Rule-based generalization and peak shift in the presence of simple relational rules

After discrimination learning between two stimuli that lie on a continuum, animals typically exhibit generalization on the basis of similarity to the physical features of the stimuli, often producing a peak-shifted gradient. However, post-discrimination generalization in humans usually resembles a monotonically increasing (e.g., linear) gradient that is better characterized as following a relational rule describing the difference between the stimuli. The current study tested whether rule-based generalization could be disrupted by reducing the applicability of a relational rule on test. We compared generalization following a difficult categorization task between a group who could use their rule consistently throughout test (Group Consistent), and a group who could only apply their rule effectively on 50% of test trials and thus could only use it inconsistently (Group Inconsistent). Across two experiments, a peak shift was found in the Inconsistent group and a monotonic gradient in the Consistent group. A post-hoc sequential analysis revealed that the Inconsistent group produced both peak-shifted and monotonic gradients as a function of whether or not the relevant rule was applicable on the previous trial. Reducing the applicability of a rule on test thus appeared to lead participants to revert to generalizing on the basis of similarity. Our results suggest that humans learn about the physical features of the stimuli alongside relational rules, and that rule- and similarity-based learning can interact in determining generalization.

Note. All training stimuli values were set between the restricted minimum and maximum values while all test stimuli were allowed to vary between the extended minimum and maximum values. The base prototype values varied along the whole dimension in the High Variability group, but was restricted to the middle 52% of values in the Low Variability group. The category prototype and all test stimuli were created by multiplying the multiplier by the distortion level and adding or subtracting to the base prototype value.
For each participant, a 'base' prototype was first created which contained randomly chosen hue and size values for each of the 9 circles (in Figure 2, the base prototype would be at the intersection of the two dimensions). The color and size values for the base prototype could be any value within the respective ranges. The category prototypes (P1 and P2 in Figure 2) were then constructed from the base prototype. For each participant, one category (left/right) was randomly chosen to have larger and the other smaller circles, and one category was chosen to have bluer and the other greener circles. From the base prototype, the color (hue) and size values of all circles were distorted in opposing directions to the same degree, such that one category's size values were all larger than the other category's, and one category's hue values were all larger than the other category's (see Figure 2). The exact degree of distortion was determined by multiplying the distortion level (arbitrary level of 0.8) by the feature multiplier (see Table 1) and then adding or subtracting these values from the base prototype values to form the category prototypes.
These category prototypes then formed the basis for creating the 120 training stimuli, which all contained the same color and size values but different (randomized) location coordinates within its cell in the stimulus grid. Thus, all the category exemplars were unique but similar to each other in terms of the relevant dimensions. To make the training phase even more difficult so that participants would not immediately work out a rule on both dimensions, each training stimulus had 2/9 of its color values and 2/9 of its size values (randomly and independently selected for each stimulus) swapped with values in the other category prototype. This effectively meant that the category exemplars seen during training were more similar to each other than the Train stimuli seen on test. Randomizing the locations of the circles and swapping the color and size values served to discourage participants from focusing on a single circle in discrimination, and also added some noise to make the initial discrimination harder. For all groups, the color and size values for the category prototypes and training stimuli were restricted to the middle 52% of values (see Table 1).
This was done because we wanted to ensure that the test stimuli were more extreme than the training stimuli to allow an adequate assessment of generalization along each dimension.
The test stimuli (Train, Near1, Near2, Near3 and Far, see Figure 2 and In the Inconsistent group, participants either saw non-diagnostic information on the nonvaried dimension that was highly variable (color: some very blue and some very green circles, or size: some very large and some very small circles) or less variable (color: all circles mostly blueygreen, or size: all circles a medium size). For Experiment 1 and 2, all participants had High Variability in their base prototype, to provide the best chance of our test manipulation disrupting rule use.
The key difference between test groups was whether the non-varied dimension (*) contained information about the correct category (Consistent) or not (Inconsistent). Numbers indicate the degree of distortion from the base prototype (distortion level 0), which is the stimulus in the middle of the categories, with negative numbers indicating that the stimulus belongs to the left category and positive numbers indicating that the stimulus belongs to the right category. In Experiments 1 and 2 there was an attended and unattended dimension which could either be color or size of the circles. The key difference between Experiments 1 and 2 was that the critical test stimuli varying the attended dimension was equated between Consistent and Inconsistent groups in Experiment 2 but not in Experiment 1.
An additional pilot experiment was conducted with no instructional manipulation. This experiment was exploratory since it was unknown whether participants would easily form relational rules on both dimensions, on one of the dimensions, or neither dimension. A detailed questionnaire was included at the end of the experiment to assess which dimensions participants found useful and whether they could readily identify differences between the categories in terms of color and size.

Participants
One hundred and thirty-three University of Sydney students (M age = 20.6, SD = 4.52, 105 females) participated in this experiment in exchange for partial course credit or payment (AUD$15/hour). Participants were randomly allocated to the Consistent (n = 68) or Inconsistent group (n = 65). Participants who indicated that they were colorblind were excluded from the analyses (6 participants).

Procedure
The stimuli, apparatus, and procedure were identical to the experiments reported in the manuscript except there was no attention manipulation, no manipulation check, and the order of the 3AFC and 2AFC questions were asked in randomized order (color and then size, or size and then color).

Exclusion Criteria
To ensure that the data analysis included only participants who learned something about the categories, similar to Livesey and McLaren (2009), participants who scored less than or equal to 55% in the last half of training were excluded (36 participants, 27.1% of the sample). After applying this criterion, a total of 91 participants remained (49 in the Consistent and 42 in the Inconsistent group).

Fig A. Self-report answers from the questionnaire in the pilot experiment.
Proportion of participants who selected each option reporting when they noticed a difference between the categories for each dimension in the pilot experiment.

Fig C. Categorization accuracy (A) and typicality ratings (B) for test stimuli varying the "more attended" dimension.
Error bars represent the standard error of the mean.

Fig D. Categorization accuracy (A) and typicality ratings (B) for test stimuli varying the "less attended" dimension.
Error bars represent the standard error of the mean.