Non-linear Multiple-Cue Judgment Tasks

Transcription

1 Non-lnear Multple-Cue Tasks Anna-Carn Olsson Department of Psychology, Umeå Unversty SE-09 87, Umeå, Sweden Tommy Enqvst Department of Psychology, Uppsala Unversty Box 225, 7 2 Uppsala, Sweden Peter Jusln (peter.jusln@psyk.uu.se) Department of Psychology, Uppsala Unversty Box 225, 7 2 Uppsala, Sweden Abstract In two experments we examned whch cogntve processes people use n a non-lnear multple cue judgment task, proposng that people are not able to use explct cue abstracton when judgng objects wth a non-lnear structure between the cues of the objects and the crteron and therefore they are forced to use exemplar-based processes. Consstent wth the results reveal strong exemplar effects n the non-lnear condton. Introducton How do we judge f there seems to be no underlyng structure or pattern n the thngs or events we have to judge? What knowledge representatons do we use n these stuatons? Our ablty to categorze the envronment s mportant to us and dfferent envronmental categorzaton stuatons nvolve dfferent processes, ether n the form of analytc or ntutve thought. Multple cue judgment tasks typcally nvolve a probe defned by a number of bnary or contnuous cues and requre a contnuous judgment. Prevous research suggests that by manpulatng the task envronment we can nduce a shft between dstnct cogntve processes (Jusln, Olsson, & Olsson, 2003). Two cogntve models of general concern n cogntve scence, are the explct, rule-based and analytc knowledge versus mplct, slent knowledge based on personal experence (Hahn & Chater, 998; Hammond, 996; Sloman, 996; E. E. Smth, Patalano, & Jondes, 998). Research on multple cue judgment has prmarly stressed controlled ntegraton of cues that have been abstracted n tranng (Enhorn, Klenmuntz, & Klenmuntz, 979) whle exemplar models are emphaszed by research on categorzaton (Nosofsky & Johansen, 2000; Nosofsky & Palmer, 997). In ths artcle we address the followng queston: what structure of a categorzaton task trggers analytc thnkng and what structure trggers ntuton? We hypothesze that partcpants are not able to use cue abstracton n a non-lnear task, because the mnd s constraned to lnear addtve ntegraton of abstracted cue-crteron relatons (Jusln, Karlsson, & Olsson, 200). Cogntve models and processes In general, multple cue judgments are well captured by multple lnear regresson models (Brehmer, 99, Cooksey, 996). These regresson models are statstcal descrptons rather than process models, but mplctly ths research s often commtted to the dea that people use controlled processes n workng memory to mentally ntegrate cues accordng to a lnear addtve rule. In contrast, exemplar models assume that people make judgments by retrevng smlar stored exemplars from memory (Medn & Schaffer, 978; Nosofsky & Johansen, 2000). The exemplar representatons n memory are the holstc concrete experenced nstances often emphaszed n categorzaton. The generalzed judgment model Σ mplcates a sequental adjustment process, compatble both wth a lnear, addtve cue-ntegraton rule, and the addtve combnaton of exemplars n exemplar models (Jusln et al., 200), capturng the addtve character of a judgment. By consderng the structural propertes of a task envronment, we are able to predct representatonal shfts n the process that support multple cue judgment. In addtve envronments cue abstracton domnate, whle exemplar memory domnates n multplcatve envronments (Jusln et al., 200). In the Σ model the prevous estmate s adjusted every tme a new pece of evdence s presented. If no new evdence s presented cues correspond to an a pror estmate. In Σ, cue abstracton nvolves sequental adjustment based on sngle cues, where the cue weght s relatve to the cues presented so far. In contrast, the weght assgned to an exemplar s gven by ts smlarty relatve to the smlarty to all exemplars retreved so far. Because of the ntegraton of stored crteron values nstead of cue-crteron relatons, Σ s not confned to any partcular task structure. Therefore, the model can represent any task structure as long as smlar exemplars have smlar crtera, whch allows judgments also n non-lnear and multplcatve envronments (Jusln et al., 200). task and Model Predctons The partcpants used four bnary cues to nfer a contnuous crteron n ether a lnear or a non-lnear multple-cue judgment task. The exotc (but fcttous) Death bug s the cover story n the task that nvolves judgments of the toxcty of 672

2 subspeces of the bug, whch can be nferred from four cues of the subspeces (leg length, nose length, spots or no spots on the fore back and dfferent patterns on the buttock). The concentraton of poson vares from to ppm n each subspeces (Jones et al., 2000; Jusln et al., 2003; Jusln et al., 200). The task structure s summarzed n Table. Table : Structure of the judgment task wth the constraned tranng set for the lnear and non-lnear condtons. Exemplar Cues Crtera Role Constr. # C C2 C3 C Lnear Nonlnear Tr. set E 2 0,6 T 3 0, T 0 0, O 5 0, N 6 0 0,6 N N ,6 T 9 0,6 O O 0 0,6 T , T 3 0 0, T 0 0 0, T ,6 T E Note: T= tranng exemplar; O= tranng exemplar that serves as old exemplar n the nterpolaton comparson; E= new exemplar that only occurs n the test phase for measurng extrapolaton; N= new exemplar that only occur n the test phase for nterpolaton comparson. The values of the bnary cues take on or 0. The functon of the cue values to judge the toxcty c L of a subspeces s lnear and addtve n the lnear judgment task: c L = + C + 3 C2 + 2 C3 + C () A quadratc functon of the crtera n the lnear condton was made as the functon n the non-lnear judgment task wth the same range ( to ppm) and maxmum value ( ppm) wth a non-lnear relatonshp between the cues and the crteron nstead of the lnear functon. c NL = 2 c L ²/5 + c L (2) The functons c s the level of poson n the bug and C C are weghted bnary features of the bug. For example a bug wth features [0] has a toxcty level of n the lnear condton and a toxcty level of. n the non-lnear condton, see Table. The crteron c s computed by assgnng the most mportant cue, C, and has the largest weght wth the least mportant cue that has the smallest weght, C. If the bnary cue has value, t suggests hgh toxcty level, and f the cue has value 0, t suggests low toxcty level, see Table. In erment a normally and ndependently random error was added. The varance of the random error would produce a.9 correlaton (see Jusln et al, 2003). Fgure llustrates the relatonshp between the two functons n a plotted dagram. The non-lnear functon s a more complex envronment than the lnear functon. The ncrease of complexty n a functon or envronment s when a functon goes from an addtve lnear functon to be a multplcatve nonlnear one. When a partcpant can not use a smple addtve rule to calculate what correct crteron s n a judgment. The non-lnear functon can also be plotted aganst the crteron values for the non-lnear functon on the x-axs (Fgure 2), where the functon only has sx crteron values and two or four examples matchng each crteron (see Table ). Fgure. The addtve lnear functon and the non-lnear functon plotted aganst the crtera n the lnear condton. Both functons have the same range on the x-axs (crteron) and the same range on the y-axs (judgment)..6 Fgure 2. The non-lnear functon plotted aganst the crteron values for the non-lnear functon on the x-axs, wth only sx crteron values, where two or four examples matchng each crtera. Cogntve Models Cue abstracton model. The applcaton of the cue abstracton model to contnuous crtera mples that partcpants abstract cue weghts ω, whch specfy the mportance and the sgn of the relaton of the cue (= ). In tranng partcpants abstract the cue weghts and use those to compute an estmate of the crteron when a new probe s presented. The estmate of c s adjusted accordng to the cue weght and the fnal estmate ĉ R of c s a lnear addtve functon of the cue values C. For example, after tranng the rule for cue C may specfy that C goes wth a large ncrease n toxcty. Ths corresponds to the standard applcaton of a lnear addtve equaton to model multple cue judgment, cˆ R = k + = ω C where k = +.5 (0 ω ). If ω =, ω 2 =3, ω 3 =2, and ω =, Equatons and 2 are dentcal and the model produces perfectly accurate judgments. The ntercept k con-...6 (3) 673

3 strans the functon relatng judgments to crtera to be regressve around the mdpont () of the nterval [, ] specfed by the task nstructons. Exemplar model. The applcaton of exemplar model to a contnuous crtera we assume that judgments are made by retrevng smlar exemplars from memory and the estmate of the crteron c s a weghted average of the crtera c j stored for the J exemplars, where the smlartes S(p,x j ) are the weghts, cˆ J j= E = J S( p, x ) c j= S( p, x ) j j j. () Eq. 5 s the context model (Medn & Schaffer, 978) appled to a contnuum (see Delosh et al., 997; Jusln & Persson, 2000; E. R. Smth & Zarate, 992). The applcaton of an exemplar model to multple-cue judgment s llustrated n Fgure 3. The smlarty between probe p and exemplar x j s computed accordng to the multplcatve smlarty rule of the orgnal context model (Medn & Schaffer, 978): S ( p, x j ) = d, (5) = where d s an ndex that takes value f the cue values on cue dmenson concde (.e., both are 0 or both are ), and s f they devate (.e., one s 0, the other s ). s are four parameters n the nterval [0, ] that capture the mpact of devatng cues values (features) on the overall perceved smlarty S(p,x j ). s close to mples that a devatng feature on ths cue dmenson has no mpact on the perceved smlarty and s consdered rrelevant. s close to 0 means that the smlarty S(p,x j ) s close to 0 f ths feature s devatng, thus assgnng crucal mportance to the feature. The parameters s capture the smlarty relatons between stmul and the attenton pad to each cue dmenson, where a low s sgnfes hgh attenton. In effect, for low s, only dentcal exemplars have a profound effect on the judgments. For example, wth all s =.00 dentcal exemplars receve weght n Eq., but exemplars wth just one devatng feature receve weght.00. Wth s close to, on the other hand, all exemplars receve the same weght, regardless of the number of devatng features. Predctons by the Models Predctons by the cue abstracton and exemplar model are summarzed n Fgure 3. When partcpants are traned wth all 6 exemplars t s mpossble to dscrmnate the dfferent models from each other, because they predct the same accurate judgments (see Fgure 3, panel A and B). Ths perfect accuracy depends on correct knowledge of the cue weghts and error-free ntegraton of ths knowledge nto a judgment n the cue abstracton model whle t derves from retreval of stored exemplars, where only dentcal exemplars are allowed to have a strong effect on the judgment n the exemplar model. A C No Nose Nose Nose B D s=.000 s=. s=. Fgure 3: Predctons for the contnuous task. Panel A: Cue abstracton models wth no nose and nose for the complete tranng set. Panel B: Exemplar model wth all smlarty parameter s equal to.000 and. for the complete set. Panel C: Cue abstracton model wth nose for the constraned set. Panel D: Exemplar model wth smlarty parameter s=. for the constraned set. The dscrmnaton of the models nvolves the exstence or no exstence of extrapolaton and nterpolaton, n other words, the ablty to make accurate judgments of new exemplars. When a constraned set of subspeces are omtted n tranng phase (two extreme exemplars, e.g. [ ] and [ ] and three mddle exemplars, [ 0 ], [0 0], [ 0 0 ], see Table for new N, and old, O exemplars) and the partcpants are presented wth the complete set of subspeces n the test phase, the cue abstracton model affords extrapolaton and nterpolaton, because the model allows the addng functon of ntegrated cue values that produce accurate judgments. The partcpants wll fgure out that the extreme subspeces wth all cues presented and all cues absent should have the most extreme value, even f they have never been presented to these subspeces n the tranng phase, also no old-new dfferences between the ntermedate exemplars omtted n tranng should exst n the cue abstracton model. In contrast, the exemplar model s not able to extrapolate or nterpolate, when the responses for the new exemplars are determned by retreval of dentcal exemplars (Delosh et al., 997; Erckson & Kruschke, 998). Because the exemplar model nvolves lnear combnaton of the crtera observed n tranng that range between the toxcty levels and, t can never produce a judgment outsde ths range, as extrapolaton requres. The exemplar model also predcts old-new dfferences wth more accurate judgments for old exemplars than for new when the toxcty level can be retreved from memory (Jusln et al., 2003). 67

4 Method Partcpants Ffty-four undergraduate students from the unversty partcpated n the two experments (thrty partcpants n erment and twenty-four partcpants n erment 2). Twenty-three males and twenty-seven females. The average age was 2.25 years. Each partcpant receved 70 SEK (approxmately 8$) n payment for partcpaton n the study. Desgn and procedure The desgn of the two experments was a between-subjects desgn, where all partcpants made the judgments n tranng and test wth same stmul presentaton format. Each experment contaned two condtons each. In erment, a probablstc lnear judgment task condton and a probablstc non-lnear judgment task condton The wrtten nstructons nformed the partcpants that there were dfferent subspeces of a Death Bug and that the task was to estmate the toxcty (poson level) of the subspeces as a number between and. The experments contaned two phases, where the frst phase was a tranng phase whch provded tral-by-tral outcome feedback about the contnuous crteron ( Ths bug has toxcty.7% ). In the tranng phase tranng exemplars of the bug were presented 20 tmes each, a total of 220 trals wth crteron values between and n the lnear condton and between.6 and n the non-lnear condton. The remanng fve exemplars were omtted n the tranng phase and frst presented n the test phase. Partcpants were traned and tested wth analogue stmul n the form of pctures of the bug speces, presented on the computer screen. The subspeces vared wth regard n four bnary cues; leg length (short or long), nose length (short or long), spots or no spots on the fore back and two dfferent patterns on the buttock and dfferent colors were used for the cue values to strengthen ther salence. The abstract cues n Table were randomly assgned to new vsual features for each partcpant. The cue values had the weghts, 3, 2 and whch determne the porton of toxcty that each cue adds to the total amount. The queston asked on the computer screen was What s the toxcty of ths subspeces. In the test phase, all 6 exemplars were presented, ncludng the fve exemplars omtted n the tranng phase. The test phase went over 32 trals, where the 6 exemplars were presented twce n a random order. The partcpants made same judgments as n the tranng phase but receved no outcome feedback. The whole experment took 5 mnutes. erment 2 nvolved the same stmul and non-lnear structure as erment. However, n contrast to erment, the cue crteron-relaton was changed from a probablstc task to a determnstc task. The length of the tranng phase was also changed from 220 trals to 0 trals. The nstructons were the same n the frst condton (Oldnstructons condton) as n erment but was changed n the second condton ( nstructons condton), and encourage the partcpants to memorze each bug exemplar. We expected the determnsm and the exemplarmemorzaton request to ncrease the performance and the use of exemplar memory. Dependent Measures The dependent measures reported nvolve: performance, the representaton ndex, and model ft. The performance measures are, Root Mean Square Error (RMSE) of judgments (between judgment and crtera), and consstency (correlaton between the two judgments made for the same exemplar n the test phase). A Representaton ndex (RI) was calculated to see what level of representaton that domnates the judgments n the two dfferent tasks. The nterpolaton measure s obtaned by takng the dfference between absolute devaton between judgment and crtera for old exemplars and new exemplars as computed by the followng formula, I = [d(old) n d() n ]/6 (6) Where n s refers to crteron, and n the lnear condton and crteron.,.6 and n the non-lnear condton denoted ether as Tranng(Old) or Interp.() n Table. The exemplars n the extrapolaton are also consdered. Extrapolaton s measured by the devaton from an expected lnear extrapolaton, based on mean judgment of the old exemplars n the nterpolaton range. The dfference between actual judgment and expected value s the extrapolaton measure. A 0 n the extrapolaton ndex mples that the judgements for the extreme exemplars are as extreme as the expected regresson-based extrapolaton for the old exemplars, and the judgments are correct by all lnear transformatons of the correct judgments. Ths suggests approprate extrapolaton. When extreme exemplars do not receve as extreme judgments as expected from extrapolaton and the ndex s negatve. Ths n turn, supports the exemplar model and ts nablty to extrapolate. In other words, the exemplar model does not allow accurate extrapolaton whle cue abstracton does (see Fgure, for a concrete llustraton). To ncrease the statstcal power n the analyss of the data and for ease of exposton of the data the nterpolaton ndex and extrapolaton ndex are combned whch gves us the RI. The RI ndcates f partcpants have based ther judgments on exemplar model or cue abstracton. A RI of 0 mples the regresson-based paradox wth the ablty to extrapolate whle a negatve ndex mples the nablty to extrapolate, exactly the same as mentoned before, n the extrapolaton ndex. By usng the data from the tranng phase n each task to create predctons of the two models was performed n a computer smulaton. The Model ft measures are the coeffcent of determnaton (r 2 ) and Root Mean Square Devaton (RMSD) between predctons and the computed test phase data. Results erment Performance. The probablstc lnear task was lower (0.) than for the probablstc non-lnear task (0.85)n con- 675

5 sstency, wth a sgnfcant dfference (F(.28)=.; p=0.0; MSE=0.37). The errors n the judgment as measured by RMSE (Root Mean Square Error) was sgnfcant dfferent n performance of the two tasks (F(.28)=37.0; p=0.0000; MSE=0.), where the probablstc addtve lnear task showed a lower RMSE value (.) than the probablstc non lnear task (3.02). In Fgure the test mean judgment from the probablstc addtve lnear task and the probablstc non-lnear task are plotted aganst the correct crtera. A Mean of judgment n addtve lnear condton B Fgure. Test mean judgment and exemplar ndex for the probablstc addtve lnear task (Panel A) and the probablstc non-lnear task (Panel B) plotted aganst the correct crtera. The full drawn lnes show the correct judgment. Representaton. The lnear judgment task has a lower RI value (-0.67) than the non-lnear judgment task (-.33), but not sgnfcant dfferent (F(, 28)=0.79; p=0.38; MSE=.08). The RI n the non-lnear condton s sgnfcantly separated from zero but not n the lnear condton. Ths suggests that EBM s the model n use n the non-lnear condton, whle both CAM and EBM s n use n the lnear condton. Model ft. The models n Eq. 3 and were ftted to the mean judgments computed for the constraned tranng set wth subspeces of the bug across the last 0 trals. Cue abstracton model allows analytc dervaton of the bestfttng parameters that corresponds to logstc regresson. By the Quas-ton method n the MathCAD software the parameters for the exemplar model wth the mnmal squared sum of error were obtaned. The model fts (see Fgure 5) show that n the probablstc addtve lnear condton both EBM and CAM have hgh correlaton ndces and about the same RMSD values (EBM; r 2 =.92 and RMSD=0. and CAM; r 2 =.96 and RMSD= 0.). In the non lnear condton EBM and CAM presents approxmately the same RMSD (EBM; RMSD= 0. and CAM; RMSD = 0.) as n the lnear condton, whle the correlaton s very low (EBM; r 2 =. and CAM; r 2 =.2), wth some advantage for EBM. When lookng at the tranng data we see that the curve of learnng was very flat, whch mply that the partcpants may not have learned to asymptote n the tranng phase. Ths could has effected the results and therefore erment 2 was constructed to provde a better settng for testng the hypothess that partcpants are not able to use cue abstracton n a non-lnear task. Mean of judgment n non lnear condton RMSD 0,6 0,5 0, 0,3 0,2 0, 0,0 EBM CAM Lnear addtve task Model Ft Non-lnear task Condtons Fgure 5. Model fts of exemplar model and cue abstracton model n the lnear addtve task and the non-lnear task. Results erment 2 Performance. The consstency for the two condtons was nearly the same (.7 for the regular nstructon condton and.8 for the exemplar nstructon condton). The RMSE showed a lower value for the exemplar nstructon condton (.93) than the regular nstructon condton (2.), wth sgnfcant dfference (F(.0)=9.28; p=0.002; MSE=6.70). In Fgure 6 the mean test judgment from the regular nstructon condton and the exemplar nstructon condton are plotted aganst the crtera. A Mean judgment n Exemplar nstructon condton Mean judgment n Old nstructon condton Fgure. Test mean judgment and exemplar ndex for the regular nstructon condton (Panel A) and the exemplar nstructon condton (Panel B) plotted aganst the correct crtera. The full drawn lnes show the correct judgment. Representaton.The RI shows that the regular nstructon condton has a hgher RI value (-2.68) than the exemplar nstructon condton (-2.79), but the dfference s not sgnfcant (F(.0)=0.30; p=0.86; MSE=2.0). The RI s separated from zero whch suggests that EBM s used n both condtons. Model ft. The same procedure for model ft was used as n erment. The model fts (see Fgure 6) show that EBM has hgh correlaton n both regular nstructon condton (EBM; r2=.68 and RMSD=.03) and exemplar nstructon condton (EBM; r2=.87 and RMSD=0.7) compared to CAM n the condtons (regular nstructon condton, CAM; r2=.6 and RMSD=.09 and exemplar nstructon condton; CAM; r2=.39 and RMSD=.27) whch suggests a domnated use of EBM n both condton, but strongest n the exemplar nstructon condton. 676

6 RMSD,3,2,,0 0,9 0,8 0,7 0,6 0,5 0, 0,3 0,2 0, 0,0 EBM CAM Model Ft Old nstructons Condtons Exemplar nstructon Fgure 7. Model fts of exemplar model and cue abstracton model n the old nstructon condton and the exemplar nstructon condton. Dscusson The queston addressed n ths artcle s what structure of a categorzaton task trggers analytc thnkng and what structure trggers ntuton? We hypotheszed that partcpants are not able to use the cue abstracton model, because t s too dffcult to nduce the equaton that underles the crteron. Overall, the results n erment the non-lnear condton shows lttle ft to the rule-based model CAM. The hgher RI, the poor results on old and new exemplars n the nterpolaton and extrapolaton range, all support EBM. The results of the RMSD that s almost dentcal to the lnear condton can not suggest whch model that s used, f any of the models was. One possble explanaton of the low RMSD and the poor consstency correlaton s that the judgment curve has a flack nclnaton, and the task s thus to complex to learn. Because of the lack of learnng for the partcpants n erment, we nvestgated n erment 2 f longer tranng phase and a change from a probablstc to a determnstc non-lnear task could allow the partcpants to use any of the processes. The results clearly showed the use of EBM n both condtons suggestng that determnsm and more tranng make t possble to adopt exemplar-based process and use t more effectvely. The use of cue abstracton seems to be nonexstent n the experments whch support the hypotheses that the non-lnear judgment task s too complex for judgments based on mental ntegrated cues and the dea that exemplarbased processes wll serve as a back-up system when cue abstracton s mpossble. Acknowledgements Bank of Sweden Tercentenary Foundaton sup ported ths research. References Brehmer, B. (99). The psychology of lnear judgment models. Acta Psychologca, 87, 37-. Cooksey, R. W. (996). analyss: Theory, methods, and applcatons. Academc Press: San Dego. DeLosh, E. L., Busemeyer, J. R., & McDanel, M. A. (997). Extrapolaton: The sne qua non for abstracton n functon learnng. Journal of ermental Psychology: Learnng, Memory, and Cognton, 23, Enhorn, J. H., Klenmuntz, D. N., & Klenmuntz, B. (979). Regresson models and process tracng analyss, Psychologcal Revew, 86, Erckson, M. A., & Kruschke, J. K. (998). Rules and exemplars n category learnng. Journal of ermental Psychology: General, 27, Hahn, U., & Chater, N. (998). Smlarty and rules: dstnct? Exhaustve? Emprcally dstngushable? Cognton, 65, Jones, S., Jusln, P., Olsson, H., & Wnman, A. (2000). Algorthm, heurstc or exemplar: Process and representaton n multple-cue judgment. In L. Gletman, & A. K. Josh (Eds.), Proceedngs of the Twenty-Second Annual Conference of the Cogntve Scence Socety (pp. 2-2). Hllsdale, NJ: Erlbaum. Jusln, P, & Karlsson, L., Olsson, H. (200) Exemplar memory n Multple-cue : A Dvson of Labor Hypothess (n press) Jusln, P., Olsson, H., & Olsson, A-C. (2003). Exemplar effects n categorzaton and multple-cue judgment. Journal of ermental Psychology: General, 32, 33-. Jusln, P., & Persson, M. (2003). PROBabltes from Exemplars (PROBEX): A lazy algorthm for probablstc nference from generc knowledge. Cogntve Scence, 26, 3-7. Medn, D. L., & Schaffer, M. M. (978). Context theory of classfcaton learnng. Psychologcal Revew, 85, Nosofsky, R. M., & Johansen, M. K. (2000). Exemplarbased accounts of multple-system phenomena n perceptual categorzaton. Psychonomc Bulletn & Revew, 7, Nosofsky, R. M., & Palmer, T. J. (997). An exemplarbased random walk model of speeded classfcaton. Psychologcal Revew, 0, Smth, E. E., Patalano, A. L., & Jondes, J. (998). Alternatve strateges of categorzaton Cogn -ton, 65,