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Dissecting the risky-choice framing effect: Numeracy as an individual-difference factor in weighting risky and riskless options

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Using five variants of the Asian Disease Problem, we dissected the risky-choice framing effect by requiring each participant to provide preference ratings for the full decision problem and also to provide attractiveness ratings for each of the component parts, i.e., the sure-thing option and the risky option. Consistent with previous research, more risky choices were made by respondents receiving negatively framed versions of the decision problems than by those receiving positively framed versions. However, different processes were evident for those scoring high and low on numeracy. Whereas the choices of the less numerate showed a large effect of frame above and beyond any influence of their evaluations of the separate options, the choices of the highly numerate were almost completely accounted for by their attractiveness ratings of the separate options. These results are consistent with an increased tendency of the highly numerate to integrate complex numeric information in the construction of their preferences and a tendency for the less numerate to respond more superficially to non-numeric sources of information.
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Judgment and Decision Making, Vol. 3, No. 6, August 2008, pp. 435–448
Dissecting the risky-choice framing effect: Numeracy as an
individual-difference factor in weighting risky and riskless options
Ellen Peters?
Irwin P. Levin
Decision Research
Department of Psychology
Eugene, OR
University of Iowa
Abstract
Using ?ve variants of the Asian Disease Problem, we dissected the risky-choice framing effect by requiring each
participant to provide preference ratings for the full decision problem and also to provide attractiveness ratings for
each of the component parts, i.e., the sure-thing option and the risky option. Consistent with previous research, more
risky choices were made by respondents receiving negatively framed versions of the decision problems than by those
receiving positively framed versions. However, different processes were evident for those scoring high and low on
numeracy. Whereas the choices of the less numerate showed a large effect of frame above and beyond any in?uence of
their evaluations of the separate options, the choices of the highly numerate were almost completely accounted for by
their attractiveness ratings of the separate options. These results are consistent with an increased tendency of the highly
numerate to integrate complex numeric information in the construction of their preferences and a tendency for the less
numerate to respond more super?cially to non-numeric sources of information.
Keywords: numeracy, framing, individual differences, risky choices, attribute framing.
1 Introduction
outcomes are framed as losses (deaths). Later attempts to
replicate this phenomenon and extend it to other domains
Tversky and Kahneman’s (1981) introduction of the
such as money gained or lost rather than lives did not al-
Asian Disease Problem was among the earliest exam-
ways duplicate the literal preference reversal, but a gen-
ples of the malleability of human decision making. At
eral preference shift of more risky choices to avoid losses
the heart of this problem is the choice between a risk-
than to achieve gains is one of the most solid ?ndings
less option and a risky option of equal expected value.
in judgment and decision making research (see reviews
Because the current study will dissect the components of
by Kühberger, 1998; Levin et al., 1998). Later research
the prototypical risky-choice paradigm as exempli?ed by
uncovered task characteristics and individual difference
the Asian Disease Problem, we now describe these com-
factors that moderated the reliability and magnitude of
ponents. The Sure-Thing option offers a ?xed (riskless)
the risky-choice framing effect (Fagley & Miller, 1997;
outcome. In the Positive framing condition it is “save
Highhouse & Paese, 1996: Levin et al., 2002; Wang,
200 (out of 600) lives” whereas in the Negative condition
1996). The present study focuses on one such individual-
it is “400 will die.” The Risky option offers a “one-third
difference factor.
chance of saving all the lives and a two-thirds chance of
The aim of the current study is to dissect the risky-
saving no lives” in the Positive condition and a “one-third
choice framing effect into its component parts and to ex-
chance that no one will die and a two-thirds chance that
amine the moderating effect of an important individual-
all will die” in the Negative condition.
difference variable, numeracy, de?ned as the ability to
In response to this choice problem, the majority of de-
understand probabilistic and mathematical concepts. We
cision makers choose the riskless or “Sure-Thing” op-
asked participants in each framing condition to judge the
tion over the Risky option when potential outcomes are
full scenario and also to separately judge both the Sure-
framed as gains (lives saved) but choose the Risky option
Thing component and the Risky component. In that way,
over the Sure-Thing option when the exact same objective
we can assess the extent to which the Full Scenario fram-
?We thank Joshua Weller, Paul Windschitl, Paul Slovic, two anony-
ing effect is driven by framing of the separate compo-
mous reviewers, and Jon Baron for their helpful comments. This
nents, and we can compare this for individuals differing
work was supported by grants from the National Science Foundation
on a variable known to be associated with more super?-
(0517770 and 0350984) to the ?rst and second authors, respectively.
Address: Ellen Peters, Decision Research, 1201 Oak Street, Suite 200,
cial vs. more complex processing of numeric information
Eugene, Oregon, 97401. Email: empeters@decisionresearch.org
in decisions.
435

Judgment and Decision Making, Vol. 3, No. 6, August 2008
Numeracy and risky-choice framing
436
1.1 Numeracy moderates framing effects
cisions in important ways not captured by other measures
of achievement or ability.
Numeracy refers to the ability to understand and use
Although Peters et al. (2006) did not examine risky-
mathematical and probabilistic concepts. Based on the
choice frames, an unpublished Master’s Thesis by Gar-
National Adult Literacy Survey, almost half of the gen-
cia (2006, supervised by Peters), using a risky-choice
eral U.S. population has dif?culty with relatively simple
paradigm, found no effect of numeracy on risky-choice
numeric tasks such as calculating the difference between
framing problems.
We were curious about this lack
a regular price and sales price using a calculator or esti-
of ?nding given the robust nature of numeracy’s in?u-
mating the cost per ounce of a grocery item. These in-
ence on attribute framing, and our speculation that risky
dividuals do not necessarily perceive themselves as “at
choices in such problems were based on evaluations of
risk” in their lives due to limited skills; however, the re-
the two options comprising the choice: the Sure-Thing
search reviewed below demonstrates that having inade-
option and the Risky option. In prior studies of numeracy,
quate numeric skills is associated with lower comprehen-
highly numerate individuals have demonstrated deeper
sion and use of numeric information in health and ?nan-
processing of numeric information by showing smaller
cial domains.
framing effects (presumably caused by transforming the
Not surprisingly, greater ability with numbers leads
given numeric frame to its normative equivalent) and by
to more comprehension of numeric information in im-
being more likely to draw meaning from number com-
portant decisions (e.g., mammograms; Schwartz et al.,
parisons in judgments (Peters et al., 2006). The highly
1997). Numeracy relates in somewhat less intuitive ways
numerate appeared to integrate more sources of informa-
to a variety of cognitive and affective biases in deci-
tion than the less numerate. In a separate study, the highly
sion making (Peters et al., 2006). For example, Dehaene
numerate were more likely to be sensitive to numeric in-
(1997) suggests that, while children spend a lot of time
formation in judgments of the attractiveness of a hospital
learning the mechanics of math, they may not really un-
whereas the less numerate were insensitive to provided
derstand how to apply those mechanics even in adult-
numeric information and appeared to misattribute their
hood. We propose that those high in numeracy will be
current mood to the judgment instead (Peters et al., un-
more likely to do so. As a result, they should, for ex-
der review). Thus, it was curious in Garcia (2006) that
ample, ?nd alternative frames of the same number more
numeracy did not in?uence risky-choice framing effects
accessible and more in?uential in decisions.
in a similar manner with greater effects of the provided
Peters et al. (2006) examined numeracy’s effect on
frame on the less numerate. However, as pointed out by
framing of a single attribute by presenting participants
Levin et al. (1998), the risky-choice framing paradigm is
with the exam scores of ?ve psychology students and
more complex than the attribute-framing paradigm which
asking them to rate the performance of each student on
has been the source of previous work on the in?uence
a 7-point scale from –3 (very poor) to +3 (very good).
of numeracy. In attribute framing, a single attribute of
The framing of the exam scores was manipulated as ei-
an object is alternatively labeled in positive or negative
ther percent correct or percent incorrect so that “Emily,”
terms (e.g., success rate versus failure rate of a medical
for example, was described as having received either 74%
treatment) and its effect on the evaluation of that object is
correct on her exam or 26% incorrect. In a repeated-
assessed. No manipulation of risk is involved. In risky-
measures analysis of variance of the rated performance,
choice framing, the labeling of outcomes is manipulated
the usual framing effect was shown such that the more
and the element of risk is added by creating choice op-
positive frame elicited more positive ratings. Further-
tions of varying risk level.
more, the interaction of numeracy with the frame was
We developed the following hypotheses:
also signi?cant, with the less numerate participants show-
1. In order to replicate the basic Risky-Choice Framing
ing a stronger framing effect. These ?ndings are con-
effect, we expect that the Risky option will be pre-
sistent with high-numerate participants being more likely
ferred more than the Sure-Thing option in the neg-
to retrieve and use appropriate numerical principles and
ative framing condition than in the positive framing
transform numbers presented in one frame into a differ-
condition.
ent frame, and the less numerate responding more to the
affect communicated by the single given frame of the in-
2. Based on Garcia’s (2006) ?ndings, individuals high
formation. We believe that less numerate decision makers
and low in numeracy will demonstrate similar fram-
are left with information that is less complete and lacks
ing effects in risky choices.
the complexity and richness available to the more numer-
ate. Controlling for a proxy measure of intelligence (self-
3. Because the Sure-Thing option is similar to an
reported SAT scores) did not alter the results. Actual
attribute-framing problem (i.e., a single attribute is
number ability appears to matter to judgments and de-
manipulated such as “400 of 600 lives will be lost”),

Judgment and Decision Making, Vol. 3, No. 6, August 2008
Numeracy and risky-choice framing
437
we expect the frame to in?uence evaluations of the
full scenario was repeated each time a response was re-
Sure-Thing option more for the less numerate than
quired for either the full risky-choice problem or one of
the highly numerate.
its components.
4. Because the less numerate appear to integrate fewer
pieces of information and to respond more than the
2.4 Procedure
highly numerate to non numeric sources of informa-
tion such as mood states, they would be expected
Participants rated the ?ve Full Scenarios, the ?ve Sure-
to focus on a single favorable statement such as the
Thing options in two formats, and the ?ve Risky options
sure gain provided by the Sure-Thing option in the
in two formats, each presented in separate blocks in their
Positive frame or the possibility of no loss provided
booklet to ensure that the separate ratings for each part of
by the Risky option in the Negative frame. By con-
the same scenario were spaced far enough apart to reduce
trast, the highly numerate who use numeric informa-
memory effects. Participants made four other responses
tion more completely are expected to be more capa-
(the other four scenarios) before revisiting the same sce-
ble of integrating all the information from both op-
nario. Each response took about one minute. Further-
tions in their choices. Thus, choices of the highly
more, the response scales were varied between the full
numerate should be more in?uenced by their evalu-
risky-choice problem and the components.
ations of the separate options.
Each participant received the same frame throughout
the experiment. The Full Scenarios were always pre-
sented ?rst. Participants were not allowed to look back
2 Method
at earlier responses. Each participant then responded to
both the Sure-Thing and Risky options in two separate
formats in different blocks of trials. The Sure-Thing op-
2.1 Participants
tion was presented in one block as a numerical count
Participants were 108 students (42% female) ful?lling a
(e.g., 200 people will be saved) and in another block as
research participation component of an introductory mar-
a fraction (1/3 of the 600 people will be saved) in or-
keting course at the University of Iowa.
der to examine whether different effects of frame were
produced; the Risky option was presented once with the
better outcome (e.g., 1/3 chance of saving all lives) ?rst
2.2 Design
and again with the worse option (2/3 chance of saving no
lives) presented ?rst. Four different orders of presenta-
Participants were randomly assigned to a Positive frame
tion of these four blocks were constructed and counter-
group (N = 53) or a Negative frame group (N = 55).1
balanced across participants.
Within each group, participants rated their degree of pref-
In the Full Scenario, participants were asked to check
erence between the options in the Full Scenario task and
one of seven boxes labeled from “Much prefer A” (the
then provided separate ratings of the attractiveness of the
Sure-Thing option) to “Much prefer B” (the Risky op-
Sure-Thing and Risky options. They did this for each
tion) with a midpoint of “A and B are equal.” This expan-
of ?ve scenarios. Further procedural variations are de-
sion of the usual dichotomous choice was done in order
scribed below.
to provide continuous numerical data for the statistical
(regression) analyses (see Levin et al., 2002). Responses
2.3 Materials
were scored such that higher numbers represent greater
preferences for the Risky option.
Five scenarios were constructed, each patterned after the
To evaluate the Sure-Thing option and the Risky option
Asian Disease Problem but different in content domain
separately, participants were asked to circle a number be-
and in the expected value of the options. The Positive
tween –3 (Very bad) and +3 (Very good) with a midpoint
and Negative frame versions of the scenarios are repro-
of 0 (Neither bad nor good) to indicate their evaluation
duced in Appendix A. Brie?y, one is an exact replication
of that particular option.
of the Asian Disease Problem except that it was simply
called an “unusual disease” from Sweden, one involves
animals endangered by wild?res, one involves crop de-
2.5 Individual difference measures
struction from a severe drought in another country, one
involves loss of medical bene?ts in another country, and
After completing the ratings tasks, participants were
one involves investment losses. The introduction to the
asked to complete the following: a demographic informa-
tion sheet including age, gender, GPA, and ACT scores;
1The positive group is Versions 1–4 in the accompanying data ?le.
the 18-item Need for Cognition scale (Cacioppo et al.,

Judgment and Decision Making, Vol. 3, No. 6, August 2008
Numeracy and risky-choice framing
438
1984); and the 11-item Numeracy scale shown in Ap-
less numerate will respond more to the given frame of in-
pendix B (Lipkus et al., 2001).2
formation (which was the same across the formats for a
All inferential statistics used mean-deviated continu-
given participant) rather than the numbers. We retained
ous scores (Irwin & McClelland, 2001). A median split
only the usual formats (the count format for the Sure-
on numeracy was used for descriptive statistics and to
Thing option and the better outcome ?rst for the Risky
identify low and high scorers so that inferential analyses
option) in further analyses.
could be conducted separately within each group.
3.2 Separate analyses of Full Scenarios and
3 Results
components
In contrast to some previous studies, men and women
We next examined the Full Scenarios to test for the usual
scored about the same on numeracy (scores = 9.5 and 9.2,
risky-choice framing effect and to verify that numeracy
respectively, t(106) = 1.0, p = .32). Higher numeracy was
again did not moderate the effects of framing at this level.
associated with higher self-reported GPA and higher ACT
A repeated-measures ANOVA of choice preferences was
scores (r = .16, p < .10 and .28, p < .01). Numeracy and
conducted with the ?ve problems as the repeated mea-
Need for Cognition were not signi?cantly related (r = .10,
sures and frame (–1, 1), numeracy (continuous and mean-
ns).
deviated), and their interactions as the predictors. Con-
sistent with Hypothesis 1, an overall effect of frame was
found, F(1, 104) = 18.9, p < .001, with individuals pre-
3.1 Analysis of the dual formats for the
ferring the sure-thing option in the domain of gains —
sure-thing and risky options
the positive frame — and the risky option in the domain
We ?rst examined whether the two formats of the
of losses — the negative frame (average choice prefer-
Sure-Thing options (counts versus proportions) and,
ences = 3.51 and 4.31, respectively, where a response
separately, of the Risky option (the two orders)
of 4 indicates no preference between the two options
produced different framing effects on evaluations.
and lower numbers indicate a preference for the sure op-
A repeated-measures analysis of variance (repeated-
tion). Numeracy and its interaction with frame were non-
measures ANOVA) was conducted of the attractiveness
signi?cant (F(1, 104) = .04, p = .85 and F(1, 104) = 1.8,
ratings for the Sure-Thing options in the ?ve scenar-
p = .28, respectively). The effects of frame differed by
ios with format, frame, numeracy (continuous, mean-
scenario, F(4, 416) = 2.9, p < .05, with framing effects
deviated), and their interactions as predictors. A sim-
being nonsigni?cant in the scenarios in which the risky
ilar analysis was conducted of attractiveness ratings of
option had a higher expected value than the sure-thing
the Risky options. Format did not signi?cantly in?uence
option (the Spanish drought and Delta’s medical-bene?t
the attractiveness ratings as a main effect or in interac-
crisis). In the positive frame of both scenarios (where de-
tion with frame or numeracy for either the Sure-Thing or
cision makers are generally risk-averse), preferences for
Risky options.
the risky option were noticeably stronger for both high-
Correlations between responses to the two formats
and low-numerate participants. See Table 1 for prefer-
were similar for the low and high numerate (average r =
ence means by frame and numeracy.
.58 and .60 between the two Sure-Thing formats, respec-
As a result of this initial analysis, we dropped the
tively, for individuals low and high in numeracy across
two non-signi?cant framing problems and focused fur-
the ?ve scenarios and average r=.41 and .53, respectively,
ther data analysis on the three scenarios that showed sig-
between the two Risky formats). This consistency might
ni?cant effects of frame on risky choices (but see the
be considered puzzling from the standpoint that individ-
tables for results with the two dropped scenarios). A
uals lower in numeracy presumably have more dif?culty
repeated-measures ANOVA of those three scenarios re-
using numbers in judgments and decisions and therefore
vealed a stronger overall effect of frame, F(1, 104) =
should perhaps be less consistent. The consistency is not
30.5, p < .001, with average choice preferences of 3.0
puzzling, however, from the standpoint that the less nu-
and 4.2, in the positive and negative frames, respectively.
merate may process different information than the highly
In this analysis, the main effect of numeracy remained
numerate, with the less numerate processing numeric in-
nonsigni?cant, but the highly numerate demonstrated a
formation more super?cially. Our expectation is that the
marginal tendency towards smaller framing effects than
the less numerate, F(1, 104) = 3.3, p = .07.
2An example of an easy item is “Which of the following numbers
As suggested in Hypothesis 2, the effects of frame
represents the biggest risk of getting a disease? 1 in 100, 1 in 1000, 1
by numeracy were not conventionally signi?cant in the
in 10.” An example of a hard item is “In the Acme Publishing Sweep-
stakes, the chance of winning a car is 1 in 1,000. What percent of tickets
risky-choice frame (the Full-Scenario decision), and the
of Acme Publishing Sweepstakes wins a car?”
effects of frame were signi?cant in separate analyses of

Judgment and Decision Making, Vol. 3, No. 6, August 2008
Numeracy and risky-choice framing
439
Table 1: Preference means by frame and numeracy.
Overall
Less numerate (5–9)
Higher numerate (10–11)
Main
Main
Main
effects of
effects of
effects of
Full Scenario (1–7 scale)
Neg
Pos
frame in
Neg
Pos
Neg
Pos
frame in
frame in
ANOVA
ANOVA
ANOVA
F (1,106)
15.4,
1. Sweden - disease
4.45
3.32
4.65
3.00
p < .001
4.28
3.63
ns
p < .001
17.0,
2. Stock
4.18
2.96
4.12
3.08
p < .05
4.24
2.85
p < .001
p < .001
12.0,
3. Wild?re season
3.84
2.77
3.85
2.69
p < .05
3.83
2.85
p < .05
p < .001
4. Drought in Spain
4.58
4.38
ns
4.77
4.15
ns
4.41
4.59
ns
5. Delta SS
4.47
4.09
ns
4.38
4.08
ns
4.55
4.11
ns
Average preference across the
5 scenarios
18.9,
4.31
3.51
4.35
3.40
p < .001
4.26
3.61
p < .05
(repeated-measures results of
p < .001
Scenarios 1–5)
Average preference across the
?rst 3 scenarios that showed
30.5,
signi?cant framing effects
4.20
3.00
4.21
2.92
p < .001
4.11
3.11
p < .01
p < .001
(repeated-measures results of
Scenarios 1-3)
Note: Higher numbers represent greater preference for the risky option.
N = 108; n = 52 and 56 for less and higher numerate, respectively.
low and high numerate groups. Previous studies, how-
frame (attractiveness means = .53 and –.28, respectively,
ever, have shown that more and less numerate decision
F(1, 104) = 10.8, p < .01). As hypothesized, less numer-
makers appear to use different sources of information in
ate individuals showed stronger framing effects than did
decisions, setting the stage for our hypotheses concern-
the highly numerate, interaction F(1, 104) = 3.9, p = .05.
ing different information-processing mechanisms under-
Examination of the means by frame separately within
lying risky-choice framing effects for those low and high
low and high numerate groups (based on a median split,
in numeracy. Thus, we turn to an analysis of framing in
the highly numerate scored 10 or 11 correct out of 11 pos-
evaluations of the separate options next.
sible, whereas the low-numerate group scored between
A repeated-measures ANOVA was conducted of the at-
5 and 9 correct3) revealed a signi?cant framing effect
tractiveness ratings of the remaining three Sure-Thing op-
among the less numerate (attractiveness means in the pos-
tions with frame, numeracy (mean-deviated and contin-
itive and negative frame were .73 and -.54, respectively,
uous) and their interaction as predictors. Previous re-
p < .001) and a non-signi?cant effect for the highly nu-
search has demonstrated that individuals lower in numer-
merate (means = .35 and –.05, respectively, ns). In no
acy show stronger attribute-framing effects than those
case was the framing effect for a scenario greater for the
higher in numeracy. As stated in Hypothesis 3, we ex-
highly numerate than for the less numerate. See Table 2
pected that ratings of the Sure-Thing option would be
for attractiveness means by frame and numeracy.
similar to an attribute frame. The overall effect of frame
3Individuals in this study were fairly numerate overall, with only
was signi?cant with Sure-Thing options in the positive
13% of them scoring between 5 and 7 correct, 12% scoring 8 correct,
frame rated as more attractive than those in the negative
23% with 9 correct, and 26% each scoring 10 and 11 correct.

Judgment and Decision Making, Vol. 3, No. 6, August 2008
Numeracy and risky-choice framing
440
Table 2: Attractiveness means of the Sure-Thing options by frame and numeracy.
Overall
Less numerate (5–9)
Higher numerate (10–11)
Main
Main
Main
effects of
effects of
effects of
Sure-thing component
Neg
Pos
frame in
Neg
Pos
Neg
Pos
frame in
frame in
ANOVA
ANOVA
ANOVA
F (1,106)
3.8,
1. Sweden - disease
– 0.27
0.32
– 0.65
0.38
p = .02
0.07
0.26
ns
p = .06
8.2,
2. Stock
– 0.44
0.40
– 0.69
0.69
p < .01
– 0.21
0.11
ns
p < .01
12.5,
3. Wild?re season
– 0.13
0.89
– 0.27
1.12
p < .001
0.00
0.67
ns
p < .001
Average rating across three
10.8,
scenarios (repeated-measures – 0.28
0.53
– 0.54
0.73
p < .001
– 0.05
0.35
ns
p < .001
results)
Scenarios not included:
4. Drought in Spain
– 0.13 – 0.09
ns
– 0.31 – 0.08
ns
0.03 – 0.11
ns
5. Delta SS
– 0.62 – 0.40
ns
– 0.77 – 0.15
ns
– 0.48 – 0.63
ns
Average rating across ?ve
5.2,
scenarios (repeated-measures – 0.32
0.22
– 0.54
0.39
p < .01
– 0.12
0.06
ns
p < .05
results)
We conducted a similar repeated-measures ANOVA
3.3 Full-Scenario risky-choice framing ef-
with attractiveness ratings of the Risky option. In this
fects as a function of evaluations of the
case, we were not sure what to expect because two di-
separate options
ametrically opposite effects are theoretically possible.
First, negative frames could lead to the gamble being
Thus far, we have found different effects of frame for the
perceived as more attractive, consistent with Prospect
more and less numerate in evaluations of the separate op-
Theory’s psychological shift towards risk seeking in this
tions. This result may explain the overall lack of effect of
frame compared to the positive frame. Second, negative
numeracy on risky-choice framing. Speci?cally, the less
frames could lead to poorer evaluations of the gamble
numerate showed stronger framing effects in their evalua-
compared to positive frames, consistent with an attribute-
tions of the Sure-Thing option; the highly numerate were
framing effect. Results of the analysis demonstrated no
not in?uenced by frame in their evaluations of the Sure-
overall effects of frame or its interaction with numer-
Thing option but appeared to be somewhat in?uenced by
acy. However, an analysis by scenario shown in Table
a framing effect consistent with Prospect Theory in their
3 indicated a nonsigni?cant tendency for the highly nu-
evaluations of the Risky option. Thus, we turn to an ex-
merate to rate the attractiveness of the Risky component
amination of the extent to which the frame, the evalua-
as higher in the negative frame than the positive frame
tions of both options, and their interactions with numer-
condition in every scenario, whereas the less numerate
acy in?uenced the full risky-choice framing effect. Hy-
tended to show the opposite pattern. This pattern was
pothesis 4 stated that the frame would have a direct in-
signi?cant for the highly numerate when all ?ve scenar-
?uence on choice preferences of the less numerate with
ios were considered.
little in?uence from their evaluations of the separate op-

Judgment and Decision Making, Vol. 3, No. 6, August 2008
Numeracy and risky-choice framing
441
Table 3: Attractiveness means of the Risky options by frame and numeracy.
Overall
Less numerate (5–9)
Higher numerate (10–11)
Main
Main
Main
effects of
effects of
effects of
Risky component
Neg
Pos
frame in
Neg
Pos
Neg
Pos
frame in
frame in
ANOVA
ANOVA
ANOVA
F (1,106)
1. Sweden - disease
0.00 – 0.36
ns
– 0.31 – 0.38
ns
0.28
– 0.33
p = .10
2. Stock
– 0.04 – 0.13
ns
– 0.19
0.31
ns
0.10
– 0.56
ns
3. Wild?re season
0.02
0.04
ns
– 0.27
0.04
ns
0.28
0.04
ns
Average rating across three
scenarios (repeated-measures – 0.01 – 0.15
ns
– 0.26 – 0.01
ns
0.22
– 0.28
ns
results)
Scenarios not included:
4. Drought in Spain
0.13 – 0.15
ns
0.12
0.19
ns
0.14
– 0.48
p = .10
2.9,
5. Delta SS
– 0.04 – 0.51
– 0.15 – 0.15
ns
0.07
– 0.85
p = .02
p = .09
Average rating across ?ve
scenarios (repeated-measures
0.01 – 0.22
ns
– 0.16
0.00
ns
0.17
– 0.44
p = .05
results)
tions, suggesting a more super?cial reaction to the verbal
fer the Sure-Thing option more than those in the negative-
cues identifying the positive and negative frames. For the
frame condition (b = –.51, t(100) = –4.9, p < .001). In
highly numerate, however, we expected that the separate
addition, individuals who rated the Sure-Thing option as
evaluations of the Sure-Thing and Risky options would
more attractive were more likely to prefer it (b = –.22,
guide their choice preferences, indicating their more ana-
t(100) = –2.6, p = .01) and those who rated the Risky
lytical, componential approach to decisions involving nu-
option as more attractive were more likely to prefer the
meric information.
Risky option (b = .23, t(100) = 2.7, p < .01). Numer-
To simplify this analysis, we constructed average rat-
acy did not have a direct in?uence on risky-choice prefer-
ings across the three problems for evaluations of the Sure-
ences. However, the main effects were quali?ed by three
Thing option and the Risky option, as well as average
interactions. Individuals lower in numeracy were in?u-
choice preferences for the Full-Scenario risky choices. A
enced more than those higher in numeracy by the frame
regression analysis was conducted of the average choice
in the risky-choice scenarios (interaction b = .19, t(100)
preference in the Full-Scenario risky choices; indepen-
= 2.6, p = .01). Choice preferences of individuals high
dent variables were the direct effects of frame, the aver-
in numeracy were in?uenced more than those low in nu-
age mean-deviated Sure-Thing attractiveness rating, the
meracy by the attractiveness of the Sure-Thing option as
average mean-deviated Risky attractiveness rating, nu-
well as the Risky option (interaction b = –.17, t(100) =
meracy (mean-deviated and continuous), and the three
–3.1, p < .01 and b = .13, t(100) = 2.1, p < .05, respec-
two-way interactions with numeracy (model F(7, 100)
tively). A similar pattern of results was shown in each of
= 8.3, p < .001, R-squared = .37). Although the frame
the three scenarios with strongest effects in the Swedish
was associated with the component ratings, correlations
disease problem (see Table 4).
among the predictors were small enough that multi-
In order to examine these numeracy interactions in
collinearity did not appear to be a problem (the tolerances
more detail, a median split was performed on numer-
were acceptable; range = .72-.96).
acy. Regression analyses of the average choice prefer-
The results indicated a main effect of frame such that
ence were conducted with frame, the average Sure-Thing
individuals in the positive-frame condition tended to pre-
attractiveness rating, and the average Risky attractiveness

Judgment and Decision Making, Vol. 3, No. 6, August 2008
Numeracy and risky-choice framing
442
Table 4: Regression results (b-values) predicting the Full-Scenario risky-choice preferences by scenario, on average
across the three scenarios, and on average across the ?ve scenarios.
Sure-
Sure-
Risky
Overall
Risky
Thing
Thing
Frame *
option
model,
Frame
option
Numeracy
option
option
numeracy
rating *
F (7, 100)
rating
rating *
rating
numeracy
numeracy
1. Sweden - disease
6.8***
–.55***
–.15
.26**
.02
.26**
–.24***
.18*
2. Stock
4.1***
–.61***
–.15
.17
–.01
.19
–.17*
.02
3. Wild?re season
2.8**
–.41**
–.25*
.08
–.06
.10
–.03
.10
Average of 3 scenarios
8.3***
–.51***
–.22**
.23**
.00
.19**
–.17**
.13*
4. Drought in Spain
1.1
–.09
–.17
.19
–.02
.13
–.14
.06
5. Delta SS
2.4*
–.10
–.17
.37**
.12
–.07
–.09
.01
.12
Average of 5 scenarios
7.1***
–.34***
–.24**
.30***
.02
–.15**
.07
(p=.07)
Note: * p < .05; ** p < .01; *** p < .001.
rating as predictor variables, separately within each me-
Another way to examine the extent to which partici-
dian split of numeracy (see Table 5 for detailed results).
pants were consistent in their evaluations is to compare
For individuals high in numeracy, the model strongly pre-
the inferred choice preference calculated from the attrac-
dicted their preferences (model F(3, 52) = 11.1, p < .001,
tiveness ratings of the separate options and the actual
R-squared = .39). Frame was the smallest in?uence on
choice preference made when the two options were pre-
their preferences compared to the ratings of the separate
sented jointly. To do this, for each pair of ratings, we
options; the average attractiveness ratings of the Sure-
subtracted the rating for the Sure-Thing option from the
Thing and Risky options were both highly predictive.
rating for the Risky option. Thus, higher values meant
Among the less numerate, the model was less predic-
greater inferred preference for the Risky option over the
tive of their preferences but still highly signi?cant (model
Sure-Thing option, just as it does for the actual choice
F(3, 48) = 6.8, p < .001, R-squared = .30). Frame was the
preference. Correlations by scenario were then calculated
strongest predictor by far of their preferences; average at-
between the choices inferred from these differences and
tractiveness ratings of the Sure-Thing and Risky options
the actual choice responses, separately within the low-
did not predict their choice preferences.
and high-numerate groups. Finally, the average correla-
tion (computed across the ?ve scenarios) was calculated
We partialled out the effects of ACT scores and Need
within each group.
For Cognition (NFC) from numeracy scores using regres-
The results demonstrated greater consistency of in-
sion and then used the resulting numeracy residuals to
ferred and actual preferences among the high numerate
conduct again the analyses of the Full-Scenario risky-
than among the less numerate. Speci?cally, the aver-
choice problems above. The results did not change in any
age correlation between inferred and actual choice pref-
substantial way. A second analysis, after partialling out
erences across the ?ve scenarios was .45 and .09 for
the effects of GPA from numeracy instead, also showed
the high and low numerate, respectively (the respective
no substantial changes from the original analysis. Nu-
ranges of correlations were .36 to .63 and .01 to .15).4
meracy appears to in?uence framing effects over and
above these proxy measures of intelligence and prefer-
4
ence for thinking harder (see Appendix C for more de-
An analysis with participants and (all ?ve) items as crossed random
effects (Baayen, Davidson, & Bates, in press) supported these results.
tailed results). A similar set of analyses was conducted
Speci?cally, the dependent variable was risky choice, and the ?xed-
using NFC scores in place of numeracy (with and with-
effect predictors were the Risky-Sure attractiveness difference, frame
out partialling out the effects of numeracy, ACT scores,
(positive/negative), numeracy, and the interactions of numeracy with the
and GFA). Neither NFC nor its interactions with frame or
attractiveness difference and with frame. Of primary interest, both in-
teractions with numeracy were signi?cant (as assessed by MCMC sam-
ratings of the component options signi?cantly predicted
pling, p = .0012 and .0228, respectively). That is, high numeracy
choices in any analysis.
was associated with a stronger relationship between the Full-scenario

Judgment and Decision Making, Vol. 3, No. 6, August 2008
Numeracy and risky-choice framing
443
Table 5: Regression results (b-values) predicting the risky-choice preferences in low and high numeracy groups.
Low numerate (n = 52)
High numerate (n = 56)
(scores = 5–9 correct out of 11)
(scores = 10–11 correct out of 11)
Sure-
Sure-
Overall
Risky
Overall
Risky
Thing
Thing
model
Frame
option
model
Frame
option
option
option
F (3, 48)
rating
F (3, 52)
rating
rating
rating
1. Sweden - disease
6.2**
–.92***
.18
.01
11.7*** –.10
–.47***
.58***
2. Stock
2.1
–.61*
.10
.08
7.7*** –.54**
–.43**
.27
3. Wild?re season
2.5
–.46
–.16
–.06
3.9**
–.35
–.34*
.21
Average of 3 scenarios
6.8*** –.66***
.02
.04
11.1*** –.31*
–.44***
.43***
4. Drought in Spain
0.7
–.32
.06
.09
2.9*
.16
–.41**
.34*
5. Delta SS
1.4
–.17
.06
.29
4.5**
–.04
–.33*
.45**
Average of 5 scenarios
5.7**
–.50***
.01
.18
11.3*** –.16
–.44***
.43***
Note: * p < .05; ** p < .01; *** p < .001.
4 Discussion
mial distribution), but the size of the difference was small
with standard deviations of responses ranging from 1.38
These results help us understand both the risky-choice
to 1.75 for the less numerate and 1.32 to 1.68 for the
framing phenomenon and the role of numeracy in deci-
highly numerate. Finally, the less numerate may attend
sion making. Risky-choice framing effects appear robust
to the individual-option information in the risky choice
(with the highly numerate demonstrating a tendency to-
and may translate it appropriately, but fail to integrate the
wards a smaller effect of frame compared to the less nu-
information in their choices, suggesting that choices are
merate). These effects, however, do not appear to be a
not always based on integrating judgments of provided
singular effect as previously thought, but rather have dif-
options. This suggestion is supported by the lower ob-
ferent underlying mechanisms for different people.
served consistency among the less numerate between in-
In particular, in analyses of the less numerate, they
ferred preferences based on component ratings and actual
demonstrated a direct effect of frame on their choices
preferences.
and no signi?cant effects of how attractive they found
The less numerate also demonstrated a larger fram-
the separate options. This lack of in?uence may be due
ing effect on their evaluations of the Sure-Thing options.
to the less numerate attending more super?cially to the
These results are consistent with previous numeracy re-
individual options and attempting less to evaluate them;
sults with attribute framing (Peters et al., 2006). In the
they may focus, for example, on the sure gain in the
analysis of the Full-Scenario framing effect, these Sure-
Sure-Thing option when choosing in the positive frame
Thing evaluations in?uenced their choice preferences less
and the possibility of no loss provided by the Risky op-
relative to the preferences of those higher in numer-
tion in the negative frame. It may also be, however, that
acy. At face value, this result was somewhat inconsis-
the less numerate are less able to translate information
tent with the robust numeracy effect in attribute-framing
about a given option into an attractiveness rating. If this
studies; it was, however, consistent overall with Peters
is the case, then the ratings of the less numerate should
et al.’s (2006) interpretation of their attribute-framing re-
be less reliable (more variable) than those of the highly
sults. Speci?cally, they argued that the less numerate re-
numerate. There was some evidence consistent with this
spond more to the given frame of information whereas the
hypothesis, with the standard deviation of responses to
highly numerate demonstrated more complex processing
the ?ve Sure-Thing components, ?ve Risky components,
of the same information.
and ?ve Full scenarios being higher for the less numer-
Consistent with this argument, the model was more
ate compared to the highly numerate in 11 of the 15 re-
predictive of preferences of the highly numerate than the
sponses (p = .059, based on a sign test using the bino-
less numerate. The highly numerate showed only a small
direct effect of frame over and above their evaluations of
risky-choice preference and the attractiveness difference, and with rela-
tively more risky choice in the positive frame (and relatively less in the
the individual options. Evaluations of both options in-
negative frame).
?uenced their choice preferences as if the highly numer-

Judgment and Decision Making, Vol. 3, No. 6, August 2008
Numeracy and risky-choice framing
444
ate were able to keep on-line both pieces of information
judgment and decision making. In a number of choice
during the choice process whereas the less numerate may
situations with options varying on dimensions such as
not be able to keep as much numeric information on-line;
complexity, there are opposing models for the decision-
pilot data suggest that the highly numerate have some-
making process that pit an algebraic comparison of the
what greater working-memory capacity compared to the
integrated values of the choice options against a heuris-
less numerate (Peters, 2006) which could exacerbate this
tic solution based on which option “feels” better. For
effect. Overall, the highly numerate nonetheless appear
example, performance on ambiguity-probability tradeoff
to consider both the possibility of ensuring some saving
tasks originating with Ellsberg’s (1961) two-color prob-
of lives (their evaluations of sure-thing options) and the
lem has been explained as either a process where the un-
possibility of avoiding loss of life (evaluations of risky
known probability for an ambiguous option is estimated
options). One caveat on this argument is the correlational
and then compared to the known probability of the unam-
nature of the design combined with all participants re-
biguous option or as a process where the option with the
sponding to the Full Scenarios ?rst and the component
most information is preferred (see Lauriola et al., 2007).
options second. Given this, it is possible that participants
By having each respondent rate both ambiguous and un-
simply attempted to be consistent from the ?rst choice
ambiguous options separately as well as making a forced
response to the later attractiveness ratings, and the highly
choice, and by including individual-difference measures
numerate were more successful at this consistency. Fur-
such as those used by Lauriola et al., comparisons can be
ther studies should explore both these working-memory
made between those individuals who are more apt to be
and order issues.
in?uenced by component evaluations and those who are
These results provide additional evidence that the
more apt to use simplifying heuristics in forced-choice
highly numerate not only understand numbers better, as
situations. As another example, the well-known “decoy”
shown in past studies, but they use them more frequently
or “asymmetric dominance effect” in consumer behav-
as well. This is important for two reasons. First, it
ior involves a change in preference between two options
again highlights the distinction between comprehension
when a third option is introduced which is inferior to one
and use of information. Decision makers all may know
option on all attributes but is superior to the other option
that 200 is one-third of 600, but bringing this information
on one attribute. The result is an increase in preference
(e.g., evaluations of the Sure-Thing option) to bear on de-
for the dominating option (Huber et al., 1982). Using the
cisions is another matter. Compared to the less numerate,
present approach to unravel the processes used by differ-
the highly numerate appeared to use more complex pro-
ent consumers, respondents would rate each of the sepa-
cessing of given numeric information in the construction
rate options as well as make both the two-option choice
of their preferences. Second, the results are a reminder
and the three-option choice. In other words, the approach
that information providers can, and perhaps should, pro-
described in the current paper can be used to catego-
vide additional assistance to decision makers by paying
rize individuals in terms of whether choices are based
careful attention to how numeric information is provided
on the separate evaluations of choice options or whether
in order to help the less numerate make better use of nu-
new, usually simplifying, factors emerge in head-to-head
merical information (e.g., Fagerlin et al., 2007). Testing
forced choices.
formats may be critical, as the intuitions of information
providers concerning best ways to present information
References
may not be adequate. As Fischhoff (in press) argued,
“One should no more release untested communications
Baayen, R. H., Davidson, D. J., & Bates, D. M. (in press).
than untested pharmaceuticals.”
Mixed-effect modeling with crossed random effects for
These results demonstrate that the well-known risky-
subjects and items. Journal of Memory and Language.
choice framing effect is more than one effect. Debiasing
Cacioppo, J. T., Petty, R. E., & Kao, C. F. (1984). The
these effects, therefore, may require different methods for
ef?cient assessment of need for cognition. Journal of
different people. For the less numerate, providing nu-
Personality Assessment, 48, 306–307.
meric information in formats that are easier to evaluate
Dehaene, S. (1997). The number sense: How the mind
(and allow them access into the meaning of the informa-
creates mathematics. New York: Oxford University
tion that can then be used in decisions; Peters et al., 2007)
Press.
may be best. The highly numerate may already access
Ellsberg, D. (1961). Risk, ambiguity and the savage ax-
the meaning of the information, and debiasing may in-
ioms. Quarterly Journal of Economics, 75, 643–669.
volve encouragement to think harder about the individual
Fagerlin, A., Ubel, P. A., Smith, D. M., & Zikmund-
options.
Fisher, B. A. (2007). Making numbers matter: Present
At a broader level, we feel that the paradigm intro-
and future research in risk communication. American
duced here can be applied to other phenomena in human
Journal of Health Behavior, 31, S47-S56.

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