In order to test the model, its computer implementation was run on the 24 random sequences of stimuli used in the psychological experiment. To mimic the effect of the introductory demonstration, the magnitudes of the anchors were initialized as follows. Anchor 9 was set to 0.800 -- a compromise value between the longest stimulus presented on the demonstration (675 pixels) and the total width of the screen (1000 pixels). Anchor 1 was initialized to 0.150 and the remaining anchors were evenly spaced in between. The other parameters were set as reported in the previous section. The model generated 24 sequences of responses which were then analyzed in the same way as the psychological data.
===================================================================== | *** HUMAN DATA *** | ***** MODEL ***** Statistic | min mean max std | min mean max std -----------------+-------------------------+------------------------- accuracy (R2) | .65 .80 .91 .070 | .65 .76 .84 .046 response std.dev | 1.20 1.96 2.44 .28 | 1.58 1.81 2.57 .21 multiple R2 | .67 .825 .93 .067 | .73 .78 .84 .031 increase in R2 | .004 .020 .061 .013 | .003 .021 .101 .023 beta for S(t) | .80 .90 .93 .041 | .80 .87 .92 .030 beta for S(t-1) |-.53 -.25 -.075 .097 |-.47 -.225 -.10 .092 beta for R(t-1) |+.15 +.30 +.55 .102 |+.13 +.25 +.525 .101 =====================================================================
Table 1. Comparison of the performance the model and the psychological data. See text for details.
Table 1 summarizes the outcome of these various analyses and compares the performance of the model with the human data. The overall accuracy of the model, operationalized as the squared correlation between stimuli and responses, ranges from 0.65 to 0.84 in the sample of 24 runs, with mean 0.76 and standard deviation 0.046. The mean R2 for the psychological data is 0.80. The degree of non-uniformity of the response distribution is reflected in the standard deviations reported in the second row of Table 1.
The remainder of Table 1 summarizes the multiple regression analysis of the response Rt on the current stimulus St, previous stimulus St-1, and previous response Rt-1. The model shows the same pattern of sequential effects as the psychological data.
Overall, the results of the simulation experiment suggest that the ANCHOR model closely matches human category-rating behavior. The biggest discrepancy between the two data sets is that the model responses are less variable. The human data, however, includes both within-subject and between-subject variability whereas the parameter settings of the model were fixed for all 24 runs. Individual differences can be modeled by using different parameter settings for the different runs.
The fact that a model fits the data indicates that its computational mechanisms hang together and can be brought in line with the empirical observations. A much more acid test for the utility of the model, however, is the degree to which it contributes to the theoretical understanding of the psychological phenomena. This closing section discusses the empirical effects in light of the Anchor model.
Nonuniformity of the Response Distribution. The model shifts the level of theorizing from aggregate scale values to individual responses. At that level of granularity the entire response distribution becomes important. Two salient features of this distribution appear to be the predominance of responses in the middle of the scale and the relative infrequency of extreme responses (Figure 2). Several factors conspire to produce such distributions in the model. The base-level learning mechanism (Eq. 6a/b) tends to differentiate the response frequencies -- more frequent anchors build up strength which in turn makes them more likely to be retrieved in the future. This makes flat distributions unstable -- small differences tend to grow. This self-reinforcing dynamics cannot go out of hand, however, because of three stabilizing factors. First and foremost, the immediate stimulus controls about 75% of the response variance and hence the responses cannot stray too much from the stimuli. Second, the correction mechanism redistributes the strength among neighboring anchors. This inhibits the formation of isolated spikes or gaps in the distribution, making the smooth unimodal shape the most stable configuration. The third stabilizing factor is related to the context effects discussed below.
Context Effect. If the stimuli control 75% of the response variance and the base-level learning tends to amplify inequalities, what happens when the stimuli are unevenly distributed themselves? It may appear that the model would produce responses that are even more skewed. This would directly contradict the finding of several studies (Parducci, 1965; Parducci & Wedell, 1986; Schifferstein & Frijters, 1992). Empirically, the responses tend to be less skewed than the stimuli, not more so. However, simulation experiments with the Anchor model that are too long to be detailed here indicate that it produces context effects consistent with the empirical data. In a nutshell, this is due to the anchor adjustment Equation 5. Because the anchors are prototypes, they tend to cluster in those regions of the magnitude continuum that are densely populated with stimuli. In turn, this reduces the skewness of the response distribution.
Sequential Effects. The positive autocorrelation between responses on successive trials is a direct consequence of the recency component of base-level activations (Eq. 6a/b). When a particular response is given, the BLA of its corresponding anchor goes up, which in turn improves the probability of retrieving the same anchor on the next trial. This produces assimilation towards the previous response. However, the increase of the activation level matters only when the two successive stimuli are similar enough (cf. Eq. 2). If they are too far apart, the response on the first trial primes an anchor that is too remote from the target on the second trial to have any influence on the final outcome. The closer the two consecutive stimuli, the stronger the assimilation.
Another sequential effect is the negative correlation between the response Rt on a given trial and the stimulus St-1 on the previous trial. Part of this effect is probably due to the perceptual subsystem and its tendency to enhance contrasts. The ANCHOR model, however, has a deliberately simplified front end that precludes any interaction between the stimuli at the perceptual level. Still, the model exhibits contrast effects due to the plasticity of anchor magnitudes (Eq. 5) and the discrepancy penalizing aspect of the partial matching mechanism (Eq. 2). The magnitude of the past stimulus St-1 is averaged into the magnitude of one of the anchors, which then serves as a proxy of that stimulus on subsequent trials. The anchor magnitudes Ai are subtracted from the new target magnitude M during the partial matching process. In other words, one of the Ai terms in Eq. 2 is positively correlated with St-1, M is positively correlated with Rt, and Ai and M are subtracted from each other. This creates negative relationship between the response Rt and the previous stimulus St-1.
Memory-Related Effects. The anchors are stored in memory and decay only slowly with time. Therefore, the mapping from stimuli to responses implicit in these anchors can influence the performance hours and even days later.
This paper argues in favor of the hypothesis that category ratings are produced in a memory-based manner. A range of category-rating phenomena seem to arise naturally from a set of principles that are also consistent with a large body of memory research. In so far as the ANCHOR model is successful, it illustrates the advantages of its integrative methodology and the utility of general architectures for cognitive modeling.
This research is supported in part by grant AFOSR F49620-99-10086 awarded to John Anderson. The authors thank Stefan Mateeff, Stephen Gotts, and two anonymous reviewers for their valuable comments on the paper. The contribution of Stefan Mateeff is especially gratefully acknowledged.
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