Edited by: Cheng-Ta Yang, National Cheng Kung University, Taiwan
Reviewed by: Fabian A. Soto, University of California, Santa Barbara, USA; Jonathan R. Folstein, Florida State University, USA; Mario Fific, Grand Valley State University, USA
*Correspondence: Daniel R. Little, Melbourne School of Psychological Sciences, The University of Melbourne, Parkville, Melbourne, VIC 3010, Australia e-mail:
This article was submitted to Quantitative Psychology and Measurement, a section of the journal Frontiers in Psychology.
†These authors have contributed equally to this work.
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A variety of converging operations demonstrate key differences between separable dimensions, which can be analyzed independently, and integral dimensions, which are processed in a non-analytic fashion. A recent investigation of response time distributions, applying a set of logical rule-based models, demonstrated that integral dimensions are pooled into a single coactive processing channel, in contrast to separable dimensions, which are processed in multiple, independent processing channels. This paper examines the claim that arbitrary dimensions created by factorially morphing four faces are processed in an integral manner. In two experiments, 16 participants completed a categorization task in which either upright or inverted morph stimuli were classified in a speeded fashion. Analyses focused on contrasting different assumptions about the psychological representation of the stimuli, perceptual and decisional separability, and the processing architecture. We report consistent individual differences which demonstrate a mixture of some observers who demonstrate coactive processing with other observers who process the dimensions in a parallel self-terminating manner.
Understanding how our perceptual systems process multidimensional stimuli provides fundamental insights into basic cognitive operations such as categorization (Ashby and Gott,
Although many stimulus dimensions have been studied in the information processing literature, research demonstrating the integrality of stimulus dimensions has focused primarily on the dimensions of brightness and saturation of Munsell colors for visual stimuli (Shepard and Chang,
More recently, Goldstone and Steyvers (
There are a number of converging operations suggesting that integral dimensions are processed differently from separable dimensions (Garner,
The distances between stimuli derived from proximity estimates (e.g., similarity ratings, identification confusions and so on) using multidimensional scaling (MDS) are better described by an Euclidean distance metric if the dimensions are integral but by a city-block distance metric if the dimensions are separable (Attneave,
People tend to sort integral-dimensioned stimuli based on overall similarity but separable-dimensioned stimuli based on individual dimensions (Imai and Garner,
Learning to attend to important attributes takes place more efficiently for separable-dimensioned stimuli (Shepard et al.,
Integral dimensions, but not separable dimensions, tend to interfere with each other if one of the dimensions must be ignored, but tend to facilitate one another if the dimensions are varied in a correlated manner (Lockhead,
Each of these operations suggests that integral dimensions are processed as an entire object (Lockhead,
Despite this wealth of converging operations, Cheng and Pachella (
One possible set of dimensions that might satisfy the criteria of being both integral and having no identifiable dimensional structure, are the factorially-generated morph dimensions shown in Figure
Goldstone and Steyvers (
Despite the large number of converging operations to identify integrality, we argue that these operations are, in fact, somewhat equivocal with regard to the actual theoretical mechanism underlying the processing of integral dimensions. For example, there have been suggestions that integrality is a continuum from completely integral to completely separable (Torgenson,
Furthermore, some converging operations, such as finding slower RTs in Garner's (
Determining whether the arbitrary morph dimensions are, in fact, processed coactively is a fundamental question, as a number of important learning results are predicated on this assumption (e.g., Goldstone and Steyvers,
This finding is somewhat controversial as other researchers have found that differentiation does not occur with other integral dimensioned stimuli (e.g., “blobs” created via the convolution of sine waves in polar coordinates varying in amplitude and frequency; Op de Beeck et al.,
In this paper, we investigate whether the morph stimuli used to demonstrate differentiation (Goldstone and Steyvers,
General Recognition Theory (Ashby and Townsend,
GRT provides a theoretical unification of differing ideas about
By contrast, separability and integrality are constructs which refer to collections of stimuli. To explain,
These constructs are important and useful because they provide a quantitative framework which can be used to predict some of the different empirical operations which differentiate performance with integral and separable dimensions; though predicting the response time effects in, for instance, Garner's (
Nosofsky and Palmeri (
In summary, in the present work, we utilize the representational assumptions defined in GRT but couple these with processing-based assumptions that allow us to predict RTs for each item in the task. This is a novel departure from GRT because it allows a theoretical definition of integrality which is not based on the representation of the stimulus dimensions but on how those dimensions are processed. In the following section, we present coactivity (i.e., the pooling of information from all stimulus dimensions into a common processing channel) as a plausible theoretical definition of how integral dimensions are processed.
A novel, theoretically-driven definition of integrality can be achieved by directly contrasting the information processing of multidimensional stimuli. In particular, by using factorial experiments and analyzing full RT distributions, one can differentiate between processing which analyzes each of the dimensions independently (i.e., either in serial or in parallel) and processing which pools the dimensions together into a single processing channel (hereafter, termed
Using a combination of non-parametric analyses and parametric response time models, Little et al. (
The logical rule-based models (Fifić et al.,
The four stimuli in the upper right quadrant, which are assigned to the
Like GRT, the logical rule-based models (Fifić et al.,
The possible combinations of separate random-walk processes can be described using three mental architectures (i.e., serial, parallel, and coactive). For serial and parallel processes, two separate random walks occur, each driven by samples from each separate dimension. These independent random walks can occur in a serial or parallel fashion. In the case of a self-terminating stopping rule, the dimension that finishes first determines the final categorization decision and RT. In the case of an exhaustive stopping rule, however, final categorization decisions and RTs are determined by the output of both random walks.
In contrast to serial and parallel processing, coactive processing assumes that a single random walk model is driven by samples from a joint bivariate normal distribution on both dimensions X and Y. At each time step, a sample is drawn from the bivariate distribution representing the particular stimulus. If the sample falls in the Category A region, the model will take a step toward the decision criterion +A. However, if the sample falls in the Category B region, the random walk will take a step toward the decision criterion −B. This single, pooled random-walk process continues until one of the criteria is reached.
As described by Fifić et al. (
Piloting of the experimental stimuli revealed that most participants demonstrated a violation of stochastic dominance, even after extended categorization training. Consequently, the current experiments will not report the SFT analyses to differentiate between information processing architectures. Instead, we will only fit RT distributions to the logical-rule models, and utilize model comparison to differentiate between mental architectures. (Further information about these analyses is available from the authors upon request).
To date, a number of different dimensions and stimulus manipulations have been analyzed using this logical-rules framework. Across experiments, the largest differences in processing have been observed between separable-dimensioned and integral-dimensioned stimuli. For instance, when the stimulus dimensions were separable and located in spatially-separated locations (Fifić et al.,
To highlight the large effects of separability and integrality on processing, it is worthwhile noting that several manipulations had very little effect on processing (Fifić et al.,
In previous studies, the application of the logical rule models has always assumed perceptual independence, perceptual separability, and decisional separability. In those studies, the full RT distributions from the entire collection of stimuli from both categories could be accounted for by varying only the architecture used to determine how the information from each dimension was integrated over time. Little et al. (
Nonetheless, it is reasonable that less systematic shifts in stimulus location might require allowing for violations of perceptual separability and decisional separability. In the following, we analyze the RT distributions from individual categorization responses using the face morph stimuli shown in Figure
Finally, we also assumed that the decision boundaries might be either orthogonal to the decision axes or rotated to capture the optimal discrimination between stimuli from the target and contrast categories. Consequently, for each of the mental architectures, we tested three different sets of the assumptions about the perceptual representation:
By assuming perceptual separability (represented by using stimulus coordinates found using a constrained MDS solution) and decisional separability (orthogonal decision bounds).
By assuming violations of perceptual separability (by using an unconstrained MDS solution) and decisional separability (orthogonal decision bounds).
By assuming violations of both perceptual and decisional separability (represented by using stimulus coordinates found using an unconstrained MDS solution and by allowing optimal decision boundaries).
We examined a set of purportedly integral stimuli created from arbitrary morph dimensions. By using the conjunctive category design shown in Figure
Eight participants from the University of Melbourne community with normal or corrected-to-normal vision were randomly assigned into the
A category space was created using a field morphing technique (Steyvers,
Each participant completed a series of 1-h sessions on consecutive or near consecutive days for five sessions. At the beginning of each session, participants were shown experimental instructions, including example stimuli relevant to their condition (i.e., upright or inverted faces).
Each session consisted of 819 trials (9 practice trials and 810 experimental trials, divided into 9 blocks of 90 trials). Although each stimulus was presented 10 times during each block, presentation of stimuli was randomized. In between each block, participants were instructed to take a short break and were given feedback on their percentage accuracy. Participants advanced to the next block by pressing any button on the RT box. During each trial a fixation cross was presented for 1170 ms. After 1070 ms a warning tone was presented for 700 ms. A face was then presented and the participant was required to decide whether the face belonged to Category A or Category B. Faces were presented until a response was made. Feedback was provided only after incorrect responses; feedback “too slow” was provided for RTs greater than 5000 ms.
We ran a similarity rating study using Amazon Mechanical Turk to obtain similarity ratings for the faces shown in Figure
On each trial, a pair of stimuli was presented in the upper-left and upper-right of the screen. Subjects rated the similarity of each pair from 1, “least similar” to 8 “most similar.” Subjects were instructed to try to use the full range of ratings, and were given examples of high, medium, and low similarity pairs using a different set of upright faces before commencing the task. For each condition, there were 36 unique pairings of the 9 stimuli. Each pair was presented six times for each subject; the order of presentation was completely randomized as was the left-right presentation of each face. The experiment was self-paced.
For the categorization task, any trials with RTs less than 200 ms or greater than 3 SDs above the mean were removed from the analysis. No trials were removed using this method. The first session was considered practice and discarded from these analyses. Mean RTs and error rates for each participant are reported in Table
U1 | 846.65 | 1045.90 | 1092.50 | 1083.70 | 749.29 | 757.55 | 1034.40 | 834.62 | 739.03 |
U2 | 726.59 | 824.60 | 928.07 | 762.62 | 779.13 | 781.31 | 946.26 | 787.91 | 707.82 |
U3 | 643.29 | 812.05 | 766.35 | 741.77 | 626.60 | 627.55 | 708.08 | 604.06 | 525.17 |
U4 | 504.02 | 570.28 | 582.97 | 535.11 | 492.51 | 497.92 | 520.67 | 524.22 | 450.07 |
I1 | 820.75 | 840.61 | 888.08 | 949.38 | 910.41 | 815.71 | 858.14 | 769.13 | 746.85 |
I2 | 764.86 | 851.34 | 878.78 | 924.04 | 1036.00 | 830.62 | 752.88 | 740.32 | 685.86 |
I3 | 1362.50 | 1656.30 | 1459.90 | 1847.00 | 1534.70 | 1653.50 | 1624.00 | 1528.20 | 1551.00 |
I4 | 978.05 | 1424.90 | 1198.40 | 1394.00 | 1292.90 | 1152.80 | 1341.50 | 1376.10 | 966.11 |
U1 | 0.03 | 0.08 | 0.06 | 0.23 | 0.10 | 0.01 | 0.16 | 0.00 | 0.00 |
U2 | 0.03 | 0.18 | 0.14 | 0.20 | 0.12 | 0.04 | 0.12 | 0.18 | 0.01 |
U3 | 0.00 | 0.00 | 0.04 | 0.03 | 0.01 | 0.01 | 0.01 | 0.00 | 0.00 |
U4 | 0.02 | 0.03 | 0.12 | 0.08 | 0.04 | 0.01 | 0.09 | 0.07 | 0.00 |
I1 | 0.05 | 0.06 | 0.21 | 0.17 | 0.03 | 0.00 | 0.10 | 0.30 | 0.03 |
I2 | 0.01 | 0.02 | 0.03 | 0.10 | 0.11 | 0.01 | 0.06 | 0.02 | 0.00 |
I3 | 0.01 | 0.15 | 0.07 | 0.08 | 0.11 | 0.09 | 0.09 | 0.03 | 0.00 |
I4 | 0.00 | 0.03 | 0.05 | 0.08 | 0.04 | 0.02 | 0.09 | 0.01 | 0.01 |
U5 | 738.05 | 769.17 | 790.8 | 853.55 | 792.6 | 696.99 | 821.56 | 701.51 | 642.98 |
U6 | 842.46 | 890.03 | 962.3 | 957.15 | 867.32 | 830.68 | 866.29 | 906.27 | 777.98 |
U7 | 577.65 | 641.28 | 711.48 | 782.32 | 653.74 | 635.23 | 677.29 | 637.93 | 549.97 |
U8 | 1105.4 | 1514 | 1662.4 | 1713.8 | 1302.1 | 1097 | 1311.1 | 1643 | 983.33 |
I5 | 861.98 | 1000.6 | 1021.1 | 1056.7 | 787.89 | 717.57 | 940.36 | 1048.7 | 813.29 |
I6 | 752.23 | 916.65 | 918.9 | 1202.3 | 998.28 | 737.51 | 1020.9 | 821.31 | 664.97 |
I7 | 917.49 | 1410 | 1183.3 | 1696.5 | 1082.7 | 1188.7 | 1235.6 | 1121.1 | 951.46 |
I8 | 829.3 | 1038.3 | 1006.8 | 1140.5 | 965.36 | 1038.9 | 1016.6 | 921.6 | 910.21 |
U5 | 0.01 | 0.02 | 0.08 | 0.09 | 0.01 | 0.00 | 0.03 | 0.02 | 0.00 |
U6 | 0.00 | 0.03 | 0.12 | 0.01 | 0.02 | 0.01 | 0.11 | 0.02 | 0.00 |
U7 | 0.01 | 0.08 | 0.12 | 0.03 | 0.02 | 0.10 | 0.07 | 0.03 | 0.00 |
U8 | 0.00 | 0.04 | 0.05 | 0.01 | 0.01 | 0.02 | 0.02 | 0.02 | 0.00 |
I5 | 0.09 | 0.24 | 0.25 | 0.58 | 0.47 | 0.14 | 0.34 | 0.10 | 0.04 |
I6 | 0.07 | 0.19 | 0.26 | 0.60 | 0.47 | 0.13 | 0.14 | 0.06 | 0.01 |
I7 | 0.01 | 0.12 | 0.08 | 0.29 | 0.13 | 0.18 | 0.18 | 0.07 | 0.03 |
I8 | 0.00 | 0.29 | 0.06 | 0.40 | 0.19 | 0.09 | 0.14 | 0.21 | 0.02 |
We first sought to identify participants who utilized the entire rating scale as instructed; consequently, we computed the multinomial likelihood of the counts of each rating value 1 to 8 (i.e., across all pairs) assuming that responses were (a) generated uniformly for each rating value, (b) assuming that responses were sampled primarily from only one rating value and (c) assuming that responses were sampled primarily from only two rating values. That is, each of these assumptions was used to generate a prior probability of selecting each of the response options [e.g., (a) with equal probability for each response option, (b) with most of the probability on one response option, or (c) with most of the probability spread across two response options]. Using these prior probability distributions and a multinomial likelihood, we computed the posterior probability for each hypothesis given the observed distribution of counts across rating values, using Bayes' rule. We then removed any observer with a posterior probability less than 0.5 for the uniformly distributed rating hypothesis. This resulted in the removal of two participants from the upright condition and six participants from inverted condition
We computed the averaged similarity rating for each pair of stimuli and found the two-dimensional scaling solutions for each condition. This was done by fitting the averaged ratings using a model which assumed a negative linear relationship between the predicted similarity ratings and the Euclidean distance between the estimated coordinates. To find the best fitting coordinates, we minimized the sum-of-squared deviations between the predicted and observer ratings from 100 starting points chosen to span the coordinate space. There were 20 parameters in total (the nine coordinate values, and the slope and the intercept of the negative linear distance-to-similarity function) used to fit the 36 similarity ratings. The estimated two-dimensional-scaling solution accounted for 97 and 99% of the variance in the averaged ratings for the upright and inverted conditions, respectively. To display the scaling solutions, we first performed a Procrustes rotation (Borg and Groenen,
For each condition, we also fitted a scaling solution that constrained each of the nine co-ordinates to a 3 × 3 grid. This model only had six free parameters and allowed only the distance between values on the A,B and C,D morph dimensions to vary. This constrained scaling solution accounted for 85 and 79% of the variance in the averaged ratings for the upright and inverted conditions, respectively. As explained above, the constrained and unconstrained scaling solutions allow for the examination of whether changing the perceptual representation affects the model fitting.
Finally, we fitted additional scaling solutions that assumed city-block distance instead of Euclidean distance between the estimated coordinates. The unconstrained model accounted for 94 and 98% of the variance in the averaged ratings for the upright and inverted conditions, respectively. In contrast, the constrained model accounted for 77 and 73% of the variance in the upright and inverted conditions. As illustrated in Table
Upright | Full | 4.73 | −27.80 | 0.94 | 2.51 | 0.97 | |||
Constrained | 17.25 | −31.40 | 0.77 | 11.45 | −46.15 | 0.85 | |||
Inverted | Full | 1.00 | −66.97 | 0.98 | 0.59 | 0.99 | |||
Constrained | 12.86 | −24.99 | 0.73 | 9.88 | −34.48 | 0.79 | |||
U5 | Full | 1.06 | 10.26 | 0.82 | 1.21 | 15.01 | 0.79 | ||
Constrained | 2.92 | −3.43 | 0.50 | 2.82 | 0.52 | ||||
U6 | Full | 2.76 | −15.76 | 0.91 | 3.17 | −10.76 | 0.90 | ||
Constrained | 12.23 | −12.29 | 0.61 | 9.27 | 0.70 | ||||
U7 | Full | 6.52 | −28.17 | 0.94 | 4.44 | −41.98 | 0.96 | ||
Constrained | 21.38 | −35.57 | 0.80 | 10.10 | 0.90 | ||||
U8 | Full | 4.88 | −31.62 | 0.94 | 3.27 | −45.99 | 0.96 | ||
Constrained | 21.04 | −29.18 | 0.76 | 12.07 | 0.86 | ||||
I5 | Full | 1.37 | 0.99 | 1.55 | −78.31 | 0.98 | |||
Constrained | 11.38 | −56.72 | 0.89 | 9.65 | −62.65 | 0.90 | |||
I6 | Full | 4.57 | −27.17 | 0.94 | 2.93 | −43.22 | 0.96 | ||
Constrained | 13.85 | −37.43 | 0.81 | 9.90 | 0.86 | ||||
I7 | Full | 11.32 | −21.13 | 0.92 | 8.92 | −29.73 | 0.94 | ||
Constrained | 35.59 | −30.07 | 0.76 | 28.51 | 0.81 | ||||
I8 | Full | 3.94 | −16.67 | 0.91 | 2.93 | −27.30 | 0.94 | ||
Constrained | 11.83 | −27.29 | 0.74 | 8.26 | 0.82 |
Having established the coordinate values from the scaling analysis, we then estimated, for each model, the variances of the perceptual distributions, the decision boundaries, and the random walk parameters. For simplicity, we assumed equal variance across all levels of a given dimension, but allowed for differences in the variances between dimensions. As illustrated in Figure
We fitted three sets of models, each set containing the five possible logical-rule models, which accounted for violations of perceptual and/or decisional separability. The first set of models allowed violations of perceptual separability but maintained the assumption of decisional separability; we label this set of models MSI and DS for
We fitted the models simultaneously to the correct-RT distributions and the error rates for each item by using quantile-based maximum likelihood estimation (Heathcote et al.,
where
U1 | 290.07 | 616.04 | 471.95 | 979.8 | 380.75 | 797.41 | 308.04 | 655.97 | ||
U2 | 437.07 | 910.04 | 448.34 | 932.58 | 489.27 | 1014.4 | 443.56 | 927 | ||
U3 | 462.32 | 960.55 | 533.06 | 1102 | 452.12 | 940.13 | 441.19 | 922.27 | ||
U4 | 314.04 | 663.98 | 442.85 | 921.6 | 384.25 | 804.39 | 341.48 | 722.84 | ||
I1 | 376.03 | 787.96 | 267.22 | 570.33 | 514.06 | 1064 | 280.39 | 600.67 | ||
I2 | 400.75 | 837.4 | 256.6 | 549.09 | 531.55 | 1099 | 253.04 | 545.97 | ||
I3 | 306.93 | 649.76 | 288.92 | 613.75 | 366.95 | 769.8 | 258.42 | 556.73 | ||
I4 | 413.1 | 862.1 | 236.11 | 508.11 | 542.26 | 1120.4 | 268.18 | 576.24 | ||
U1 | 418.40 | 872.69 | 307.71 | 651.31 | 349.33 | 734.56 | 309.52 | 658.93 | ||
U2 | 486.76 | 1009.40 | 419.94 | 875.78 | 470.07 | 976.04 | 425.09 | 890.06 | ||
U3 | 579.67 | 1195.20 | 444.50 | 924.90 | 462.10 | 960.09 | 446.36 | 932.61 | ||
U4 | 387.66 | 811.23 | 263.47 | 562.84 | 396.34 | 828.59 | 344.65 | 729.19 | ||
I1 | 536.66 | 1109.20 | 435.17 | 906.25 | 639.92 | 1315.70 | 457.87 | 955.64 | ||
I2 | 616.82 | 1269.50 | 481.39 | 998.68 | 716.87 | 1469.60 | 498.94 | 1037.80 | ||
I3 | 515.23 | 1066.40 | 494.15 | 1024.20 | 561.77 | 1159.40 | 504.63 | 1049.20 | ||
I4 | 558.00 | 1151.90 | 424.44 | 884.77 | 608.31 | 1252.50 | 503.77 | 1047.40 | ||
U1 | 277.46 | 590.82 | 380.43 | 796.76 | 302.37 | 640.64 | 894.47 | 1828.84 | ||
U2 | 413.13 | 862.16 | 472.74 | 981.38 | 459.87 | 955.64 | 1026.32 | 2092.53 | ||
U3 | 373.81 | 783.52 | 541.82 | 1119.54 | 369.01 | 773.92 | 1115.98 | 2271.84 | ||
U4 | 275.87 | 587.64 | 419.63 | 875.16 | 358.00 | 751.90 | 1011.85 | 2063.59 | ||
I1 | 456.14 | 948.18 | 427.00 | 889.90 | 569.69 | 1175.28 | 586.19 | 1212.28 | ||
I2 | 610.14 | 1256.18 | 605.07 | 1246.04 | 726.03 | 1487.96 | 791.77 | 1623.42 | ||
I3 | 588.70 | 1213.30 | 585.75 | 1207.40 | 727.57 | 1491.04 | 786.34 | 1612.56 | ||
I4 | 645.95 | 1327.80 | 631.75 | 1299.40 | 817.53 | 1670.96 | 808.54 | 1656.97 |
U1 | PS and DS | ParallelST | 255.26 | 546.41 | 1.35 | 1.59 | 0.76 | 0.19 | 5 | 3 | 6.38 | 0.36 | 38.71 | ||
U2 | MSI and OP | ParallelST | 392.59 | 821.08 | 1.34 | 1.73 | 0.46 | 0.25 | 4 | 3 | 6.15 | 0.23 | 45.07 | ||
U3 | MSI and DS | Coactive | 358.91 | 753.72 | 1.40 | 1.27 | 0.19 | 0.24 | 3 | 2 | 5.58 | 0.38 | 99.09 | ||
U4 | MSI and DS | Coactive | 234.86 | 505.62 | 1.43 | 1.28 | 0.13 | 0.13 | 3 | 2 | 5.81 | 0.16 | 50.40 | ||
I1 | PS and DS | Coactive | 231.62 | 499.14 | 1.18 | 0.50 | 4.17 | 2.98 | 4 | 3 | 6.13 | 0.14 | 37.60 | ||
I2 | PS and DS | Coactive | 215.65 | 467.20 | 1.44 | 0.50 | 1.61 | 6.12 | 3 | 3 | 6.04 | 0.16 | 53.86 | ||
I3 | PS and DS | ParallelST | 258.91 | 553.72 | 1.47 | 1.51 | 2.27 | 2.78 | 6 | 7 | 6.87 | 0.49 | 15.68 | ||
I4 | MSI and DS | Coactive | 208.42 | 452.73 | 0.76 | 1.08 | 1.83 | 7.87 | 7 | 7 | 6.34 | 0.17 | 18.36 | ||
U5 | PS and DS | Coactive | 289.90 | 615.70 | 2.20 | 1.51 | 4.25 | 2.98 | 9 | 5 | 6.31 | 0.17 | 7.48 | ||
U6 | PS and DS | ParallelST | 253.97 | 543.83 | 2.50 | 2.48 | 2.02 | 0.95 | 3 | 3 | 6.09 | 0.14 | 56.98 | ||
U7 | MSI and DS | ParallelST | 244.73 | 525.37 | 1.68 | 1.57 | 1.89 | 1.57 | 8 | 8 | 5.97 | 0.02 | 9.48 | ||
U8 | MSI and DS | ParallelST | 338.97 | 713.84 | 1.66 | 1.43 | 1.53 | 1.24 | 6 | 6 | 5.69 | 0.07 | 48.81 | ||
I5 | MSI and OP | ParallelST | 304.50 | 644.89 | 1.66 | 1.59 | 2.88 | 2.89 | 5 | 6 | 6.19 | 0.09 | 18.44 | ||
I6 | MSI and DS | Coactive | 290.12 | 616.13 | 1.46 | 1.16 | 1.35 | 0.46 | 5 | 4 | 5.96 | 0.14 | 47.22 | ||
I7 | PS and DS | ParallelST | 261.63 | 559.17 | 2.50 | 2.50 | 1.98 | 1.37 | 5 | 5 | 5.93 | 0.06 | 46.72 | ||
I8 | PS and DS | ParallelST | 299.58 | 635.06 | 2.50 | 2.38 | 1.52 | 0.99 | 6 | 5 | 6.24 | 0.18 | 26.94 |
Table
Overall, there was a consistency of the best fitting model (parallel self-terminating or coactive) within each set of models. That is, we can rule out serial processing and, for the most part, any exhaustive processing, which accords with previous findings regarding integral dimensioned stimuli (Little et al.,
For the inverted condition, the coactive model was the best fitting model for all participants in the two sets of models that assume perceptual integrality (regardless of decisional separability or integrality). For the set of models that assume both PS and DS, the coactive model was the best fitting model for participant I1, I2, and I4 but the parallel self-terminating model was the best model for I3.
Examining the best model across all model sets, participants I1 (BIC = 499.14) and I2 (BIC = 467.20) demonstrated coactive processing under the assumption of PS and DS. Under the same assumptions, the parallel self-terminating model was the best model for I3 (BIC = 553.72). Finally, I4 (BIC = 452.73) demonstrated coactive processing under the assumptions of MSI and DS. The predictions of the best fitting parameters are plotted against individual RT distributions in Figure
In each of the logical rule models there are two key components which determine the types of predictions that are generated. The first component is the architecture of the model. The second component is the psychological representation of the stimuli, which can vary based on the nature of perceived similarity between each of the stimuli. For the current set of stimuli, we fitted a series of models by varying the assumption of perceptual and decisional separability. It is clear that changing these assumptions affects the best model for each participant. A benefit of the parametric approach taken here is that we are able to test these different assumptions in a systematic fashion.
Experiment 1 highlighted two important findings. First, there were individual differences in the processing of the face morph dimensions. In the general, participants in the upright and inverted conditions were best explained by either the coactive or parallel self-terminating models. Specifically, two of four participants processed the face morphs coactively in the upright condition, and three of four participants showed coactivity in the inverted condition.
Second, the best fitting model for each participant varied with changes in the perceptual representation of the stimuli. In the upright condition for example, the coactive model provided the best fit for all participants when the perceptual representation was not assumed to conform to a 3 × 3 grid-layout (see Figure
A potential caveat on this interpretation is that the scaling solution was obtained from averaged similarity ratings of online participants. Given the individual differences in processing architecture, it is highly possible that there are also individual differences in the psychological representation of the face morphs shown in Figure
Experiment 2 replicated the upright and inverted conditions of Experiment 1 with two important alterations. First, a different stimulus space was created by swapping the positions of the two of the base faces from the set used in Experiment 1. The result of this change in base faces is that all of the stimuli except for EY, LL, and EX are different in Experiment 2 than in Experiment 1 (though similar because they are comprised of the same four base faces). Second, each participant completed a session of similarity ratings following their categorization sessions. Thus, participant-specific scaling solutions were used in the computational modeling.
Eight participants from the University of Melbourne community with normal or corrected-to-normal vision were randomly assigned into the
The apparatus was identical to Experiment 1. The base faces used to create the stimulus space were also identical to those used in Experiment 1, however, the positions of base faces A and C were swapped. This led to a morph sequence between faces A and D, and B and C. This resulted in a different stimulus space, which was nonetheless similar as it comprised the same base faces (see Figure
The procedure was identical to the categorization sessions of Experiment 1. Each participant completed five 1-h sessions on consecutive or near consecutive days, and only the final four sessions of categorization were used for analysis. In order to improve overall performance accuracy, participants were first shown the entire stimulus space with decision boundaries removed and were instructed take some time to study these faces to improve their performance during the experiment.
After completing the categorization sessions, participants were asked to return for a subsequent 1 h session in which they rated the similarity of the morphed faces used in the categorization task. There were 36 unique combinations of these stimuli, which were presented to participants 20 times each. On each of the 720 trials, a fixation cross was presented for 500 ms, then one of the combinations of faces was presented (i.e., two faces appeared on the screen, one face in the center of the upper right quadrant and the other in the center of the upper left quadrant of the monitor) and participants were then asked to rate the faces on the number pad using a scale of 1–8, where 1 was least similar and 8 was most similar. The presentation order of each unique pair was counterbalanced across the 20 repetitions. Comparisons were randomized for each participant. Participants in the
For the categorization task, any trials with RTs less than 200 ms or greater than 3 SDs above the mean were removed from the analysis. This resulted in the removal of less than 1% of trials. The mean RTs and error rates are shown in Table
The scaling solutions for participants in the upright and inverted conditions are presented in Figure
Similar to Experiment 1, unconstrained and constrained models assuming city-block and Euclidian distance between the estimated coordinates were fitted for each participant. A summary of the two sets of scaling solutions is provided in Table
The model fits for each subject in the upright and inverted conditions are shown in Table
U5 | 492.62 | 1021.14 | 361.32 | 758.54 | 517.10 | 1070.09 | 409.43 | 858.75 | ||
U6 | 272.71 | 581.32 | 376.66 | 789.23 | 454.94 | 945.78 | 354.61 | 749.11 | ||
U7 | 495.81 | 1027.51 | 282.97 | 601.85 | 525.06 | 1086.02 | 402.50 | 844.89 | ||
U8 | 502.23 | 1040.37 | 368.20 | 772.31 | 491.54 | 1018.98 | 458.31 | 956.52 | ||
I5 | 472.66 | 981.22 | 657.95 | 1351.79 | 744.24 | 1524.38 | 492.54 | 1024.97 | ||
I6 | 737.07 | 1510.05 | 365.19 | 766.28 | 716.51 | 1468.91 | 533.73 | 1107.35 | ||
I7 | 412.70 | 861.30 | 518.87 | 1073.64 | 514.43 | 1064.76 | 376.34 | 792.58 | ||
I8 | 372.39 | 780.67 | 368.31 | 772.52 | 410.63 | 857.17 | 353.91 | 747.72 | ||
U5 | 552.57 | 1141.04 | 455.15 | 946.19 | 591.10 | 1210.13 | 447.10 | 934.10 | ||
U6 | 408.12 | 852.14 | 361.11 | 758.12 | 364.58 | 765.06 | 433.45 | 894.82 | ||
U7 | 294.90 | 625.71 | 531.11 | 1098.12 | 519.29 | 1066.50 | 286.60 | 613.10 | ||
U8 | 515.94 | 1067.78 | 538.03 | 1111.97 | 508.53 | 1044.99 | 406.77 | 853.43 | ||
I5 | 439.36 | 914.62 | 553.70 | 1143.29 | 602.31 | 1232.55 | 328.97 | 697.84 | ||
I6 | 737.84 | 1511.59 | 350.37 | 736.65 | 693.59 | 1415.10 | 468.98 | 977.84 | ||
I7 | 568.12 | 1172.14 | 610.76 | 1257.43 | 650.95 | 1329.82 | 474.17 | 988.22 | ||
I8 | 446.98 | 929.87 | 393.25 | 822.40 | 467.70 | 963.32 | 366.24 | 772.36 | ||
U5 | 731.59 | 1499.09 | 1055.75 | 2147.40 | 807.25 | 1650.41 | 482.32 | 1004.54 | ||
U6 | 496.00 | 1027.90 | 490.14 | 1016.19 | 570.76 | 1177.42 | 870.17 | 1780.23 | ||
U7 | 395.85 | 827.60 | 488.78 | 1013.45 | 550.07 | 1136.03 | 1000.76 | 2041.40 | ||
U8 | 507.92 | 1051.75 | 590.52 | 1216.93 | 568.15 | 1172.21 | 1288.88 | 2617.65 | ||
I5 | 455.77 | 947.44 | 512.42 | 1060.74 | 596.91 | 1229.73 | 1202.71 | 2445.32 | ||
I6 | 720.69 | 1477.27 | 860.90 | 1757.70 | 865.15 | 1766.20 | 1588.40 | 3216.69 | ||
I7 | 860.90 | 1757.70 | 803.03 | 1641.97 | 837.51 | 1710.93 | 1108.51 | 2256.90 | ||
I8 | 833.26 | 1702.42 | 784.05 | 1603.99 | 827.49 | 1690.87 | 1385.56 | 2811.00 |
Inspection of Table
Individually, participant U5 demonstrated coactive processing under all three different assumptions of perceptual representation, but the model that assumes PS and DS was the overall best fitting model (BIC = 615.70). The parallel self-terminating model best fitted U6 (BIC = 543.83) with the same assumptions of perceptual representation. The parallel self-terminating model best fitted U7 (BIC = 525.37) and U8 (BIC = 713.84) under the assumption of MSI and DS. The predictions of the best fitting models are plotted against individual RT distributions in Figure
The model fits of the inverted condition present a clear picture. The parallel self-terminating model best fitted the data for participants I5, I7, and I8 under all three different assumptions of perceptual representations. Participant I5 (BIC = 644.89) was best fitted with the assumption of MSI and OP, but participants I7 (BIC = 559.17) and I8 (BIC = 635.06) were best fitted with the assumption of PS and DS. For participant I6, the coactive model with the assumption of MSI and DS was the overall best fitting model (BIC = 616.13). The predictions of the best fitting parameters are plotted against individual RT distributions in Figure
In sum, parallel self-terminating processing was observed for three of the four participants in both the upright and inverted conditions of Experiment 2. This is in contrast to Experiment 1 in which a majority of participants demonstrated coactive processing of upright and inverted face morphs dimensions. Taken together with Experiment 1, and given the small number of observers, our conclusion is that there are individual differences in the manner in which the face morph dimensions are processed. Regardless of whether the morphs are presented in an upright or inverted fashion, processing may be coactive or parallel depending on the individual observer. Similar to Experiment 1, Experiment 2 showed that changing the assumption of the underlying perceptual representations affects the best fitting model.
In this paper, we examined processing of purportedly integral, arbitrary morph dimensions, comparing both upright and inverted face morphs. Our primary finding was that some individuals process the dimensions in a parallel self-terminating fashion and others process the dimensions coactively for both upright and inverted face morphs.
A strength of the present study is the comparison of the model fits under different assumptions of the underlying perceptual representation. The scaling solutions from both experiments reveal deviations from the 3 × 3 grid-layout outline in Figure
Overall, more participants used a coactive strategy in Experiment 1 compared to Experiment 2. There are two possible reasons for this difference. Firstly, participants may have perceived the face morphs differently since the visual angle and the face morph dimensions were altered between experiments (i.e., the position of two base faces were swapped). Secondly, model fitting for Experiment 1 utilized the averaged scaling solution of independent participants, but model fitting for Experiment 2 utilized individual scaling solutions after categorization training. In general, there is high variability in the perceptual representation of these face morphs between individuals and thus the average scaling solution may not have adequately represented the perceptual representation of each participant in Experiment 1.
The finding of individual differences in processing face morph stimuli implies that previous studies employing these stimuli on the assumption that they are processed in an integral fashion need to be interpreted with caution. On the one hand, the stimuli clearly satisfy one of the empirical operational definitions of integrality in that for most observers, the best fitting scaling metric was Euclidean. On the other hand, only half of the observers required assuming a violation of perceptual separability. Furthermore, only half of the observers were best fit by a coactive processing architecture, and of those, only two observers from Experiment 2, where individual scaling solutions were used, were found to be coactive. Consequently, the evidence that the face morph stimuli provide consistent and converging evidence of coactive processing is rather weak.
In their study of perceptual differentiation, Goldstone and Steyvers (
An alternative interpretation of our result would be to assume that differentiation is not precluded by training a category boundary on both stimulus dimensions, and that our observation that some observers processed the dimensions independent (in a parallel, self-terminating fashion) is evidence of that differentiation. In support of this idea, the MDS solution from Experiment 1, which was the only data collected prior to category learning (concerns about averaging notwithstanding; Ashby et al.,
Finally, a further caveat on the implications of the present research is that we tested a relatively small number of individuals. This is a consequence of the experimental design which necessitates collecting large numbers of observations from each observer. Nevertheless, we can clearly rule out a large number of models including all serial models and all exhaustive models. This leaves coactivity and parallel self-termination as the remaining candidate processing models for the present face morph stimuli. That we found, essentially, the same sorts of individual differences in both experiments suggests that the individual differences are real and not due to small idiosyncratic differences between subjects.
Here we have shown that stimuli which were previously thought to be integral on the basis of one empirical test of integrality, do not necessarily meet all other tests of integrality (cf. Cheng and Pachella,
Yet, one may question why additional theoretical definitions of integrality are necessary. GRT offers a theoretical definition of perceptual representation, which rigorously defines violations of perceptual independence, perceptual separability and decisional separability, so is there any need to posit coactivity as a theoretical representation for integrality? As a background consideration, it is worthwhile to note that GRT does not predict RTs without additional mechanisms, and aside from the logical rule models presented here, only the distance-from-boundary hypothesis has been applied to explain some of the empirically observable definitions of integrality (Ashby and Maddox,
There are two somewhat orthogonal ideas that might be considered when addressing the question of whether aligning integrality with coactivity is necessary. The first is that defining integrality as coactivity might confound integrality at the perceptual and decisional stages. For instance, one could imagine that perceptual separable dimensions might be pooled together at a decisional stage. While this is a conceptually possible, we do not consider this to be very plausible in the present case. This hypothesis would capture ideas present in many two-stage, salience-based models of visual search (Neisser,
A second issue arising from consideration of the mechanisms used to generate the RTs is that to the extent that integrality is aligned with the notion of holism and to what extent coactivity captures what is typically meant by that latter concept. For instance, in a task similar to the task used here, Fifić and Townsend (
Fifić and Townsend's (
Finally, although the logical rules framework that we adopt here combines many existing approaches to studying integrality and separability, it is worth considering whether some deeper theoretical insight can be used to understand the variety of converging operations. Three converging operations are worth considering: the MDS metric (Attneave,
With regard to the efficiency of selective attention, in the logical rules models, there are at least two possible ways by which selective attention might influence processing. One mechanism is to increase the processing rate of attended dimensions and decrease the rate of less attended dimensions (see for example, Nosofsky and Palmeri,
As noted in the introduction, Garner's (
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This work was supported by ARC Discovery Project Grant DP120103120. Portions of this work were completed as part of an honors project completed by the Anthea G. Blunden and a PhD project by the David W. Griffiths.
1We do not examine variance shift integrality (or other violations of perceptual separability) in this paper because when coupled with the decision boundary, the effect of changing the mean or changing the variance of a perceptual distribution in the logical rule models is to change the probability that the random walk takes a step up or down toward the +A or −B boundary. We considered it unlikely that we would be able to differentiate these two accounts using the present design and instead leave that for future research.