Navigating Increasing Levels of Relational Complexity: Perceptual, Analogical, and System Mappings

Abstract

Relational thinking involves comparing abstract relationships between mental representations that vary in complexity; however, this complexity is rarely made explicit during everyday comparisons. This study explored how people naturally navigate relational complexity and interference using a novel relational match-to-sample (RMTS) task with both minimal and relationally directed instruction to observe changes in performance across three levels of relational complexity: perceptual, analogy, and system mappings. Individual working memory and relational abilities were examined to understand RMTS performance and susceptibility to interfering relational structures. Trials were presented without practice across four blocks, and participants received feedback after each attempt to guide learning. Experiment 1 instructed participants to select the target that best matched the sample, whereas Experiment 2 additionally directed participants' attention to same and different relations. Participants in Experiment 2 demonstrated improved performance when solving analogical mappings, suggesting that directing attention to relational characteristics affected behavior. Higher performing participants—those with above-chance performance on the final block of system mappings—solved more analogical RMTS problems and had greater visuospatial working memory, abstraction, verbal analogy, and scene analogy scores compared to lower performers. Lower performers were less dynamic in their performance across blocks and demonstrated negative relationships between analogy and system mapping accuracy, suggesting increased interference between these relational structures. Participant performance on RMTS problems did not change monotonically with relational complexity, suggesting that increases in relational complexity places nonlinear demands on working memory. We argue that competing relational information causes additional interference, especially in individuals with lower executive function abilities.

Full text

Navigating Increasing Levels of Relational Complexity:
Perceptual, Analogical, and System Mappings
Matthew J. Kmiecik
1
, Rodolfo Perez
1
, and Daniel C. Krawczyk
1,2
Abstract
■Relational thinking involves comparing abstract relationships
between mental representations that vary in complexity; however,
this complexity is rarely made explicit during everyday comparisons.
This study explored how people naturally navigate relational
complexity and interference using a novel relational match-to-
sample (RMTS) task with both minimal and relationally directed
instruction to observe changes in performance across three levels
of relational complexity: perceptual, analogy, and system mappings.
Individual working memory and relational abilities were examined
to understand RMTS performance and susceptibility to interfering
relational structures. Trials were presented without practice across
four blocks, and participants received feedback after each attempt
to guide learning. Experiment 1 instructed participants to select
the target that best matched the sample, whereas Experiment 2
additionally directed participants’attention to same and different
relations. Participants in Experiment 2 demonstrated improved
performance when solving analogical mappings, suggesting that
directing attention to relational characteristics affected behavior.
Higher performing participants—those with above-chance perfor-
mance on the final block of system mappings—solved more analog-
ical RMTS problems and had greater visuospatial working memory,
abstraction, verbal analogy, and scene analogy scores compared to
lower performers. Lower performers were less dynamic in their
performance across blocks and demonstrated negative relation-
ships between analogy and system mapping accuracy, suggesting
increased interference between these relational structures.
Participant performance on RMTS problems did not change
monotonically with relational complexity, suggesting that in-
creases in relational complexity places nonlinear demands on
working memory. We argue that competing relational information
causes additional interference, especially in individuals with lower
executive function abilities. ■
INTRODUCTION
The foundation of human thinking and reasoning relies on
relational thought. When we think relationally, we are able
to understand and appreciate how mental representations
relate to each other despite their featural discrepancies. For
instance, consider an analogy between Ohm’slawand
water pressure. In an electrical circuit, Ohm’slaw(V=
IR)statesthatvoltage(V) is directly proportional to the
amount of current (I) given a constant resistance (R), there-
fore implying that an increase in resistance, given a constant
current, results in an increase in voltage. Similarly, the water
pressure of a constant flow of water through a hose will
increase if the hose becomes narrower. In this case, a rela-
tional mapping of concepts (water pressure-to-voltage;
water flow-to-current; hose width-to-resistance) facilitates
the understanding of their relationships, despite their fea-
tural differences (e.g., concepts like current, voltage, and
resistance are impossible to see with the naked eye). Our
ability to extract, compare, and integrate relationships
enables myriad cognitive abilities ubiquitous in daily life
(Hofstadter & Sander, 2013; Holyoak & Thagard, 1995;
Gentner, 1983) that include learning (Vendetti, Matlen,
Richland, & Bunge, 2015; Gentner, 2010; Richland, Zur, &
Holyoak, 2007; Goswami, 1992), problem solving (Gick &
Holyoak, 1980, 1983), creative thinking (Green, Kraemer,
Fugelsang, Gray, & Dunbar, 2012), and discovery (Hofstadter
& Sander, 2013; Gentner, 2002).
Analogical reasoning is a specific form of relational rea-
soning defined by the mapping of objects between knowl-
edge structures that prioritizes relationships rather than
attributes (Gentner, 1983). Object attributes and their
degree of featural similarity are allowed to vary; however,
the objects and their relations must maintain a systematic
structure in which the relations between objects are the
same. In essence, analogies involve the comparisons of
relationships—water “flowing through”ahoseislike
current “flowing through”a circuit. Analogies can also be
compared to one another. For instance, consider again
the Ohm’s law hydraulic analogy. There are often two
different forms of this analogy offered that differ in how
the water receives the energy necessary to flow through
the hose either via (1) an electric water pump or (2) poten-
tial energy given to a bucket of water raised on an elevated
surface. Despite these analogies not differing in their in-
herent relational structure, the reasoner in developing a
This article is part of a Special Focus, Relational Reasoning,
deriving from a symposium at the 2019 annual meeting of the
Cognitive Neuroscience Society, organized by Silvia Bunge and
Keith Holyoak.
1
The University of Texas at Dallas,
2
The University of Texas
Southwestern Medical Center
© 2020 Massachusetts Institute of Technology Journal of Cognitive Neuroscience 33:3, pp. 357–376
https://doi.org/10.1162/jocn_a_01618

preference for either analogy, or even merely appreciating
their relational similarities, has engaged in comparing
higher-order relations (e.g., relations of other relations).
Therefore, relational structures between comparable repre-
sentations can vary in their complexity. Understanding how
humans are able to perform relational comparisons
of increasing complexity has implications for ontogeny
(Halford, Wilson, & Phillips, 1998; Halford, 1992), phylog-
eny (Premack, 2010; Penn, Holyoak, & Povinelli, 2008), and
cognitive neuroscience (Holyoak & Kroger, 1995; Robin &
Holyoak, 1995).
Patient and fMRI research across the past two decades has
solidified the role of the pFC in integrating relational infor-
mation of increasing complexity (see Krawczyk, 2012).
Frontotemporal dementia patients with selective pFC
degeneration (frontal variant) demonstrated pronounced
deficits in integrating multiple relationships and inhibiting
distracting information compared to those with selective
anterior temporal damage (Krawczyk et al., 2008; Morrison
et al., 2004; Waltz et al., 1999). When solving nonsemantic
Raven’s Progressive Matrices (Raven, 1941), the process of
integrating increasingly complex relations in the face of dis-
tracting elements selectively activated more anterior pFC
regions, including the rostrolateral and dorsolateral pFC
(Kroger et al., 2002; Christoff et al., 2001; Prabhakaran,
Smith, Desmond, Glover, & Gabrieli, 1997). Voxel-based
morphometry and symptom lesion mapping techniques
have more specifically implicated the left rostrolateral
pFC as an essential region for relational integration
(Aichelburg et al., 2016; Urbanski et al., 2016) with addi-
tional support from both semantic (Green, Kraemer,
Fugelsang, Gray, & Dunbar, 2010; Bunge, Wendelken,
Badre, & Wagner, 2005) and nonsemantic (Volle, Gilbert,
Benoit, & Burgess, 2010; Bunge, Helskog, & Wendelken,
2009) relational mappings.
Inhibiting distracting or irrelevant information from
entering working memory is also important for successful
relational reasoning. Individuals with greater general fluid
intelligence—as measured via Raven’s Advanced
Progressive Matrices—were more successful in overcom-
ing distracting lure trials during an n-back task and exhib-
ited increased lateral pFC activation during interference
control, suggesting relational integration and inhibitory
control are closely related and rely on lateral pFC (Gray,
Chabris, & Braver, 2003). However, factorially manipulat-
ing relational complexity and the number of distracting
elements emphasized the role of the lateral frontal pole
during relational integration, whereas inhibitory control
relied more on lateral pFC regions (i.e., middle and inferior
frontal gyri; Cho et al., 2010). Together, these results sug-
gest that integrating increasingly complex relations relies
on anterior regions of pFC and competes with cognitive re-
sources in working memory, especially when encoded dis-
tracting elements need inhibition for successful relational
reasoning (see also Cho, Holyoak, & Cannon, 2007).
Halford et al. (1998) argued that working memory capacity
is defined not by the number of items requiring processing
but rather by their relational complexity. They defined rela-
tional complexity as the “number of related dimensions or
sources of variation”(p. 803), such that unary relations have
a single relational argument (e.g., fruit[apple]), binary rela-
tions have two arguments (e.g., opposite[black, white]),
ternary relations have three arguments (e.g., taller[John,
Mark, Luke]), and so on. Higher-order relations and rela-
tional structures are composed by nesting these first-order
relations as arguments in second-order relations. For
example, an analogy under this formulation is stated: same
[opposite(black, white), opposite(noisy, quiet)]. We
borrow from Robin and Holyoak (1995) to characterize
higher-order relational complex into three levels:
1
(1) Attribute/perceptual mappings contain a single dimen-
sion same(triangle, triangle)
(2) Relational/analogical mappings contain two dimensions
same[same(triangle, triangle), same(square, square)]
(3) System mappings contain three dimensions
same(same[same(triangle, triangle), same(square,
square)], same[different(circle, rectangle), different
(star, diamond)])
Studies of comparative psychology have employed match-
to-sample (MTS) tasks to probe the relational abilities of var-
ious animal species. MTS tasks present a “sample”stimulus
composed of shapes or objects, and participants/subjects are
instructed or trained to select a matching “target”stimulus
given alternative options. Identical perceptual matches
(i.e., matching A to A and not B) are easily performed by ver-
tebrates, including pigeons (e.g., Blaisdell & Cook, 2005),
monkeys (e.g., baboons; Bovet & Vauclair, 2001), crows
(Smirnova, Zorina, Obozova, & Wasserman, 2015), chimpan-
zees (Thompson & Oden, 2000; Premack, 1983), humans
(Kroger, Holyoak, & Hummel, 2004), and even the inverte-
brate honeybee (Giurfa, Zhang, Jenett, Menzel, & Srinivasan,
2001). Importantly, these species have demonstrated the
ability to perform same/different discriminations on stimuli
not presented during training, also known as transfer, sug-
gesting their ability to acquire a rudimentary conceptual un-
derstanding of “sameness”that extends beyond perceptual
features. Relational MTS (RMTS) conditions require the
selection of targets that are relationally similar to the sample.
Analogical matches (i.e., matching AA to BB and not CD)
present a critical condition where many animals—other
than chimpanzees, crows, and humans—fail to demon-
strate relational abilities (Holyoak & Thagard, 1995), such
as rhesus monkeys (Flemming, Beran, Thompson,
Kleider, & Washburn, 2008). Only after intensive language
training using a symbol-based system do chimpanzees
demonstrate analogical abilities (Premack, 1983); however,
hooded crows have remarkably demonstrated spontaneous
analogical transfer during RMTS (Smirnova et al., 2015).
Simply put, RMTS tasks have demonstrated that no other
animal species, other than humans, can produce complex
relational solutions as easily with minimal instruction or
training. Whether analogical ability is a uniquely human
358 Journal of Cognitive Neuroscience Volume 33, Number 3

ability is highly debated (see Premack, 2010; Penn et al.,
2008) and is not the focus of this investigation.
Rather, we were interested in the relational abilities of hu-
mans that extend beyond analogical matches that Holyoak
and Thagard (1995) surmised as exceeding the limit of chim-
panzee relational abilities. Comparing the relations between
relations, defined above as a system mapping, requires
thinking about second-order relations, in all likeliness a un-
ique human ability. Only one study to date has specifically
addressed human RMTS performance across the three levels
of increasing relational complexity: perceptual, analogical,
and system mappings. Kroger et al. (2004) demonstrated
that increases in relational complexity increased processing
demandsasmeasuredviaRTsforcorrectlysolvedproblems.
This pattern of results was seen even after controlling for
working memory demands and iconic memory effects.
In the interest of isolating processing demands in work-
ing memory, Kroger et al. (2004) carefully trained partici-
pants on the relational structure in each condition. We
developed a novel task that removed this training require-
ment to observe the unstructured learning rates of RMTS
problems with increasing relational complexity and inter-
ference. RMTS problem stimuli are composed of fractal-
like patterns to reduce the influence of semantic processing.
Participants were not instructed on how to solve the
problems but received feedback after each attempt to facil-
itate learning. Because of the increased working memory
demands imposed with increasing relational complexity,
as well as interference across differing relational structures,
we hypothesized that increases in relational complexity
would negatively affect participant accuracy. Given that
participants received feedback after each trial, we also
hypothesized that participants would learn the relational
structure of each condition at different rates depending
on their relational complexity. Therefore, we predicted
that participants would learn the perceptual matches
fastest, followed by analogical matches, and then system
mappings. In the pursuit of these ideas, two experiments
were conducted to further understand factors that con-
tribute to variability, as well as improvements, in participant
performance when solving RMTS problems.
EXPERIMENT 1
Methods
Participants
Thirty-nine undergraduate students attending The University
of Texas at Dallas participated in the experiment in exchange
for course credit. One participant was excluded from anal-
yses because of extremely long RTs (> 10 sec on average)
compared with the rest of the sample; therefore, 38 partic-
ipants were included in the analyses presented below (age:
M= 20.70 years, SD =3.14years;14men;threeleft-
handed; education: M= 14.90 years, SD =2.17years).
All participants gave informed consent, and experimental
procedures were carried out in accordance with the
Declaration of Helsinki and approved by the university’s
institutional review board.
Materials
RMTS task. We developed a novel RMTS task that uniquely
features custom-generated fractal-like images presented
as stimuli with no practice and limited instruction. Each
RMTS problem was composed of nonsemantic fractal-like
patterns that were generated using a modified algorithm
based on the work of Miyashita, Higuchi, Sakai, and Masui
(1991) in MATLAB (R2014b; algorithm and task code is avail-
able in an online GitHub repository: https://github.com
/mkmiecik14/fractal-rmts). Four hundred ninety-five unique
fractal-like images were created such that each problem pre-
sented new stimuli that were never repeated throughout the
experiment. The fractal-like images were randomly colored
across the grayscale spectrum to provide contrast among
overlapping elements. To reduce the influence of semantic
representations, each problem was visually inspected to
ensure stimuli did not resemble shapes seen in everyday life.
The generated fractal-like images were used to create
three problem types: perceptual, analogical, and system
mapping. Each problem type was constructed using three
2 × 2 configurations of fractal images placed against a
black background. A single sample set was placed above
two target sets that were separated by a white dotted line
resembling the traditional format of an MTS task (see
Figure 1A). Matches between sample and target sets were
determined using only two relationships of “same”and
“different.”The three problem types were identical in their
stimulus presentation but differed in the relational com-
plexity required to select the correct target.
Correct responses to perceptual mapping problems
required participants to regard the sets as whole images
and select the target set that contained perceptually iden-
tical stimuli (i.e., relationship of sameness) represented in
the sample set.
Analogical mapping problems required participants to
regard sets as separable top and bottom pairs of fractals that
were either the same or different when evaluating the rela-
tionship between the top and bottom pairs of fractals within
each set. Within-pair fractals, or fractals that appeared on
the same row of the 2 × 2 configuration, on analogical map-
ping problems never differed and were always identical im-
ages. Correct analogical maps between sample and target
sets required identifying identical relationships shared
between top and bottom pairs of fractals (see Figure 1A).
These relationships varied between both “same”and “dif-
ferent,”thus making it possible to match based on different-
ness as well as sameness.
System mapping problems, like analogical mapping prob-
lems, required pairwise comparisons between top and bot-
tom pairs of stimuli within each set; however, within-pair
fractal stimuli now also varied between either being “same”
or “different.”Relationships now varied within pairs as well
Kmiecik, Perez, and Krawczyk 359

as between pairs. Correct responses to system mapping
problems required participants to evaluate systems of rela-
tionships, evaluating the relationships shared within and be-
tween pairs as well as across source and target sets (see
Figure 1A).
Thirty-two problems were constructed for each of the
three conditions, for a total of 96 problems. All problems
were designed such that inference of relational structures
other than those outlined above would result in chance per-
formance. In other words, vertical (i.e., column-wise compar-
isons) or diagonal comparisons within each set would not
provide sufficient information to solve the problems above
chance. Targets were counterbalanced so that each condi-
tion had an equal number of both left- and right-sided correct
responses. This resulted in two problem types for perceptual
mapping. Analogical mapping problems were represented
across four problem types because of equating left and right
target response probabilities and counterbalancing between
sample–target matching across both relations of “same”and
“different.”System mapping problems were represented
across 32 unique problem types by counterbalancing the
following aspects: (1) left and right target response probabil-
ities, (2) equating the probabilities of “same”and “different”
relational matching across samples and targets, and (3) en-
suring within-pair “same”or “different”relations appeared
equally in both top and bottom slots in each of the three sets.
Additional assessments. The RMTS task depicted above
was part of a larger study investigating reasoning abilities.
Therefore, additional tasks assessing visuospatial working
memory, relational abstraction, and analogical reasoning
were administered.
Participants’visuospatial working memory was assessed
using the Wechsler Memory Scale–Fourth Edition Symbol
Span task (Wechsler, 2009). The Symbol Span task required
participants to observe a series of nonsemantic shapes for
5 sec and then recall them, in order, on a presented sheet
of options. The task progressively increases in difficulty by
increasing the number of shapes to memorize within the
5-sec encoding period. Participants’visuospatial working
Figure 1. RMTS task conditions and trial procedure. (A) Examples of perceptual, analogical, and system mapping problems (see text for description
of counterbalancing and problem types). We intended for participants to infer relationships of same (filled arrows) and different (open circles) that
differed based on relational complexity. Correct target choices contained the same relational structure as featured in the sample; therefore, matches
were based on same (shown here in analogy and system) and different (not shown). By this definition, perceptual matches were only allowed to
match based on same. Correct and incorrect choices are denoted by green checkmarks and red Xs, respectively. (B) Participants received feedback
after each of the 96 trials and breaks between blocks every 24 trials. All trials were presented on a black background (white background shown here
for illustrative purposes).
360 Journal of Cognitive Neuroscience Volume 33, Number 3

memory ability was quantified using the Wechsler Memory
Scale–Fourth Edition Symbol Span total raw score (maxi-
mum score possible was 50).
Participants’relational abstraction ability was assessed
using the Abstraction and Working Memory (AIM) task
(Glahn, Cannon, Gur, Ragland, & Gur, 2000). In this com-
puterized task, participants were shown five shapes on-
screen that were distorted slightly to reduce verbalization:
two images at the top left, two images at the top right, and
one at the center bottom. Participants were instructed to
select via button press whether the left or right pair of
images best belonged with the bottom image. Five different
relational structures based on combinations of color and
shape determined correct matches (see Glahn et al.,
2000). In the AIM plus Memory subtest, participants were
shown the target image for 500 msec that was followed by
a blank screen for 2.5 sec before the four shapes appeared at
the top. In the AIM simple condition, this additional working
memory maintenance requirement was reduced by present-
ing the target together with the four shapes. Participants’re-
lational abstraction ability was quantified using the number
of correct responses out of a possible 20 trials for each con-
dition: (1) AIM and (2) AIM plus memory.
Participants’analogical reasoning ability was assessed
using separate verbal and scene-based tasks. The verbal
analogies task (Jones, Kmiecik, Irwin, Unsworth, &
Morrison, in preparation) presented participants with
four-term analogies in the form A:B::C:?, and participants
were instructed to select a D term out of two options pre-
sented at the bottom of a computer screen. The verbal
analogies were manipulated based on semantic distance
shared between the source (A:B) and the target (C:D) word
pair, thus creating semantically near (JANUARY:MONTH::
WINTER:SEASON) and far (LOG:FORT::CERAMIC:JAR)
analogies. Furthermore, the options of D and D’were
manipulated for distractor salience by varying the associa-
tion between C and D/D’. Incorrect choices that were
highly associated with the C term were more distracting
(BOWL:DISH::SPOON:SILVERWARE/FORK) than those
with a lower association (OATMEAL:COOKIE::BANANA:
MUFFIN/KIWI). We quantified verbal analogy performance
as overall task accuracy as percentage correct out of 60 total
trials collapsing across semantic distance and distracter
salience conditions.
Scene analogy performance was assessed using the
Similar Situations Task (SST) developed in-house
(Martinez et al., in preparation; Kmiecik, Schauer,
Martinez, & Krawczyk, 2016). Briefly, participants were
shown 48 line-art scene analogy problems. Each source
scene was presented for 5 sec and comprised two sets of
items (humans, animals, or objects) that interacted in dis-
tinct areas within the scene. One or two arrows directed par-
ticipants to encode and remember specific items and their
relational roles. The target scenes comprised two matching
items that interacted analogously to one set of items in the
source, whereas two distractor items interacted in a super-
ficially similar manner to the alignable items. No-Match
problems were identical to matchable problems but did
not contain an analogous match. Participants were tasked
with determining which item, if any, was in a similar situa-
tionasoneofthosepointedtointhesourcebyclickingtheir
selection on-screen. Participants were instructed to choose
“No Match”if they did not find an analogous match.
Therefore, the SST had four conditions: one-arrow match,
two-arrow match, one-arrow no match, and two-arrow no
match. We quantified scene analogy performance as overall
accuracy out of 48 trials by collapsing performance across
these conditions.
Procedure
After providing consent, participants were first seated
alone in a quiet testing room and administered the SST,
verbal analogies, AIM, and Symbol Span tasks. The comput-
erized tasks (i.e., all tasks except for the Symbol Span) were
displayed on a 17-in. Dell LCD monitor (resolution: 1280 ×
1024). Participants were then administered the RMTS task
and were instructed to place their right index and middle
fingers on the “1”and “2”keys of the keyboard number
pad, respectively. The task was administered using
E-Prime 2.0 (SP1; Schneider, Eschman, & Zuccolotto,
2012). The experimenter read the following instructions:
“In this experiment, you will be shown three sets of images.
One set of images will be shown at the top of the screen,
while the other two sets will be shown at the bottom left
and bottom right of the screen. Your job is to determine
which of the bottom two sets best matches the top set of
images.”Participants were instructed to press the “1”and
“2”keys to choose the bottom left and right images, respec-
tively, and to utilize the feedback after each trial to get as
many problems correct as possible.
Each trial began with a randomly jittered fixation cross
lasting between 500 and 1000 msec. RMTS problems
were presented and remained on-screen until a selection
was made. Feedback was presented for 500 msec as a
green checkmark for correct responses or a red “X”for
incorrect responses (see Figure 1B). A 500-msec intertrial
interval separated each trial. All 96 problems were pre-
sented in a pseudorandom order across four blocks with
24 problems per block. Each block contained an equal
number of left and right correct responses, problems
from each condition (i.e., eight), and problem types for
perceptual and analogical mapping problems. The 32 un-
ique system mapping problems appeared randomly
across the blocks and were never repeated.
Statistical analyses were performed in R (Version 3.5.3; R
Core Team, 2015) and RStudio (Version 1.1.456) using
the dplyr (Wickham, François, Henry, & Müller, 2018),
broom (Robinson, Hayes, & Couch, 2018), psych (Revelle,
2019), and lmSupport (Curtin, 2018) packages, and figures
were prepared using ggplot2 (Wickham, 2016) and patch-
work (Pederson, 2019). Participant performance was mod-
eled using multilevel modeling (see Judd, McClelland, &
Ryan, 2017). In Level 1, each participant’s performance
Kmiecik, Perez, and Krawczyk 361

was modeled individually using a priori orthogonal contrast
codes for the within-participant factors of block and condi-
tion as well as all their interactions. Because of the four
blocks, we modeled three separate block contrasts: linear
(−3/2, −1/2, 1/2, 3/2), quadratic (1/2, −1/2, −1/2, 1/2),
and cubic (−1/2, 3/2, −3/2, 1/2) changes in performance
across time/block. Because of the three different condi-
tions, we modeled two separate condition contrasts: per-
ceptual + analogy versus system mapping (perceptual =
1/3, analogy = 1/3, system = −2/3) and perceptual versus
analogy (perceptual = 1/2, analogy = −1/2, system = 0).
The contrasts were constructed to first evaluate differences
between the highest level of relational complexity (system
mappings) and conditions with lower relational complexity
(perceptual and analogical matches). Although people
solve perceptual and analogical RMTS problems easily
(Holyoak & Thagard, 1995), we were further interested in
performance differences between these conditions when
other relational structures may have interfered with learn-
ing rates. Constructing the two contrasts in this manner al-
lowed us to answer questions of theoretical interest while
maintaining orthogonality of contrasts that increased the in-
terpretability of the unstandardized regression coefficients
(i.e., weighted mean difference). The dependent variables
in Level 1 models were proportion correct rates and RTs
for correctly solved problems (correct RT). For the correct
RT models, the contrast codes stated above were reversed in
sign (i.e., multiplied by −1) to reflect predictions that better
performance was characterized by reduced RT (i.e., faster
responses). The participants’regression estimates (i.e.,
mean differences between condition contrasts) for each
contrast were used as dependent variables in Level 2 models
that estimated participant performance in each condition,
block, and their interactions (i.e., intercept term).
Results and Discussion
Participant performance on Experiment 1 was well differ-
entiated based on the condition and progress (i.e., block)
through the task. The participants learned the perceptual
matching problems the fastest with near-ceiling perfor-
mance throughout the task, whereas analogy and system
mapping problems were more slowly learned with vari-
able performance (see Figure 2). Descriptive statistics
of condition- and block-wise performance are described
in Table 1.
Multilevel modeling results demonstrated several signif-
icant within-participant effects for block, condition, and
their interactions for both proportion correct and correct
RT models (see Table 2). The participants’proportion
Figure 2. Participant solving performance across block and condition for Experiment 1 (left column) and Experiment 2 (right column) with respect
to (top row) proportion correct rates and (bottom row) RT for correct solutions (correct RT ). Transparent lines indicate individual participant
performance and are slightly jittered to reveal individual differences. Points with SEM error bars represent mean performance.
362 Journal of Cognitive Neuroscience Volume 33, Number 3

correct rates across the four blocks when collapsing across
conditions were described by both a linear increase and a
fluctuating cubic function, whereas correct RT performance
demonstrated linear, quadratic, and cubic changes. When
collapsing performance across block, participants were more
accurate and faster at solving perceptual and analogy prob-
lems (combined) compared to system mapping problems.
Furthermore, participants were more accurate and faster
at solving perceptual problems compared to analogies.
These mean differences of condition interacted with pre-
sentation block. The participants improved in their analogy
performance with both linear (proportion correct and cor-
rect RT) and quadratic (correct RT only) changes across
blocks, whereas perceptual matching performance did
not change from high-ceiling performance. In addition, a
quadratic effect of block for proportion correct rates was
observed that depended on the perceptual and analogy
(combined) versus system mapping condition contrast.
Experiment 1 results suggest markedly different perfor-
mance patterns across blocks depending on the relational
complexity of the problem. For proportion correct rates, sys-
tem mapping problems were characterized by a quadratic
function, with a decrease before an increase in performance
across time, whereas performance for solving analogies
tended to linearly increase across the task blocks.
Perceptual matching performance remained at ceiling across
all four blocks. Correct RTs mainly differentiated perceptual
from analogy performance across blocks, demonstrating
improvements in learning of analogies via faster RTs with
both linear and rapid initial improvement (i.e., quadratic)
compared to perceptual matches that were quickly learned.
Taken together, these results support our hypothesis
that increasing levels of relational complexity differentially
affect participant learning rates for relational structures.
Despite minimal instructions, participants immediately
learned the perceptual matches. Near-perfect perfor-
mance on perceptual matchings was expected and further
demonstrates the salience that perceptual “sameness”
exerts on human decision-making (Holyoak & Thagard,
1995), especially given that each perceptual RMTS trial is
test of transfer (i.e., all trials used unique, never repeated,
stimuli). Analogy and system mapping performance was
characterized by a positive linear slope and a nonlinear
quadratic function, respectively, across trial blocks. The
difference in these learning rates suggests that increasing
relational complexity does not result in monotonic changes
to working memory demand; otherwise, participant perfor-
mance across increasing levels of relational complexity
should have resulted in similar linear patterns across blocks.
We believe our experimental design introduced addi-
tional interference of relational structures by presenting
all conditions equally and randomly within each block.
Given that participants received minimal instructions
and no practice trials before RMTS task administration, it
is likely that the analogical relational structure interfered
with participants’ability to learn the system mappings.
The initial decrease in system mapping performance is
perhaps explained either by the participants’application
of the analogical relational structure to system mapping
problems or by the application of other incorrect strate-
gies. For example, the application of a vertical or diagonal,
rather than the imposed horizontal, solution strategy to
any of the three conditions would result in chance levels
of performance (see Methods). In addition, incorrectly ap-
plying a relational structure from a different condition
would also result in chance performance. Therefore, we
Table 1. RMTS Task Performance between Experiments
Measure Condition Experiment
Performance across Blocks, M (SD)
1234
Accuracy (%) Perceptual 1 95.39 (8.92) 99.67 (2.03) 98.03 (5.46) 97.04 (7.37 )
2 96.7 (7.27) 97.57 (7.46) 96.7 (10.7) 95.14 (9.74)
Analogy 1 71.71 (24.26) 83.88 (24.12) 84.21 (19.43) 85.53 (20.24)
2 61.98 (21.03) 77.6 (22.39) 83.33 (20.56) 82.12 (19.55)
System 1 61.84 (23.42) 56.91 (13.22) 54.28 (14.61) 70.39 (19.8)
2 58.85 (21.75) 52.6 (15.55) 49.48 (18.7) 63.19 (20.22)
Correct RT (msec) Perceptual 1 2890 (1422) 1871 (826) 1815 (852) 1535 (573)
2 2551 (1199) 1753 (666) 1772 (667) 1455 (529)
Analogy 1 4894 (1656) 2995 (865) 2823 (871) 2633 (687)
2 6245 (3420) 3490 (2124) 2883 (1419) 2807 (1428)
System 1 6230 (2550) 4722 (1981) 4693 (2140) 4171 (1796)
2 6039 (2974) 4700 (2576) 4678 (3032) 4347 (2755)
One participant was excluded from correct RT analysis in Experiment 2. Correct RT = RT for correct solutions.
Kmiecik, Perez, and Krawczyk 363

surmise that the additional relational interference across
conditions contributed to the nonmonotonic differences
between levels of increasing relational complexity.
EXPERIMENT 2
The system mapping problems in Experiment 1 were the
most relationally complex problems. Solving the system
mapping problems required the participants to perform
an analogy of analogies. However, we were still surprised
at the participants’rather low performance on the system
mapping problems in Experiment 1. Participants achieved
a mean accuracy of only 70.39% (SD = 19.8%) in the final
block, despite having 24 learning trials across the first
three blocks. Given the minimal instructions presented
before the task, it is likely that some participants may have
used nonrelational strategies (e.g., perceptual strategies
based on color matching or orientation of fractal-like
Table 2. Experiment 1 Regression Results
Measure Source b SS MSE F p PRE
Proportion
correct
Between-participant 0.799 24.263 0.011 2231.398 * .984
Linear Block 0.023 0.020 0.001 14.968 * .288
Quadratic Block 0.008 0.003 0.006 0.420 .521 .011
Cubic Block 0.012 0.005 0.001 6.564 .015 .151
Perceptual + Analogy vs. System 0.286 3.103 0.006 556.048 * .938
Perceptual vs. Analogy 0.162 0.997 0.031 31.816 * .462
Linear Block × Perceptual +
Analogy vs. System
* * 0.005 0.002 .964 *
Linear Bock × Perceptual vs. Analogy −0.038 0.056 0.006 8.742 .005 .191
Quadratic Block × Perceptual +
Analogy vs. System
−0.146 0.805 0.031 25.752 * .410
Quadratic Block ×
Perceptual vs. Analogy
0.028 0.030 0.019 1.551 .221 .040
Cubic Block × Perceptual +
Analogy vs. System
−0.007 0.002 0.005 0.355 .555 .009
Cubic Block × Perceptual vs. Analogy −0.006 0.001 0.004 0.411 .525 .011
Correct RT Between-participant 3439 449,525,947 938,032 479.22 * .928
Linear Block 576 12,612,017 156,241 80.72 * .686
Quadratic block −572 12,443,213 603,149 20.63 * .358
Cubic block 164 1,015,956 56,565 17.96 * .327
Perceptual + Analogy vs. System 2272 196,161,789 1,807,925 108.50 * .746
Perceptual vs. Analogy 1308 65,047,834 441,803 147.23 * .799
Linear Block × Perceptual +
Analogy vs. System
67 168,493 353,708 0.48 .494 .013
Linear Bock × Perceptual vs. Analogy 283 3,051,728 330,301 9.24 .004 .200
Quadratic Block × Perceptual +
Analogy vs. System
119 536,851 1,487,774 0.36 .552 .010
Quadratic Block ×
Perceptual vs. Analogy
−485 8,943,544 1,004,730 8.90 .005 .194
Cubic Block × Perceptual +
Analogy vs. System
51 97,759 320,330 0.31 .584 .008
Cubic Block × Perceptual vs. Analogy 55 116,768 71,226 1.64 .208 .042
All effects are the intercept term; degrees of freedom for each source and error are 1 and 37, respectively. PRE = proportional reduction in error (η
p
2
).
* Values < .001.
364 Journal of Cognitive Neuroscience Volume 33, Number 3

shapes) leading to poor performance and learning. One
method thought to foster relational reasoning and invite
deeper relational comparisons is to utilize relational lan-
guage (Gentner, 2016; Vendetti et al., 2015). Gick and
Holyoak (1980, 1983) demonstrated that spontaneous an-
alogical transfer between semantically disparate domains
is difficult but improves when reasoners are given explicit
hints to compare these domains. Although the fractal-like
stimuli are minimally semantic, using explicit relational
language in the instructions may encourage relational
comparisons and discourage more perceptual strategies.
Therefore, to improve system mapping performance,
we presented a new sample of participants with the same
task as described in Experiment 1 except for a slight in-
structional manipulation. In Experiment 2, participants
were given an additional line of instructions that read
“Be sure to attend to how the images are same or differ-
ent”before beginning the task. We hypothesized that this
subtle additional line of instructions would encourage re-
lational thinking, while reducing the use of distracting
nonrelational solving strategies. We predicted this hint
might be sufficient to raise performance in Experiment
2, especially for relationally complex problems like the
analogy and system mappings, in the form of faster learn-
ing rates, overall higher accuracies, and faster correct RTs
compared to Experiment 1.
Methods
Participants
Seventy-two undergraduate students (age: M= 21.10 years,
SD = 3.04 years; 34 men; seven left-handed, one ambidex-
trous; education: M= 14.60 years, SD =1.45years)attend-
ing The University of Texas at Dallas participated in the
experiment in exchange for course credit. One participant
was excluded from the correct RT analysis because of incor-
rectly answering all system mapping problems in Block 3;
therefore, the final sample for the proportion correct anal-
ysis was n= 72, whereas that for the correct RT analysis was
n=71. All participants gave informed consent, and exper-
imental procedures were carried out in accordance with the
Declaration of Helsinki and approved by the university’s
institutional review board.
Materials and Procedure
All materials and procedures were identical to those
described in Experiment 1 (see above) except for an addi-
tional line of instructions presented before the participants
beginning the task. The participants were read the same
instructions in the same order as described above. After
these instructions, the participants were read, “Be sure to
attend to how the images are same or different.”The task
began immediately after the participants indicated they
understood these instructions.
The same statistical procedure used to model Experiment 1
performance was replicated when estimating Experiment 2
performance. In addition, we modeled the effect of
Experiment 1 (−0.5) versus Experiment 2 (+0.5) as a
between-participant factor in Level 2 of the multilevel model
to examine whether the instructional manipulation af-
fected participant performance. The signs of these con-
trast codes were reversed (i.e., multiplied by −1) for
the correct RT analysis to reflect decreased RT associated
with better performance.
Results and Discussion
Similar to Experiment 1, the participants’performance
across blocks also depended on the relational complexity
of the problems in Experiment 2 (see Figure 2 and Table 1
for descriptive statistics) that was well described with both
linear and quadratic changes (see Table 3). In contrast to
perceptual matches, which were solved at near ceiling
throughout the experiment, analogical comparisons were
best characterized by a linear increase and quadratic
change in both proportion correct rates and correct RTs,
suggesting dramatic initial performance increases between
Blocks 1 and 2 (see Perceptual vs. Analogy contrasts in
Table 3). In comparison to perceptual and analogical com-
parisons, the more relationally complex system mapping
problems were better characterized by a quadratic function
than a linear decrease in proportion correct rates, suggest-
ing participants performed worse before improving at the
fourth block. When comparing the proportional reduction
in error (also known as η
p
2
), these results suggest that
system mapping performance was better characterized
by a quadratic change over block, whereas analogy perfor-
mance is better characterized as a linear improvement. The
results of Experiment 2 replicate those of Experiment 1
by further demonstrating that participants learn the rela-
tional structures of perceptual, analogy, and system
mapping problems at different rates when given minimal
instruction.
Furthermore, we compared whether participant perfor-
mance across Experiments 1 and 2 was modulated by
the additional line of instructions given in Experiment 2
(i.e., “Be sure to attend to how the images are same or
different.”)byadding“Experiment”as a between-
participant factor in a regression model. Participants across
Experiments 1 and 2 did not differ in age, t(108) = 0.61,
p=.55,sex,χ
2
(1) = 0.71, p= .40, and years of education,
t(108) = −0.86, p=.39.
We observed three between-participant interactions of
experiment, suggesting reliable differences in participant
performance across tasks that is likely attributable to the
instructional manipulation (multilevel modeling results
are presented in Table 4 for proportion correct and
Table 5 for correct RTs). More specifically, the effect of
experiment interacted with the linear differences between
perceptual versus analogy conditions in both proportion
correct rates and correct RTs, such that solving analogies
improved at a faster rate (i.e., steeper linear slope) when
participants were directed to attend to the relational
Kmiecik, Perez, and Krawczyk 365

properties of the problems (Experiment 2) compared to
when these additional instructions were not given
(Experiment 1;see Figure 3). We also observed a quadratic
difference between perceptual and analogy correct RTs,
suggesting that the relational hint impacted participant
solving strategies resulting in facilitated initial acquisition
of analogies compared to perceptual matches. Together,
improved proportion correct rates and faster correct RTs
suggest that our instructional manipulation may have fa-
cilitated relational comparisons by directing participants’
attentiontorelationallycompare items. However, this
interpretation is limited given the between-participant
nature of this design (e.g., Experiment 2 participants
performed worse in Block 1 so they had more opportunity
to improve). Our primary aim was to examine learning
rates of complex relational structures over time with
Table 3. Experiment 2 Regression Results
Measure Source b SS MSE F p PRE
Proportion
correct
Between-participant 0.763 41.887 0.008 5329.753 * .987
Linear Block 0.023 0.040 0.002 22.047 * .237
Quadratic Block 0.001 * 0.007 0.014 .907 *
Cubic Block 0.006 0.003 0.001 2.887 .094 .039
Perceptual + Analogy vs. System 0.304 6.637 0.011 590.282 * .893
Perceptual vs. Analogy 0.203 2.958 0.028 105.422 * .598
Linear Block × Perceptual +
Analogy vs. System
0.020 0.030 0.007 4.332 .041 .058
Linear Bock × Perceptual vs. Analogy −0.072 0.370 0.005 67.605 * .488
Quadratic Block × Perceptual +
Analogy vs. System
−0.148 1.577 0.048 33.160 * .318
Quadratic Block ×
Perceptual vs. Analogy
0.072 0.374 0.025 14.855 * .173
Cubic Block × Perceptual +
Analogy vs. System
−0.012 0.010 0.007 1.478 .228 .020
Cubic Block × Perceptual vs. Analogy −0.002 * 0.004 0.074 .786 .001
Correct RT Between-participant 3560 899,928,002 2,210,954 407.03 * .853
Linear Block 643 29,345,783 233,422 125.72 * .642
Quadratic Block −695 34,267,984 716,922 47.80 * .406
Cubic Block 147 1,524,606 75,138 20.29 * .225
Perceptual + Analogy vs. System 2072 304,678,497 2,823,922 107.89 * .607
Perceptual vs. Analogy 1973 276,523,546 2,017,098 137.09 * .662
Linear Block × Perceptual +
Analogy vs. System
−200 2,830,433 778,895 3.63 .061 .049
Linear Bock × Perceptual vs. Analogy 765 41,561,072 836,341 49.69 * .415
Quadratic Block × Perceptual +
Analogy vs. System
286 5,819,810 3,719,954 1.56 .215 .022
Quadratic Block ×
Perceptual vs. Analogy
−1099 85,766,297 2,708,282 31.67 * .311
Cubic Block × Perceptual +
Analogy vs. System
24 40,787 376,437 0.11 .743 .002
Cubic Block × Perceptual vs. Analogy 46 152,896 191,734 0.80 .375 .011
All effects are the intercept term; degrees of freedom for each source and error are 1 and 71 (proportion correct) and 1 and 70 (correct RT ),
respectively.
* Values < .001.
366 Journal of Cognitive Neuroscience Volume 33, Number 3

minimal instruction; therefore, we would argue that a
within-participant design would conflict with this aim
such that repeated exposure to the task to examine in-
structional manipulations would confound with previous
experience with the task. Although limited in inference, a
between-participant design was a necessary concession to
examine the navigation of relational structures with limited
prior experience and instruction.
When collapsing across experiments, the participants
demonstrated several repeated-measures interactions as
indicated by significant intercepts. Linear and quadratic
learning rates across blocks well described the differences
between analogy and perceptual matches for both propor-
tion correct rates and correct RTs. System mapping pro-
portion correct rates, but not correct RTs, were again
described by quadratic changes—first decrease before in-
crease in performance—compared to the combined per-
ceptual and analogy performance. These results further
suggest that relational complexity interacts with learning
rates, however, not in a monotonic pattern. System map-
pings, which are the most relationally complex, were bet-
ter characterized by initial difficulties in learning, whereas
Table 4. Regression Results for Proportion Correct Rates between Experiments
Source Term b SS MSE F p PRE
Between-participant Intercept 0.781 60.670 0.009 6823.138 * .984
Between-participant Experiment −0.036 0.033 0.009 3.693 .057 .033
Linear Block Intercept 0.023 0.053 0.002 32.495 * .231
Linear Block Experiment 0.001 * 0.002 0.010 .922 *
Quadratic Block Intercept 0.005 0.002 0.007 0.324 .570 .003
Quadratic Block Experiment −0.007 0.001 0.007 0.184 .669 .002
Cubic Block Intercept 0.009 0.008 0.001 9.279 .003 .079
Cubic Block Experiment −0.006 0.001 0.001 1.065 .304 .010
Perceptual + Analogy vs. System Intercept 0.295 8.640 0.009 928.692 * .896
Perceptual + Analogy vs. System Experiment 0.018 0.008 0.009 0.850 .359 .008
Perceptual vs. Analogy Intercept 0.182 3.308 0.029 113.350 * .512
Perceptual vs. Analogy Experiment 0.041 0.041 0.029 1.411 .238 .013
Linear Block × Perceptual +
Analogy vs. System
Intercept 0.010 0.010 0.006 1.612 .207 .015
Linear Block × Perceptual +
Analogy vs. System
Experiment 0.021 0.011 0.006 1.776 .185 .016
Linear Bock × Perceptual vs. Analogy Intercept −0.055 0.302 0.006 52.018 * .325
Linear Bock × Perceptual vs. Analogy Experiment −0.033 0.027 0.006 4.727 .032 .042
Quadratic Block × Perceptual +
Analogy vs. System
Intercept −0.147 2.144 0.042 51.061 * .321
Quadratic Block × Perceptual +
Analogy vs. System
Experiment −0.002 * 0.042 0.004 .953 *
Quadratic Block × Perceptual vs. Analogy Intercept 0.050 0.249 0.023 10.769 .001 .091
Quadratic Block × Perceptual vs. Analogy Experiment 0.044 0.048 0.023 2.093 .151 .019
Cubic Block × Perceptual +
Analogy vs. System
Intercept −0.009 0.008 0.006 1.397 .240 .013
Cubic Block × Perceptual +
Analogy vs. System
Experiment −0.005 0.001 0.006 0.101 .751 .001
Cubic Block × Perceptual vs. Analogy Intercept −0.004 0.002 0.004 0.464 .497 .004
Cubic Block × Perceptual vs. Analogy Experiment 0.004 * 0.004 0.131 .718 .001
Degrees of freedom for each source and error are 1 and 108, respectively.
*p< .001.
Kmiecik, Perez, and Krawczyk 367

analogies were characterized by both gradual (linear) and
rapid initial (quadratic) increases in performance.
PERFORMANCE TYPES
When further examining the raw data across the different
experiments (see Figure 2), we noticed that several par-
ticipants achieved rather high proportion correct rates in
the final block on system mapping problems than the
group mean suggested. The pattern of participant perfor-
mance suggested two groups of individuals: (1) those
who learned the system mappings and (2) those who
failed to learn their relational structure. To further exam-
ine this idea, we divided participants into two groups
based on their system mapping performance on the
fourth block. Those participants who correctly answered
a minimum of five Block 4 system mapping trials (i.e.,
above-chance performance) were labeled as higher
Table 5. Regression Results for Correct RTs between Experiments
Source Term b SS MSE F p PRE
Between-participant Intercept 3500 1,212,732,614 1,770,785 684.86 * .865
Between-participant Experiment −121 361,079 1,770,785 0.20 .652 .002
Linear Block Intercept 610 36,781,173 206,733 177.92 * .624
Linear Block Experiment −67 110,440 206,733 0.53 .466 .005
Quadratic Block Intercept −633 39,732,299 677,580 58.64 * .354
Quadratic Block Experiment 122 371,399 677,580 0.55 .461 .005
Cubic Block Intercept 155 2,379,436 68,716 34.63 * .244
Cubic Block Experiment 17 7130 68,716 0.10 .748 .001
Perceptual + Analogy vs. System Intercept 2172 466,991,510 2,472,595 188.87 * .638
Perceptual + Analogy vs. System Experiment 201 995,095 2,472,595 0.40 .527 .004
Perceptual vs. analogy Intercept 1641 266,595,707 1,472,369 181.07 * .629
Perceptual vs. analogy Experiment −665 10,950,917 1,472,369 7.44 .007 .065
Linear Block × Perceptual +
Analogy vs. System
Intercept −67 438,333 631,868 0.69 .407 .006
Linear Block × Perceptual +
Analogy vs. System
Experiment 266 1,754,686 631,868 2.78 .099 .025
Linear Bock × Perceptual vs. Analogy Intercept 524 27,210,489 661,355 41.14 * .278
Linear Bock × Perceptual vs. Analogy Experiment −482 5,743,519 661,355 8.68 .004 .075
Quadratic Block × Perceptual +
Analogy vs. System
Intercept 203 4,063,250 2,948,078 1.38 .243 .013
Quadratic Block × Perceptual +
Analogy vs. System
Experiment −167 693,982 2,948,078 0.24 .629 .002
Quadratic Block ×
Perceptual vs. Analogy
Intercept −792 62,121,759 2,119,203 29.31 * .215
Quadratic Block ×
Perceptual vs. Analogy
Experiment 614 9,329,818 2,119,203 4.40 .038 .040
Cubic Block × Perceptual +
Analogy vs. System
Intercept 37 138,079 357,036 0.39 .535 .004
Cubic Block × Perceptual +
Analogy vs. System
Experiment 27 17,715 357,036 0.05 .824 *
Cubic Block × Perceptual vs. Analogy Intercept 51 256,709 150,063 1.71 .194 .016
Cubic Block × Perceptual vs. Analogy Experiment 9 2017 150,063 0.01 .908 *
Units for b,SS, and MSE are in milliseconds; degrees of freedom for each source and error are 1 and 107, respectively.
* Values < .001.
368 Journal of Cognitive Neuroscience Volume 33, Number 3

performers; otherwise, participants at chance or lower
performance were characterized as lower performers.
The same statistical procedure used to model the above
Experiment 1 versus Experiment 2 differences was repli-
cated here by replacing the “Experiment”factor with a
between-participant “Performance type”factor (lower
performers = −0.5, higher performers = +0.5). Again,
the signs of these contrast codes were reversed (i.e.,
multiplied by −1) for the correct RT analysis to reflect
decreased RT associated with better performance.
Methods
Participants
From the 110 participants, 60 were classified as lower
performers (age: M=21.20years,SD =3.47years;
24 men; six left-handed, one ambidextrous; education:
M= 14.80 years, SD = 1.57 years) and 50 were classified as
higher performers (age: M= 20.60 years, SD = 2.49 years;
24 men; four left-handed; education: M= 14.60 years,
SD = 1.91 years). These two groups did not differ in age,
t(108) = 1.05, p= .30, sex, χ
2
(1) = 0.42, p= .52, and years
of education, t(108) = 0.46, p= .65. The participant who
was excluded from the correct RT analysis in Experiment 2
was a lower performer; therefore, the correct RT analyses
compared 59 lower performers to 50 higher performers,
whereas proportion correct analyses compared 60 lower
performers to 50 higher performers. Because of a computer
malfunction, a small subset of participant data was lost for
the AIM and AIM + memory (n= 3 higher performers;
n= 5 lower performers), verbal analogy (n=1lower
performer), and scene analogy (n= 1 higher performer)
assessments (see Methods under Experiment 1 for task
descriptions).
Results and Discussion
Descriptive statistics of task performance are presented
in Table 6 for performance types. Multilevel modeling re-
sults demonstrated several performance type interactions
for proportion correct rates (see Table 7) and correct RTs
(see Table 8). Although performance type was only
Figure 3. Effect of experiment on (top row) proportion correct rates and (bottom row) RT for correct solutions (correct RT ). Before the task,
Experiment 2 presented an additional line of instructions cueing participants to pay attention to how stimuli were same or different. Participants in
Experiment 2 demonstrated elevated performance as evidenced by steeper linear slopes in the analogy condition for both outcome measures. An
additional quadratic fit for correct RTs suggests this initial acquisition for analogies was more rapid in Experiment 2.
Kmiecik, Perez, and Krawczyk 369

determined by system mapping accuracy on the fourth
block, this factor differentiated performance on the entire
task such that higher performers, despite responding
902 msec slower, correctly solved 13% more RTMS problems
than lower performers. Furthermore, higher and lowerper-
formers were differentiated in RMTS analogy performance.
Perceptual matching performance was comparable
between the groups, although higher performers correctly
answered more analogy problems than lower performers
(see Figure 4). In contrast, perceptual and analogy condi-
tions did not differ between performance types for correct
RTs; rather, correct RTs differentiated system mappings
from analogy and perceptual matches (combined) such
that higher performers took longer to solve system map-
pings compared to lower performers (see Figure 4). We
interpret these effects to mean that higher performers
were generally more relationally minded than lower per-
formers and were better able to discern the difference
between analogy and system mapping relational structures
despite interference; however, this increase in accuracy for
higher performers resulted in increased RTs that likely
reflect a speed–accuracy tradeoff driven by the relational
comparisons.
Importantly, block and condition effects interacted,
resulting in performance type interactions in proportion
correct rates with linear and quadratic changes in
perceptual + analogy (combined) versus system mapping
performance (see Figure 4). When comparing the effect of
performance type to the effect of experiment (i.e., instruc-
tional manipulation), as presented above, it is clear that
performance types were stronger in differentiating perfor-
mance than the instructional manipulation presented in
Experiment 2. Higher performers—those who achieved
above-chance performance on system mapping problems
by the end of the experiment—learned the various rela-
tional structures at a different rate than lower performers.
This ability to learn the system mapping relational struc-
ture also benefited in the analogy condition, with higher
performers achieving greater solving rates than lower
performers.
When inspecting correct RTs, performance type inter-
acted with the cubic effect of block on perceptual + analogy
(combined) versus system mappings (see Figure 4). While
solving system mappings, higher performers responded
slower in Block 3 after an initial improvement with faster
correct RTs between Blocks 1 and 2. In contrast, lower
performers did not perform slower for correct problems
during Block 3 system mappings but responded increas-
ingly faster throughout the course of the experiment.
Combined with the low performance in Block 3 system
mappings (inaccurate and slow RTs), higher performers
appear affected by interference of competing relational
structures, likely the analogy structures. To further explore
this possibility, we computed Pearson correlations
with bootstrapped 95% confidence intervals (CIs) be-
tween analogy and system mapping performance across
block and performance type for proportion correct rates
and correct RTs. The presence of negative correlations
would support the idea that increases in analogy solving
would interfere and result in decreased system mapping
performance.
The correlations of higher performers for proportion
correct rates demonstrated all positive correlations (see
Figure 4): Block 1, r= .52, 95% CI [.28, .70]; Block 2, r=
.18 [−.10, .44]; Block 3, r= .5 [.25, .68]; and Block 4, r=.37
[.10, .59]. Meanwhile, lower performers demonstrated both
Table 6. RMTS Task Performance between Higher and Lower Performers
Measure Condition Performer
Proportion Correct across Blocks, M (SD), %
1234
Accuracy (%) Perceptual Lower 95.42 (8.6) 97.71 (7.8) 95 (12) 93.75 (10.92)
Higher 97.25 (6.82) 99 (3.43) 99.75 (1.77) 98.25 (5.06)
Analogy Lower 57.29 (20.1) 72.08 (22.83) 75.42 (21.21) 75.83 (21.57)
Higher 75 (21.72) 89 (19.99) 93.5 (13.18) 92.25 (12.6)
System Lower 52.92 (18.75) 50.63 (16.02) 46.04 (16.19) 49.79 (12.71)
Higher 68.25 (23.45) 58.25 (12.27) 57.25 (17.14) 84.75 (6.33)
Correct RT (msec) Perceptual Lower 2345 (1061) 1685 (668) 1663 (696) 1377 (505)
Higher 3053 (1424) 1923 (772) 1934 (755) 1609 (565)
Analogy Lower 5279 (2801) 3275 (1986) 2709 (1337) 2584 (1311)
Higher 6359 (3125) 3367 (1569) 3043 (1129) 2939 (1087)
System Lower 5221 (2695) 4135 (2397) 3505 (2346) 3475 (2418)
Higher 7150 (2629) 5384 (2185) 6074 (2536) 5243 (2155)
One lower performer was excluded from correct RT analysis.
370 Journal of Cognitive Neuroscience Volume 33, Number 3

positive and negative correlations: Block 1, r= .43 [.19,
.61]; Block 2, r=−.24 [−.47, .01]; Block 3, r=−.06
[−.31, .20]; and Block 4, r= .31 [.06, .52]. Despite the neg-
ative correlations not reaching significance (i.e., 95% CIs
crossed zero) for lower performers, we observed a trending
negative correlation for Block 2 ( p= .06) and characteris-
tically different patterns in correlations between higher and
lower performers, especially in Block 3, providing support-
ive evidence of interference in lower performers. In con-
trast, analogy and system mapping correct RTs were all
positively correlated and significant for higher performers
—Block 1, r= .46 [.20, .65]; Block 2, r= .47 [.22, .66];
Block 3, r= .58 [.36, .74]; and Block 4, r= .62 [.41, .77]
—and lower performers—Block 1, r= .52 [.31, .69];
Block 2, r= .49 [.27, .67]; Block 3, r= .68 [.52, .80]; and
Block 4, r= .77 [.64, .86]. These similar positive correla-
tions suggest that RTs, when correctly solving analogy and
system mapping RMTS problems, shared similar solving
patterns between higher and lower performers; however,
correct RTs speak less toward interference effects because
of their inclusion of only correctly solved trials and there-
fore, by definition, have fewer trials.
Table 7. Regression Results for Proportion Correct Rates between Performance Types
Source Term b SS MSE F p PRE
Between-participant Intercept 0.781 66.539 0.005 12753.797 * .992
Between-participant Performance 0.126 0.430 0.005 82.361 * .433
Linear Block Intercept 0.024 0.064 0.001 43.196 * .286
Linear Block Performance 0.024 0.015 0.001 10.305 .002 .087
Quadratic Block Intercept 0.006 0.004 0.006 0.632 .429 .006
Quadratic Block Performance 0.051 0.072 0.006 11.859 .001 .099
Cubic Block Intercept 0.008 0.007 0.001 7.968 .006 .069
Cubic Block Performance −0.001 * 0.001 0.046 .830 *
Perceptual + Analogy vs. System Intercept 0.294 9.443 0.008 1165.029 * .915
Perceptual + Analogy vs. System Performance −0.071 0.137 0.008 16.931 * .136
Perceptual vs. Analogy Intercept 0.182 3.621 0.024 147.892 * .578
Perceptual vs. Analogy Performance −0.142 0.549 0.024 22.421 * .172
Linear Block × Perceptual +
Analogy vs. System
Intercept 0.011 0.012 0.005 2.261 .136 .021
Linear Block × Perceptual +
Analogy vs. System
Performance −0.058 0.092 0.005 17.164 * .137
Linear Bock × Perceptual vs. Analogy Intercept −0.060 0.387 0.006 64.453 * .374
Linear Bock × Perceptual vs. Analogy Performance 0.014 0.005 0.006 0.911 .342 .008
Quadratic Block × Perceptual +
Analogy vs. System
Intercept −0.154 2.600 0.036 72.995 * .403
Quadratic Block × Perceptual +
Analogy vs. System
Performance −0.159 0.687 0.036 19.298 * .152
Quadratic Block × Perceptual vs. Analogy Intercept 0.057 0.355 0.024 15.100 * .123
Quadratic Block × Perceptual vs. Analogy Performance 0.006 0.001 0.024 0.039 .843 *
Cubic Block × Perceptual +
Analogy vs. System
Intercept −0.011 0.012 0.006 2.071 .153 .019
Cubic Block × Perceptual +
Analogy vs. System
Performance −0.015 0.006 0.006 1.037 .311 .010
Cubic Block × Perceptual vs. Analogy Intercept −0.004 0.001 0.004 0.383 .537 .004
Cubic Block × Perceptual vs. Analogy Performance −0.003 * 0.004 0.065 .799 .001
Degrees of freedom for each source and error are 1 and 108, respectively.
* Values < .001.
Kmiecik, Perez, and Krawczyk 371

Individuals classified as higher performers above also
performed better on separate tasks of visuospatial work-
ing memory (i.e., symbol span), abstraction (i.e., AIM and
AIM + memory), and verbal and scene-based analogical
reasoning (i.e., verbal analogies and the SST, respectively;
see Table 9). This widespread differentiation in perfor-
mance types suggests that the RMTS task used in this
investigation provided an excellent measure of reasoning
that was sensitive to individual differences and general fluid
intelligence abilities. Higher performers demonstrated
competence and understanding of system mappings
given their elevated Block 4 performance (M= 85%,
SD = 6%) compared to lower performers who hovered
around chance performance (M=50%,SD =13%).
Given higher performers additionally performed better
across all additional assessments that required attention
to and maintenance of relational information, even across
temporal delays in the case of AIM + memory and SST, it is
likely that these individuals have elevated working
memory capacities to (1) process relationally complex sit-
uations that are novel and (2) face interference. Although
hardly comprehensive, our cognitive battery contained
Table 8. Regression Results for Correct RTs between Performance Types
Source Term b SS MSE F p PRE
Between-participant Intercept 3555 1,368,415,062 1,568,305 872.54 * .891
Between-participant Performance −902 22,026,419 1,568,305 14.04 * .116
Linear Block Intercept 623 42,022,828 206,025 203.97 * .656
Linear Block Performance −83 186,285 206,025 0.90 .344 .008
Quadratic Block Intercept −661 47,315,312 668,803 70.75 * .398
Quadratic Block Performance 220 1,310,519 668,803 1.96 .164 .018
Cubic Block Intercept 161 2,803,064 58,168 48.19 * .311
Cubic Block Performance −205 1,135,748 58,168 19.53 * .154
Perceptual + Analogy vs. System Intercept 2202 524,876,801 1,938,997 270.70 * .717
Perceptual + Analogy vs. System Performance −1465 58,090,135 1,938,997 29.96 * .219
Perceptual vs. analogy Intercept 1746 329,974,807 1,572,021 209.90 * .662
Perceptual vs. analogy Performance −103 288,182 1,572,021 0.18 .669 .002
Linear Block × Perceptual +
Analogy vs. System
Intercept −117 1,486,202 632,439 2.35 .128 .021
Linear Block × Perceptual +
Analogy vs. System
Performance 250 1,693,559 632,439 2.68 .105 .024
Linear Bock × Perceptual vs. Analogy Intercept 599 38,891,035 714,306 54.45 * .337
Linear Bock × Perceptual vs. Analogy Performance −54 77,816 714,306 0.11 .742 .001
Quadratic Block × Perceptual +
Analogy vs. System
Intercept 245 6,512,447 2,909,933 2.24 .138 .020
Quadratic Block × Perceptual +
Analogy vs. System
Performance −420 4,775,496 2,909,933 1.64 .203 .015
Quadratic Block × Perceptual vs. Analogy Intercept −897 87,096,970 2,185,321 39.86 * .271
Quadratic Block × Perceptual vs. Analogy Performance 289 2,255,201 2,185,321 1.03 .312 .010
Cubic Block × Perceptual +
Analogy vs. System
Intercept 46 230,144 332,838 0.69 .408 .006
Cubic Block × Perceptual +
Analogy vs. System
Performance −310 2,606,838 332,838 7.83 .006 .068
Cubic Block × Perceptual vs. Analogy Intercept 53 306,006 148,144 2.07 .154 .019
Cubic Block × Perceptual vs. Analogy Performance −88 207,351 148,144 1.40 .239 .013
Units for b,SS, and MSE are in milliseconds; degrees of freedom for each source and error are 1 and 107, respectively.
*p< .001.
372 Journal of Cognitive Neuroscience Volume 33, Number 3

Figure 4. Effect of participant performance types on overall performance and learning rates (data shown are collapsed across both experiments).
Higher performers achieved above-chance performance during Block 4 system mappings. Top row depicts condition-wise mean performance for
(left) proportion correct rates and (right) RTs for correct solutions (correct RT ) with SEM error bars.(Center row) Block-wise proportion correct
rates were differentiated both linearly (left) and quadratically (right) between performance types when comparing perceptual and analogy conditions
(combined) to system mappings. Points indicate individual participant performance and are slightly jittered to reveal individual differences, whereas
lines indicate predicted regression fits. (Bottom left) Correct RTs for higher performers continuously fluctuated throughout the experiment (cubic
fit) with slower RTs in Block 3 despite poor performance. (Bottom right) Correlations with bootstrapped 95% CIs between analogy and system
mapping performance suggest that lower performers likely encountered interference from competing relational structures with a trending negative
correlation in Block 2 for proportion correct rates ( p= .06). Correct RT correlations were all positive and not different between performance types.
Table 9. Additional Assessment Differences between Performance Types
Measure
Performance Type
tdfpLower, M (SD) Higher, M (SD)
Symbol span 26 (7) 29 (7) 2.05 108 .04
AIM 17 (2) 18 (2) 2.11 100 .037
AIM + memory 17 (2) 18 (1) 3.14 100 .002
Verbal analogies 87% (7%) 92% (8%) 2.85 107 .005
Scene analogies (SST) 57% (19%) 69% (18%) 3.26 107 .001
Differences in degrees of freedom across assessments resulted from computer malfunction-related data loss (see Methods under Performance
Types).
Kmiecik, Perez, and Krawczyk 373

several challenging tasks that required participants to face
novel situations/stimuli to detect relational patterns and
develop strategies for accurate performance.
Conclusion
This investigation explored how humans reason through
increasingly complex relationships. Using a novel variant
of the RMTS task, we demonstrated that increasingly
complex relational structures differentially affected accu-
racy rates and correct RTs when participants were given
problems without practice, minimal instructions, and
feedback after each selection. Participants quickly
learned perceptual and analogical mappings; however,
not all participants learned the most relationally complex
structure of system mappings. To improve task perfor-
mance, we directed the participants’attention to the re-
lational properties of “sameness”and “difference”in
Experiment 2. This had a modest effect on improving
analogy performance but did not translate into improving
system mapping performance. The effect of instructions
notwithstanding, whether participants performed above
chance on system mappings in the experiments’final
block highly differentiated participant performance on
not only the RMTS task but also tasks of visuospatial
working memory, abstraction, and verbal/scene-based
analogy. Given the differentiating linear, quadratic, and
even cubic effects on the analogy and system mapping
conditions, respectively, we argue that discerning in-
creasingly complex relational structures places nonmono-
tonic demands on working memory. We believe this is
likely a function of our experimental design with partici-
pants, especially those who have lower working memory
abilities and place less emphasis on relational properties
(i.e., lower performers), needing to resolve interference
because of differences in relational structures; however,
our experiments were not designed to fully test theories
of interference in working memory. Therefore, future
studies would be well served to further explore how in-
terference competes with relational complexity in work-
ing memory.
Our two-experiment study demonstrated that perfor-
mance on a nonsemantic RMTStaskisquitevariable.
Even young college-educated people do not always effec-
tively solve the system mapping problems, despite when
provided immediate feedback on their performance. This
ability to learn and understand complex relational struc-
tures relies on a variety of cognitive abilities that support
relational thinking, including working memory and inter-
ference control. System mapping performance remains a
dividing line among primates (Penn et al., 2008; Holyoak
& Thagard, 1995), and further understanding the cogni-
tive substrates of this divide, as well as the individual dif-
ferences within humans, will aid our understanding on
the basis of relational comparison and related cognitive
abilities that contribute to intellect.
Acknowledgments
We thank David Martinez for his assistance in developing the
study protocol; Ekarin Pongpipat for his assistance with statistical
procedures; and Mina Kim, Garrett Virgin, Alex Martin, Roberto
Espinoza, Kimberly Brennan, Pranali Kamat, Brandon Pires, and
Niki Allahyari for their help with data collection.
Reprint requests should be sent to Matthew J. Kmiecik, Department
of Obstetrics and Gynecology, NorthShore University HealthSystem,
2650 Ridge Ave., Suite 1507, Evanston, IL 60201, or via e-mail:
mkmiecik@uchicago.edu.
Note
1. We recognize that there are an infinite number of levels to
higher-order relational complexity, but for simplicity and cohe-
siveness with the current investigation, we limit ourselves to
these three levels: perceptual, analogy, and system mappings.
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