Baserate Judgment in Classification Learning: A Comparison of Three Models
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1 Baserate Judgment in Classification Learning: A Comparison of Three Models Simon Forstmeier & Martin Heydemann Institut für Psychologie, Technische Universität Darmstadt, Steubenplatz 12, Darmstadt sforst@hrz1.hrz.tu-darmstadt.de, heydemann@hrz1.hrz.tu-darmstadt.de Abstract. The baserate neglect effect is a stable phenomenon which can be observed in classification learning and in other contexts of human judgment and decision making. An experiment by Gluck and Bower (1988) and their explanation of the baserate neglect effect with the delta rule explanation is described. Their results can also be accounted for by two variants of Bayes model, the baserate equalizing hypothesis and the single symptom interpretation hypothesis. The current study aimed at comparing the three models regarding their explanatory power of the baserate neglect effect. 60 subjects saw combinations of six symptoms and were asked to predict the correct disease. At the end of the experiment, they estimated the probabilities of each disease in the presence of certain symptoms. The pattern of results are best accounted for by the single symptom interpretation hypothesis and not by the two other models. The baserate neglect effect The baserate neglect effect is a phenomenon which occurs in different contexts of human learning and decision making. It has been extensively investigated in the field of classification learning (e.g. Gluck & Bower, 1988; Kruschke, 1996; Shanks, 1990b). In a typical experiment, in which the effect occurs, subjects are asked to learn to associate stimuli and categories. The categories have different baserates and, thus, are called R (rare) and C (common). The stimuli consist of several features. The task is to predict a category (e.g. diseases) when knowing some features (e.g. symptoms). The baserate neglect effect can be observed with those features which appear equally often with R and C in the learning phase. When presented with such a feature in the test phase, subjects predict most often the rare category R. The aim of this study is to test the predictions of three explanatory models of the baserate neglect effect. Before turning to these explanations, the experiment by Gluck and Bower (1988, exp. 1) is described. The experiment by Gluck and Bower (1988, exp. 1). The subjects were asked to imagine a world, in which only two diseases exist: a rare 268
2 disease R with a baserate of p(r) = 0.25 and a common disease C with a baserate of p(c) = There are only four symptoms (s 1, s 2, s 3, s 4 ). Not every patient, however, who suffers from one of the diseases, exhibits each of the symptoms which are associated with this disease. Therefore, these symptoms occur with different probabilities in the presence of the diseases R and C (p(s i D j ), where s i refers to the symptoms and D j to the diseases). Every symptom can either be present or absent. The conditional probabilities of the symptoms p(s i D j ) are shown in Table 1. From the baserates of the diseases and the conditional probabilities of the symptoms follows that the diseases occur only with a certain probability in the presence of the symptoms (see Table 1 for p(r s i )). As it can be seen, the rare and common disease (R and C) have the same probability in the presence of symptom 1, because p(r s 1 ) = For all other symptoms, C is more likely than R (see p(r s i )). Table 1. Conditional probabilities p(s i D j) and p(r s i), mean ratings r p(r s i), and predictions by three explanatory models for the experiment by Gluck and Bower (1988, exp. 1). Symptoms p(s i R) p(s i C) p(r s i ) r p (R s i ) Predictions for p(r s i ) observed Delta a BEH b SSIH c s s s s a Delta rule explanation: These are values for the weights w 1 through w 4. Weights bigger than 0 indicate predictions for p(r s i) bigger than 0.50 (i.e. preference of R), weights smaller than 0 indicate predictions for p(r s i) smaller than 0.50 (i.e. preference of C). b Predictions by the baserate equalizing hypothesis with modified baserate p(r) = 0.40 instead of the correct value p(r) = c Single symptom interpretation hypothesis: The values correspond to p(r s i.alone ) The subjects were shown descriptions of patients in the 250 learning trials and were asked to diagnose the disease. After their diagnosis, they received feedback about the correct answer. After learning, the subjects were asked to estimate for every symptom, with which probability a patient has got the disease R or C. Gluck and Bower (1988) assumed that this estimate is a rating of p(d j s i ). Comparing the observed mean ratings r p (R s i ) with p(r s i ), we see that each of the observed values are somewhat larger than the normative values p(r s i ). The critical symptom is symptom 1: When it is present, the diseases R and C occur with equal probability. However, the subjects overestimate the conditional probabilities of the rare disease R: r p (R s 1 ) = This is called the baserate neglect effect. Three explanatory models Three models are proposed which can explain the baserate neglect effect. Delta rule explanation. Since the experiment by Gluck and Bower (1988), the dominating explanation of the baserate neglect effect in the literature is the connectionist delta rule explanation. This explanation can be described with reference to the network model of Gluck and Bower (1988). Their network consists 269
3 of four input nodes and one output node. Each of the four input nodes represents one of the symptoms, different values of the output node represent the two diseases. During learning, the connection weights w i are changed according to Equation 1. w i = β (d o) a i. (1) a i is the activation on input node i, which represents the symptom (a i = 1 if the symptom is present, otherwise a i = 0). d is the value on the output node which is reinforced in the learning trial (d = 1 if disease R is the correct response, d = -1 if disease C is the correct response). o is the value which is predicted by the network on the basis of the current weights, calculated by Equation 2. o = n w i i= 1 When using the net to predict a disease after learning, o is calculated. If the value for o is bigger than 0, disease R is preferred, if it is smaller than 0, disease C is preferred. For learning according to Equations 1 and 2, asymptotic values can be calculated. The asymptotic weight for symptom 1 is w 1 = (Gluck & Bower, 1988, p. 233). Thus, the network clearly predicts disease R when symptom 1 alone is present, since in this case o = w 1 = When presented with symptom 2, 3, or 4, disease C is predicted by the network (see Table 1, Delta). Why is, according to the delta rule explanation, the rare disease R preferred for symptom 1 which is equally likely with R and C: p(c s 1 ) = p(r s 1 ) = 0.50? Disease R is the correct answer in only few cases (for it is the rare disease), disease C is the correct answer in most cases. Since C is correct in most cases, it is (at first) mostly predicted by the net. When suddenly R is the correct answer, the error is quite big. Since the errors with disease R are bigger than with disease C, weights are more strongly changed with R. Remember now that both diseases are equally probable in the presence of symptom 1. Since this is the case, the bigger errors (and stronger weight changes) with R lead to a positive weight of the connection from symptom 1 to the output node. Since a positive weight w 1 is associated with R, the net predicts R in the presence of symptom 1 at the end of learning. The baserate equalizing hypothesis. A simple alternative hypothesis assumes that subjects learn the approximate values of the conditional probabilities of the symptoms p(s i D j ) and of the baserates p(d j ). Having this information, the probabilities of the diseases p(d j s i ) can be calculated with Bayes theorem. The baserate equalizing hypothesis is based on the assumption that the actual baserates are equalized and these distorted baserates are put in the Bayes theorem. To explain the results of Gluck and Bower (1988), baserates of p(r) = 0.40 and p(c) = 0.60 could be used instead of the actual baserates of p(r) = 0.25 and p(c) = Using these distorted values, one gets predictions which are similar to the observed values (Table 1, BEH). a i. (2) 270
4 Single symptom interpretation hypothesis. Shanks (1990a, 1990b) suggests a third explanation. According to the single symptom interpretation hypothesis, subjects did not rate the conditional probability of the diseases, given the target symptom alone and in all possible combinations with other symptoms (p(d j s i )), but the conditional probability of the diseases, given the symptoms alone while other symptoms being absent (p(d j s i.alone )). For, say, p(r s 1.alone ), this probability corresponds to the mathematical formulation p(r s1 s2 s3 s4 ). In this case, values for p(d j s i.alone ) must be computed when using the Bayes theorem, and these are very similar to the observed values, e.g. p(r s 1.alone ) = 0.67 (Table 1, SSIH). Shanks (1990b, exp. 3) compared the single symptom interpretation hypothesis with the delta rule explanation in one experiment. To do this, he used the conditional probabilities in a way that not only p(r s 1 ) = p(c s 1 ) = 0.50, but also p(r s 1.alone ) = p(c s 1.alone ) = Despite this change in the probabilities, which exclude the single symptom interpretation hypothesis as an explanation for the baserate neglect effect, an effect could be observed (see Shanks, 1990b, p. 232). The experiment The aim of the present study is to compare the three different explanations of the baserate neglect effect within a single experiment. Method. 60 subjects participated in this experiment, all of them students at Darmstadt University of Technology, Germany. The design exhibits two extensions compared to the experiments of Gluck and Bower (1988) and Shanks (1990b). (1) Six symptoms are used which are clustered into two sets of symptoms. The symptoms in the first set are called 1, 2, 3, the symptoms in the second set 1, 2, 3 (these are the abstract names, the concrete names used in the experiment are realistic symptoms). The symptoms 1, 2, 3 occur with two diseases: a rare disease R and a common disease C. The symptoms 1, 2, 3 occur with two other diseases: a rare disease R and a common disease C (see Table 2). The baserates used for the diseases were p(r) = p(r') = and p(c) = p(c') = For symptoms 1, 2, 3, only p(r s 1 ) = p(c s 1 ) = 0.50, but p(r s 1,alone ) > p(c s 1,alone ). For symptoms 1, 2, 3, p(s i R ) and p(s i C ) are chosen so that p(r s 1 ) = p(c s 1 ) and p(r s 1,alone ) = p(c s 1,alone ) (see Table 2). Table 2. Conditional probabilities in the present experiment. Sympt. p(s i R) p(s i R') p(s i C) p(s i C') p(r s i ) p(r' s i ) p(r s i,alone) p(r' s i,alone) s s s s s s
5 (2) The design involves two groups of subjects called four-diseases-group and two-diseases-group. They differ from each other in two respects. First, in the fourdiseases-group, the abstract diseases R, C, R, and C are assigned to four different disease names. In the two-diseases-group, the abstract diseases R and C are assigned to the same disease name, and R and C got the same disease name. Second, the conditional baserates of the diseases are different in the two groups. In the fourdiseases-group, the four disease names have the baserates mentioned above (p(r) = p(r ) = and p(c) = p(c ) = 0.375). In the two-diseases-group, there are only two disease names, and both of them have a baserate of 0.50 (since p(r) + p(c ) = 0.50 and p(r ) + p(c) = 0.50). On the 320 learning trials, the subjects saw one, two, three, or no symptom(s). It was either a combination of the symptoms 1, 2, 3 or a combination of the symptoms 1, 2, 3. They were asked to press a key according to their diagnosis, feedback was given. On the test trials, subjects were asked to rate the conditional probability of the diseases for each of the 15 possible symptom combinations. Averaged across all subjects, we get the mean ratings r p (D j sss). Predictions for the experiment. The three explanatory models make different predictions for the two sets of symptoms and the two groups. The critical symptoms are 1 and 1, therefore this explanation is restricted to these symptoms. (1) The baserate equalizing hypothesis explains the baserate neglect effect by stating that the actual baserates are equalized and these distorted baserates are put in the Bayes theorem. In the four-diseases-group, the baserates of the two rare diseases are These baserates are distorted towards 0.25, because 0.25 is the baserate which each disease should have, if all diseases have the same baserate. Therefore, a baserate neglect effect is predicted (Table 3). In the two-diseases-group, the baserates of the two diseases are already Therefore, the baserate cannot be distorted anymore, the correct baserates are put in the Bayes theorem, and no baserate neglect effect is predicted. (2) The delta rule explanation: The delta-rule predicts for all conditions a baserate neglect effect (see Table 3), which is independent of the value of the parameter β. This was verified by simulations, using a wide range of different parameters. (3) The single symptom interpretation hypothesis: For this hypothesis, the probabilities p(r s 1,alone ) are used. Therefore, the effect should arise only for symptom 1 (in both groups) and not for symptom 1 (Table 3). Table 3. Predictions and results of the present experiment. Two-diseases-group Four-diseases-group s 1 s 1 s 1 s 1 Delta rule explanation Effect Effect Effect Effect Baserate equalizing hypothesis - - Effect Effect Single symptom interpr. hyp. Effect - Effect - Results of the experiment Effect: Baserate neglect effect is predicted. -: No baserate neglect effect is predicted. 272
6 Results. Table 3 shows the results for the mean ratings r p (R s 1 ) and r p (R s 1 ). Values larger than 0.5 indicate a baserate neglect effect. A statistically significant effect for symptom 1 in both groups could be observed. For symptom 1, however, there is no baserate neglect effect. We even see a tendency towards a reversed baserate effect, i.e. a tendency towards the common disease. Discussion To summarize the results, we observed a pattern of results which is neither compatible with the delta rule network model of Gluck and Bower (1988), nor with the baserate equalizing hypothesis. The only model which predicts the data is the single symptom interpretation hypothesis (see Table 3). This is contradicting to the results obtained by Shanks (1990b), who got in a context where p(r s 1,alone ) = p(c s 1,alone ) = 0.50 a baserate neglect effect. In his study, the effect was not as pronounced as in the study by Gluck and Bower (1988). Yet in the current experiment the effect is not even small, but it is reversed. This reversion of the baserate neglect effect is quite meaningful, for it can be observed in two groups of subjects who are treated partially different. Conclusions. Research in classification learning is faced with our surprising results which support the single symptom interpretation model and not the simple network model of Gluck and Bower (1988). The failure of the delta rule explanation does not imply that connectionist models are inappropriate for classification learning tasks. More elaborate connectionist models may be compatible with a missing or even with a reversed baserate neglect effect in certain conditions. Further research is necessary, motivated by our data, in at least two regards: On the one hand, the reversed baserate neglect effect should be replicated. On the other hand, connectionist network models or other theoretical models must be modified in order to account for the pattern of results. References Gluck, M.A. & Bower, G.H. (1988). From conditioning to category learning: An adaptive network model. Journal of Experimental Psychology: General, 117, Gluck, M.A. & Bower, G.H. (1990). Component and pattern information in adaptive networks. Journal of Experimental Psychology: General, 119, Kruschke, J.K. (1996). Base rates in category learning. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22, Shanks, D.R. (1990a). Connectionism and human learning: Critique of Gluck and Bower (1988). Journal of Experimental Psychology: General, 119, Shanks, D.R. (1990b). Connectionism and the learning of probabilistic concepts. Quarterly Journal of Experimental Psychology, 42A,
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