Normative and descriptive accounts of the influence of power and contingency on causal judgement

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1 Q1134 QJEP(A)12101 / Jul 1, 03 (Tue)/ [31 page 7 Table 4 Figure 5 Footnote 2 Appendice].. Centre ingle caption. Edited from dic - MATHS EQUATIONS THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2003, 56A (6), Normative and decriptive account of the influence of power and contingency on caual judgement Joé C. Perale Univeridad de Granada, Spain David R. Shank Univerity College London, UK The power PC theory (Cheng, 1997) i a normative account of caual inference, which predict that caual judgement are baed on the power p of a potential caue, where p i the caue effect contingency normalized by the bae rate of the effect. In three experiment we demontrate that both caue-effect contingency and effect bae-rate independently affect etimate in caual learning tak. In Experiment 1, caual trength judgement were directly related to power p in a tak in which the effect bae-rate wa manipulated acro two poitive and two negative contingency condition. In Experiment 2 and 3 contingency manipulation affected caual etimate in everal ituation in which power p wa held contant, contrary to the power PC theory prediction. Thi latter effect cannot be explained by participant conflation of reliability and caual trength, a Experiment 3 demontrated independence of caual judgement and confidence. From a decriptive point of view, the data are compatible with Pearce (1987) model, a well a with everal other judgement rule, but not with the Recorla Wagner (Recorla & Wagner, 1972) or power PC model. M w d M l t 3 d t Human and animal readily acquire knowledge about the caual texture of their environment (Tolman & Brunwik, 1935) and are very adept at judging whether and to what extent two event, or a ingle repone and an event, are caually related. Thi i an important capacity a many area of judgement and deciion making rely on accurate etimate of Requet for reprint hould be ent to David R. Shank, Department of Pychology, Univerity College London, Gower St., London WC1E 6BT, UK. d.hank@ucl.ac.uk The reearch decribed here wa upported by grant from the Miniterio de Educacion y Cultura (Programa FPI, Programa de Ayuda para Etancia Breve en el Extranjero), Spain, to the firt author, and from the United Kingdom Economic and Social Reearch Council (ESRC) and Biotechnology and Biological Science Reearch Council to the econd author. The work i part of the programme of the ESRC Centre for Economic Learning and Social Evolution, Univerity College London. We thank André Catena, Klau Lober, Francico López, and Edward A. Waerman for their many helpful comment and Paloma Gómez Moare for her aitance in conducting the firt and econd experiment The Experimental Pychology Society DOI: /

2 978 PERALES AND SHANKS cauation (e.g., medical expert drawing inference about drug ide-effect). Recent reearch on caual learning ha focued on determining the cognitive mechanim by which etimate of caual trength can be derived from covariation information and ha led to the development of a number of model. The main aim of the preent work are (1) to obtain parametric data on caual etimate acro a fairly wide range of condition in which certain experimental variable (in particular, power and contingency) are manipulated, which can then be ued (2) to determine how well everal current formal model can account for the influence of thee variable, which in turn (3) hould allow u to evaluate whether people ue of covariation information can be conidered normatively accurate. Thee are iue that have been at the heart of much recent empirical and theoretical work (Cheng, 1997; Shank, Holyoak, & Medin, 1996). The tructure of the paper i a follow. In the firt part we decribe even model of caual judgement ( P, Power PC, Recorla Wagner, Van Hamme & Waerman, Pearce, D, and Belief Adjutment), and for each of thee we preent a formula tating it aymptotic prediction (in the cae of two of thee model, analytic olution have not previouly been derived). We then decribe how experiment in which contingency and bae-rate are manipulated allow thee model to be ditinguihed. In the econd part of the paper we report new reult that challenge everal of thee model. Normative account: P and the power PC theory Mot computational theorie of caual learning in ituation in which the participant ha to judge the magnitude of a caual relationhip between two or more dicrete event are baed on the tatitic P, or contingency, defined a the unidirectional tatitical dependency between two dichotomou variable, uually named the cue and outcome. 1 Imagine a imple cae in which a potential caue (the ingetion of a certain chemical ubtance) i either preent or abent, and the target effect (a digetive illne) i either preent or abent. In thi ituation, P i calculated a the difference between the probability of the effect when the potential caue i preent and the probability of the effect when the potential caue i abent. Conequently, according to thi rule the chemical hould be judged a an actual caual agent when the probabilitie are unequal and a a noncaual factor when they are equal. In other word, the chemical will be judged to have a caual influence if it increae or decreae the probability of the occurrence of the digetive illne. Conditional probabilitie can be calculated on the bai of the frequencie of the different type of trial that can occur during the tak (ee Table 1): namely, Type A, in which both the caue and effect are preent; type B, in which the caue i preent, and the effect i abent; Type C, in which the caue i abent, and the effect i preent; and Type D, in which both caue and effect are abent. P=P(E/C) P(E/~C) =A/(A+B) C/(C+D) 1 1 In mot laboratory tak it i aumed that the cue preented firt repreent the potential caue, whoe influence on the outcome or effect the participant have to evaluate. For an extenive dicuion of the role of cue and outcome a potential caue and effect ee Waldmann and Holyoak (1992) and Shank and López (1996).

3 CONTINGENCY AND CAUSAL POWER EFFECTS 979 TABLE 1 Combination of the preence or abence of a caue and an effect Effect Potential caue Preent Abent Preent A B Abent C D Note: A D repreent the frequencie of the event. where A, B, C, and D are the abolute frequencie of the four type of trial, and P(E/C) and P(E/~C) are the probability of the effect in the preence and abence of the caue, repectively. The meaure P can be calculated from the univeral et of trial or it can be calculated from a ubet of them. Normatively, it eem more rational to compute the probabilitic contrat over a reduced et of event in which all other poible caue of the ame effect are held contantly preent or abent in order to identify the pecific influence of the target caue. Thi aumption i at the core of the probabilitic contrat model (PCM: Cheng & Novick, 1992). Moreover, it ha been demontrated (Spellman, 1996) that in ome ituation human caual judgement qualitatively match conditional contrat computed in thi way. The power PC theory (Cheng, 1997) trie to overcome ome perceived limitation of the probabilitic contrat model (ee Cheng, 1997, pp ), etablihing the normative boundary condition in which P can be ued a an etimator of the trength of the caual relationhip between two dicrete event. According to thi theory (1) caual influence (or caual power) can only be properly etimated in a et of event in which the to-be-judged factor occur independently of any other potential caue of the ame effect, (2) the generative caual power of the to-be-judged factor cannot be etimated if the effect occur in every cae in which the target factor i abent, and (3) the preventative caual power of the target factor cannot be etimated if the effect doe not occur in any cae in which the target factor i abent. Although the power PC theory i in ome apect quite complex, the equation that pecify the degree of caual power (p) for a concrete potential caue are rather imple if certain boundary condition are atified. Two equation are propoed, one for generative caue (Equation 2) and another for preventative one (Equation 3). The appropriate equation i choen according to whether P i poitive (Equation 2) or negative (Equation 3). p= P/[1 P(E/~C)] 2 p= P/P(E/~C) 3 Therefore, the power PC theory ugget that, when the boundary condition are atified, P i a conervative etimator of caual power. For generative caue, the higher P(E/~C) i, the more conervative P i; for preventative caue, the lower P(E/~C) i, the more conervative P i.

4 980 PERALES AND SHANKS Decriptive account Aociative model Aociative algorithm, developed to account for phenomena in animal conditioning, have been demontrated to be highly predictive in a huge range of trial-by-trial caual and covariation learning preparation. Experimental data have often indicated that caual and contingency judgement are affected by the ame manipulation a are conditioned repone in animal learning. The imilar effect of contingency, contiguity, and cue competition in phenomena uch a blocking, predictive validity, and overhadowing (ee Allan, 1993; Shank, 1995; Shank et al., 1996, for review) have led reearcher to propoe that human covariation/caual judgement relie on a imilar mechanim to that which control conditioning. Recorla and Wagner (1972) model (henceforth RW) ha for many year been the mot prominently dicued candidate for thi mechanim. Applied to caual learning tak, the model i expreed a follow: V C = α C β (λ ΣV) 4 where V C repreent the change in the aociative trength of the link between the cue or potential caue and the outcome or effect in the current trial, α C and β are learning rate parameter directly related to the alience of the cue and the outcome, repectively, λ i the maximum trength that an aociative link can upport (the learning aymptote), and ΣV i the total amount of aociative trength of all cue preent on the current trial. Taking into account the fact that the total amount that a link can hold i limited, learning will proceed gradually to a maximum aymptotic value a the error term (λ ΣV) get maller. The retricted verion of the RW model aume that β i aigned the ame value when the outcome i preent (β E ) and when it i abent (β noe ). However, if the β parameter i aumed to depend on outcome alience, it can be argued that outcome preence and abence are differentially alient depending on tak factor. Under thi aumption the unretricted RW model allow β E and β noe to be aigned different value. A problem that arie when aociative algorithm are compared with theorie uch a Cheng power PC or probabilitic contrat model (Cheng & Holyoak, 1995; Cheng & Novick, 1990, 1992) i that the latter are formulated at the computational level of cognitive analyi, but the former are algorithmic theorie, elaborated to explain the mechanim that human ue both to acquire and to proce caual information. To overcome thi problem, everal reearcher have tried to derive aymptotic prediction from the RW model. For example, Cheng (1997) ha demontrated that when there i a contant background, the RW model i in certain cae computationally equivalent to a conditional computation of P. More generally, in the imple cae in which there i only a ingle potential caue to evaluate, and auming the preence of a contant background, the aymptotic learning level predicted by the RW model can be expreed a follow (Waerman, Elek, Chatloh, & Baker, 1993): V aymp = β E A /(β E A+β noe B) β E C /(β E C+β noe D) 5 which i equal to P if β E = β noe. It i important to note that thi derivation aume that aymptote ha been reached.

5 To our knowledge uch derivation effort have been made only with the RW model. No other aociative algorithm ha been analyed in order to compare it aymptotic prediction againt thoe of computational model uch a the power PC theory. A the original RW model ha been hown to have eriou limitation in accounting for human caual learning data, it i worthwhile making imilar derivation for other algorithm that have been hown to accommodate phenomena that the RW model cannot account for. Aymptotic prediction of aociative model One influential reviion of the RW model wa propoed by Van Hamme and Waerman (1994: henceforth VHW). Appendix A give a fuller decription of the background to thi model. The original formulation of the model did not take background cue into account. Thi fact could be problematic, a it leave ome context-related effect unexplained. Hence, we have derived the aymptotic prediction for the model auming that background cue are incorporated into the learning proce (ee Appendix A): VC + VX = ( AαCβE + CαnoCβE)( CαXβE + DαXβnoE) ( AαXβE + CαXβ E) ( CαnoCβE + DαnoCβ ) 6 noe ( Aαβ + Bαβ )( Cα β + Dα β ) ( Aα β + Bα β )( Cα β + Dα β ) C E C noe X E X noe CONTINGENCY AND CAUSAL POWER EFFECTS 981 X E X noe noc E noc noe Aα XβE + Cα XβE ( VC + VX)( Aα XβE + Bα XβnoE) VX = 7 Cα XβE + Dα XβnoE where A, B, C, and D are the abolute frequencie of the four poible type of trial; β E and β noe repreent the alience parameter for the preence and the abence of the outcome, repectively; α C and α noc are the alience parameter for the preence and the abence of the candidate caue; and α X i the alience parameter for the preence of the background. It i important to note that the alience parameter for the preence of the candidate caue and the background take poitive value, but the alience parameter for the abence of the target caue take a negative value. Equation 6 and 7 allow u to calculate, repectively, the predicted maximum learning level for the context (V X ) and for the context and target cue together (V C +V X ). We aume that judgement are baed on V C that i, the difference between (V C +V X ) and V X. A noted in Appendix A, the only theoretical difference between the original RW model and Van Hamme and Waerman (1994) revied model i that the mental repreentation of a cue can be negatively activated by the abence of that cue, if it i expected in a given context. However, Equation 6 and 7 have a trikingly counterintuitive implication, which ha not previouly been noted: For imple cue outcome tak the VHW modification to RW make almot no difference. A with the RW model, the aymptotic prediction from Van Hamme and Waerman model are identical to P if β E = β noe, and even without thi retriction they are qualitatively inditinguihable from thoe of RW in any other cae in which only one cue and one outcome are conidered. We have followed a imilar procedure to derive the aymptotic formulation of Pearce (1987) model (ee Appendix B). Thi model can explain claic cue competition effect uch a blocking and overhadowing a well a certain cae in which catatrophic forgetting i predicted by the RW model but not actually found (López, Shank, Almaraz, & Fernández, 1998). The main aumption of thi model i that, when more than one cue i preented, the participant procee them a if they form a unique compound cue, and when a ingle tet cue

6 982 PERALES AND SHANKS i preented the repone to that cue i a function of generalization from the compound cue. For our derivation we have aumed the preence of two different compound cue, one compoed of the background cue (X) and the other compoed of the ame background cue and the target cue combined (CX). It i alo aumed that the repone to the ingle cue i a direct monotonic function of the aociative value of the latter compound cue. 2 AC ( + D) C( Ax1 + Bx1) VCX = 8 A( C + D) + B( C + D) Cx2 ( Ax1 + Bx1) Dx2 ( Ax1 + Bx1) Here, V CX i the maximum aociative trength for the target cue/background compound. The judgement i then aumed to be a function of that value, weighted by the imilarity between the compound cue and the ingle cue. The parameter x 1 and x 2 are indice of generalization from the target/background compound to the background and from the background to the compound. The other parameter have the ame meaning a before (ee Appendix B for a detailed decription of the model). Rule-baed model Rule-baed model originated in the eminal work of Edward (1954) who potulated that human behave a if they calculate probabilitie and act a intuitive tatitician. According to uch model, human ue tatitical rule to make judgement. Mot of them potulate an algorithm baed on the frequencie of the trial type from the contingency table, or on conditional probabilitie baed on thoe frequencie, and claim that the algorithm i intuitively ued by caual reaoner. Initial rule-baed model, uch a the algorithmic verion of the P rule, have been reformulated everal time to account for new data and unexplained effect. For example, the exitence of cue competition effect in caual learning tak forced a reformulation of the conditional verion of the P rule into the probabilitic contrat model. Another verion of thi rule (henceforth the weighted P rule) aume that the two conditional probability term in Equation 1 are differentially weighted (ee Lober & Shank, 2000). On thi rule, judgement are generally expected to vary more with variation in P(E/C) than with variation in P(E/~C); that i, the former receive more weight. Following Lober and Shank, in all ubequent dicuion of thi rule we aume that P(E/C) i given a weight of (i.e., i multiplied by) 1.0 and P(E/~C) a weight of.6 in the calculation of P. The D rule (ee Allan, 1993) hare everal feature with P and make very imilar prediction in imple ituation in which only one cue and one outcome are preented. D i a heuritic, intuitively impler than P, baed on a contrat between confirmatory and diconfirmatory information about the relationhip between the cue and the outcome: D =(A B C+D)/N 9 2 Pearce (1987) model include two independent procee for excitatory and inhibitory conditioning. In order to implify the derivation, inhibitory learning ha been interpreted here a unlearning that i, a weakening of the aociative trength between the cue and the outcome. Thi aumption doe not affect the ordinal pattern of data predicted for our experiment.

7 CONTINGENCY AND CAUSAL POWER EFFECTS 983 where N i the number of trial preented during the tak. D and P are almot undifferentiable in mot imple tak. However, D i applicable in ome ituation in which P i not, for example when P(E/C)orP(E/~C) are not calculable. The main problem for the D model to overcome i that, to our knowledge, no conditional verion ha been potulated, and hence it i not applicable in complex ituation in which more than one cue i preented. Catena, Maldonado, and Cándido (1998; ee alo Maldonado, Catena, Cándida, & García, 1999) have recently developed a tatitical model partially baed on D that i the mot elaborated rule-baed model to date. According to Catena et al., two type of cognitive mechanim are reponible for caual belief and judgement. The hierarchically higher one i an information integration ytem that i activated whenever the participant i aked to judge the relationhip between event. Thi ytem behave according to the following equation: J i =J i 1 + β (NewEvidence J i 1 ) 10 where J i tand for the current judgement, J i 1 i the prior judgement, and β i a growth rate parameter the value of which will only affect trialwie adjutment to the objective contingency, thereby accounting for acquiition function and certain equential effect (ee Hogarth & Einhorn, 1992). Finally, NewEvidence i the ubjective value of the contingency evidence preented in the interval between two conecutive repone or judgement. The model alo propoe that people compute NewEvidence by mean of a lower level mechanim, uing the information from all type of trial ince the lat judgement on the bai of trial frequencie multiplied by weight, a in the weighted D rule, expreed in Equation 11: NewEvidence = (w A A w B B w C C+w D D)/ N 11 where the w repreent weight, uually aumed to be contrained by the abolute ubjective ranking w A >w B w C >w D. However, the model could alo account for idioyncratic ue of trial information (White, 1998) by changing the pattern of weight. Note that if β = 1 the model imply aume that terminal judgement are baed on D. The main problem for thi belief adjutment (BA) model i, again, that it ha not been developed for tak in which more than one potential caue i preented. An extenion of the model for more complex ituation remain to be developed, but at the moment effect uch a blocking (Shank, 1985), ignalling (Shank, 1989), and relative validity (Baker, Mercier, Vallée-Tourangeau, Frank, & Pan, 1993) cannot be accounted for. A imple olution i to aume that people can calculate NewEvidence in a conditional way. The contingency effect and the outcome bae-rate effect a tool to dicriminate between theorie A dicued above, dicriminating between different model can be quite complicated. Theorie ituated at different analyi level can only be compared with repect to aymptotic performance, and in complex ituation ome model cannot be applied wherea other become extremely veratile, a further aumption about the model parameter have to be taken into account. There have been everal attempt to directly contrat theorie ituated at different analyi level: for example, the power PC theory and the PCM againt the RW model (Buehner &

8 984 PERALES AND SHANKS Cheng, 1997; Cheng, Park, Yarla, & Holyoak, 1996; Lober & Shank, 2000; Shank, 2002; Vallée-Tourangeau, Murphy, Drew, & Baker, 1998) and the power PC theory againt Pearce model (Vallée-Tourangeau, Murphy, & Drew, 1997). The uual trategy ha been to compare theorie one-againt-one, uing different procedure to tet the prediction of one theory againt thoe of an alternative theory. However, none of thoe report ha taken into account the full range of poible explanation for the obtained pattern of data, and the evidence ha often been inconcluive. Additionally, nonaociative algorithmic explanation have often been ignored. Neverthele, it i important to note that the deign decribed in ome of thee tudie are imple enough to tet a huge range of learning model, and o they become appropriate tool for reducing the et of theorie that hould be taken into account in further reearch. In the three one-caue one-outcome trial-by-trial experiment reported by Lober and Shank (2000), a fairly large range of condition wa explored. In the firt experiment P wa manipulated acro three condition (.23,.47, and.70), while power (p) wa held contant. In the econd experiment p wa held contant again, and P wa manipulated acro four poitive condition (.25,.50,.75, and 1.00). Finally, acro three of the four condition of Experiment 3, power wa manipulated in a range from.40 to 1.00, and P wa held contant. The reult were in accord with thoe from earlier experiment by Vallée-Tourangeau et al. (1997) and Buehner and Cheng (1997), a both P and power had direct, ignificant effect on mean judgement. Table 2 how the different conditional probabilitie and the mean final judgement for each condition in Lober and Shank (2000) experiment. Additionally, we have computed the prediction made by Pearce (1987) model, Van Hamme and Waerman (1994) verion of the RW model, and Catena et al. (1998) belief adjutment model in order to contrat their prediction acro the full et of reult. A can be een, the belief adjutment model, Pearce model, the unretricted RW model, and the VHW model fit the qualitative pattern of data from the three experiment quite well. 3 Lober and Shank alo howed that the weighted P model reproduced the data. However, the power PC theory doe not predict the oberved effect of the contingency manipulation on caual etimate in Experiment 1 and 2, and the (unweighted) P model doe not predict the different etimate acro Condition 1 3 of Experiment 3. Rationale of the experiment The initial aim of the work reported here i to preent ome very imple tak, imilar to thoe of Lober and Shank (2000), involving one potential caue and one effect, in which all of the model we have decribed can be applied without making further aumption, but in which different model are able to make very different prediction. Our final aim i two-fold: (1) to compare caual judgement to computational rule uch a P and p in order to tet their 3 In the model prediction reported here (and later) we have not undertaken exhautive parameter earche to optimize free parameter a our interet i in ordinal prediction. In cae where we conclude that a parameterized model fail to predict the correct ordinal pattern thi i true acro all parameter value. All model prediction have been confirmed by imulation. An Excel preadheet containing aymptotic formulae for all model, and Viual Baic program for imulating the Pearce, RW, and VHW model are available from the econd author on requet.

9 TABLE 2 Condition and final etimate in Lober and Shank (2000) Experiment 1 3 and prediction of the power PC, Recorla Wagner, Pearce, Van Hamme and Waerman, and belief adjutment model Final Experiment Condition P(E/C) P(E/~C) P p RW(β E < β noe ) Pearce VHW BA judgement Note: P(E/C) and P(E/~C) are the probabilitie of the effect in the preence and abence of the target potential caue, repectively. p = power, RW = Recorla- Wagner, VHW = Van Hamme and Waerman, BA = belief adjutment. The following parameter were ued: RW: β E =.3, β noe =.6; Pearce: x 1 =.8; x 2 =.5; x 3 =.5; VHW: β E =.3, β noe =.6, α C =.8, α noc =.3, α X =.5; BA: w A =1,w B =.7, w C =.7, w D =.5, β =1. 985

10 986 PERALES AND SHANKS normative accuracy, and (2) to find out which contemporary learning algorithm are compatible with poible behavioural deviation from uch rule. To achieve thi goal, we have extended the trategy ued in the recent work by Lober and Shank (2000). In that paper the author preented everal experiment (ee Table 2) in which both the manipulation of P and the manipulation of caual power p eemed to affect caual judgement. Thoe reult were interpreted a evidence againt the power PC model, a it predict judgement to be contant in ituation in which p i held contant, other factor being equal. However, that paper leave everal quetion unanwered. For example, Lober and Shank (2000) interpreted the global pattern of reult a compatible with the RW model. However, thi i only true for the unretricted, not the retricted, verion. If P i held contant, judgement increae with increae in generative power and decreae (become more negative) with increae in preventative power (Buehner & Cheng, 1997) a pattern of reult that can only be explained by the RW model if β for a preent effect i aumed to be lower than β for an abent effect in poitive contingency condition, and the oppoite et of value i aumed for negative contingency condition. In previou reearch thi pattern ha been confounded with difference in the cover torie ued in generative and preventative condition (ee Lober & Shank, 2000), and it i not unreaonable to peculate that the cover tory may affect the relative parameter value. The RW model could be effectively rejected, however, if thi pattern of reult were obtained in a ingle experiment with a ingle cover tory in which equality of parameter acro condition i enured. Second, the range of model taken into account by Lober and Shank (2000) i not exhautive. A noted above (ee Table 2), the Catena et al., VHW, and Pearce model are compatible with the pattern of reult. Finally, alternative explanation of the Lober and Shank (2000) data, compatible with the power PC theory, have been uggeted. Buehner and Cheng (1997) have argued that participant judgement might deviate from the actual caual power p becaue of difference in ample reliability acro condition. Thi can be illutrated by conidering the firt experiment by Lober and Shank (2000; ee Table 2). The eential point i that the target caue ha more trial on which to prove it generative power the larger the abolute value of P.Inthe P =.7 condition, the effect never occurred in the 30 trial in which the target caue wa abent, while it occurred on 21 of the 30 trial on which the caue occurred. Thi mean that the caue had 30 trial to prove it generative power, and it did o on 21 of them. In the P =.23 condition, by contrat, the effect occurred on 20 out of the 30 trial on which the caue wa abent, and it occurred in 27 out of the 30 trial on which the caue wa preent. Thu there were only 10 trial on which the caue had an opportunity to prove it power, a the effect would have been expected to occur anyway on the remaining 20 trial. Thi mean that the P =.7 condition ha three time the amount of evidence or effective ample ize a the P =.23 condition. If participant tend to conflate their caual judgement with their confidence in thoe judgement then an artifactual effect of contingency on judgement would appear in condition of contant caual power. Therefore, the effect of manipulating P i interpreted a a conequence of participant lack of confidence in their judgement in the low contingency condition, in relation to the high contingency condition (Buehner & Cheng, 1997). A that explanation cannot a yet be ruled out, our experiment have been deigned to enure that judgement are made independently of confidence.

11 EXPERIMENT 1 In the firt experiment the RW model prediction were teted againt thoe from the power PC theory, Pearce model, the VHW model, and the belief adjutment model. In a imple tak in which only one potential caue and one effect are preented, the RW model prediction are equal to P at aymptote, but only if β for the preence of the effect (β E ) and for it abence (β noe ) are aumed to have the ame value. If we accept that premie, the model doe not predict any effect of caual power when P i held contant. In contrat, if parameter inequality i allowed (the unretricted RW model) the model make different prediction about the effect of power on caual judgement depending on the direction of that inequality. Under the aumption that β E > β noe, caual judgement are predicted to be an invere function of p in poitive contingency condition, but a direct function of p in negative contingency condition. The oppoite trend are predicted when β E i aumed to be lower than β noe. Exactly the ame prediction are made by Van Hamme and Waerman model. Moreover, the β E and β noe parameter have to be contrained by the ame aumption a thoe in RW. Buehner and Cheng (1997) experiment howed judgement to be a direct function of p in both poitive and negative contingency condition, and Lober and Shank (2000) Experiment 3 confirmed thi for poitive contingencie. However, the effect of p ha never been demontrated in a ingle experiment in which the ame timuli and the ame cover tory are ued for poitive and negative contingency condition. Conequently, it can be argued that in previou experiment (Buehner & Cheng, 1997) in which judgement increaed with p for poitive contingencie and decreaed (i.e., became more negative) for negative one, participant may have elected different et of β parameter in the different condition. Our firt experiment wa deigned to tet the model prediction in a ingle experiment in which power (p) wa manipulated acro two poitive and two negative contingency condition. Method Participant and apparatu The participant were 16 Univerity College London tudent; 10 were male, 6 were female, and their age varied between 18 and 25 year, with a median of 20. They were paid 3 (approximately $US 4.30) for their participation. Participant were teted individually and required between 25 and 35 min to complete the experiment. The intruction and all of the information were preented on a PC computer creen, and participant gave their anwer by uing the computer keyboard or moue. Procedure CONTINGENCY AND CAUSAL POWER EFFECTS 987 A within-ubject deign wa ued in which every participant had to work on each of four different condition. In poitive condition (1/.6 and.4/0) contingency wa held contant at P =.4, wherea in negative condition (.6/1 and 0/.4) contingency wa held contant at P =.4. The two number deignating each condition refer to P(E/C) and P(E/~C), repectively. Caual power (p) wa manipulated acro condition in a range from 1 to 1 (ee Table 3). Each condition conited of a many 20-trial block a the participant needed to make a 100% confident judgement. The number of each trial type were held contant acro block, o neither P nor p (for each condition) depended on the number of block oberved by the participant. The order of trial wa

12 988 PERALES AND SHANKS TABLE 3 Trial type frequencie, conditional probabilitie, caual power, and contingency for each condition in Experiment 1 Trial type frequencie Condition A B C D P(E/C) P(E/~C) P p 1/ / / / Note: A, B, C, and D are the frequencie of each trial type per block. P(E/C) and P(E/~C) are the probabilitie of the effect in the preence and the abence of the potential caue, repectively. p = caual power. P = contingency. randomly elected within each block and the order of preentation of thee condition wa counterbalanced acro participant. When participant had typed in their age and ex they were aked to read the following intruction on the creen: Imagine you are working in a laboratory and you want to find out whether certain type of radiation caue or prevent genetic mutation in butterflie DNA. You will ee laboratory record from 4 tudie, with 4 different pecie of butterflie and 4 type of radiation. In one tudy, Aian butterflie were irradiated with U256 nuclear radiation, in another Monarca butterflie were irradiated with P290, in a third Common Black butterflie were irradiated with W119, and in a fourth Granada butterflie were irradiated with Z210. In each tudy ome butterflie received nuclear radiation and other did not. After ome time the butterflie DNA wa examined for genetic mutation. Of coure, DNA mutation may ometime occur pontaneouly in inect not expoed to nuclear radiation. What you mut decide i whether the radiation increae or decreae the likelihood of mutation. There are 200 butterflie in each tudy, but you can top when you think you have enough information to make a 100% reliable judgement. Each record tell you whether the butterfly wa expoed to the relevant nuclear radiation or not. You will then be aked to predict whether or not the butterflie DNA will how a genetic mutation. When you have made your prediction you will be told whether a mutation wa found or not. Ue thi feedback to try to find out whether the radiation really caue or prevent mutation. Initially, you will have to gue, but eventually you will become an expert. During each tudy you will be aked from time to time to etimate the degree to which the radiation caue or prevent mutation. Type in a number between 100 and +100, where 100 indicate that the radiation prevent mutation, 0 indicate that it neither caue nor prevent mutation, and 100 indicate that the radiation caue mutation. Intermediate number indicate intermediate level of caual or preventative influence. IF YOU HAVE ANY QUESTIONS ASK FOR MORE INFORMATION FROM THE RESEARCHER. Participant were then hown the laboratory record of the butterflie. Each record indicated whether the butterfly wa expoed to the radiation or not. To make their prediction participant had to pre the

13 button Ye, the mutation i going to occur or the button No, the mutation i not going to occur with the computer moue pointer on the creen, to indicate whether they thought that the animal DNA would or would not how a mutation. After thi prediction, they were informed about the actual occurrence of a mutation in that butterfly. To go on to the next trial, participant clicked a button on the creen with the label Pre here for another trial or preed Enter. After every block of 20 trial, participant were aked to etimate the degree to which the radiation caue mutation: Pleae give an etimate of the degree to which you think the radiation caue or prevent mutation. Select a value between 100 and +100, where 100 indicate that the radiation prevent mutation, 0 indicate that it neither caue nor prevent mutation, and 100 indicate that radiation caue mutation. Intermediate number indicate intermediate level of caual or preventative influence. The cale with which the participant had to make their judgement wa a horizontal 20 cm crollbar with a cale pointer initially located at it centre. The cale pointer could be moved in both direction by clicking on the crollbar or uing the curor key. After participant had given their judgement, they were aked to continue tudying more block from the ame tudy if they thought their judgement wa not yet 100% reliable or to top if they thought their judgement wa already a reliable a it could be. When the participant decided to top tudying more block the next condition wa preented. Reult and dicuion CONTINGENCY AND CAUSAL POWER EFFECTS 989 The.05 ignificance level wa employed in all tatitical tet for thi and the following experiment, and degree of freedom were adjuted by multiplication with Greenhoue Geier epilon (Greenhoue & Geier, 1959) where appropriate. Decimal point in the degree of freedom in ome of the ubequent analye are due to the ue of that correction. Figure 1 diplay participant mean final caual rating for the four condition. We do not report participant earlier judgement becaue the number of uch judgement i a function of the number of block tudied and hence varie acro participant. A one-factor repeated meaure analyi of variance (ANOVA) applied to the final judgement in each condition revealed a ignificant effect of treatment, F(3, 45) = 56.73, MSE = Planned t tet revealed ignificant difference between condition 1/.6 and.4/0, t(15) = 6.93, a well a between condition.6/1 and 0/.4, t(15) = A econd one-factor repeated meaure ANOVA, applied to the number of block tudied by participant in each condition (M = 1.88, 2.19, 1.88, and 2.00, and SD = 1.03, 0.98, 0.89, and 0.89 for condition 1/.6,.4/0, 0/.4, and.6/1, repectively), did not how any ignificant effect, F(3, 45) = The data are clearly conitent with the power PC theory a judgement are ordinally related to power p, and they how that judgement are not directly baed on P. What about the algorithmic account? Figure 2 how the prediction of the unretricted RW model auming that (1) parameter β E i aigned a higher value than β noe, and (2) that β E i aigned a lower value than β noe. The figure alo how the aymptotic prediction for Pearce, Van Hamme, and Waerman, and the belief adjutment model. To fit Lober and Shank (2000) data, it wa aumed that the β E and β noe parameter in the RW algorithm can be aigned different value in negative and poitive contingency condition. That aumption wa maintained on the bai that participant in different experiment (e.g., Buehner &

14 990 PERALES AND SHANKS Figure 1. Mean final judgement in condition 1/.6,.4/0,.6/1, and 0/.4 in Experiment 1. Condition are denoted by two number, the firt being P(E/C) and the econd P(E/~C). Error bar repreent the tandard deviation for each condition. Figure 2. Prediction of the unretricted Recorla Wagner, RW(1): β E > β noe and RW(2): β E < β noe, Pearce, Van Hamme and Waerman (VHW), belief adjutment (BA), and weighted P model for each condition in Experiment 1. Condition are denoted by two number, the firt being P(E/C) and the econd P(E/~C). The parameter ued for the imulation are given in Table 7. Cheng, 1997) had received ditinctive intruction that allowed them to appropriately weight caual information. Thi eemed to be epecially plauible if one conider the pecific intruction given to participant in ituation in which they had to evaluate generative or preventative caual influence (Buehner & Cheng, 1997). For example, if a peron i aked to evaluate a vaccine effectivene in preventing a dieae, the occurrence of the dieae in a vaccinated animal might be epecially alient becaue of the participant expectation about the vaccine effectivene. On the other hand if the peron ha to evaluate the degree in which a certain type of radiation caue mutation, the abence of a mutation after expoure might be more urpriing than it preence and, hence, more alient. In the preent experiment the intruction and cover tory were the ame for both generative and preventative condition. Conequently the β E and β noe value are contrained to be the ame for all condition, a there i no reaon to weight the outcome preence or abence

15 CONTINGENCY AND CAUSAL POWER EFFECTS 991 differently depending on the condition. Under thee circumtance, a noted before, the RW model wrongly predict oppoite effect of the caual power (p) manipulation in generative and preventative tak. Our data how caual judgement to be a direct function of p, independently of the ign of caual influence. However, it eem obviou that the power PC theory i not the only account capable of accommodating the preent reult. A can be een in Figure 2, both Pearce (1987) model and Catena et al. (1998) belief adjutment model predict the correct ordinal pattern of reult. The weighted P rule, a decribed in more detail later (ee Table 7), alo correctly predict the pattern. The VHW model, in contrat, fail to provide an ordinally correct account. Finally, the ign of the lat judgement of each participant in the negative condition wa revered in order to compare ymmetrically equivalent (negative v. poitive) condition. Paired ample two-tailed t tet howed judgement to be more extreme in condition 1/.6 than in condition.6/1, t(15) = The ame analyi alo revealed a ignificant difference between condition.4/0 and 0/.4, t(15) = Thu caual etimate in generative and preventative condition are aymmetrical, being more extreme (higher in abolute value) in generative than in preventative condition. Thi inequality ha been reported previouly (Maldonado et al., 1999) and i traightforwardly predicted by both Pearce and Catena et al. model. 4 In ummary, Experiment 1 how caual trength judgement to be a direct function of caual power, independent of the ign (poitive or negative) of the programmed contingency between the potential caue and the effect. Thi reult run counter to the RW model prediction: indeed, bearing in mind that only a ingle cue and outcome were involved, we ugget that thi i one of the mot imple and compelling violation of RW that ha been demontrated to date. On the other hand, the reult are qualitatively well fitted by the power PC theory. However, Pearce and Catena et al. model eem to make better quantitative prediction, a they traightforwardly predict a noticeable aymmetry between generative and preventative caual etimation. Experiment 2 wa deigned more directly to tet the prediction from the latter model againt thoe from the power PC theory. EXPERIMENT 2 A mentioned before, everal tudie have reported evidence againt the power PC theory, a caual judgement eem to be a function of contingency acro condition in which power (p)i held contant (Lober & Shank, 2000; Vallée-Tourangeau et al., 1997). However, Buehner and Cheng (1997) have claimed that the effect of contingency in thoe report can be explained by participant conflation of caual trength and confidence. Following the reaoning decribed previouly (ee Rationale of the experiment), to enure that participant in the low 4 There i no direct evidence to decide whether the aymmetrical ue of poitive and negative information i a baic property of caual information proceing or the reult of an interaction between previou belief and new evidence. However, ome general evidence eem to make the former explanation more plauible. For example, difficulty in proceing negative information appear to be a general attribute of human information proceing (White, 1998), and the differential weighting of trial in contingency learning tak doe not eem to depend on the ign of the programmed contingency, nor on the intruction preented to the participant (Kao & Waerman, 1993; Maldonado et al., 1999; Mandel & Lehman, 1998).

16 992 PERALES AND SHANKS contingency condition reach the ame level of confidence a thoe in the high contingency condition, the trial erie for the former hould be longer than the one for the latter. The firt aim of Experiment 2 wa to tet directly the conflation hypothei, a formulated by Buehner and Cheng (1997). Participant had to work in ix different condition. In two of them caual power wa held contant at p = 1, in another two it wa held contant at p = 0, and in the lat two it wa held contant at p = 1. P wa manipulated in the range.9 to.9. The third and fourth condition were equal with regard to p and P, but they differed in the probability of the outcome, P(E). The mot important ditinction wa that, a in Experiment 1, participant were allowed to tudy a many trial a they needed to make a 100%-confident judgement and to top when they thought they had reached that level of confidence. Under thee condition, the conflation hypothei traightforwardly predict that (1) the erie of trial tudied by participant in order to reach maximum confidence hould be horter in extreme contingency (.9 and.9) condition than in intermediate contingency condition (-.3 and.3), and (2) caual etimate hould not differ when the maximum level of confidence i reached, a p i held contant acro negative, poitive, and zero contingency condition. Method Participant and apparatu The participant were 18 Univerity College London tudent; 9 were male and 9 were female, and their age varied between 17 and 34 year, with a median of 23. They were paid 3 for their participation. Participant were teted individually and required between 25 and 35 min to complete the experiment. Procedure A within-ubject deign wa ued in which every participant had to work on each of ix different condition. In poitive condition (1/.1 and 1/.7) caual power wa held contant at p = 1; in zero contingency condition (.8/.8 and.2/.2) caual power wa held contant at p = 0, and, finally, in negative condition (0/.3 and 0/.9) caual power wa held contant at p = 1 (ee Table 4). It i important to note that the third and fourth condition (.8/.8 and.2/.2) are equal with regard to p and P, but in condition.8/.8 the probability of the outcome, P(E), i much higher than in condition.2/.2 (.8 v..2). The intruction and repone meaurement detail were the ame a in Experiment 1. Reult and dicuion Figure 3 diplay participant mean final caual rating for the ix condition. A one-factor repeated meaure ANOVA applied to the final judgement in each condition revealed a ignificant effect of treatment, F(3.36, 57.21) = 69.1, MSE = Planned t tet revealed ignificant difference between condition 1/.1 and 1/.7, t(17) = 3.43, between condition.8/.8 and.2/.2, t(17) = 3.67, but not between condition 0/.3 and 0/.9, t(17) = 1.39, p = A econd one-factor repeated meaure ANOVA applied to the number of block tudied by participant in each condition (M = 1.5, 1.61, 1.78, 1.83, 1.56, and 1.35, and SD =.9, 1.09,.94, 1.04,.70, and.64 for the firt to lat condition, repectively), did not reveal any ignificant effect, F(3, 47.8) = 1.44, p =.24. Finally, the ign of the lat (global) judgement in the negative condition wa revered in order to compare ymmetrical (negative v. poitive) condition. Paired-ample two-tailed t

17 CONTINGENCY AND CAUSAL POWER EFFECTS 993 TABLE 4 Trial type frequencie, conditional probabilitie, caual power, and contingency for each condition in Experiment 2 Trial type frequencie Condition A B C D P(E/C) P(E/~C) P p 1/ / / / / / Note: A, B, C, and D are the frequencie of each trial type per block. P(E/C) and P(E/~C) are the probabilitie of the effect in the preence and the abence of the potential caue, repectively. p = caual power. P = contingency. Figure 3. Mean final judgement for each condition in Experiment 2. Condition are denoted by two number, the firt being P(E/C) and the econd P(E/~C). Error bar repreent the tandard deviation for each condition. tet did not how judgement to be more extreme in condition 1/.7 and.8/.8 in relation to condition 0/.3 and.2/.2, t(17) =.047 and.27, repectively. However, the ame analyi howed the difference between condition 1/.1 and 0/.9 to be marginally ignificant, t(17) = 1.89, p =.076. One poible concern with thi tudy might be that terminal judgement may not have been at aymptote. Since account uch a the power PC theory require difference to be teted at aymptote, it i important to etablih that judgement are at or near that level, at leat in thoe condition involved in key comparion. Our procedure wa deigned to enure that the maximum level of confidence wa reached, and it eem reaonable to uppoe that aymptote i met before participant reach uch a maximal confidence level (a it eem implauible that a participant would be completely confident in a judgement that ha yet to reach aymptote). In addition, however, there are empirical reaon to think that the poibility of pre-aymptotic judgement i not a problem: The following analyi ugget that judgement were at or very cloe to aymptote.

18 994 PERALES AND SHANKS In the preent experiment, participant in each condition can be divided into thoe who aw only a ingle block and thoe who tudied more than one block. For participant who tudied more than one block, the mean lat (J) and lat-but-one (J-20) judgement are almot identical in condition 1/.1, 1/.7, and.2/.2 and differ lightly in the other condition (ee Table 5). None of the difference i tatitically ignificant by a paired t tet, p >.13 in each cae. Note that thee comparion are very conervative a judgement could only be made at the end of each 20-trial block, and hence, even in thoe condition (e.g.,.8/.8) where judgement at J and J-20 differed omewhat, aymptote could have been reached during the final 20-trial block. Latly, participant who aw only a ingle block gave rating very imilar to the terminal rating of participant who aw more than one block, o thee judgement too appear to be aymptotic. Overall, then, it eem likely that judgement were very cloe to aymptote in thi tudy. Thi i conitent with the finding of Lober and Shank (2000) and Shank (2002), who found that aymptote wa reached within about 30 trial in a imilar procedure. How do the other theorie fare with the data from thi experiment? A we how later (ee Table 7), the concluion are imilar to thoe of Experiment 1. The pattern of reult, including the difference between judgement in the.8/.8 and.2/.2 condition, i compatible with the Pearce, BA, and weighted P model but not with VHW. The VHW model, a well a the unweighted P theory, predict equal judgement in thee noncontingent condition. To conclude, the main reult from thi experiment clearly contradict the power PC theory prediction: Judgement differ in condition of equal power. Although the magnitude of the effect eem to have been lightly reduced in relation to previou experiment, the effect i till highly ignificant in poitive and zero contingency condition, and how the expected (although not ignificant) trend in negative condition. The difference between the zero contingency condition not only contradict the power PC theory, but alo the RW model. The ame outcome bae-rate effect or denity bia (Allan & Jenkin, 1983) ha been recently reported by Vallée-Tourangeau et al. (1998) and Buehner and Cheng (1997). Buehner and Cheng offered an explanation in term of the reduced reliability of the cae oberved by participant that wa due to the preentation of only 16 trial. Thi interpretation TABLE 5 Final and previou caual judgement for thoe ubject who tudied more than one block in each condition of Experiment 2 Condition J-20 J 1/ / / / / / Note: J: Mean final judgement. J-20: Mean judgement in the previou block.