . , 2011 , 2015 ; redelmeier & tversky , 1996 ; vyse , 1997 ) .
given the societal costs of these and other 
 biases , cognitive psychologists have started to develop a number of interventions 
 and guidelines for protecting people from cognitive biases ( barbera et al . , 2013 ; lewandowsky , ecker , seifer , schwarz , & cook , 
 2012 ; lilienfeld , ammirati , 
 & landfield , 2009 ; schmaltz & lilienfeld , 2014 ) .
interventions designed to reduce 
 biases can only be successful to the extent that they are based on an accurate view 
 of their underlying mechanisms .
if the underlying information was somehow encoded in 
 a bias - free format , as suggested by dual - process models of contingency learning , 
 these beliefs should be relatively easy to modify .
teaching people how to use more 
 rationally the information and intuitions they already have should suffice to 
 overcome these biases .
this prediction stands in stark contrast with the well - known 
 fact that superstitions are difficult to modify or eradicate ( arkes , 1991 ; lilienfeld et al .
, 2009 ; nyhan , reifler , richey , & freed , 2014 ; pronin , gilovich , & ross , 2004 ; 
 smith & slack , 2015 ) .
we 
 think that the persisting effect of biases is more consistent with the view that 
 these beliefs are hardwired in the way people encode information about the 
 relationship between events , as suggested by single - process models .
based on the 
 evidence we have discussed so far , it seems safe to suggest that attempts to debias 
 superstitions and misperceptions of contingency should include teaching people to 
 look for unbiased information .
 
 abstract . 
 decades of research in causal and contingency learning 
 show that people s estimations of the degree of contingency between two 
 events are easily biased by the relative probabilities of those two events . 
 if 
 two events co - occur frequently , then people tend to overestimate the strength of 
 the contingency between them . 
 traditionally , these biases have been explained in 
 terms of relatively simple single - process models of learning and reasoning . 
 
 however , more recently some authors have found that these biases do not appear 
 in all dependent variables and have proposed dual - process models to explain 
 these dissociations between variables . in the present paper 
 we review the 
 evidence for dissociations supporting dual - process models and we point out 
 important shortcomings of this literature . 
 some dissociations seem to be 
 difficult to replicate or poorly generalizable and others can be attributed to 
 methodological artifacts . 
 overall , we conclude that support for dual - process 
 models of biased contingency detection is scarce and inconclusive . 
Biased Contingency Detection and Illusory Correlations
Single-Process Models of Cue- and Outcome-Density Biases
Dissociations and Dual-Process Models
Cue- and Outcome-Density Biases in Trial-by-Trial Predictions
Signal Detection Theory Analyses of Trial-by-Trial Predictions
Different Processes or Different Strategies?
Illusory Correlations in the Implicit Association Test
General Discussion