10.1109/TVCG.2019.2934536 1119 Research Evaluation Melissa A. Schoenlein,
Psychology and Wisconsin Institute for Discovery, University of
Wisconsin–Madison. Email<EMAIL_ADDRESS>Johnny Campos, Cognitive
Science, University of California, Merced. Email<EMAIL_ADDRESS>Kevin J. Lande, Philosophy and Centre for Vision Research, York University.
Email<EMAIL_ADDRESS>Laurent Lessard, Mechanical and Industrial Engineering,
Northeastern University. Email<EMAIL_ADDRESS>Karen B. Schloss,
Psychology and Wisconsin Institute for Discovery, University of
Wisconsin–Madison. Email<EMAIL_ADDRESS>Schoenlein et al.: Unifying
Effects of Direct and Relational Associations for Visual Communication

#  Unifying Effects of Direct and Relational Associations for Visual
Communication

Melissa A. Schoenlein  Johnny Campos  Kevin J. Lande  Laurent Lessard  and
Karen B. Schloss

###### Abstract

People have expectations about how colors map to concepts in visualizations,
and they are better at interpreting visualizations that match their
expectations. Traditionally, studies on these expectations (inferred mappings)
distinguished distinct factors relevant for visualizations of categorical vs.
continuous information. Studies on categorical information focused on direct
associations (e.g., mangos are associated with yellows) whereas studies on
continuous information focused on relational associations (e.g., darker colors
map to larger quantities; dark-is-more bias). We unite these two areas within
a single framework of assignment inference. Assignment inference is the
process by which people infer mappings between perceptual features and
concepts represented in encoding systems. Observers infer globally optimal
assignments by maximizing the “merit,” or “goodness,” of each possible
assignment. Previous work on assignment inference focused on visualizations of
categorical information. We extend this approach to visualizations of
continuous data by (a) broadening the notion of merit to include relational
associations and (b) developing a method for combining multiple (sometimes
conflicting) sources of merit to predict people’s inferred mappings. We
developed and tested our model on data from experiments in which participants
interpreted colormap data visualizations, representing fictitious data about
environmental concepts (sunshine, shade, wild fire, ocean water, glacial ice).
We found both direct and relational associations contribute independently to
inferred mappings. These results can be used to optimize visualization design
to facilitate visual communication.

###### keywords:

Visual reasoning, information visualization, colormap data visualizations,
visual encoding, color cognition

In this study, participants inferred which region of colormaps (left/right)
represented more of a domain concept (e.g., ocean water). Inferences can be
predicted by simulating assignment inference using a weighted combination of
multiple (sometimes competing) sources of “merit”: direct associations and
relational associations (dark-is-more bias).

Introduction Imagine you are interpreting a bar chart and need to infer which
colors map to which concepts represented in the chart. Now, imagine instead
interpreting a colormap data visualization111Literature on visualizing
continuous data using color has inconsistent terminology. In this paper,
“colormap” refers to a visualization that maps gradations of colors to
quantities (e.g., weather maps, neuroimaging maps, correlation matrices).
“Color scale” refers to the color gradient used to construct a colormap. and
you need to infer which colors map to which quantities represented in the
colormap.

Traditionally, researchers studying the role of color semantics for visual
communication have treated these cases as two distinct problems. One involves
mapping colors to different categories in categorical information [18, 41, 38,
36, 22] and the other involves mapping gradations of color to gradations of
quantity in continuous data [8, 21, 35, 43]. In both cases, a key goal is to
understand people’s expectations about the mappings between colors and
concepts in visualizations (called inferred mappings) because visualizations
designed to match people’s expectations are easier to interpret [26, 51, 50,
14, 18, 38, 35, 36, 43, 22].

Studies on visualizations of categorical information focus on direct
associations—the degree to which each color is associated with each concept
represented in the visualization. Methods have been developed to use direct
associations to optimize mappings between discrete colors and concepts to
facilitate visualization interpretability [18, 41, 38, 36, 22].

Studies on visualizations of continuous data focus on relational
associations—correspondences between relational properties of visual features
and relational properties of concepts. For example, observers have a dark-is-
more bias, inferring that darker colors map to larger quantities [21, 8, 4,
35, 43]. This bias is relational because it depends on relative lightness,
rather than particular colors in visualizations. Although empirical studies of
colormaps have focused on relational associations and explicitly tried to
avoid potential effects of direct associations [21, 35, 43], direct
associations likely play an important role (see Samsel et al.’s [34] intuitive
colormaps for environmental visualizations).

In this paper, we aim to unite the study of direct and relational associations
under a single framework of assignment inference. Assignment inference is the
process by which people infer mappings among visual features and concepts in
visual encoding systems [38]. Previous work on assignment inference focused on
visualizations of categorical information, showing that observers infer
optimal assignments (i.e., mappings) that maximize the total “goodness” of
each possible color-concept pair [38, 36, 22]. This “goodness” is called
merit.

We propose that assignment inference also governs inferences about the
meanings of colors in visualizations of continuous data. In testing this
possibility, our work makes the following contributions: (1) We broaden the
notion of “merit” in assignment inference to include relational associations,
and show that both relational and direct associations influence inferred
mappings for colormap visualizations. (2) We develop a method for combining
multiple (sometimes conflicting) sources of merit for simulating assignment
inference, and show that our method effectively predicts inferred mappings for
colormap visualizations.

##  1 Background

In this section, we review previous work on color semantics in information
visualization. Following tradition, we discuss effects of direct associations
for visualizations of categorical information and relational associations for
visualizations of continuous data. We will unite these two areas in Section 3
on our approach in the present study.

###  1.1 Direct associations and assignment inference

Direct associations (a.k.a. color-concept associations) are the degree to
which a color is associated with a concept. They are estimated using various
measures, including human judgments [1, 27, 39, 15, 49, 31, 36, 38, 24], image
statistics [18, 41, 19, 31], and language corpora [41, 13].

Although direct associations influence inferred mappings between colors and
concepts in visualizations of categorical information [18, 38, 36, 22], direct
associations and inferred mappings are not the same thing. Cases arise in
which people infer that a concept maps to a weakly associated color, even when
there are more strongly associated colors in a visualization. This distinction
is shown in Fig. 1A. The bipartite graph (left) represents association
strengths between each of two colors (purple and white) and each of two
concepts (trash (T) and paper (P)) in an encoding system for recycling
bins[38]. The thickness of the edges connecting colors and concepts represents
direct association strength (thicker indicates stronger association). Trash is
more associated with white than with purple (thicker edges). Yet, when asked
which colored bin is for trash (Fig. 1A right), people choose purple. Why?

Evidence suggests the reason is that people approach this problem using
assignment inference, a process that considers all colors and concepts in the
scope of the encoding system [38]. Assignment inference is analogous to
solving an assignment problem in optimization [23]. In Fig. 1A, the scope of
the encoding system includes trash and paper, even though paper was not
relevant on this particular trial. Assignment inference does not simply assign
a color to the concept with the strongest merit (for now, think of merit as
direct association strength). Instead, the process selects the combination of
color-concept pairs that maximizes total merit across all pairings. The total
merit for the T-purple/P-white assignment is greater than the alternative,
T-white/P-purple. Thus, observers infer that trash maps to purple, despite
trash being more strongly associated with white.

Fig. 1: (A) Example of dissociation between associations and inferred mappings
from [38] (figure adapted from [22]). Bipartite graph represents associations
for concepts trash (T) and paper (P) with colors purple and white (thicker
edges mean stronger associations). Observers infer trash maps to purple even
though trash is more strongly associated with white, which is the optimal
global assignment. (B) Color pairs with high vs. low semantic discriminability
for concepts watermelon (W) and mango (M) from [36]. $\Delta S$ indicates
semantic distance.

The ability to perform assignment inference depends on semantic
discriminability of the colors, given the concepts in the encoding system.
Semantic discriminability can be understood by analogy to perceptual
discriminability. Perceptual discriminability concerns how well one can
distinguish the appearance of different colors, whereas semantic
discriminability concerns how well one can distinguish the meaning of
different colors in the context of an encoding system [36, 22]. In assignment
inference, semantic discriminability is the degree to which one assignment has
greater merit than the alternative assignment(s). For example, Fig. 1B shows
color sets that differ in semantic discriminability for the concepts mango (M)
and watermelon (W) (data from [36]). The red and orange set (left) has high
semantic discriminability because the W-red/M-orange assignment has far
greater merit than the alternative. In contrast, the red and green set (right)
has low semantic discriminability because the W-red/M-green assignment is only
slightly better than the alternative. In their semantic discriminability
theory, Mukherjee et al. [22] specified constraints on the ability to design
semantically discriminable color palettes for a given set of concepts.

Semantic discriminability can be operationalized through a metric called
semantic distance ($\Delta S$) [36, 22], which uses merit to quantify the
degree to which any one assignment is better than alternative assignment(s),
while accounting for uncertainty in the system. We reproduce the details for
calculating semantic distance defined in [36] in Supplementary Material
Section S.2 of the present paper.

Simulating assignment inference. To simulate the outcome of assignment
inference, it is necessary to (a) determine which assignment is optimal
according to an assignment problem [23] and (b) estimate the probability of
inferring any one assignment over all alternative assignment(s), which is
given by semantic distance. The combination of these two pieces of information
indicates which colors observers will map to which concepts in assignment
inference, and the probability that they will infer that assignment. This
method is effective for predicting how people map colors to concepts for
visualizations of categorical information (e.g., recycling bin signage [38],
bar charts [36, 22]), although earlier work did not yet refer to “semantic
distance” by name [38]. This approach may also extend to inferences about
properties of food and beverage products based on coloring in package design
[46].

Definitions of merit for direct associations. So far, we have treated merit
merely as direct association strength. However, there are multiple methods to
specify merit for direct associations, with some more effective than others
[38]. These different methods reduce to the same outcome in encoding systems
with two concepts and two colors, such as those modeled in the present paper.
Thus, we will withhold further discussion of metrics for computing merit for
direct associations here, and we refer the interested reader to [38] and [22].

###  1.2 Relational associations

Relational associations are correspondences between relational properties of
visual features (e.g., darkness, opacity, spatial arrangement) and relational
properties of concepts (e.g., concepts of greater or lesser quantities). A
fundamental aspect of relational associations is that they are structure-
preserving. Structure preservation arises when structural properties between
visual features correspond to structural properties among the concepts to
which they are mapped [29, 42, 3, 20, 14, 11, 50].

If particular relations among visual features are salient and certain
relations among represented features are salient, then correspondences between
these relations can be exploited to constrain the number of potential inferred
mappings. For example, people are sensitive to the natural progression from
lighter to darker shades and to the natural progression from smaller to larger
quantities. Lightness can be mapped to quantities in many ways (see Fig. 2A
for four of many possibilities), but only two ways are structure preserving:
darker colors map to larger quantities (dark-more assignment) or lighter
colors map to larger quantities (light-more assignment). From the perspective
of structure preservation, both assignments are equally “good.” Any assignment
that scrambles the mapping of lightness values to quantities is not structure-
preserving and thereby is less “good.”

Yet, not all structure-preserving assignments are equally good in peoples’
inferred mappings. People have biases prioritizing one structure-preserving
assignment over another [21, 8, 35, 43], discussed below.

Dark-is-more bias. The dark-is-more bias is the expectation that darker colors
map to larger quantities (“more” of what is being measured) [21, 8, 35, 43].
People have a robust dark-is-more bias when interpreting colormaps without
legends [8, 21]222Although legends are a central part of colormap
visualization grammar, Christen et al. [7] found that journal articles often
leave out legends. Thus, studying colormaps without legends is relevant for
real-world visualizations, while also providing a direct window into people’s
inferred mappings. and with legends [35, 43]. Studying visualizations without
legends, McGranaghan [21] asked participants to interpret maps of U.S. states
colored in shades of blue, and found that participants inferred that darker
blues mapped to “more.” McGranaghan [21] was purposefully ambiguous about the
concept represented in the visualization, stating that the maps represented
different amounts of “data” to avoid effects of direct color-concept
associations. Studying visualizations with legends, Schloss et al. [35]
presented participants with colormaps representing alien animal sightings,
with the assumption that people would not have direct associations with these
novel concepts. The legend either indicated dark-more encoding (greater animal
sightings mapped to darker colors) or light-more encoding (greater animal
sightings mapped to lighter colors). Overall, participants were faster at
correctly interpreting the visualizations when legends indicated dark-more
encoding, compared to light-more encoding, providing further evidence for the
dark-is-more bias.

Opaque-is-more bias. The opaque-is-more bias is the expectation that regions
appearing more opaque represent larger quantities. This bias is only
applicable when visualizations appear to vary in opacity [35, 2], such as in
value-by-alpha maps [33]. When the opaque-is-more bias is activated, it aligns
with the dark-is-more bias on light backgrounds but conflicts with the dark-
is-more bias on dark backgrounds. Under such conflicts, the opaque-is-more
bias can cancel or even override the dark-is-more bias, leading observers to
infer that lighter colors map to larger quantities [35, 2]. When the opaque-
is-more bias is non-applicable (i.e., a visualization does not appear to vary
in opacity), the dark-is-more bias leads observers to infer that darker colors
map to larger quantities on both dark and light backgrounds [35, 2].

Hotspot-is-more bias. The hotspot-is-more bias is the expectation that spatial
regions that look like hotspots represent larger quantities in data. Hotspots
emerge in datasets like fMRI, EEG, and meteorological data, in which extreme
values are neighbored by less extreme values in concentric ring-like patterns
[40]. Sibrel et al. [43] found that the dark-is-more bias dominated over the
hotspot-is-more bias unless the hotspot was highly salient. Still, when
colormaps contained hotspots that encoded larger quantities, they were easier
to interpret when the hotspot was dark than when it was light (i.e., dark-is-
more bias)[43].

##  2 Current approach

Previous work on assignment inference focused on visualizations of categorical
information, where merit depends on direct associations [38, 22, 36]. We
propose that assignment inference also governs inferred mappings for
visualizations of continuous data, where merit may depend on both direct and
relational associations. As such, assignment inference would operate over
multiple (sometimes competing) sources of merit to determine inferred
mappings.

To test this possibility, we asked participants to infer the meanings of
colors in colormaps (Fig.  Unifying Effects of Direct and Relational
Associations for Visual Communication), and then predicted their responses
using simulations of assignment inference. We studied inferred mappings for
colormaps without legends, similar to [8, 21].333This method requires fewer
within-subject trials per condition than methods assessing inferred mappings
for colormaps with legends, which require counterbalancing legend conditions
(see [35, 43]). We assessed the proportion of times participants inferred the
darker region mapped to “more,” depending on the domain concept and the color
scales used to construct the colormap. In Fig.  Unifying Effects of Direct and
Relational Associations for Visual Communication, the domain concept is ocean
water, and participants indicated whether there was more ocean water on the
left or right of the maps. Colormaps were displayed on a white background and
avoided hotspot spatial structure to prevent cases in which the dark-is-more
bias conflicted with the opaque-is-more bias [35] and hotspot-is-more bias
[43]. In the General Discussion, we discuss extending our approach to handle
these additional biases.

Next, we consider how direct and relational associations can serve as sources
of merit for visualizations of continuous data, and how multiple sources
combine to produce inferred mappings in assignment inference.

###  2.1 Direct associations as a source of merit

Representing merit for direct associations in assignment inference for
visualizations of continuous data (Fig.  Unifying Effects of Direct and
Relational Associations for Visual Communication) is analogous to representing
merit for direct associations for visualizations of categorical information
(Fig. 1). In the examples in Fig.  Unifying Effects of Direct and Relational
Associations for Visual Communication, merit from direct associations for the
colormaps is illustrated in the bipartite graphs under the label “direct
associations.” In the bipartite graphs, circles represent the endpoint
concepts (more ocean water; +O, and less ocean water; -O) and squares
represent the endpoint colors of color scales used to create the colormaps.
Edge thickness represents association strength between each endpoint color and
concept. From the perspective of merit from direct associations alone,
assignment inference simulations for the colormaps in Fig.  Unifying Effects
of Direct and Relational Associations for Visual Communication predict that
more ocean water should map to darker blues in the top row and should map to
lighter blues in the bottom row.

Although the colormaps represent continuous data (more vs. less ocean water)
with a continuous gradation of color, we simplify the assignment problem by
focusing on only the the endpoint concepts and endpoint colors. As described
in Section 1.1, merit for direct associations can be computed in multiple
ways, but they simplify to the same outcome when there are two colors and two
concepts [38]. By limiting our simulations to the two endpoint colors and two
endpoint concepts, we can think about merit for direct color-concept
associations simply as association strength. This simplification assumes that
colors between the endpoints vary monotonically in association strength with
the domain concept (e.g., ocean water in Fig.  Unifying Effects of Direct and
Relational Associations for Visual Communication).

###  2.2 Relational associations as a source of merit

To consider how relational associations can be represented as sources of merit
in assignment inference for visualizations of continuous data, we first turn
to Figs. 2B-C. In these bipartite graphs, edges connect each possible color
(shades from white to black) to each possible concept (numeric values from 1
to 4). As indicated in Fig. 2B, only two possible sets of edges are structure-
preserving with respect to the natural orderings of quantity and lightness:
the set representing dark-more assignment (colored black) and the set
representing light-more assignment (colored blue). Edges within each
structure-preserving assignment receive more merit than edges that are not
structure-preserving (colored gray), assuming that each set of structure-
preserving edges is bound together (e.g., all blue or all black edges) and
never a mix (e.g. some blue and some black edges). Based on structure
preservation alone, dark-more and light-more assignments have equal merit, and
thus should not be semantically discriminable.

Fig. 2:  Illustrations of structure preservation. (A) A colormap assigning
lightness (light to dark) to quantity (1-4), with legends specifying
structure-preserving assignments (natural progressions of lightness correspond
to a natural ordering of quantities) vs. not structure-preserving assignments
(assignment of lightness to quantity is scrambled). Bipartite graphs can code
merit in terms of (B) structure preservation and (C) structure preservation
plus the dark-is-more bias.

Fig. 2C shows merit from the combination of structure preservation and the
dark-is-more bias. The dark-is-more bias places additional merit on structure-
preserving edges representing dark-more assignment (thicker edges in Fig. 2C).
With greater merit on the dark-more assignment than light-more assignment,
these two structure preserving assignments should be semantically
discriminable. Given that treating the dark-is-more bias as a source of merit
also implies structure preservation, here forward, we focus on the dark-is-
more bias.

Fig.  Unifying Effects of Direct and Relational Associations for Visual
Communication shows the dark-is-more bias represented as a source of merit for
the example colormaps about ocean water. From the perspective of merit from
the dark-is-more bias alone, assignment inference simulations for these
colormaps predict that more ocean water should map to darker blues in the top
row (consistent with direct associations) and darker browns in the bottom row
(conflicting with direct associations).

Like for direct associations, we reduced the problem to model only the
endpoint colors and concepts. This simplification ensured that the edges from
each potential structure-preserving set (dark-more and light-more) are not
mis-matched during simulations of inferred mappings. Our approach assumes the
colors in color scales used to construct colormaps vary monotonically in
lightness, which was true in the present study (we return to this issue in the
General Discussion).

###  2.3 Combining direct and relational sources of merit

We propose that assignment inference for visualizations of continuous data can
be simulated using a weighted sum over multiple sources of merit. With
knowledge on how much weight to put on merit from direct associations
($W_{A}$) and the dark-is-more bias ($W_{D}$), we can combine these sources of
merit (combined merit bipartite graph in Fig.  Unifying Effects of Direct and
Relational Associations for Visual Communication) and use established methods
for simulating assignment inference [38, 36, 22] to predict inferred mappings.
In the top row, these sources of merit are consistent, and simulating
assignment inference over combined merit predicts observers will infer that
darker colors map to larger quantities. In the bottom row, these two sources
of merit are conflicting. Depending on the relative weight given to each
source, they might cancel each other out, or one might dominate over the
other. The weights used in Fig.  Unifying Effects of Direct and Relational
Associations for Visual Communication are based on the results of Exp. 3, with
greater weight on direct associations than on the dark-is-more bias (see Exp.
3 for details).

In this study we asked whether direct and relational associations
independently contribute to merit in assignment inference for colormap data
visualizations, and if so, what is their relative contribution? Answering
these questions enabled us to create a model that predicts people’s inferred
mappings, which can be used to help design colormaps that facilitate visual
communication.

##  3 Experiment 1

Experiment 1 investigated whether both direct and relational color-concept
associations contribute to inferred mappings for colormaps. We addressed this
question using colormaps depicting fictitious data about two domain concepts,
shade and sunshine. We chose these concepts because the dark-is-more bias and
direct associations would be consistent for shade and conflicting for
sunshine, allowing us to test for independent effects of each factor.

###  3.1 Methods

We began by collecting direct color-concept association data for the domain
concepts shade and sunshine. We then used these data to generate colormap
stimuli to assess inferred mappings. Data, code, and color coordinates for all
experiments in this paper can be found at
https://github.com/SchlossVRL/assign-infer-colormaps.

####  3.1.1 Measuring direct color-concept associations

In the color-concept association task, participants were presented with a
concept word at the top of the screen (sunshine or shade) and a colored square
centered below (Fig. 3A). They rated how much they associated the given
concept with the given color by moving a slider along a scale ranging from
“not at all” ($-200$) to “very much” ($200$), and clicking “continue” to begin
the next trial. Each concept was rated for each of the UW-71 colors [22] shown
in Fig. 3B (see Table S.3 in Supplementary Material for CIELAB coordinates).
The UW-71 colors include 58 colors uniformly sampled from CIELAB space (UW-58
from [31, 36]), plus 13 additional colors sampled at a higher lightness plane
to incorporate more saturated yellows and greens [22].

Our target sample size was $n=30$ and we collected data from 35 Amazon mTurk
workers given we expected several participants would be excluded for failing
the attention check, described below (35 total, 3 excluded). The final sample
was $n=32$ (mean age = 40 years old; 11 women, 21 men; gender assessed using
free-response here and in all subsequent experiments). All participants
indicated normal color vision when asked if they had difficulty distinguishing
between colors relative to the average person and if they considered
themselves colorblind. All participants of this and all subsequent experiments
gave informed consent and the UW–Madison IRB approved the protocol.

Fig. 3: (A) Example association rating trial. The slider indicates a slight
association for the given purple color with the concept shade. (B) The UW-71
colors as seen during the association task instructions.

Before beginning the task, participants were shown the domain concept words
and all 71 colors (Fig. 3B). They were asked to identify which color they
associated most and least with each concept to anchor the endpoints of the
scale [30].The experiment was blocked by concept, with shade and sunshine
presented in a random order within the first two blocks. The 71 colors
appeared in a random order within each block. Given our plan to use
association ratings from this task to generate stimuli for the colormaps task,
we sought to include associations only from participants who made careful
judgments. Thus, we included a third, attention check block for all
participants and set an a priori exclusion criterion (see Section S.3 in the
Supplementary Material).

The displays of this and all subsequent experiments were created using jsPsych
[9]. All participants completed the experiments on their own devices so the
color coordinates were calculated using standard assumptions about RGB
displays. Thus, as is typical in color experiments in visualization, which aim
to be robust to variations across displays [47, 48, 12, 22], the precise
colors each participant saw varied with the specifications of their monitors.
This experiment took approximately 30 min. and participants were compensated
$3.63. The mean color-concept associations for sunshine and shade are shown in
Fig. S.4 of the Supplementary Material.

####  3.1.2 Generating colormaps

To generate the colormaps for this experiment, we (1) specified eight pairs of
endpoint colors, (2) interpolated between the eight endpoint colors to create
color scales, and then (3) applied the colorscales to 10 underlying datasets
to create colormap data visualizations (Fig. 4).

(1) We selected endpoint colors such that one color was lighter (L) and the
other was darker (D). For four endpoint pairs, association difference was
high—sunshine was far more associated with the light than the dark endpoint,
and shade was far more associated with the dark than the light endpoint. For
four other pairs, association difference was low, which occurred when both
colors were either weakly or moderately associated with the domain
concept.444Overall, mean association ratings increased with lightness for
sunshine (CIELAB L*) ($r(69)=.71,p<.001$) and decreased with L* for shade
($r(69)=-.79,p<.001$), but some light colors were moderately associated with
sunshine, and some dark colors were moderately associated with shade. These
properties enabled us to generate colormaps that varied in association
difference. Within each level of association difference, two endpoint pairs
had a lightness difference of $L^{*}=38$, and two had a difference of
$L^{*}=50$. We tested multiple color pairs for each condition to ensure our
results were not specific to any one color pair. We checked if the colors
interpolated between the endpoints varied (approximately) monotonically in
direct association strength for both sunshine and shade (i.e., the domain
concept was not more associated with the intermediate colors than with either
endpoint)555Two color scales for shade did not meet our statistical criterion,
due to a coding error treating hue angle as radians instead of degrees.
However, our statistical criterion is a heuristic, and visual inspection
suggested that the intermediate colors still varied monotonically between the
endpoints (Supplementary Material Fig. S.6A), so we kept data for these color
scales in the analysis.. See Supplementary Material Section S.4 and Fig. S.5
for details.

(2) Using these endpoint colors, we created eight color scales by linearly
interpolating eight steps between the light and dark endpoints (interpolation
computed in CIELAB space). The resulting color scales had 10 steps, as in the
stimuli from [35].

(3) Finally, we applied each of the eight color scales to 10 underlying
datasets, producing 80 colormap data visualizations. The underlying datasets
produced colormaps appearing as an $8\times 8$ grid, where one side was biased
to be lighter and the other side was biased to be darker. Within the 10
underlying datasets, half produced colormaps in which the left side was darker
than the right side (as in Fig. 4), and the other half produced colormaps in
which the right side was darker (as in Fig.  Unifying Effects of Direct and
Relational Associations for Visual Communication). A full set of 10 colormaps
from one color scale are shown in Fig. S.8 in the Supplementary Material.

The underlying datasets we used were previously used to generate colormaps in
[35]. The data ranged from 0-1, with values sampled from eight discrete points
along an arctangent curve with added noise. The eight points corresponded to
the eight columns of the colormaps. The samples at each point were used to
assign values to the rows within each column of the colormap (see
Supplementary Material Section S.1 for further details). One endpoint of the
color scale was assigned a data value of 0 and the other endpoint a data value
of 1, such that the color scales corresponded to the full range of the
underlying data. Given that the data were evenly sampled along the arctangent
curve, the data represented in the colormaps evenly span the full data range.
This method of generating stimuli mitigates concerns about the dynamic range
of data variability being hidden in the data visualization [10, 52].

Fig. 4: Color scales and corresponding colormaps in Exp. 1.

####  3.1.3 Assessing inferred mappings for colormaps

In the colormaps task, participants were presented with colormaps along with a
domain concept (sunshine or shade). They were told that the colormaps
represented amounts of sunshine (or shade) from various counties in a state.
In some counties, there was more sunshine (shade) on the left side of the
county; in other counties, there was more sunshine (shade) on the right side.
Their task was to indicate whether there was more sunshine (shade) on the
left/right of the map by pressing the left/right key on their keyboard.

Domain concept and color scale varied between-subjects, and participants were
randomly assigned to one of 16 groups (8 color scales $\times$ 2 domain
concepts). Each participant judged all 10 colormaps for their assigned domain
concept and color scale, one at a time in a random order. Trials were
separated by a 500-ms inter-trial interval. The colormaps (approx. 4cm
$\times$ 4cm) appeared on a white square (approx. 9cm $\times$ 9cm) in the
center of a medium gray screen (size estimates using a 15.6in, 1920 $\times$
1080 pixel monitor). Below each half of the colormap was a horizontal line
labeled “Left”/“Right” (Fig.  Unifying Effects of Direct and Relational
Associations for Visual Communication).

Our target sample size was $n=192$, $n=12$ per group (sample size based on a
power analysis reported in Supplementary Material Section S.6). The final
sample was 187 mTurk workers (mean age = 38.9 years old, 105 women, 82 men),
after excluding $n=3$ for atypical color vision and $n=2$ for not completing
the experiment. The groups ranged from $9-13$ participants due to how the
experiment code automated assignments to conditions while managing exclusions.
The experiment took approx. 5 min. and participants were compensated with
$0.60.

Fig. 5: (A) Predicted patterns of results in Exp. 1 if inferred mappings are
influenced by only the dark-is more bias (left), only direct associations
(center), or both (right). (B) Results of Exp. 1, showing the mean proportion
of times the darker side was selected for maps about sunshine (circles) and
shade (triangles) as a function of signed association difference. Mark colors
represent endpoint colors in the color scales and error bars represent
standard errors of the means.

###  3.2 Results and Discussion

Fig. 5A shows potential patterns of inferred mappings if there was an effect
only of direct associations, only of the dark-is-more bias, or both. The
y-axis represents the proportion of times the darker side would be chosen over
all trials, as a function of signed association difference. Positive/negative
association differences indicate the darker/lighter color is more associated
with the domain concept, respectively. Direct associations only predicts the
probability of choosing the darker side would increase monotonically as the
darker side becomes more associated with the domain concept. Dark-is-more only
predicts participants would always choose the darker side, regardless of
association difference. If both have an effect, then participants would choose
the darker side when the two biases are consistent (positive association
differences) but less likely to choose the darker side as the lighter side
becomes more associated with the domain concept (negative association
differences).

Fig. 5B shows the mean proportion of times the darker side was chosen,
averaged over the 10 repetitions within each participant, and then averaged
over participants. The pattern of responses resembles the predicted pattern if
both the dark-is-more bias and direct associations influenced inferred
mappings. Participants almost always chose the darker side for shade
(association differences greater than zero), and their likelihood of choosing
the darker side decreased as the lighter side became more associated with
sunshine.

To test for independent effects of each potential source of merit, we used a
mixed-effect logistic regression model. Although we plot the data in terms of
the proportion of times participants chose the darker side (Fig. 5B), this way
of coding the data poses a problem for including the dark-is-more bias as a
predictor in a regression model, given that there is no variability in the
predictor (it predicts a response of ‘1’ on every trial). Thus, we conducted a
model to predict whether participants chose the left side on each trial (1 =
left, 0 = right), from a predictor coding whether the left side was darker (1
= left darker, $-1$ = right darker), and a predictor coding which side was
more associated, and by how much (scaled to range from $-1$ to 1; x-axis
values in Fig. 5B). Conducting models with respect to the left side is a
standard approach in psychophysics research, and is valid as long as the
stimuli are left/right balanced, as in the present stimulus set (see Section
3.1.2).

Participants were more likely to select the left side if it was more strongly
associated with the domain concept than the right side
($B=4.51,SE=0.21;z=21.22,p<.001$) and if it was dark than light
($B=1.33,SE=0.09;z=15.46,p<.001$) (dark-is-more bias). Thus, both direct and
relational associations influenced inferred mappings. See Section S.7 in
Supplementary Material for an additional analysis that includes concept as a
factor in the model.

Summary. Exp. 1 showed that direct associations and the dark-is-more bias
contribute independently to people’s inferred mappings. When these two factors
conflict (the domain concept is more associated with the light endpoint than
the dark endpoint) and the direct association difference is large, direct
associations override the dark-is-more bias.

##  4 Experiment 2

Given evidence that direct associations can override the dark-is-more bias
when they conflict and direct associations are strong, we conducted Exp. 2 to
test how much association difference was needed for direct associations to
fully override the dark-is-more bias. The results led us to study effects of
semantic distance for predicting inferred mappings.

###  4.1 Methods

The methods were the same as Exp. 1, except for two changes. First, we only
tested sunshine as the domain concept in order to focus on cases where direct
associations and the dark-is-more bias conflict. Second, we included eight new
color scales of intermediate association difference, in addition to the
original eight from Exp. 1 (16 color scales) (Fig. 6).

For the new color scales, we selected endpoint color pairs using the
association data from Exp. 1 with the same selection criteria (Supplementary
Material Section S.4). Two of the new color scales did not meet our
statistical criterion for monotonicity due to a coding error treating hue
angle as radians instead of degrees (Supplementary Material Fig. S.6B), and
visual inspection showed that intermediate colors were more associated with
sunshine than the endpoints. Thus, we excluded data from these two color
scales from analysis.

Fig. 6: Color scales used in Exp. 2. From right to left, sunshine is
increasingly more associated with the light endpoint of the color scale. The
four leftmost and four rightmost color scales were used in Exp. 1.

We focused on association difference and allowed association strength to vary
(e.g., the same value of association difference could be achieved if sunshine
was moderately associated with the light endpoint and weakly associated with
the dark endpoint, or strongly associated with the light endpoint and
moderately associated with the dark endpoint). As in Exp. 1, each participant
judged 10 colormaps constructed from one of the 16 color scales (between-
subjects).

Our target sample size was $n=640$ ($n=40$ per condition) based on a power
analysis (see Supplementary Material Section S.6). We collected data until
each condition reached at least 40 participants after excluding those with
atypical color vision (696 Amazon mTurk workers in total, 41 excluded). We
assessed color vision using the two self-report questions in Exp. 1, plus
responses to six digital Ishihara plates. Participants were excluded if they
answered yes to either question and/or answered incorrectly for more than one
of the six plates. The final sample included 655 participants (mean age = 38.6
years old; 294 women, 358 men, 2 non-binary, 1 no report). The groups ranged
from $40-44$ participants due to how the experiment code automated assignments
to conditions while managing exclusions. The experiment took approximately 5
min. and participants were compensated $0.60.

###  4.2 Results and Discussion

Fig. 7A shows the mean proportion of times participants selected the darker
side for colormaps generated from each color scale (averaged over the 10
colormaps judged by each participant, and then averaged over participants). As
in Exp. 1, direct associations were more likely to override the dark-is-more
bias as association difference increased ($r(12)=.85,p<.001$). But, once
association difference reached about $-.55$, participants almost always
inferred that the lighter side of the colormaps mapped to more sunshine.
Direct associations fully overrode the dark-is-more bias, so further
increasing association difference could not further influence inferred
mappings (floor effect). This observation led us to ask, why would inferred
mappings level off at around $-.55$?

One possibility is that participants approached this task using assignment
inference, comparing each possible assignment (dark-more or light-more), and
inferring the assignment with greater merit. Once the merit of one assignment
is sufficiently greater than the alternative, the colors reach maximal
semantic discriminability. Then, further increasing association difference has
no further effect on assignment inference. This limit may have been reached at
an association difference of around $-.55$. If so, then the plateauing
function in Fig. 7A may become linear when we replace the x-axis (signed
association difference) with simulations of assignment inference (Section
1.1).

To simulate assignment inference for each color scale, we first calculated
semantic distance for each pair of endpoint colors (using equation 1 defined
in [36] and reproduced in Supplementary Material Section S.2 of this paper).
We then determined the optimal assignment (i.e., which assignment had greater
merit), and coded the outcome as dark-more = $+1$ and light-more = $-1$. Last,
we multiplied this coding by semantic distance to compute signed semantic
distance, which gave positive values to the probability of inferring dark-more
assignments and negative values to the probability of inferring light-more
assignments.

Fig. 7: Mean proportion of times the darker side was selected for each color
scale, plotted as a function of (A) signed association difference, and (B)
signed semantic distance using direct associations as merit. Mark colors
indicate endpoint colors. Error bars indicate standard error.

Computing signed semantic distance required specifying merit based on direct
associations between each endpoint color and each endpoint concept (“no
sunshine” vs. “a lot of sunshine”), as in Fig.  Unifying Effects of Direct and
Relational Associations for Visual Communication. From Exp. 1, we had
association data for domain concept “sunshine,” but not the endpoint concepts.
Thus, we collected data from additional participants ($n=31$), who rated the
association strength between each endpoint color and endpoint concepts “no
sunshine” and “a lot of sunshine” (see Supplementary Material Section S.5). We
used the mean ratings as merit to compute signed semantic distance.

As shown in Fig. 7B, inferred mappings were predicted by simulations of
assignment inference: signed semantic distance was strongly correlated with
the proportion of times participants chose the darker side
($r(12)=.97,p<.001$). This correlation was stronger than the correlation for
signed association difference reported above ($z=1.96,p=.05$). The trail of
points that plateaued in Fig. 7A now compress onto signed semantic distance
values near $-1$ in Fig. 7B.

From this strong linear relation, one may suppose that only merit from direct
associations is needed to simulate inferred mappings for colormap
visualizations. But, Exp. 2 only included conditions in which direct
associations and the dark-is-more bias conflicted, and Exp. 1 suggested that
when they were consistent, the dark-is-more bias dominated regardless of
association strength difference. To understand the relative contribution of
these two potential sources of merit in assignment inference, it is necessary
to model data sampled from multiple points along the full range of signed
semantic distance (see Exp. 3).

Summary. As in Exp. 1, Exp. 2 showed that direct associations override the
dark-is-more bias when the association difference between the light and dark
colors was sufficiently large. The pattern of inferred mappings was strongly
predicted by simulations of assignment inference using merit from direct
associations (signed semantic distance).

##  5 Experiment 3

In Exp. 3, we developed and tested a new method to combine multiple (sometimes
conflicting) sources of merit to simulate assignment inference (Fig.  Unifying
Effects of Direct and Relational Associations for Visual Communication). The
experimental task was the same as Exps. 1-2, but we tested three new domain
concepts (ocean water, wild fire, and glacial ice), and sampled 21 points
along the full range of direct association-based signed semantic distance for
each concept. Our goals were: (1) determine the optimal weighting on direct
associations ($W_{A}$) relative to the dark-is-more bias ($W_{D}$) when
computing combined merit, and (2) test whether simulations of assignment
inference using the optimal combined merit predicted people’s inferred
mappings better than simulations using each source of merit alone.

###  5.1 Methods

We first collected direct association ratings for the domain concepts and then
used the mean ratings to generate colormaps to assess inferred mappings. We
also collected additional data to quantify merit for direct associations and
the dark-is-more bias.

####  5.1.1 Measuring direct color-concept associations

We collected direct association ratings for five domain concepts relevant to
environmental data: wild fire, ocean water, glacial ice, ground soil, and tree
foliage (inspired by [34]), using the same methods as in Exp. 1. The data are
shown in Supplementary Material Fig. S.7.

The target sample was $n=35$ to match Exp. 1, and we collected data in batches
until reaching this target after excluding those with atypical color vision
($n=15$) and who failed the attention check ($n=17$); 70 Amazon mTurk workers
total. We shortened the attention check block but used the same a priori
exclusion criterion (see Supplementary Material Section S.3). Our final sample
was $n=38$ (mean age = 42.8 years old, 18 women, 20 men). The experiment took
approximately 60 min. and participants were compensated $7.25.

####  5.1.2 Generating colormaps and computing merit

Based on the direct association data, we created colormaps for three concepts:
wild fire, ocean water, and glacial ice. These domain concepts enabled
spanning the full range of signed semantic distances within each concept. We
generated colormaps using the methods in Exp. 1 (Fig. 4) and Supplementary
Material Section S.4. For each domain concept, we chose 21 pairs of endpoint
colors that spanned the full range of association differences from strongly
negative (light color was more associated with the domain concept) to strongly
positive (dark color was more associated with the domain concept). Fig. 8
shows the resulting 21 color scales for each domain concept. All color scales
satisfied the criterion for monotonicity. As in Exps. 1 and 2, each color
scale was applied to 10 underlying datasets to produce 10 colormaps per color
scale, with darker side left/right balanced (Fig. S.8).

After selecting the color pairs, we collected additional data to estimate
merit for each endpoint color paired with each endpoint concept, with respect
to direct associations and the dark-is-more bias (Fig.  Unifying Effects of
Direct and Relational Associations for Visual Communication).

Merit for direct associations. A new set of 30 participants rated the
association strength between each endpoint of each domain concept (e.g., “a
lot of ocean water,” “no ocean water”) and each corresponding endpoint color
(details in Supplementary Material Section S.5 ). As in Exp. 2, we used these
associations to estimate merit derived from direct associations for each
color-concept endpoint pairing (Fig.  Unifying Effects of Direct and
Relational Associations for Visual Communication).

Fig. 8: Color scales used to create colormaps in Exp. 3. Rightward of center,
the domain concept increases in direct association strength with the darker
endpoint (consistent direct and relational associations). Leftward of center,
the domain concept becomes increasingly more associated with lighter
(conflicting direct and relational associations).

Merit for the dark-is-more bias. So far, we have discussed the dark-is-more
bias as binary—dark-more assignments have greater merit than light-more
assignments. However, the dark-is-more bias can also be considered
continuous—the degree to which dark-more assignments have greater merit
depends on the degree to which one endpoint appears clearly darker than the
other endpoint. One might consider quantifying merit of the dark-is-more bias
using lightness (L*) difference between the two endpoint colors of the color
scale that varied monotonically in lightness. However, we reasoned that the
dark-is-more bias would be activated if one side appeared clearly darker than
the other, and adding additional lightness difference may not increase
activation of the bias.

Thus, we used an empirical approach to quantify merit for the dark-is-more
bias. For each endpoint color pair, volunteers with expertise in color
perception ($n=4$) rated the degree to which one color was clearly darker than
the other color (referred to as darkness difference ratings). They judged each
pair twice (left/right balanced), and made their ratings on continuous slider
scale from “left color is clearly darker” to “right color is clearly darker.”
The middle was labeled “equal darkness” (see Supplementary Material Section
S.5 for details). For each color scale, we coded dark-more edges to have merit
= 1 and light-more edges to have merit = 0, and then multiplied these values
by the darkness difference ratings. As a result, differences in total merit of
dark-more vs. light-more assignments scaled with the degree to which it was
obvious that the dark endpoint appeared darker than the light endpoint.

####  5.1.3 Colormap interpretation task

This task was the same as in Exps. 1 and 2, except the domain concepts were
wild fire, ocean water, and glacial ice and there were 21 color scales per
domain concept (3 domain concepts $\times$ 21 color scales = 63 groups of
participants). Participant judged 10 colormaps for their assigned domain
concept and color scale. They were told that the colormaps represented data
about [domain concept] in different counties. Their task was to indicate
whether there was more [domain concept] on the left/right side of the county
(Fig.  Unifying Effects of Direct and Relational Associations for Visual
Communication).

The target sample size was $n=1260$ ($n=20$ per condition) based on a power
analysis (see Supplementary Material Section S.6). We collected data in
batches until each condition had at least $n=20$ after excluding those with
atypical color vision as assessed in Exp. 2 (1391 Amazon mTurk workers total,
107 excluded). The final sample was $n=1284$ (mean age = 40.3 years old, 1 no
reported age; 672 women, 598 men, 9 non-binary, 5 no reported gender). The
groups ranged from $20-22$ participants due to how the experiment code
automated assignments to conditions while managing exclusions. The experiment
took approximately 5 min. and participants were compensated $0.60.

###  5.2 Results and Discussion

In the following analyses, we determined the optimal relative weighting on
direct and relational associations, and then assessed whether assignment
inference simulations using the optimal weighting predicted inferred mappings
better than simulations using each source of merit alone. We split
participants into a training set to determine the optimal weighting, and a
test set to compare the optimal weighting with each source of merit alone.
Each set had $10-12$ participants per color scale for each domain concept. We
simulated assignment inference with varying relative weight on each source of
merit as follows:

(1) Computing combined merit. First, we specified merit of each color-concept
pairing within each source of merit (Section 5.1.2). Then, we calculated
combined merit by computing the weighted sum over bipartite graphs for each
source of merit (Fig.  Unifying Effects of Direct and Relational Associations
for Visual Communication). We used each combination of weights on direct
associations ($W_{A}$) and the dark-is-more bias ($W_{D}$) in increments of
.05, such that their sum was 1. Each weight was a multiplicative factor on
each edge of the respective bipartite graphs. For instance, a weight pairing
of (1,0) placed all the weight on direct associations, (0,1) placed all the
weight on the dark-is-more bias, and (.5, .5) placed equal weight on both
sources of merit.

(2) Computing signed semantic distance. We computed signed semantic distance
over combined merit for each weight pairing, following the procedure in Exp.
2. First, we computed semantic distance between the endpoint colors for each
domain concept. Next, we determined the optimal assignment (which assignment
had greater overall merit), coded as $+1$ for dark-more and $-1$ for light-
more. Last, we multiplied this coding by semantic distance to obtain signed
semantic distance.

To determine the optimal weighting, we used mean squared error (MSE) to
compare assignment inference simulations with human judgments. For each of the
21 color scales for each of the three domain concepts, we computed MSE between
signed semantic distance and the mean probability that participants in the
training set chose the darker side of the colormaps. When computing MSE, we
scaled the proportion chosen data to range from -1 to 1, corresponding with
the scale of signed semantic distance. Fig. 9A shows MSEs averaged over the 21
color scales for each domain concept, plotted as a function of weight pairs,
along with the average over domain concepts. On average, the best performing
weight pair yielding the lowest MSE had a weight of $W_{A}=.7$ on direct
associations and $W_{D}=.3$ on the dark-is-more bias.

Fig. 9: Mean squared error (MSE) predicting inferred mappings from assignment
simulations with varying weights on dark-is-more bias ($W_{D}$) and direct
associations ($W_{A}$) for the (A) training set and (B) testing set. MSEs are
shown separately for colormaps representing ocean water (blue diamonds),
glacial ice (gray squares), and wildfire (yellow triangles), plus the average
of all three concepts (black circles). Error bars represent standard error of
the means. The gray bar indicates the best pair, determined from the training
set.

Using data from the held out testing set, we evaluated whether this optimal
weight pair was better for predicting assignment inference than each source of
merit alone. For each color scale for each domain concept, we computed MSE
between mean responses (scaled to range from $-1$ to $1$) and signed semantic
distance with the optimal weighting identified from the training set (.7, .3),
with all weight on direct associations (1,0), and with all weight on the dark-
is-more bias (0,1) (Fig. 9B). To test effects of relative weighting, we used a
linear mixed effects model predicting MSE for each color scale, with fixed
effects for relative weighting, domain concept, and their interaction (using
Helmert contrasts). The model also included a by-color scale random intercept
and random slope for relative weighting. Here we focus on the main effect of
relative weighting ($F(2,53.45)=9.18,p<.001$), and we report on further
details of this model in Supplementary Material Section S.7. Planned
independent samples t-tests indicated that the optimal weight pair fit
inferred mappings better than direct associations alone ($t(124)=-2.55,p=.01$)
and dark-is-more alone ($t(124)=-3.53,p=.001$).

Fig. 10 shows the relation between participant responses and simulations of
assignment inference using the optimal weight pairings for each color scale.
Points would fall along the diagonal line if the simulations perfectly
predicted inferred mappings. Signed semantic distance was significantly
correlated with inferred mappings for all three domain concepts, but to
varying degrees: strong correlation for wild fire ($r=.83,p<.001$), moderately
strong for ocean water ($r=.72,p<.001$), and moderate for glacial ice
($r=.55,p=.01$). Preliminary exploration suggests this weaker relation for
glacial ice might be due to some colormaps appearing to vary in opacity,
activating the opaque-is-more bias. The opaque-is-more bias aligns with the
dark-is-more bias on light backgrounds (as used here), and the two relational
associations may have combined to jointly override effects of direct
associations. Our study was not designed to test the opaque-is-more bias, so
future work is needed to study these effects more directly.666This weaker
correlation for glacial ice is not likely due to concept glacial ice, but
rather the colormaps appearing to vary in opacity happened to be in the
glacial ice condition. The applicability of the opaque-is-more bias depends on
the combination of background and colors of the color scale [35], and would be
applicable for these colormaps if they represented any other domain concept.

Fig. 10: Mean proportion of times the darker side was selected for the held-
out testing set for each color scale as a function signed semantic distance
using the optimal weight pair. Mark colors indicate color scale endpoint
colors, and error bars indicate standard error.

Supplementary Material Section S.7 includes an additional analysis showing
independent effects of direct and relational associations on inferred mappings
(as in Exp. 1). It also includes plots of inferred mappings as a function of
association difference and semantic distance from direct associations (Fig.
S.9), analogous to Fig. 7 in Exp. 2. The strong plateau for sunshine colormaps
in Exp. 2 was less apparent in Exp. 3, and we consider possible explanations
in Section S.7.2.

Summary. Exp. 3 showed that inferred mappings for colormaps were well-
predicted using a simulation of assignment inference with combined merit. The
optimal combined weighting resulted in predictions with less error than
predictions simulated with weight on direct associations or the dark-is-more
bias alone.

##  6 General discussion

A central problem in visual communication is understanding how people infer
meaning from visual features. By anticipating people’s expectations about how
visual features should map onto concepts, designers can create visualizations
that align with those expectations, thereby facilitating communication [26,
51, 50, 14, 18, 38, 35, 36, 43, 22].

We approached this problem by bridging work on inferred mappings for
visualizations of categorical information [18, 41, 38, 36, 22] and
visualizations of continuous data [8, 21, 35, 43] to understand both within a
single framework of assignment inference. Doing so required broadening the
notion of merit in assignment inference to include not only direct
associations as in [38, 36, 22], but also relational associations (e.g., dark-
is-more bias). Exp. 1 showed that direct and relational associations
contribute independently to inferred mappings for colormaps. Exp. 2 showed
that inferred mappings for colormaps were predicted by simulations of
assignment inference (signed semantic distance) using merit from direct
associations. Exp. 3 showed that simulating assignment inference using a
weighted sum over merit from direct and relational associations was better at
predicting inferred mappings than simulations using each source of merit
alone.

This study is an initial step towards comprehensively modeling the effects of
multiple sources of merit in assignment inference. Here, we began with direct
associations and one type of relational association, the dark-is-more bias. In
future work, we will extend our approach to include additional sources of
merit, including the opaque-is-more bias and hotspot-is-more bias. To quantify
merit for the opaque-is-more bias, it will be necessary to estimate the degree
to which colors in the colormap appear to vary in opacity depending on the
background color (see [35]), and ensure that this source of merit falls out of
the equation when colormaps do not appear to vary in opacity. To quantify
merit for the hotspot-is-more bias, it will be necessary to quantify the
degree to which hotspots are salient in the colormap, and again ensure that
this source of merit falls out of the equation when colormaps do not appear to
have hotspots. Our approach for estimating inferred mappings not only has
potential to accommodate known sources of merit, but can also scale as
additional sources of merit are discovered.

We also expect our approach to extend to abstract concepts. Evidence suggests
that sets of abstract concepts previously considered “non-colorable” (e.g.,
sleeping, driving, safety, comfort) can be meaningfully encoded using color as
long as their association distributions are sufficiently different from one
another (semantic discriminability theory [22]). In the present framework, as
long as the colors in the color scale vary in association strength with the
domain concept, then merit from direct associations will influence combined
merit with the dark-is-more bias. If the associations do not vary in
association strength (low semantic distance), then merit from direct
associations will have little effect on combined merit, and the dark-is-more
bias should dominate inferred mappings. These patterns should hold regardless
of whether the concepts are abstract/concrete. If a concept has no systematic
color-concept associations, regardless of whether it is abstract/concrete,
then it will not be possible to create a color scale with large direct
association difference, so the dark-is-more bias (and any other sources of
merit) would dominate inferred mappings.

Overall, our findings can be translated to incorporate color semantics into
tools that generate colors for information visualizations (e.g., Colorgorical
[12], Color Crafter [45], and CCC-Tool [25]). These tools already allow
designers to balance different factors, such as perceptual discriminability
and aesthetics. With a comprehensive model of assignment inference combining
multiple sources of merit, it will be possible to incorporate semantic
discriminability into algorithms that optimize color selection for
visualization design.

Limitations. This study has limitations for future work to address.

Linearly interpolated color scales. We used color scales that were linearly
interpolated between two endpoints in CIELAB space, which supported the goals
of this study. Interpolated color scales allowed us to compare merit of dark-
more vs. light-more assignments using direct color-concept associations from
only the endpoint colors. Using only the endpoints was possible because the
intermediate colors varied approximately monotonically in association strength
between the endpoint colors (see Supplementary Material Section S.4).
Monotonicity would be violated if the domain concept was more/less associated
with intermediate colors of a color scale than the endpoints (e.g., using a
color scale for sunshine that interpolated between a red and yellowish-green,
resulting in more strongly associated yellows in the middle).

Monotonicity would also likely be violated in industry standard color scales
that spiral through color space [45, 5, 16]. Yet, Smart et al. [45] showed
that such color scales that spiral produce colormaps that are more
interpretable and aesthetically preferable than linear colormaps like the ones
in the present study. Indeed, many criteria determine whether color scales
(also referred to as ramps) are effective for visualizing continuous data [34,
6, 53, 44, 45, 32, 25] and our color scales were not designed to meet those
criteria. Thus, the color scales in the present study are not meant to be used
for visualizations of real data. To apply our modeling approach to more
complex color scales, it will be necessary to quantify merit for color-concept
pairings sampled in multiple steps between the two endpoints, and use a method
for computing the optimal assignment that accounts for many colors and many
concepts.

Sequential color scales. The present work, and most previous work on inferred
mapping for colormaps [35, 43, 21], has focused on sequential color scales,
where encoded data ranged from small to large. Questions remain concerning how
this work extends to diverging scales, where encoded data has a neutral point.
The dark-is-more and opaque-is-more biases imply that more extreme data
(furthest from the neutral point) should map to darker, more opaque regions,
respectively. Future work is needed to test these hypotheses, and to
investigate people’s expectations about which colors represent data values
above/below the neutral point. Future work is also needed to determine whether
the relative weightings on sources of merit established in Exp. 3 for
sequential color scales generalize to diverging color scales.

Task type. Our study and much of the previous studies on inferred mappings for
colormaps [35, 43, 21] used tasks that asked participants to interpret where
“more” was represented in a colormap. However, people use colormap data
visualizations for a wide variety of other tasks, such as those studied by
Padilla et al. [28]: finding a specific value of a concept, comparing values
across regions, and averaging values across regions. Future research is needed
to test whether the present framework modeling combined merit to estimate
inferred mappings predicts performance in these other kinds of tasks.

Conclusion. This work builds a new bridge for understanding how direct and
relational associations combine to influence inferences about the meanings of
colors in visualizations. We have laid the groundwork to develop a more
comprehensive model of assignment inference that accounts for additional
sources of merit that we know of, and can scale to accommodate new sources of
merit as they are discovered. Our findings can be translated directly to
design visualizations that align with people’s expectations about the meanings
of colors, thereby making visualizations that are easy to interpret.

###### Acknowledgements.

We thank Kushin Mukherjee, Chris Thorstenson, Anna Bartel, Clementine
Zimnicki, Madeline Parker, and Audrey Wang for helpful discussions on this
work. This project was supported by the UW–Madison Office of the Vice
Chancellor for Research and Graduate Education, Wisconsin Alumni Research
Foundation, NSF award BCS-1945303 to KBS, and NSF award $\\#$2050782\.

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##  MyAppx S Supplementary Material

###  S.1 Underlying data distribution for generating colormaps

The following description of the process for generating the colormaps,
including Fig. S.1, is reproduced directly from pp. 813-814 in [35]:

The data used to generate the colormaps were sampled from an arctangent curve
with added normally-sampled noise (Fig. S.1). To generate the data for each
row of the colormap, we discretized the arc tangent curve into eight
bins,corresponding to the eight columns in the colormap display. We centered
the arctangent curve between the fourth and fifth bins, such that half of the
display was biased to have larger values than the other half. We then
perturbed each arctangent value by sampling from a normal distribution with
the mean equal to the arctangent value and the standard deviation equal to
0.25. When the values fell outside the [0,1] interval, we re-sampled until
they were all within the correct range. For half of the datasets, the
arctangent curve was oriented as shown in Fig. S.1, and for the other half, it
was left/right reversed. This enabled a left/right balance of the darker
region (i.e., half of the colormaps contained the darker region on the left
and the other half contained the darker region on the right).

Fig. S.1: Distribution used to sample values at each time point to generate
the data used to construct the colormap images. Figure reproduced from Figure
4 in Schloss et al. [35].

###  S.2 Calculating semantic distance

We calculated semantic distance as specified in [36, 22]. The following
definition is reproduced directly from p. 700 in [22]:

Semantic distance is a way to operationalize semantic discriminability in the
case where there are $n=2$ features and concepts [36]. Fig. S.2 illustrates an
example in which we have concepts {M,W} and colors {1,2}. The color-concept
associations between all possible pairs are $x_{1},\dots,x_{4}$, as shown in
Fig. S.2. We assume each $x_{k}$ is normally distributed with mean
$\bar{x}_{k}$ equal to the corresponding $a_{ij}$ and standard deviation
$\sigma_{k}=1.4\cdot\bar{x}_{k}(1-\bar{x}_{k})$, which was found to be a good
fit to experimental data [36]. The outcome of the assignment problem is
determined by the quantity $\Delta x:=x_{1}-x_{2}+x_{3}-x_{4}$. The optimal
assignment is: (M-1 and W-2 if $\Delta x>0$) and (M-2 and W-1 if $\Delta
x<0$).

Semantic distance is defined by the equation

$\Delta S=\bigl{|}\text{Prob}(\Delta x>0)-\text{Prob}(\Delta x<0)\bigr{|}.$
(1)

Since the $x_{k}$ are assumed to be normally distributed, so is $\Delta x$,
and the probabilities in (1) can be computed analytically:

$\text{Prob}(\Delta
x>0)=\Phi\Biggl{(}\frac{(\overline{x}_{1}+\overline{x}_{4})-(\overline{x}_{2}+\overline{x}_{3})}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}+\sigma_{4}^{2}}}\Biggr{)},$
(2)

and $\text{Prob}(\Delta x<0)=1-\text{Prob}(\Delta x>0)$, where $\Phi(\cdot)$
is the cumulative distribution function (cdf) of the standard normal
distribution. When $\Delta S$ is close to $0$, $\Delta x$ has a similar
probability of being positive or negative, so the assignment is fragile. When
$\Delta S$ is close to $1$, $\Delta x$ is almost always positive or almost
always negative, so the assignment is robust. This notion of semantic distance
can be used even when the features are not colors, by replacing the color-
concept associations with feature-concept associations, and adjusting the
formula for $\sigma_{k}$ as appropriate.

Fig. S.2: Diagram showing how association ratings between concepts {M,W} and
colors {1,2} produce a distribution for $\Delta x$. Semantic distance is the
absolute difference of the area under the curve to the left and right of zero.
Figure reproduced from Figure 3 in Mukherjee et al. [22], which was adapted
from Figure 4 in Schloss et al. [36].

###  S.3 Attention check in color-concept association task

At the end of color-concept association tasks in Exps. 1 and 3, we included an
additional block of trials to serve as an attention check. The goal was to
collect color-concept association data for a concept for which participants
should have strong, specific associations if they were making careful ratings
during this task. We chose the concept celery, because we knew from previous
data [22] that participants (on average) had strong specific color-concept
associations for celary (Fig. S.3). That is, shades of greens were strongly
associated with celery, and shades of reds and purples were weakly associated
with celery. Using data from [22], we identified the six strongest and six
weakest associated colors with celery (See Fig. S.3). We then defined an a
priori exclusion criterion that the participants of the present study must
have an association rating greater than .5 (on a scale from 0 to 1) for at
least 5 out of 6 strongest celery colors, and an association rating less than
.5 for at least 5 out of 6 weakest celery colors. We chose to put the
attention check as the last block to evaluate performance at the end of the
task when participants were most fatigued.

Fig. S.3: Mean association ratings for celery from Mukherjee et al. [22]. The
six strongest and six weakest colors were used as the basis for attention
check exclusion criteria in Experiments 1 and 3 of the present work. In Exp.
3, only these twelve colors and the twelve moderate colors were shown during
the association ratings task attention check.

In Exp. 1, participants rated the association between each of the UW-71 colors
and the concept celery. In Exp. 3 we reduced the length of the attention check
by having participants rate a subset of colors, rather than all 71 colors. We
included the six colors most-associated celery colors, the six least-
associated colors, and 12 moderately associated colors that straddled the
middle of the ratings scale (around .5), determined from data in [22] shown in
Fig. S.3.

###  S.4 Selecting endpoint colors to create color scales

In all three experiments, we used the same general process for selecting pairs
of colors to serve as endpoints for creating color scales (Figs. S.4 and S.7).
Below, we explain this approach in detail for Exp. 1, and explain any changes
to this approach for Exps. 2 and 3.

Experiment 1. To generate the color scales, we needed pairs of endpoint colors
that varied in lightness and varied in their direct associations with shade
and sunshine. We began with the mean color-concept association data (Fig.
S.4). From the 71 individual colors, there were 2485 possible pairs. To ensure
there was obvious lightness variation to activate the dark-is-more bias, we
reduced this number of pairs by excluding pairs of the same lightness (637
pairs) or pairs that had a small lightness difference (difference of L* $<38$
in CIELAB space) (812 pairs). It is not yet known how much lightness
difference is needed to activate the dark-is-more bias, but this should
provide a conservative amount given prior work on color discriminability along
the L* axis [48, 47]. We also removed pairs that included white or black (96
pairs) to reduce the possibility that the opaque-is-more bias would be
activated [35] . After this exclusion, we limited the set by only working with
pairs that had a lightness difference of either 38 or 50 (143 pairs excluded).

Fig. S.4: Mean direct association ratings for (A) sunshine and (B) shade with
each of the UW-71 colors (fill of bar). Colors are sorted from high to low
association strength.

For the remaining color pairs, we linearly interpolated between the light and
dark colors in CIELAB space to create a color scale with 10 total steps (Fig.
4). Ideally, association strengths between the domain concepts and the
interpolated colors would fall between association strengths for endpoint
colors. To estimate the direct associations for intermediate colors, we used a
color space regression model as in Schloss et al. [37]. Color space regression
models predict a dependent measure (e.g., color preferences in [37]) from
coordinates of each color in a well-specified color space. Schloss et al. [37]
tested several kinds of color space models and found that a cylindrical model
in CIELCh space with two harmonics best predicted color preference data, so we
use that model here. The model includes seven predictors for each color:
lightness (L*), chroma (C*), first harmonic of hue angle ($\sin(h)$ and
$\cos(h)$), second harmonic of hue angle ($\sin(2h)$ and $\cos(2h)$), and an
additive constant. We note that CIELCh space is a cylindrical representation
of the coordinates in CIELAB space. Lightness (L*) coordinates are the same in
both spaces, whereas a* and b* in CIELAB (roughly) represent red/green and
blue/yellow Cartesian axes, C* and h in CIELCh represent radius and angle.

We used the following procedure to predict associations for colors for which
we did not have direct association ratings. First, for each concept, we
conducted a multiple linear regression model to estimate the mean direct
association ratings for the colors we did have (UW-71). This analysis provided
weights for the seven predictors and the additive constant for each concept.
We checked that the models fit the data well, and found strong fits for both
sunshine (multiple $R=.81$) and shade (multiple $R=.88$). Second, we used
these weights and constant to define the regression equation for each concept.
Finally, we plugged the CIE L*, C*, and $h$ coordinates for the new,
interpolated colors into the regression equation for each concept to estimate
the direct color-concept associations.

Once we had the estimated direct associations for interpolated colors between
the endpoints of each color scale, we used another linear regression analysis
to estimate monotonicity. For each color scale (two endpoints plus eight
intermediate colors), we fit a regression line estimating predicted direct
association ratings from color sequence (1-10) in the color scale (Fig. S.5).
If model estimates for interpolated colors were not intermediate between
associations strengths for endpoint colors (non-monotonic), there should be a
poor fit (low $R^{2}$). We set the exclusion criterion for color scales to be
to be $R^{2}<.8$. We note that this estimate of monotonicity is a heuristic,
and it is possible for model estimates to vary monotonically while being
poorly fit by a regression line. Thus, our criterion may have been overly
conservative in excluding candidate color scales.

A total of 429 pairs of endpoint color pairs were excluded for not passing
this criterion. Fig. S.5A shows an example color scale that meets this
criterion of approximate monotonicity for the concept sunshine. The estimated
associations for the interpolated colors fall between the associations
expected for the endpoints colors, meaning the lightest color is predicted to
be most associated with sunshine, and the darkest color is predicted as the
least associated. The drop-off that happens for the last point (darkest gray)
could be due to chroma equaling zero, and consequently dropping out of the
equation. This drop-off for an achromatic endpoint emerges for other color
scales (see Fig. S.6A). Fig. S.5B shows a color scale that is estimated to be
non-monotonic for sunshine. Here, some interpolated colors are expected to be
more associated with sunshine than either endpoint color of the scale.

Fig. S.5: Predicted association ratings (circles filled according to color
scale) for colors within two color scales representing amounts of sunshine. In
(A) the predicted associations are approximately monotonic, whereas in (B),
the predicted associations are non-monotonic). Color scales ordered with more
associated endpoint on the left and less associated endpoint on the right.

When we originally performed this linear regression analysis to estimate
monotonicity, there was an error in the code, in which hue angle was treated
as radians, rather than degrees. This resulted in slightly different estimated
associations and fits. Once corrected, two color scales for shade did not meet
our statistical criterion of a fit of $R^{2}>.8$ in Exp. 1. Our statistical
criterion is just a heuristic, and visual inspection suggested that the
intermediate colors still vary monotonically such that the interpolated colors
were not more or less associated than the endpoints, so we kept them in the
analyses (see Fig. S.6A).

Fig. S.6: Predicted association ratings (circles filled according to color
scale) for colors within color scales that did not pass the monotonicity
heuristic criterion. (A) Predicted associations for shade (Exp. 1) do still
appear to vary monotonically in association strength. (B) Predicted
associations for sunshine (Exp. 2) vary such that intermediate colors are more
strongly associated than the endpoints. Color scales ordered with more
associated endpoint on the left and less associated endpoint on the right.

We selected our final eight color scales (Fig. 4) by identifying four high
association difference color scales and four low association difference color
scales. The high association difference color scales were those that had the
largest association difference for both sunshine and shade. The low
association difference color scales had association differences near zero for
both concepts. When making this final selection, we chose color scales that
appeared different from one another to provide variability in our stimulus
set.

We applied each color scale to 10 underlying datasets to create a set of 80
colormaps. Half the datasets featured the darker color on the left side (as in
Fig. S.8 top) and the other half featured the darker color on the right side
(as in Fig. S.8 bottom).

Experiment 2. We used the same approach as in Exp. 1, with two exceptions.
First, the only domain concept in this experiment was sunshine, so we only
checked monotonicity for sunshine, and not shade.

After exclusions, we examined the association differences between the endpoint
colors for the remaining color scales and selected 8 new color scales that
spanned the full range of association differences between the high and low
color scales used in Exp. 1. We also checked that colormaps generated using
these colors scales appeared different from each other. To create the
colormaps for this experiment, we applied these new color scales to the same
10 underlying datasets.

After collecting the data for Exp. 2, we discovered that two of the color
scales in this experiment did not meet our statistical criterion for
monotonicity due to the same coding error treating hue angle as radians
instead of degrees as described for Exp. 1. Visual inspection showed that
these color scales strongly deviated from the best fit line, and in one case
the intermediate colors were more associated with sunshine than the endpoints
(Fig. S.6B). Thus, we excluded data from these two color scales from our
analyses in Exp. 2.

Experiment 3. We used the same approach as in Exp. 1, with the following
exceptions. When selecting color pairs to generate the color scales, we
allowed black and white to be included to provide more possible color pairs to
choose from. Fig S.7 shows the mean direct association ratings for each of the
environmental concepts. Unique to this experiment, we investigated the full
range of signed association difference within each domain concept. This full
range of signed association difference corresponded to a full range of signed
semantic distance (when non-zero weight was placed on merit for direct
associations). To select color scales spanning this range, we grouped all
remaining color scales into three even-sized bins based on their signed
association difference. We selected seven color pairs from each bin, for
twenty-one color scales per domain concept that varied along the range of
association differences, such that for half the color scales, the direct
associations and dark-is-more bias were consistent (Fig. 8; right color
scales) and for half, the direct associations and dark-is-more bias were
conflicting (Fig. 8; left color scales).

During this process, we narrowed our set of concepts to glacial ice, ocean
water, and wild fire given there were color pairs for these concepts that
spanned nearly the full range of signed association differences, whereas the
color pairs for tree foliage and ground soil were more limited. We the checked
that the color space regression model strongly fit our chosen domain concepts
(multiple $R=.94$ for wild fire, multiple $R=.90$ for ocean water, and
multiple $R=.90$ for glacial ice). We also ensured that interpolations between
each pair of selected end point colors passed the monotonicity criterion.

Fig. S.7: Mean association ratings for (A) wild fire, (B) glacial ice, (C)
ocean water, (D) tree foliage, and (E) ground soil with each of the UW-71
colors (fill of bar). Colors are sorted from high to low association strength.
Fig. S.8: Colormap data visualizations generated using the 10 underlying
datasets in study. In the top row, the darker side is on the left, and in the
bottom row, the darker side is on the right. The examples in this figure were
created using one of the color scales from Exp. 1. Different color scales were
applied to these same 10 underlying datasets to create all the colormap
stimuli for this study.

###  S.5 Additional data for estimating merit

For each domain concept used in the colormap tasks, we collected direct
association ratings for the endpoint quantities of each domain concept (“no
[domain concept]” and “a lot of [domain concept]”), which was displayed at the
top of the screen. We used the means of these ratings to specify merit for
direct associations. For Exp. 3, we also collected darkness difference ratings
which we used to specify merit for the dark-is-more bias.

Sunshine endpoint associations. This task was the same as the direct
associations task described in Section 3.1.1, with the following exceptions.
Instead of rating associations for all UW-71 colors with the domain concept
name (e.g., sunshine), participants only rated their associations for each
endpoint color in the color scales and each endpoint quantity of the domain
concept (e.g., “no sunshine” and “a lot of sunshine”). Participants judged
each of the endpoint colors from the color scales in Exp. 2, which included
endpoint colors from Exp. 1 as well.

Before beginning the task, participants were shown the two endpoint concepts
and the set of endpoint colors, presented as individual squares. To anchor the
endpoints of the response scale, they were prompted to think about which color
they associated most and least for each endpoint concept (“no sunshine” and “a
lot of sunshine”).

After completing the block for sunshine, participants completed the attention
check as used in Exp. 3. For each endpoint concept, we transformed the ratings
from a scale of -200 to 200, to a scale of 0 to 1. We then averaged the mean
across all participants.

The target sample size was $n=30$ as in Exp. 1. We collected 30 participants
initially, determined how many participants were excluded, and iteratively
posted the experiment to collect data for the number of participants excluded
until reaching at least 30 participants. Exclusions were based on atypical
color vision (as assessed in Exp. 2) and the attention check (as used in Exp.
3) (48 Amazon mTurk workers in total, 17 excluded for atypical color vision, 0
for failing the attention check). Our final sample included 31 participants
(mean age = 39 years old; 20 women, 11 men). The experiment took approximately
10 min. and participants were compensated with $1.21.

Shade endpoint associations. We also collected endpoint associations for
shade, but they were not used in any of analyses in this paper. This task was
the same as the endpoint association ratings task for sunshine, except
“sunshine” was replaced with “shade”, and it only included the endpoint colors
from Exp. 1.

The target sample size was $n=30$. We collected 30 participants initially,
determined how many participants were excluded, and iteratively posted the
experiment to collect data for the number of participants excluded until
reaching at least 30 participants. Exclusions were based on atypical color
vision and the attention check (56 Amazon mTurk workers in total, 10 excluded
for atypical color vision, 16 for failing the attention check). Our final
sample size was 30 participants (mean age = 39 years old; 10 women, 20 men).
The average time to complete the experiment was approximately 5 min.
Participants were compensated with $0.60.

Environment concept endpoints. We collected endpoint direct association
ratings for the three domain concepts selected for Exp. 3: wild fire, glacial
ice, and ocean water. Domain concepts varied within-subject, presented in a
blocked randomized design, such that participants rated their associations for
the two endpoint concepts for a given domain concept, prior to moving on to
the next domain concept. The two endpoint concepts were presented in a random
order within each domain concept. At the start of each block, participants
were shown the endpoint concept names and the set of endpoint colors they
would be judging in that block to anchor the endpoints of the response scale.

The target sample size was $n=30$. We collected 30 participants initially,
determined how many participants were excluded, and iteratively posted the
experiment to collect data for the number of participants excluded until
reaching at least 30 participants. Exclusions were based on atypical color
vision and the attention check (37 undergraduate psychology students in total,
5 excluded for atypical color vision, 5 for failing the attention check). Our
final sample size was 30 participants (mean age = 19 years old; 25 women, 4
men, 1 non-binary). The experiment took approximately 30 min. and participants
received course credit.

Darkness difference ratings As described in the Section 5.1.2) of the main
text, we collected darkness difference ratings to quantify the degree to which
one endpoint color was clearly darker than the other endpoint color within
each color scale. The means of these ratings were used to quantify merit for
the dark-is-more bias (see Section 5.1.2).

We collected data from four volunteers with expertise in color perception.
During the task, participants saw two colored squares above the sliding scale
with the endpoints of the scale labeled “left color is clearly darker” and
“right color is clearly darker”. The middle point of the scale was labeled
“equal darkness.” Participants used the slider scale to indicate which color
in the pair they thought was darker, and the degree to which they thought that
color was darker. Participants rated each of the color pairs twice, once with
the darker color on the left and once with the darker color on the right.

At the beginning of the experiment, participants were shown an achromatic
color gradient (5 steps from black to white) and were instructed how to
interpret darkness. To indicate what we meant by darkness, participants were
told that the color gradient displayed ranged from black to white. Colors near
the left were darker, with black being the darkest, and colors near the right
were lighter, with white being the lightest. They were then told that all
colors fall somewhere along this dimension ranging for dark to light and to
keep this idea of darkness in mind during the experiment. We averaged ratings
across the two repetitions for each color pair for each participant, and then
averaged across participants. The experiment took approximately 15 min. and
participants were not compensated.

###  S.6 Power analyses for colormap tasks

To conduct power analyses for our logistic mixed-effect regression model, we
used the mixed power package in R [17].

Experiment 1. We conducted the power analysis using a logistic mixed-effects
regression model with data from a different study. The model predicted which
side of an achromatic colormap participants selected as representing more
alien animal sightings (left = 1, right = 0) from a factor coding which side
was darker (left darker = 1, right darker = -1). To achieve a power of at
least .8, we aimed to recruit 12 participants per condition (total $n=192$).

Experiment 2 We conducted the power analysis using data from participants in
Exp. 1 who judged low association difference colormaps representing sunshine.
The analysis was based on a binomial logistic mixed-effect regression
predicting the side selected (left = 1, right = 0) from which side was dark
(left darker = 1, right darker = -1), and a by-subject random intercept. The
goal was to have enough participants per condition to support stable
differences from chance, should they exist, while examining a fuller range of
association differences. To achieve a power of at least .8, recruited 42
participants per condition, which would also account for expected exclusions
for atypical color vision. Our target sample size was $n=640$.

Experiment 3. We conducted a power analysis with the data from Exp. 2, using a
binomial logistic regression in which we predicted which side was selected
(left = 1, right = 0) from which side was dark (left darker = 1, right darker
= -1) and the signed semantic distance for the pair (ranging from light-more
associated: -1 to dark-more associated: 1). To achieve a power of at least .8,
we aimed to recruit 20 participants per condition, which would also support
having enough participants to split the data into a testing set and training
set. The target sample size was $n=1260$.

###  S.7 Additional analyses

####  S.7.1 Experiment 1

An alternative analysis for evaluating the results from Exp. 1 is to use the
domain concept as a factor, and code the association difference as positive,
regardless of which color is more associated. In doing so, we can still test
for independent effects of direct associations and the dark-is-more bias.

With these predictors, we used a logistic mixed-effects regression to predict
participant’s response (left vs. right) from the concept (shade vs. sunshine),
the average (unsigned) association difference (centered), and which side was
dark (dark-left vs. dark-right). Here, we include all pair-wise interactions,
and the three way interaction, as well a by-subject random intercept (Table
S.1). Overall, participants chose the darker side, consistent with the dark-
is-more bias. The significant three-way interaction suggests responses varied
depending on the concept and color pairs used to create the colormap.

Table S.1: Logistic mixed-effects model predicting side chosen from dark side (dark), association difference (assoc), concept, and all interactions. Factor | $\boldsymbol{\beta}$ | SE | $\boldsymbol{z}$ | $\boldsymbol{p}$
---|---|---|---|---
Intercept | $.065$ | $.103$ | $.625$ | $.532$
Dark | $1.785$ | $.207$ | $8.639$ | $<.001$
Assoc | $.293$ | $.330$ | $.889$ | $.374$
Concept | $-.170$ | $.207$ | $-.823$ | $.410$
Dark*Assoc | $-4.821$ | $.660$ | $-7.303$ | $<.001$
Dark*Concept | $9.164$ | $.413$ | $22.18$ | $<.001$
Assoc*Concept | $-.201$ | $.660$ | $-.305$ | $.760$
Dark*Assoc*Concept | $11.67$ | $1.321$ | $8.833$ | $<.001$

We ran the model separately for shade and sunshine to better understand this
three-way interaction, and consequently the interactions of dark side $\times$
concept and dark side $\times$ association difference (Table S.2). For shade,
dark side was the only significant predictor. This indicates the darker side
was more likely to be selected as having more shade, regardless of direct
associations of the colors in the colormap.

For the concept sunshine, on average, participants also selected the darker
side more often. However, there was a significant interaction of dark side and
association difference. This interaction indicates that for low association
difference maps, participants were more likely to select the darker side as
having more sunshine, even though sunshine is generally associated with
lighter colors. However, for high association difference maps, participants
were more likely to select the lighter side as having more sunshine. This
pattern of results aligns with those discussed in Section 3.2 and indicates
that direct color-concept associations can override the dark-is-more bias, and
most closely resembles the prediction that both the dark-is-more bias and
direct associations influence people’s inferred mappings (Fig. 5A).

Table S.2: Logistic mixed-effects models predicting side chosen from dark side (dark), association difference (assoc), and their interaction for each concept. Concept | Factor | $\boldsymbol{\beta}$ | SE | $\boldsymbol{z}$ | $\boldsymbol{p}$
---|---|---|---|---|---
Shade | Intercept | $-.021$ | $.170$ | $-.121$ | $.904$
| Dark | $6.366$ | $.339$ | $18.772$ | $<.001$
| Assoc | $.193$ | $.542$ | $.356$ | $.772$
| Dark*Assoc | $1.012$ | $1.084$ | $.934$ | $.350$
Sunshine | Intercept | $.150$ | $.118$ | $1.268$ | $.205$
| Dark | $-2.797$ | $.236$ | $-11.86$ | $<.001$
| Assoc | $.394$ | $.377$ | $1.05$ | $.296$
| Dark*Assoc | $-10.66$ | $.753$ | $-14.15$ | $<.001$

####  S.7.2 Experiment 3

In Exp. 3 we aimed to develop and test a method for combining multiple sources
of merit. Here, we include additional details of the analysis reported in the
main text, and an additional analysis, comparable to those performed for Exp.
1 and 2, that allow for testing the independent effects of direct associations
and the dark-is-more bias.

As described in Section 5.2 of the main text, we used the data from the
testing set to evaluate whether the optimal weight pair was better for
predicting assignment inference than each source of merit alone. To examine
effects of relative weighting, we used a linear mixed effects model predicting
MSE for each color scale, with fixed effects for relative weighting, domain
concept, and their interaction (using Helmert contrasts). The model also
included a by-color scale random intercept and random slope for relative
weighting.

There was a main effect of relative weighting ($F(2,53.45)=9.18,p<.001$),
which was due to lower MSE for combined merit than for the average of each
source of merit alone ($t(53.55)=-4.24,p<.001$). Performance of each source of
merit alone did not significantly differ ($t(48.32)=-1.14,p=.26$). Our model
also revealed a main effect of concept ($F(2,60.76)=12.66,p<.001$). Fits were
significantly better for wild fire compared to the average of water and ice
($t(91.75)=-5.01,p<.001$), with no significant difference between ocean water
and glacial ice ($t(100.08)=.68,p=.50$). Concept interacted with relative
weighting ($F(4,65.63)=3.13,p=.02$), which was driven by the MSE for fire
being especially low for the optimal weight pairing compared to the average of
the other pairings ($t(66.84)=2.57,p=.01$), and being lower for direct
associations alone compared to dark-is-more alone ($t(103.15)=2.87,p=.005$).

To test for independent effects of direct associations and the dark-is-more
bias, we also used a binomial logistic regression to predict the side selected
(left vs. right) from a predictor coding which side was more associated and
the degree to which it was more associated measured by semantic distance using
association difference as merit (ranged from -1 to 1), and a predictor coding
which side was darker and the degree it was darker determined by darkness
difference ratings (scaled to range from -1 to 1), and a by-subject random
intercept and random effects for each fixed factor. Signed association
semantic distance ($B=16.91,SE=1.35;z=12.56,p<.001$) and darkness ratings
($B=13.66SE=1.25;z=10.91,p<.001$) both emerged as significant predictors,
further supporting that both contribute independent effects on inferred
mappings.

Fig. S.9: Mean proportion of times the darker side was selected, plotted as a
function of (A) signed association difference, and (B) signed semantic
distance using direct associations as merit, for each domain concept: wild
fire, ocean water, and glacial ice. Each mark represents a color scale, and
the stroke and fill colors represent the endpoint colors of the color scales.
Error bars represent standard errors of the means.

Recall that in Exp. 2, once association difference reached about $-.55$,
participants almost always inferred that the lighter side of the colormaps
mapped to more of the domain concept (sunshine). This plateau could be
explained by semantic distance reaching its limit (Fig. 7). Fig. S.9 shows
analogous plots for Exp. 3. It shows the mean proportion of times participants
selected the darker side for colormaps generated from each color scale for
each domain concept, as function of (A) association difference and (B) signed
semantic distance based on direct associations. The data for these concepts do
not appear to have the plateau around $-.55$.

We propose that the reason for this difference is that sunshine is more
inherently associated with light colors than the concepts in Exp. 3. That is,
the correlation between lightness (CIELAB L*) of the 71 colors and mean
association ratings for sunshine was strong ($r=.71$), whereas those
correlations are weak for wild fire ($r=-.03$), ocean water ($r=.05$), and
glacial ice ($r=.33$). Direct associations may be less effective at overriding
the dark-is-more bias for concepts that are less inherently light. If so, then
it may be necessary to build inherent lightness into simulations of assignment
inference in the future.

We also note that Exp. 3 included fewer color scales in the association
difference range $-.55$ to $-1$ where the plateau occurred in Exp. 2. As
discussed in the main text, glacial ice has extreme outliers in which the
dark-is-more bias strongly appears to override direct associations when they
conflict. These outliers may be due to activation of the opaque-is-more bias,
but future work designed to test the opaque-is-more bias will be necessary to
study its effects.

Table S.3: Coordinates for the University of Wisconsin 71 (UW-71) colors in CIE 1931 xyY, CIELAB, and CIELCh space. The white point used to convert between CIE 1931 xyY and CIELAB space was CIE Illuminant D65 (x = 0.313, y = 0.329, Y = 100). Table adapted from [22]. Color | x | y | Y | L* | a* | b* | C* | h
---|---|---|---|---|---|---|---|---
1 | $0.178\,13$ | $0.140\,21$ | $18.419$ | $50$ | $28.891$ | $-73.589$ | $79.057$ | $291.435$
2 | $0.1742$ | $0.082\,514$ | $4.4155$ | $25$ | $53.857$ | $-72.28$ | $90.139$ | $306.690$
3 | $0.217\,26$ | $0.135\,88$ | $18.419$ | $50$ | $53.857$ | $-72.28$ | $90.138$ | $306.690$
4 | $0.2591$ | $0.130\,88$ | $18.419$ | $50$ | $78.822$ | $-70.972$ | $106.066$ | $318.000$
5 | $0.187\,15$ | $0.191\,57$ | $18.419$ | $50$ | $2.6168$ | $-49.931$ | $50.000$ | $273.000$
6 | $0.190\,63$ | $0.1298$ | $4.4155$ | $25$ | $27.583$ | $-48.623$ | $55.902$ | $299.566$
7 | $0.231\,45$ | $0.184\,48$ | $8.419$ | $50$ | $27.583$ | $-48.623$ | $55.902$ | $299.566$
8 | $0.254\,95$ | $0.122\,84$ | $4.4155$ | $25$ | $52.548$ | $-47.315$ | $70.711$ | $318.000$
9 | $0.278\,72$ | $0.176\,35$ | $18.419$ | $50$ | $52.548$ | $-47.315$ | $70.711$ | $318.000$
10 | $0.327\,83$ | $0.1674$ | $18.419$ | $50$ | $77.514$ | $-46.006$ | $90.139$ | $329.310$
11 | $0.223\,97$ | $0.283\,99$ | $48.278$ | $75$ | $-23.657$ | $-26.274$ | $35.355$ | $228.000$
12 | $0.2081$ | $0.214\,15$ | $4.4155$ | $25$ | $1.3084$ | $-24.966$ | $25.000$ | $273.000$
13 | $0.244\,71$ | $0.253\,95$ | $18.419$ | $50$ | $1.3084$ | $-24.966$ | $25.000$ | $273.000$
14 | $0.262\,61$ | $0.273\,54$ | $48.278$ | $75$ | $1.3084$ | $-24.966$ | $25.000$ | $273.000$
15 | $0.286\,44$ | $0.198\,84$ | $4.4155$ | $25$ | $26.274$ | $-23.657$ | $35.355$ | $318.000$
16 | $0.297\,97$ | $0.240\,51$ | $18.419$ | $50$ | $26.274$ | $-23.657$ | $35.355$ | $318.000$
17 | $0.302\,72$ | $0.2622$ | $48.278$ | $75$ | $26.274$ | $-23.657$ | $35.355$ | $318.000$
18 | $0.369\,41$ | $0.181\,08$ | $4.4155$ | $25$ | $51.24$ | $-22.349$ | $55.902$ | $336.435$
19 | $0.352\,88$ | $0.2259$ | $18.419$ | $50$ | $51.24$ | $-22.349$ | $55.902$ | $336.435$
20 | $0.407\,95$ | $0.210\,59$ | $18.419$ | $50$ | $76.206$ | $-21.041$ | $79.057$ | $344.565$
21 | $0.237\,84$ | $0.356\,62$ | $72.065$ | $88$ | $-49.931$ | $-2.6168$ | $50.000$ | $183.000$
22 | $0.253\,32$ | $0.351\,08$ | $18.419$ | $50$ | $-24.966$ | $-1.3084$ | $25.000$ | $183.000$
23 | $0.269\,38$ | $0.345\,23$ | $48.278$ | $75$ | $-24.966$ | $-1.3084$ | $25.000$ | $183.000$
24 | $0.274\,73$ | $0.343\,27$ | $72.065$ | $88$ | $-24.966$ | $-1.3084$ | $25.000$ | $183.000$
25 | $0.313$ | $0.3290$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
26 | $0.312\,73$ | $0.329\,02$ | $4.4155$ | $25$ | $0$ | $0$ | $0$ | $0$
27 | $0.312\,73$ | $0.329\,02$ | $18.419$ | $50$ | $0$ | $0$ | $0$ | $0$
28 | $0.312\,73$ | $0.329\,02$ | $48.278$ | $75$ | $0$ | $0$ | $0$ | $0$
29 | $0.312\,73$ | $0.329\,02$ | $100.00$ | $100$ | $0$ | $0$ | $0$ | $0$
30 | $0.312\,73$ | $0.329\,02$ | $72.065$ | $88$ | $0$ | $0$ | $0$ | $0$
31 | $0.410\,44$ | $0.2905$ | $4.4155$ | $25$ | $24.966$ | $1.3084$ | $25.000$ | $3.000$
32 | $0.373\,53$ | $0.305\,34$ | $18.419$ | $50$ | $24.966$ | $1.3084$ | $25.000$ | $3.000$
33 | $0.3568$ | $0.311\,96$ | $48.278$ | $75$ | $24.966$ | $1.3084$ | $25.000$ | $3.000$
34 | $0.433\,76$ | $0.280\,95$ | $18.419$ | $50$ | $49.931$ | $2.6168$ | $50.000$ | $3.000$
35 | $0.491\,81$ | $0.256\,66$ | $18.419$ | $50$ | $74.897$ | $3.9252$ | $75.000$ | $3.000$
36 | $0.270\,22$ | $0.432\,68$ | $48.278$ | $75$ | $-51.24$ | $22.349$ | $55.902$ | $156.435$
37 | $0.276\,23$ | $0.418\,29$ | $72.065$ | $88$ | $-51.24$ | $22.349$ | $55.902$ | $156.435$
38 | $0.3075$ | $0.524\,35$ | $4.4155$ | $25$ | $-26.274$ | $23.657$ | $35.355$ | $138.000$
39 | $0.315\,61$ | $0.444\,08$ | $18.419$ | $50$ | $-26.274$ | $23.657$ | $35.355$ | $138.000$
40 | $0.316\,54$ | $0.410\,04$ | $48.278$ | $75$ | $-26.274$ | $23.657$ | $35.355$ | $138.000$
41 | $0.316\,52$ | $0.399\,17$ | $72.065$ | $88$ | $-26.274$ | $23.657$ | $35.355$ | $138.000$
42 | $0.417\,91$ | $0.4496$ | $4.4155$ | $25$ | $-1.3084$ | $24.966$ | $25.000$ | $93.000$
43 | $0.381\,79$ | $0.407\,28$ | $18.419$ | $50$ | $-1.3084$ | $24.966$ | $25.000$ | $93.000$
44 | $0.363\,55$ | $0.386\,34$ | $48.278$ | $75$ | $-1.3084$ | $24.966$ | $25.000$ | $93.000$
45 | $0.357\,35$ | $0.379\,28$ | $72.065$ | $88$ | $-1.3084$ | $24.966$ | $25.000$ | $93.000$
46 | $0.521\,74$ | $0.376\,56$ | $4.4155$ | $25$ | $23.657$ | $26.274$ | $35.355$ | $48.000$
47 | $0.446\,82$ | $0.369\,93$ | $18.419$ | $50$ | $23.657$ | $26.274$ | $35.355$ | $48.000$
48 | $0.410\,32$ | $0.362\,14$ | $48.278$ | $75$ | $23.657$ | $26.274$ | $35.355$ | $48.000$
49 | $0.508\,73$ | $0.333\,41$ | $18.419$ | $50$ | $48.623$ | $27.583$ | $55.902$ | $29.566$
50 | $0.566\,18$ | $0.298\,73$ | $18.419$ | $50$ | $73.589$ | $28.891$ | $79.057$ | $21.435$
51 | $0.297\,36$ | $0.577\,31$ | $18.419$ | $50$ | $-52.548$ | $47.315$ | $70.711$ | $138.000$
52 | $0.310\,49$ | $0.502\,94$ | $48.278$ | $75$ | $-52.548$ | $47.315$ | $70.711$ | $138.000$
53 | $0.313\,13$ | $0.478\,88$ | $72.065$ | $88$ | $-52.548$ | $47.315$ | $70.711$ | $138.000$
54 | $0.367\,53$ | $0.525\,08$ | $18.419$ | $50$ | $-27.583$ | $48.623$ | $55.902$ | $119.566$
55 | $0.359\,98$ | $0.471\,35$ | $48.278$ | $75$ | $-27.583$ | $48.623$ | $55.902$ | $119.566$
56 | $0.355\,93$ | $0.453\,07$ | $72.065$ | $88$ | $-27.583$ | $48.623$ | $55.902$ | $119.566$
57 | $0.436\,71$ | $0.472\,38$ | $18.419$ | $50$ | $-2.6168$ | $49.931$ | $50.000$ | $93.000$
58 | $0.4092$ | $0.439\,25$ | $48.278$ | $75$ | $-2.6168$ | $49.931$ | $50.000$ | $93.000$
59 | $0.398\,61$ | $0.426\,82$ | $72.065$ | $88$ | $-2.6168$ | $49.931$ | $50.000$ | $93.000$
60 | $0.502\,46$ | $0.421\,34$ | $18.419$ | $50$ | $22.349$ | $51.24$ | $55.902$ | $66.435$
61 | $0.4572$ | $0.407\,35$ | $48.278$ | $75$ | $22.349$ | $51.24$ | $55.902$ | $66.435$
62 | $0.563\,15$ | $0.373\,45$ | $18.419$ | $50$ | $47.315$ | $52.548$ | $70.711$ | $48.000$
63 | $0.617\,93$ | $0.329\,63$ | $18.419$ | $50$ | $72.28$ | $53.857$ | $90.139$ | $36.690$
64 | $0.300\,23$ | $0.564\,26$ | $72.065$ | $88$ | $-78.822$ | $70.972$ | $106.066$ | $138.000$
65 | $0.342\,64$ | $0.561\,47$ | $48.278$ | $75$ | $-53.857$ | $72.28$ | $90.139$ | $126.690$
66 | $0.344\,55$ | $0.532\,21$ | $72.065$ | $88$ | $-53.857$ | $72.28$ | $90.139$ | $126.690$
67 | $0.393\,81$ | $0.521\,23$ | $48.278$ | $75$ | $-28.891$ | $73.589$ | $79.057$ | $111.435$
68 | $0.388\,86$ | $0.499\,64$ | $72.065$ | $88$ | $-28.891$ | $73.589$ | $79.057$ | $111.435$
69 | $0.443\,88$ | $0.481\,31$ | $48.278$ | $75$ | $-3.9252$ | $74.897$ | $75.000$ | $93.000$
70 | $0.432\,44$ | $0.467\,14$ | $72.065$ | $88$ | $-3.9252$ | $74.897$ | $75.000$ | $93.000$
71 | $0.491\,96$ | $0.4425$ | $48.278$ | $75$ | $21.041$ | $76.206$ | $79.057$ | $74.565$