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Online Supplement
The issue addressed in this supplement is whether the poor performance observed in the
II-4(fixed) and Diag-4 conditions may have resulted from subjects’ use of suboptimal rule-based
strategies in those conditions, rather than the use of information-integration strategies. In the
main text, we considered the II-4(fixed) condition [and also the II-4(spread) condition] by
comparing a main representative of each model class. The representative from the class of rulebased models was a particular extreme-values model that had been tested by Maddox et al.
(2004). The representative from the class of information-integration models was the baseline
version of the GCM (Equations 2-4) that served to motivate the major qualitative predictions that
were tested in the present research. Those comparisons provided no evidence that subjects used
the rule-based strategy as a basis for classification in the information-integration conditions. In
particular, the GCM provided better quantitative fits (measured using the BIC statistic) than did
the extreme-values rule model for the overwhelming majority of the individual subjects.
However, it is possible that subjects used other types of rule-based strategies (although
those other strategies seem less plausible to us than the particular extreme-values model that was
tested). To investigate that possibility, we formalized a larger set of models representing a wider
selection of rule-based strategies and fitted that larger set of models to the individual subjects’
data. We provide a full description of this larger set of rule-based models below. A problem
with this approach is that the rule-based class of models is now being allowed a large number of
representatives. If one simply selects the best-fitting rule-based model, and compares it to the fit
of a single representative from the class of information-integration models (i.e., the baseline
GCM), then the rule-based class may yield better fits simply because it is has been provided with
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far more flexibility. We expect that any good resolution of this difficulty will require extremely
sophisticated statistical and model-selection techniques that go well beyond the scope of the
present research.
As a partial solution to this type of problem, Maddox and his colleagues have often
compared the fits of multiple rule-based models to multiple information-integration models. The
most usual information-integration model that they have used as a representative is one that
assumes that subjects use diagonal linear decision boundaries to partition the space (e.g., see
Figure 3 in our main text). To give that class some flexibility, they typically fit multiple versions
of the diagonal linear-decision-boundary model. A baseline version assumes that subjects place
the boundaries in ideal-observer locations that would maximize the percentage of correct
decisions of a linear classifier. In a more flexible version, however, the slopes and y-intercepts
of the linear boundaries are allowed to vary as free parameters to produce different response
biases for each category.
In the present investigation, we followed this general procedure, except using the GCM
as our representative from the class of information-integration models instead of the diagonal
decision-boundary model.1 In particular, in addition to fitting the baseline version of the GCM
to the data (Equations 2-4), we also fitted a version that allowed different response biases for
each category. The decision rule in this extended version of the model is given by
g
æ
Bö
ç å s(i, a) + ÷ * bA
Nø
è
P(A | i) = aÎA
g
æ
Bö
åç å s(i, k) + N ÷ * bK
ø
K è kÎK
2
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where bK (0 < bK) denotes the response bias for Category K. Without loss of generality the bias
parameters can be constrained to sum to one; therefore, this extended version adds three free
parameters to the model. Although this biased-GCM must provide at least as good an absolute
fit to the data as does the baseline version (because it generalizes the baseline version with
additional free parameters), it is important to note that the BIC fit it produces can be worse than
that of the baseline model. The reason is that the BIC statistic penalizes a model for its number
of free parameters.
The class of rule-based models that we fitted to the data was expanded as follows. As
already explained in the text, for the II-4 structure the main candidate model was the particular
extreme-values model that Maddox et al. (2004) had fitted to their data. To review, using Figure
3 as a guide, according to that model the subject responds A if the line is short; responds D if the
line is long; responds B if the line is medium length and high in angle; and responds C if the line
is medium length and low in angle. This model checks first for length and only second for angle,
so we refer to it as the extreme-values(length) model. An analogous extreme-values model can
be formulated, however, by assuming that the subject checks first for angle and only second for
length. We refer to that model as extreme-values(angle). Following Maddox et al., we also
tested a conjunctive-rule model. The model assumes that the subject partitions the space by
using the boundaries illustrated for the rule-based-4 structure in our Figure 1c (although these
boundaries are clearly inappropriate for the II-4 structure). Likewise, we also formulated a
variety of rule-based models for the Diag-4 structure. These included both the extremevalues(length) and extreme-values(angle) models that we fitted to the II-4 structures. In
addition, we fitted undimensional rule-based models to those data as well. In the
unidimensional(length) model, the observer was presumed to partition the space by using the
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types of vertical decision boundaries illustrated in our Figure 5a. We also fitted an analogous
unidimensional(angle) model to the data.
The summary results of the model-fitting were as follows. (Detailed tables of model fits
to the individual subjects’ data are available from the first author upon request.) For the II4(spread) condition of Experiment 1, at least one of the versions of the GCM provided a better
BIC fit than did any of the rule-based models for 17 of the 20 subjects. For the II-4(fixed)
condition of Experiment 2, at least one of the versions of the GCM provided a better BIC fit than
did any of the rule-based models for 20 of the 24 subjects. And for the Diag-4 condition of
Experiment 3, at least one of the versions of the GCM provided a better BIC fit than did any of
the rule-based models for 17 of the 22 subjects. For completeness, we also fitted the models to
the Diag-2 condition of Experiment 3, and the GCM provided a better BIC fit than did any of the
rule-based models for 18 of 23 participants. These results are extremely similar to the ones that
we reported in the main text that considered only the baseline GCM and the extremevalues(length) model. The results provide little evidence that the poor performance in the II4(fixed) and Diag-4 conditions resulted from subjects using rule-based strategies in those
conditions.
Footnotes for Online Supplement
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Because our research was motivated around qualitative predictions derived from the
GCM, it seemed ill-advised to us to switch courses midstream and fit diagonal linear decisionbound models to the individual subjects’ data. In addition, research reported by Ashby and
Waldron (1999), Nosofsky and McKinley (1995), and Nosofsky and Stanton (2005) has
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demonstrated severe limitations of the predictions stemming from the diagonal linear-decision
boundary model, even in cases involving information-integration structures. Therefore, it
seemed ill-advised to us to use it as our representative of an information-integration model.
Additional References for Online Supplement
Ashby, F.G., & Waldron, E.M. (1999). On the nature of implicit categorization. Psychonomic
Bulletin & Review, 6, 363-378.
Nosofsky, R.M., & Stanton, R.D. (2005). Speeded classification in a probabilistic category
structure: Contrasting exemplar-retrieval, decision-boundary, and prototype models.
Journal of Experimental Psychology: Human Perception and Performance, 31, 608-629.
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