Is 1 out of 20 riskier than 5%? effect of representing unlikely events as ratio versus percentage on risk perceptions

Risk perceptions, or intuitive judgments about risk that people rely on in characterizing and evaluating hazards and technologies (Slovic, 1987), are fundamental to theoretical models of health behavior (Ferrer and Klein, 2015) and behavioral decision-making (Tversky and Kahneman, 1981). In domains such as health, finance, product and service purchases, perceptions of uncertainty about the outcome and adverse consequences are a critical factor in influencing individuals’ decision processes including information search (e.g. Dowling and Staelin, 1994), willingness to adopt innovations (e.g. Shimp and Bearden, 1982), purchase intentions (e.g. Dholakia, 2001; Li et al., 2020; Nardi et al., 2020), online search and purchase behavior (Forsythe and Shi, 2003) and engaging in risk reducing behavior (e.g. Dowling and Staelin, 1994). Perceptions of risk are likely to be more powerful in explaining consumer behavior when consumers are more motivated to avoid mistakes than to maximize utility in purchasing (Mitchell, 1999). Since risk can be a crucial barrier that prevents individuals from choosing a specific brand or offering (Conchar et al., 2004), firms often provide numeric product performance information as an estimate of the risk, presumably to mitigate risk perceptions prior to purchase or encourage the purchase of risk-reducing components (e.g. warranties, insurance). Government agencies such as the National Highway Traffic Safety Administration (NHTS) and Consumer Product Safety Commission (CPSC), as well as independent entities such as the Insurance Institute for Highway Safety (IIHS), and Consumer Reports, provide numeric performance information that may be used in assessing risk prior to purchase. However, the numeric performance information is often presented as a ratio or a percentage. For example, the IIHS introduced its new side crash test in the form of a ratio by stating “In the first tests of 2020–21 vehicles, only one out of 20 small SUVs, the 2021 Mazda CX-5, earns a good rating” (Link to Small SUVs struggle in new, tougher side testLink to the Website). In contrast, an appliance retailer publishes reliability of the brands it sells by stating the percentage of units requiring service (or repair) out of the total it has sold (e.g. 3.3% of LG’s front load washers require service ^[1]). Finally, Colgate launched a campaign to help raise awareness about the association between oral health and diabetes ^[2]. The ads contained statistical information in both ratio and percentage formats (e.g. 1 out of 3 adults has prediabetes; in some states over 10% of the population has diabetes).

Consider two individuals contemplating the purchase of a new laptop computer. Both are interested in the same brand and model. While one individual comes across a product review stating that 1 out of 20 of the brand’s laptops is expected to require repairs, the other individual reads a review reporting that 5% of the brand’s laptops are expected to require repairs. Although the risk is relatively low and objectively equivalent, the frequency of failure in the two reviews is represented differently (as is the numeric description by IIHS and the appliance retailer) such that the expected performance information is provided in terms of ratio (1 out of 20) and percentage (5%) formats, respectively ^[3]. In this research we investigate whether the perceived risk and purchase intentions for a product will be different across the two presentation formats, and if yes, what is driving the effects. At first blush, it would appear that the relatively large literature examining the influence of numeric format (e.g. Kirkpatrick and Epstein, 1992; Slovic et al., 2000; Yamagishi, 1997), as well as the framing of attributes and outcomes (e.g. Tversky and Kahneman, 1981; Levin et al., 1998; Mandel, 2001), must have addressed these questions. However, a critical examination of the literature reveals that the answers to these questions are not clear, particularly for low likelihood (unlikely) events. As such, despite a large literature on the effects of numeric format, our research adds to the handful of studies that examine risk perceptions and behavioral intentions as a function of how an identical normalized frequency is represented as a ratio versus a percentage (Brase, 2002; Monahan et al., 2002; Peters et al., 2011; Slovic et al., 2000; Tan et al., 2005). We highlight how our research compares with related work in Table 1, and outline our contribution next

Our research contributes to the existing literature in several ways. First, rather than examining frequencies with identical numerators (e.g. 10 out of 100 vs 10%), we examine frequencies represented by different numerators (e.g. 1 out of 20 vs 5%; Slovic et al., 2000). Studies comparing identical numerators reported no difference between the ratio and percentage representations on risk perceptions (Monahan et al., 2002; Peters et al., 2011), whereas studies comparing frequencies with different numerators provide inconsistent evidence regarding how an identical frequency represented as a ratio versus a percentage affects risk perceptions and behavioral intentions (Slovic et al., 2000).

Second, since extant research suggests different ways in which individuals may respond to unlikely events represented as a ratio versus a percentage, a key contribution of this research is to offer evidence for the underlying reasoning for the observed effects. In the context of violent psychiatric behavior, although Slovic et al. (2000) speculated that the ratio format might have created a frightening image compared to the percentage format, they did not test for the underlying reasoning and thus did not offer any evidence to support their speculation. We argue that a low likelihood event represented as a ratio is more concrete than an equivalent percentage. We explicitly test the hypothesis that because concrete (vs abstract) representations are more vivid and easier to imagine, perceptions of risk are likely to be higher when a low likelihood event is represented as a ratio versus a percentage.

Third, an understanding of the underlying reasoning allows us to identify and test boundary conditions for the effects. To trace the reasoning that a ratio (vs an equivalent percentage) is more concrete, we show that risk perceptions are higher for a ratio versus a percentage representation when the ratio is paired with concrete processing and the percentage is paired with abstract processing, but the effects attenuate when the ratio is paired with abstract processing and the percentage is paired with concrete processing. Further, to the extent that representing a frequency as a ratio (vs percentage) facilitates spontaneous imagining of the situation, we show that the effect of frequency format attenuates when individuals are asked to imagine (or visualize) the frequency information in a “rational” manner that impairs spontaneous processing. We thus provide evidence for the underlying reasoning using both mediation and moderation.

Fourth, our research extends the domain of inquiry from personal health (Brase, 2002; Globic et al., 2022) and violent behavior (Slovic et al., 2000) to product evaluations and purchase intentions. While previous research has examined the effect of numeric format on perceptions of personal (or physical) risk, we focus on the effect of representing an identical frequency as a ratio versus a percentage on perceptions of functional (or financial) risk (Jacoby and Kaplan, 1972) which may not be as affect-laden. Further, the valence of product performance information can be varied easily such that it is described negatively as the frequency of failure (e.g. 5% or 1 out of 20 requires repairs) or positively as the frequency of success (e.g. 95% or 19 out of 20 do not require repairs). Although the performance frequency is the same across the negative and positive descriptions, altering the valence changes the corresponding magnitude of the numbers as well. Nonetheless, from an applied as well as a public policy perspective, a secondary objective of our research is to test whether valence (and thus magnitude) moderates the effect of representing an equivalent frequency as a ratio versus a percentage on risk perceptions.

It is worth noting that our research is distinct from existing studies on risky choice framing where several options differing in risk level are compared (e.g. Tversky and Kahneman, 1981). Further, unlike the Bayesian reasoning literature which suggests that judgments based on natural frequencies are more accurate and simpler than those based on probabilities (e.g. Brase et al., 1998; Gigerenzer and Hoffrage, 1995; Hoffrage et al., 2000), our focus in on whether representing a single normalized frequency as a ratio (e.g. 1 out of 20 or 2 out of 20) versus an equivalent percentage (5% or 10%) affects risk perceptions, and if so, why.

Risk perception the intuitive judgment of how individuals characterize and evaluate any threat or hazard (Slovic, 1987), has been the topic of extensive research in a variety of domains (e.g. Loewenstein et al., 2001; Hsee and Rottenstreich, 2004; Slovic, 2013). Much of this research focuses on how to improve risk assessment methods as well as on identifying ways to effectively communicate risk and encourage individuals to engage in preventive or protective behavior (e.g. Hoffrage et al., 2000; Sloan and Platt, 2011; Slovic et al., 1982). We examine how a relatively low frequency represented as a ratio versus a percentage affects risk perceptions and decision-making in the context of product performance information.

Standard economic theory predicts no difference in risk perceptions when a frequency is represented as a ratio or as an equivalent percentage. The assumption is that individuals are perfect information processors, emotionless, deliberate and thus make choices that are context-invariant (Tversky et al., 1988). However, extensive research demonstrates that individuals simplify decisions, rely on heuristics (e.g. Hamilton and Koukova, 2008) and the normative principle of descriptive invariance is commonly violated across many substantive domains (e.g. Raghubir and Srivastava, 2002; Shafir et al., 1997). In the present context, the question is how risk perceptions are affected by the manner in which an equivalent frequency is represented (e.g. Denes-Raj and Epstein, 1994; Kirkpatrick and Epstein, 1992; Yamagishi, 1997).

Based on past life experiences, individuals may intuit that any event with a ratio of a relatively small number out of a large number (e.g. 1 out of 20) is highly unlikely to occur (Denes-Raj and Epstein, 1994; Koehler and Macchi, 2004). Reports of statistical information in the popular press such as “only 1 out of 30 adults can identify all 50 states” or “only 2 out of 20 such cases are reported” appear to reinforce the relative infrequency or low incidence of such events or occurrences. In a game where participants had an opportunity to win money by drawing a red jelly-bean from two bowls, a bowl containing 1 red out of 10 jelly-beans and another containing 10 red out of 100 jelly beans, Kirkpatrick and Epstein (1992) reported that 76.9% of the participants chose to draw from the bowl containing the larger number of beans, despite both bowls containing an identical 10% of the red beans (see also Denes-Raj and Epstein, 1994). Although the frequency information was provided in both ratio and percentage formats, the findings suggest a ratio bias or a tendency of individuals to judge an unlikely event as more likely when the ratio information is presented as a ratio of relatively large numbers compared to an equivalent ratio expressed in smaller numbers.

Similarly, in single evaluation tasks (vs joint evaluation tasks such as choice between options) in the health domain, Yamagishi (1997) reported that individuals judged cancer to be riskier when the frequency was described by a ratio of large numbers compared to an equivalent ratio of small numbers (e.g. killing 2,414 out of 10,000 people vs killing 24.14 out of 100 people). In another study, individuals perceived risk to be higher when information about a medical treatment was presented as “50 out of 1,000 who follow a medical treatment will die” compared to “5 out of 100 who follow a medical treatment will die” (Pinto-Prades et al., 2006). Since both the joint and single evaluation tasks examined two explicit ratios, these findings apply only if a percentage is naturally (and implicitly) processed as out of 100. The notion that a percentage may be naturally processed as out of 100 is supported by findings which document no difference between X% and X out of 100 (Monahan et al., 2002; Oudhoff and Timmermans, 2015; Peters et al., 2011). In the context of the two individuals considering the purchase of a laptop, this stream of literature suggests that risk perceptions are likely to be lower when the risk is represented as a ratio (1 out of 20) versus an equivalent percentage (5%).

The anchoring and adjustment heuristic (i.e. individuals’ tendency to rely heavily on the first piece of information or the anchor, and then adjust insufficiently upward or downward based on other information; Tversky and Kahneman, 1974) and denominator neglect (i.e. individuals focus on the numerator of a fraction or ratio and ignore the denominator) have been offered as a joint explanation for the ratio bias. Yamagishi (1997) suggested that individuals may be more sensitive to the numerator as it serves as a natural, salient and easy-to-use anchor, whereas they are relatively insensitive to the denominator as it is likely to be underweighted or even neglected. If individuals anchor on the numerator and do not adjust adequately for the denominator (or neglect it), risk perceptions monotonically increase with the nominal value of the numerator.

There is a third possibility about how risk perceptions may differ across a relatively low frequency being represented as a ratio versus a percentage. In a study where psychologists rated the risk and likelihood of a single event (i.e. a patient suffering from a mental illness harming someone during the first several months after being discharged), Slovic et al. (2000) reported that the patient was judged as posing higher risk when the likelihood of violent behavior was presented as a ratio (10 out of 100; 20 out of 100) versus a percentage (10%; 20%). While not providing any evidence for the underlying reasoning for this effect, Slovic et al. (2000) speculated that the ratio format may have created a frightening image of one or more patients committing acts of violence in the minds of the clinicians leading to more extreme judgments relative to the percentage format. Gigerenzer (2002) attributed Slovic et al.’s (2000) findings to differences in the reference class used to assess the percentage relative to the ratio where the reference class was clearly specified (i.e. other patients like the one being assessed). However, in the same study, Slovic et al. (2000) reported that judgments of likelihood of harming someone (another measure of risk) were higher when the response scale was based on percentages than ratios. In another study about violence risk communication with a sample of American Psychological Association psychologists, Monahan et al. (2002) found no difference in risk estimates across ratio (20 out of 100) and percentage (20%) formats in the overall sample. Peters et al. (2011) also reported no difference in risk perceptions across ratio (10 out of 100) and percentage (10%) formats. The existing literature thus provides an inconsistent picture regarding the effect of representing an equivalent frequency as a ratio versus a percentage on risk perceptions. Importantly, for reported differences, there are no explicit tests and thus no evidence for the underlying reasoning.

In contrast to the ratio bias discussed earlier (e.g. Kirkpatrick and Epstein, 1992), research shows that individuals overestimate the likelihood of “1-in-X” ratios (e.g. 1 in 13; 1 in 200) compared to equivalent “N-in-X*N” ratios (e.g. 10 in 130; 5 in 1000; Pighin et al., 2011, 2015; Sirota and Juanchich, 2019; Sirota et al., 2018) or percentages (Oudhoff and Timmermans, 2015; Sirota et al., 2014). The underlying reasoning for these findings is, however, unclear. Pighin et al. (2011) noted the need for further research to pinpoint what causes the “1-in-X” effect. They speculate whether it is driven “by the increased ability to see oneself or others as that affected.” In probing the underlying reason for the difference between ratio and percentage representations, our conceptualization considers “1 in X” ratios as a subset of possible expressions of low likelihoods (e.g. 2 out of 20). In other words, our research contributes to the literature on “1-in-X” ratios as well by outlining the underlying process and generalizing the effect to other ratios with small numerators.

In the context of assessing risk based on product performance information, we argue that the numeric format affects risk perceptions because of the differences in how ratio and percentage information is processed. Frequencies in the form of ratios or counts, postulated to correspond more closely to how individuals experience and learn statistical information over their lives (Gigerenzer and Hoffrage, 1995; Hoffrage et al., 2000), have been found to be clearer and easier to process and understand (Brase, 2002; Denes-Raj and Epstein, 1994). To the extent that individuals experience statistical information via an implicit count of the number of times some event occurs or does not occur, we argue that ratios are more concrete than equivalent percentages. Concrete representations are more vivid and easier to mentally imagine than abstract representations (Loewenstein et al., 2001; Newell et al., 2008; Sherman et al., 1985; Slovic et al., 2005). Research suggests that providing individuals with concrete information enhances vividness of the imagery which then upwardly bias perceived likelihoods (MacInnis and Price, 1987). In other words, the more vividly and easily individuals envision a scenario, the higher the estimates that the scenario will occur (Kahneman et al., 1982; Sherman et al., 1985). Mental imagery is also superior in influencing purchase likelihood because it more closely resembles the actual experience than does cognitive elaboration (MacInnis and Price, 1987). Thus, a frequency in the form of a ratio (e.g. 1 out of 20 or 2 out of 20) will be more concrete, and thus more easily imagined and visualized, than an equivalent percentage (e.g. 5% or 10%), leading to higher risk perceptions and lower purchase intentions.

If a ratio is more concrete and more easily imagined, it is likely to induce higher negative affect, compared to an equivalent percentage. Negative affect or affectivity is a general dimension of subjective distress and unpleasurable engagement that subsumes a variety of aversive mood states, including anger, contempt, disgust, guilt, fear and nervousness (Watson et al., 1988). The difference in negative affect across the ratio and percentage formats is consistent with the affect as information model which suggests that individuals use the affect induced by a target stimulus (i.e. numeric representation) at the time of judgment as information in the judgement and decision task (e.g. Hsee and Rottenstreich, 2004; Schwarz and Clore, 1983). For example, Hsee and Rottenstreich (2004) reported that affect rich decisions encourage valuation by feeling such that individuals were insensitive to numeric scope information relative to valuation by calculation (e.g. willingness to pay for 5 vs 10-CD set varied for those primed to calculate but not for those primed with feelings). The differential negative affect across the two numeric formats is also consistent with the risk as feelings model (Loewenstein et al., 2001) which suggests that individuals’ reactions are immediate, instinctive, intuitive, automatic and experiential, and that risk assessments are a direct consequence of the induced emotions such as feelings of worry, fear, dread or anxiety. However, we argue that the difference in negative affect is a consequence of the ratio being more concrete, and thus more easily imagined than the equivalent percentage, particularly when the numerators are relatively small (e.g. 1 out of 20 vs 5%; 2 out of 20 vs 10%).

It is important to note that in a somewhat related research, Raghubir (2008) demonstrated that the more salient the denominator of a base rate, whether due to its smaller nominal scale (e.g. 0.903 / 1000 vs 90.3 / 100,000), smaller population unit (e.g. state = 35 million vs campus = 32,000) or the proximity of the represented population (e.g. nearby vs far away), the higher the attention paid to the numerator, and the higher the estimated risk. The author argues that smaller populations that are geographically proximate are easier to relate to than larger and more distant ones. Unlike Raghubir’s (2008) focus on examining differences across equivalent ratios, our research investigates differences across equivalent ratios and percentages (e.g. 1 / 20 vs 5%). Further, we explicitly test and demonstrate that the concreteness of a ratio drives the effect of numeric framing on risk perceptions and behavioral intentions.

Product performance information can be described with a negative or positive valence (Levin et al., 1998). For example, ground beef can be described negatively (25% fat) or positively (75% lean), the performance of a basketball player can be described negatively (20% of shots missed) or positively (80% of shots made) and medical treatments can be described negatively (mortality rate) or positively (survival rate). Similarly, the same relatively low likelihood of product failure can be framed negatively in terms of requiring repairs (i.e. 1 out of 20 and 5%) or positively in terms of not requiring repairs (i.e. 19 out of 20 and 95%). Theoretically, varying the valence (negative vs positive) implies representing the frequency with numbers of different magnitudes, as is common in the framing literature (Levin et al., 1998). Further, since products are unlikely to be successful with very high rates of failure (e.g. 19 out of 20 require repairs), from a pragmatic perspective, we examine whether valence (or frame) moderates the effect of frequency representation on risk perceptions (e.g. 5% versus 1 out of 20 and 95% versus 19 out of 20). Recognizing that negative valence is expressed with small numbers and positive valence is expressed with large numbers, we focus on the differential effect of ratio versus percentage in the context of unlikely events separately across the negative (e.g. 5% vs 1 out of 20) and positive (95% vs 19 out of 20) valence. In examining risk perceptions as a function of numeric format and valence, we also contribute to the relatively limited research on the role of valence in moderating the effects of numeric format.

The findings regarding effectiveness of attribute/outcome frame (or valence) have been somewhat inconsistent as some studies find greater persuasiveness with a positive frame (e.g. Levin and Gaeth, 1988), some studies show greater persuasiveness with a negative frame (e.g. Meyerowitz and Chaiken, 1987), and yet other studies find no differences (e.g. Beach et al., 1996; Levin et al., 1988). Nonetheless, because negative frames typically lead to more unfavorable evaluations than positive frames (Levin et al., 1998), and individuals pay more attention and react more strongly to negative information (e.g. Kahneman and Tverksy, 1984; Skowronski and Carlston, 1989), risk perceptions are likely to be higher when the valence is negative (vs positive). The more extreme reaction to negative information coupled with the argument that ratio information is more concrete than percentage information suggests that the difference in imagining oneself in the risk situation is likely to be higher when the valence is negative (i.e. 1 out of 20 vs 5%) than positive (19 out of 20 vs 95%).

Figure 1 presents our conceptual model. Based on the arguments presented so far, we state the following formal hypotheses about the effect of numeric format and outcome valence on risk perceptions, negative affect and extent of imagining:

Construal level theory (Trope and Liberman, 2010; Vilches-Montero and Spence, 2015) suggests that individuals may view an event/object at various levels of abstraction. A high-level construal is more abstract such that events are represented in terms of global, superordinate and decontextualized features, whereas a low-level construal is more concrete such that events are represented in terms of local, subordinate and contextualized features. In the context of a product purchase, the high-level, abstract representation specifies why the product is to be purchased, whereas the low-level, concrete representation specifies how the product is to be purchased (see Trope and Liberman, 2010).

We propose that the effect of numeric format on risk perceptions will be moderated by individuals’ construal level. Specifically, the effect of numeric format on risk perceptions is likely to be stronger when the ratio format is paired with a more concrete, low-level construal and the percentage format is paired with a more abstract, high-level construal. The effects are likely to attenuate when the ratio format is paired with a more abstract construal and the percentage format is paired with a more concrete construal. The rationale is that since the ratio format is more concrete, a concrete, low-level construal will facilitate processing, leading to more congruent and fluent processing, thus accentuating the effect of ratio (vs percentage) format on risk perceptions. In contrast, an abstract, high-level construal will hinder processing of the ratio information, such that the processing is less congruent and fluent, thus attenuating the effects of ratio versus percentage format on risk perceptions.

We further argue that when there is a match between numeric format and the construal level, the congruence and fluency in processing the frequency information affects the extent of imagining and the associated negative affect. In other words, the extent of imagining and negative affect are expected to be higher when the ratio format is paired with a more concrete, low-level construal and the percentage format is paired with a more abstract, high-level construal. The effects are likely to attenuate when the ratio format is paired with an abstract construal and the percentage format is paired with a concrete construal. Stated formally:

Finally, the theory of planned behavior postulates that behavioral, normative and control beliefs are antecedents to attitudes, subjective norm and perceived control, respectively, which in turn affect intentions and behavior (Ajzen, 1991). This suggests that perceptions of risk are likely to mediate the effect of numeric format on intentions and behavior. Expecting the same pattern as risk perceptions, we also test whether numeric format affects purchase intentions, an important variable for marketers.

Study 1 begins the investigation by examining the effect of numeric format on risk perceptions in both the negative and positive valence conditions. Study 2a establishes the generalizability and robustness of the findings by extending the inquiry to a slightly higher risk and to a ratio that is not represented as “1 out of any number” (i.e. 2 out of 20 vs 10%). Study 2b examines the effect of numeric format on risk perceptions at several risk levels: 20% (4 / 20), 10% (2 / 20) and 5% (1 / 20) to test for possible threshold effects. Study 2b also sheds insight into the underlying process by testing whether the extent to which individuals can imagine (or identify) themselves in the risk situation mediates the effect of numeric format and valence on risk perceptions and purchase intentions. Study 3 investigates whether construal level, or the level of concrete versus abstract thinking, moderates the effect of numeric format on risk perceptions. Finally, to provide evidence that it is the extent of imagining that underlies the effect of numeric format, Study 4 focuses on the extent to which numeric format facilitates spontaneous imagining of the situation.

For all studies, we predetermined a sample size to allow us to detect a medium effect size (i.e. f = 0.25), with a desired power of 0.80, and an alpha level of 0.05 (minimum sample size of 158 for a 2 by 2 conditions design). Further, we used between-subject design with random assignment in all studies. We implemented Study 1 with college students from a private university in the Northeast (sophomores through seniors as a part of a participant pool), while Studies 2a, 3 and 4 were executed on Amazon’s Mechanical Turk with a more diverse sample of U.S. residents (see details in each study including the gender and age distribution of the participants). Similarly, Study 2b was conducted on prolific.com to increase the external validity of the findings by using another platform popular with academic researchers. We report all items and measures collected in each individual study, and data exclusions. Further, we took measures to ensure that there was no overlap of participants across studies. Data analysis for each study was done after all the data for the study was collected. Data for the five studies are available on Open Science Framework (OSF) ^[4]. Complete stimuli and measures are available in Appendix A. This research was approved by the Institutional Review Boards of the respective institutions of the researchers.

Study 1 examines whether risk perceptions are higher when a relatively low frequency is represented as a ratio (1 out of 20) versus an equivalent percentage (5%). A secondary objective is to examine whether the effect of numeric format attenuates when the valence is positive (19 out of 20 versus 95%).

Two hundred and one undergraduate business students participated in a 2 (numeric format: ratio and percentage) × 2 (valence: positive and negative) between-subject design with random assignment for course credit. We did not collect age and gender of the participants because of a technical glitch. The sample for Study 1 is from the same population as the follow-up study briefly described in the supplementary material (the studies were implemented a few years apart with no overlap in participants). Participants were asked to imagine that they were considering the purchase of a new laptop computer and had come across an excerpt from an article about a new laptop that would soon be introduced to the market. Frequency format and valence were manipulated by citing PC World’s independent testing of the performance of the new laptop. In the negative valence, the review stated that “5% or 1 out of 20 of the computers manufactured by this manufacturer are (is) expected to require repairs.” In the positive valence, the review stated that “95% or 19 out of 20 of the computers manufactured by this manufacturer are expected to operate without any repairs.”

After reading the excerpt, participants responded to three seven-point scale items that were averaged into a risk perceptions index (α = 0.67): “Considering the possible problems associated with a laptop, how much risk would you say would be involved in purchasing the new laptop?” (1 = Very little risk; 7 = A great deal of risk), “How risky do you feel it would be for you to purchase this new laptop computer?” (1 = Not risky at all; 7 = Very risky), and “How likely is that the new laptop will require repairs in the next two years?” (1 = Very unlikely; 7 = Very likely).

Table 2 summarizes the results for all studies (means, standard errors and cell sizes). A 2×2 ANOVA on risk perceptions revealed significant effects of numeric format (⁠$MRatio$ = 3.70 vs $MPercent$ = 3.37; F(1, 197) = 3.90, p = 0.050, $ηp2$ = 0.019) and valence (⁠$MPositive$ = 3.13 vs $MNegative$ = 3.94; F(1, 197) = 23.30, p < 0.001, $ηp2$ = 0.106). The main effects were qualified by a significant two-way interaction (F(1, 197) = 14.84, p < 0.001, $ηp2$ = 0.070). Specifically, as expected, risk perceptions were significantly higher in the ratio format relative to the percentage format when the valence was negative (⁠$M1 out of 20$ = 4.43 vs M_5% = 3.45; F(1, 197) = 16.40, p < 0.001, $ηp2$ = 0.077), but there was no significant difference when the valence was positive (⁠$M19 out of 20$ = 2.97 vs M_95% = 3.28; F(1, 197) = 1.83, p = 0.178; $ηp2$ = 0.009, see Figure 1 below). These findings provide support for H1a and H1b. Although not central to our theorizing, the findings also reveal that while risk perceptions did not differ across the positive and negative valence when the frequency was described as a percentage (F(1, 197) =0.487, p = 0.486, $ηp2$ = 0.002), risk perceptions were higher in the negative versus the positive valence when the frequency described as a ratio (F(1, 197) = 36.74, p < 0.001, $ηp2$ = 0.157).

Consistent with the idea that numeric information is more concrete when represented as a ratio (vs percentage), Study 1 shows that risk perceptions were higher when the frequency was represented as a ratio (1 out of 20) than an equivalent percentage (5%). Study 1 also shows that while risk perceptions were higher in the ratio versus percentage format when the valence was negative (1 out of 20 vs 5%), there was no difference when the valence was positive (19 out of 20 vs 95%). Two aspects of the findings are noteworthy. First, the relatively large numbers represented by 19 out of 20 and 95% appear to have been perceived similarly. This finding is consistent with prior research which finds that the effect of information on product risk perceptions is greater when the information is framed negatively than positively (e.g. Grewal et al., 1994). Second, Study 1 shows that valence had a stronger effect on risk perceptions when the frequency was represented as a ratio versus percentage. The finding that risk perceptions do not vary across the negative and positive valence when the frequency was represented as a percentage (5% vs 95%) is inconsistent with prior findings demonstrating the effect of framing on evaluations (e.g. Levin and Gaeth, 1988). Our results are, however, consistent with studies which find no differences (e.g. Beach et al., 1996; Levin et al., 1988). In a review of the effects of framing, Levin et al. (1998) note that framing effects are less likely to be manifested when the percentages used are relatively extreme (e.g. Beach et al., 1996), as was the case in our studies (5% and 95%). However, since negative valence is expressed with small numbers and positive valence is expressed with large numbers in our studies, we focus on the differential effect of ratio versus percentage within the negative (5% versus 1 out of 20) and positive valence (95% versus 19 out of 20) separately.

We examined the robustness of Study 1 findings to situations where the product performance could be described as not functioning/functioning satisfactorily rather than requiring/not requiring repairs (negative vs positive outcome description, Mandel, 2001). The follow-up study, described in the supplementary material (Appendix B), corroborates the finding of Study 1. Risk perceptions were higher in the ratio format relative to the percentage format when the valence was negative (1 out of 20 vs 5%), but there was no reliable difference when the valence was positive (19 out of 20 vs 95%). The findings are robust regardless of whether the outcome is described as requiring repairs (operates without repairs) or does not function satisfactorily (functions satisfactorily). Next, Study 2 examines the generalizability of the findings for a different frequency level and extends the reasoning to a ratio that is not represented as “1 out of some number” (i.e. 2 out of 20 vs 10% and 18 out of 20 vs 90%).

Based on prior research (e.g. Gigerenzer and Hoffrage, 1995; Newell et al., 2008), we argue that numeric information in the form of ratios is more concrete than equivalent percentages. As such, ratio information is more vivid and easier to imagine than percentage information. Further, our conceptualization suggests that individuals will imagine themselves in the situation more with a ratio versus percentage format, inducing higher negative affect, resulting in higher risk perceptions, particularly when the valence is negative.

Study 2a extends the inquiry to a ratio that is not “1 out of some number,” and examines the extent to which individuals can imagine themselves or identify with the situation as a function of numeric format and valence. We also investigate the effect of numeric format and valence on purchase intentions and affect.

Two hundred and 41 participants from Amazon’s Mechanical Turk (Female = 43%; $MAge$ = 36.30, $SDAge$ = 10.73,$RangeAge$⁠: 19–70) were randomly assigned to one of four conditions of a 2 (numeric format: ratio and percentage) × 2 (valence: positive and negative) between-subject design. Participants considered the purchase of a new digital camera, extending the inquiry to another product category. They were provided with an excerpt from an article describing a new digital camera that would soon be introduced to the market. The excerpt contained information about the manufacturer, some generic attribute information about digital cameras, and the frequency format manipulation. In the negative valence, the review stated that “10% or 2 out of 20 of the cameras manufactured by this manufacturer are expected to require repairs.” In the positive valence, the review stated that “90% or 18 out of 20 of the cameras manufactured by this manufacturer are expected to operate without repairs.”

Risk perceptions were measured by averaging responses to six seven-point items (α = 0.94; we added three more items to those used in Study 1 to increase the reliability of the measure: “Given the expenses involved with purchasing cameras today, how much risk would you say would be involved in purchasing the new Profoto camera?” (1 = Very little risk; 7 = Substantial risk), “How likely is that the camera is going to provide you with trouble-free service?” (1 = Not likely at all; 7 = Very likely; reverse scaled) and “How likely is that the new Profoto camera will require major repairs in the next two years?” (1 = Very unlikely; 7 = Very likely)). A factor analysis confirmed that the six items loaded on a single factor (explaining 77.29% of the variance). Purchase likelihood (α = 0.97) was measured by averaging three items: “My likelihood of purchasing of the new Profoto camera is?” (1 = Very low; 7 = Very high), “How probable is it that you will buy the new Profoto camera?” (1 = Highly improbable; 7 = Highly probable), and “What is the chance that you will buy the new Profoto camera?” (1 = No chance at all; 7 = Very good chance). Negative affect was measured by averaging five seven-point items (α = 0.96): “The information provided about the new Profoto camera makes me feel scared/afraid/anxious/nervous/jittery; 1 = Strongly disagree; 7 = Strongly agree). Similarly, positive affect was measured by averaging five seven-point items (α = 0.96): “The information provided about the new Profoto camera makes me feel enthusiastic/assured/excited/inspired/happy; 1 = Strongly disagree; 7 = Strongly agree). An average of two nine-point scale items was used to measure the extent to which participants could imagine or identify themselves with the situation (r = 0.83): “If you were to buy the camera, to what extent can you imagine yourself as having a problem with the camera?” and “To what extent can you identify yourself as the customer who may have a problem with a camera from this company? (1 = Not at all; 9 = A lot). Finally, we asked participants to recall how many cameras were expected to require major repairs according to Consumer Reports (10%, 2 / 20, 90%, 18 / 20, other) ^[5].

We performed confirmatory factor analysis for studies 2a, 2b, 3 and 4 to estimate model fit (Appendix F: Part 1) and to establish discriminant validity (Appendix F: Part 2). As shown in Appendix F (Part 1), the model fit was acceptable for all models in all four studies with a TLI ranging from 0.953–0.983, a CFI ranging from 0.962–0.987 and RMSEA of 0.051–0.078. All the items showed significant positive factor loadings (and no cross loadings), with standardized coefficients ranging from 0.66–0.98.

For all studies, we also report discriminant validity tests (e.g. Fornell and Larcker, 1981; Voorhees et al., 2016; Wang and Shi, 2022) including chi-square difference tests between the unconstrained CFA model in each study and constrained CFA models in which the correlations between some or all constructs are constrained to 1.0. (Table F-1, Appendix F: Part 2), and AVE-SV tests based on average variance extracted and construct reliability (Table F-2, Appendix F: Part 2). Overall, the results indicate good discriminant validity for all measures.

A 2×2 ANOVA on risk perceptions revealed a significant main effect of valence (⁠$MPositive$ = 2.59 vs $MNegative$ = 3.49; F(1, 237) = 25.35, p < 0.001 $ηp2$ = 0.097) and a significant two-way interaction (F(1, 237) = 4.68, p = 0.032 $ηp2$ = 0.019). Replicating the pattern in Study 1, risk perceptions were significantly higher when the frequency was represented as a ratio than an equivalent percentage when the valence was negative (⁠$M2 out of 20$ = 3.80 vs M_10% = 3.18; F(1, 237) = 6.27, p = 0.013, $ηp2$ = 0.026), but there was no significant difference when the valence was positive (⁠$M18 out of 20$ = 2.51 vs M_90% = 2.66; F(1, 237) = 0.37, p = 0.545, $ηp2$ = 0.002). These findings provide support for H1a and H1b. Although our focus is on comparing frequency formats within the positive and negative valence conditions, we note that risk perceptions were higher across the negative versus the positive valence when the probability was described as a ratio (F(1, 237) = 25.85, p < 0.001, $ηp2$ = 0.098). In contrast to Study 1, the effect of valence emerged for percentage as well (F(1, 237) = 4.13, p = 0.043, $ηp2$ = 0.017) giving credence to the idea that valence effects are less likely when the percentages are more extreme.

A 2x2 ANOVA on purchase likelihood as the dependent measure revealed a significant main effect of valence (⁠$MPositive$ = 5.18 vs $MNegative$ = 3.93; F(1, 237) = 37.19, p < 0.001 $ηp2$ = 0.136); the two-way interaction did not reach significance (F(1, 237) = 2.74, p = 0.099 $ηp2$ = 0.011). Planned contrasts revealed that purchase intentions were significantly lower when the frequency was represented as a ratio than an equivalent percentage when the outcome valence was negative (⁠$M2 out of 20$ = 3.63 vs M_10% = 4.21; F(1, 237) = 4.24, p = 0.041, $ηp2$ = 0.018), but there was no reliable difference when the valence was positive (⁠$M18 out of 20$ = 5.23 vs M_90% = 5.13; F(1, 237) = 0.11, p = 0.743, $ηp2$ = 0.000). Further, purchase intentions were lower across the negative versus the positive valence when the probability was described as a ratio (F(1, 237) = 30, p < 0.001, $ηp2$ = 0.112). The effect of valence emerged for percentage as well (F(1, 237) = 9.89, p = 0.002, $ηp2$ = 0.040). These results are consistent with H1a and H1b and mirror the findings for risk perceptions.

A 2x2 ANOVA on the extent of imagining revealed a significant effect of valence (⁠$MPositive$ = 3.02 vs $MNegative$ = 4.69; F(1, 237) = 37.19, p < 0.001, $ηp2$ = 0.136) and a significant two-way interaction (F(1, 237) = 4.29, p = 0.039, $ηp2$ = 0.018). As predicted, planned contrasts revealed that when the valence was negative, extent of imagining was significantly higher in the ratio relative to the percentage format (⁠$M2 out of 20$ = 5.10 vs M_10% = 4.29; F(1, 237) = 4.62, p = 0.033, $ηp2$ = 0.019). However, when the valence was positive, there was no reliable difference in extent of imagining (⁠$M18 out of 20$ = 2.85 vs M_90% = 3.18; F(1, 237) = 0.67, p = 0.416, $ηp2$ = 0.003). These findings provide support for H2a and H2b. As with risk perceptions, the extent of imagining was higher in the negative versus the positive valence both when the frequency was described as a ratio (⁠$M2 out of 20$ = 5.10 vs $M18 out of 20$ = 2.85; F(1, 237) = 33.31, p < 0.001, $ηp2$ = 0.123) and a percentage (M_10% = 4.29 vs M_90% = 3.18; F(1, 237) = 8.13, p = 0.005, $ηp2$ = 0.033).

A 2x2 ANOVA on negative affect revealed a significant effect of valence only (⁠$MPositive$ = 1.90 vs $MNegative$ = 2.71; F(1, 237) = 16.03, p < 0.001, $ηp2$ = 0.063). No other effects were significant (p’s > 0.286). These findings provide support for H1b only.

We report the analysis of positive affect in this study and in Studies 3 and 4 in Appendix C (supplementary material).

Study 2b extends the inquiry to a higher risk level (20% vs 4 out of 20) to empirically explore whether there is a threshold level of low risk at which the effect of numeric format on risk perceptions attenuates. We also test whether the effect of numeric format on risk perceptions and purchase likelihood is mediated by the extent of imagining and negative affect.

Five hundred and one participants from Prolific (Female = 53.6%; $MAge$ = 39.38, $SDAge$ = 13.99,$RangeAge$⁠: 18–80) were randomly assigned to one of six conditions of a 2 (numeric format: ratio and percentage) × 3 (risk level: low [5% or 1 / 20], medium [10% or 2 / 20] and high [20% or 4 / 20]) between-subjects design. The stimuli and the procedure were identical to those used in Study 2a, the only exception being the risk level. The valence was always negative, and the review stated that “5%/10%/20% or 1 / 2/4 out of 20 of the cameras manufactured by this manufacturer are/is expected to require repairs.”

Perceptions of risk (α = 0.92), purchase intentions (α = 0.96), extent of imagining (r = 0.87) and negative affect (α = 0.97) were measured as in Study 2a. The results of a confirmatory factor analysis supporting the discriminant validity of the scale measures are reported in Appendix F (supplementary material).

A 2×3 ANOVA on risk perceptions revealed a significant main effect of numeric format (⁠$MPercent$ = 3.66 vs $MRatio$ = 4.04; F(1, 495) = 9.16, p < 0.001, $ηp2$ = 0.018), a significant main effect of risk level (⁠$MLow$ = 3.33 vs $MMedium$ = 3.85 vs $MHigh$ = 4.36; F(2, 495) = 22.67, p < 0.001, $ηp2$ = 0.084) and a significant interaction (F(2, 495) = 3.10, p = 0.046, $ηp2$ = 0.012). Planned contrasts revealed that risk perceptions were higher in the ratio than in the percentage condition when risk was low (M_5% = 2.92 vs M_{1 / 20} = 3.74; F(1, 495) = 14.52, p < 0.001, $ηp2$ = 0.029), supporting H1a. Risk perceptions were not reliably different across the ratio and percentage formats either when the risk level was medium (M_10% = 3.75 vs M_{2 / 20} = 3.95; F(1, 495) = 0.83, p = 0.363, $ηp2$ = 0.002) or when it was high (M_20% = 4.30 vs M_{4 / 20} = 4.42; F(1, 495) = 0.32, p = 0.573, $ηp2$ = 0.001).

A 2×3 ANOVA on purchase likelihood revealed a significant main effect of numeric format (⁠$MPercent$ = 4.54 vs $MRatio$ = 4.11; F(1, 495) = 8.15, p = 0.004, $ηp2$ = 0.016) and a significant main effect of risk level (⁠$MLow$ = 4.55 vs $MMedium$ = 4.62 vs $MHigh$ = 3.80; F(2, 495) = 12.10, p < 0.001, $ηp2$ = 0.047); the interaction was not significant (F(2, 495) = 1.30, p = 0.273, $ηp2$ = 0.005). Planned contrasts revealed that purchase likelihood was higher in the percentage than in the ratio condition when risk was low (M_5% = 4.86 vs M_{1 / 20} = 4.24; F(1, 495) = 5.88, p = 0.016, $ηp2$ = 0.012) and when risk was medium (M_10% = 4.91 vs M_{2 / 20} = 4.33; F(1, 495) = 4.90, p = 0.027, $ηp2$ = 0.010), consistent with H1a. Purchase likelihood was not reliably different across the ratio and percentage formats when the risk level was high (M_20% = 3.85 vs M_{4 / 20} = 3.76; F(1, 495) = 0.11, p = 0.744, $ηp2$ = 0.000).

A 2×3 ANOVA on extent of imagining revealed a significant main effect of numeric format (⁠$MPercent$ = 4.60 vs $MRatio$ = 5.16; F(1, 495) = 7.42, p = 0.007, $ηp2$ = 0.015), a significant main effect of risk level (⁠$MLow$ = 4.19 vs $MMedium$ = 4.66 vs $MHigh$ = 5.80; F(2, 495) = 21.33, p < 0.001, $ηp2$ = 0.079) and a significant interaction (F(2, 495) = 3.40, p = 0.034, $ηp2$ = 0.014). Planned contrasts revealed that extent of imagining was higher in the ratio than in the percentage condition when risk was low (M_5% = 3.53 vs M_{1 / 20} = 4.85; F(1, 495) = 13.88, p < 0.001, $ηp2$ = 0.027), supporting H2a. Extent of imagining was not reliably different across the ratio and percentage formats either when the risk level was medium (M_10% = 4.51 vs M_{2 / 20} = 4.80; F(1, 495) = 0.63, p = 0.428, $ηp2$ = 0.001) or when it was high (M_20% = 5.75 vs M_{4 / 20} = 5.84; F(1, 495) = 0.06, p = 0.80, $ηp2$ = 0.000).

A 2×3 ANOVA on negative affect revealed a significant main effect of numeric format (⁠$MPercent$ = 2.72 vs $MRatio$ = 3.11; F(1, 495) = 6.28, p = 0.013, $ηp2$ = 0.013), a significant main effect of risk level (⁠$MLow$ = 2.52 vs $MMedium$ = 2.82 vs $MHigh$ = 3.42; F(2, 495) = 11.60, p < 0.001, $ηp2$ = 0.045) and a significant interaction (F(2, 495) = 4.08, p = 0.017, $ηp2$ = 0.016). Planned contrasts revealed that negative affect was higher in the ratio than in the percentage condition when risk was low (M_5% = 2.06 vs M_{1 / 20} = 2.97; F(1, 495) = 11.62, p < 0.001, $ηp2$ = 0.023), supporting H1a. Risk perceptions were not reliably different across the ratio and percentage formats either when the risk level was medium (M_10% = 2.60 vs M_{2 / 20} = 3.04; F(1, 495) = 2.67, p = 0.103, $ηp2$ = 0.005) or when it was high (M_20% = 3.51 vs M_{4 / 20} = 3.33; F(1, 495) = 0.43, p = 0.513, $ηp2$ = 0.001).

These findings suggest that there appears to be threshold level of risk around 20% where the effect of numeric format on risk perceptions and purchase likelihood attenuates. Although risk perceptions were not reliably different across the ratio and percentage formats when the risk level was medium (2 / 20 vs 10%), the means were in the expected direction as in Study 2a. Importantly, purchase intentions were higher in the ratio than percentage format, providing support for our theoretical framework.

Our conceptualization suggests that the effect of numeric format on risk perceptions and purchase likelihood is mediated by the extent of imagining, particularly when the risk level is low and medium. We tested a moderated sequential-mediation model using Hayes (2022) bootstrapping method (Model 85; 95% confidence interval; 10,000 bootstrap resamples). The results revealed a significant indirect path (index of moderated mediation = −10., se = 0.05, CI95 [−0.20, −0.02]) such that the conditional indirect effect of numeric format on purchase likelihood through extent of imagining, and then risk perceptions was significant when the risk level was low (b = 0.19, se = 0.07, CI95 [0.07, 0.33]) and when the risk level was medium (b = 0.10, se = 0.04, CI95 [0.02, 0.17]), but not significant when the risk level was high (b = −0.01, se = 0.05, CI95 [−0.11, 0.09).

We also tested the full moderated sequential-mediation model using Hayes (2022) bootstrapping method (Model 85; 95% confidence interval; 10,000 bootstrap resamples). The results revealed a significant indirect path (index of moderated mediation = −03., se = 0.01, CI95 [−0.06, −0.005]) such that the conditional indirect effect of numeric format on purchase likelihood through extent of imagining, then negative affect and then risk perceptions was significant when the risk level was low (b = 0.05, se = 0.02, CI95 [0.02, 0.11]) and when the risk level was medium (b = 0.03, se = 0.01, CI95 [0.01, 0.05]), but not significant when it was high (b = −0.002, se = 0.02, CI95 −0.04, 0.03).

Generalizing Study 1 findings with a less extreme risk level and to a ratio that is not represented as “1 out of some number,” Study 2a showed that risk perceptions were higher in the ratio format than in the percentage format when the outcome valence was negative (2 out of 20 vs 10%), but not when it was positive (18 out of 20 vs 90%). Study 2b showed that the numeric format did not affect risk perceptions at an even higher risk level (4 out of 20 vs 20%). The results suggest that there is a threshold risk level above which the differences across the formats attenuate.

We conducted another study (see Additional Study 2 in Appendix D in supplementary material) using a different response mode (i.e. choice between options with the frequency described as ratio versus percentage) across a range of risk levels (5% to 25%). The results revealed that for risk levels ranging from 5% to 20%, the choice likelihood was higher for the option with the frequency described as a percentage (vs ratio).

Although valence did not affect risk perceptions in the percentage format in Study 1 (5% vs 95%), risk perceptions were higher when the valence was negative (vs positive) in Study 2a (10% vs 90%). Consistent with prior research (Levin and Gaeth, 1988), our findings add to the literature revealing that framing effects are more likely for moderate than extreme levels (Levin et al., 1998). Importantly, the findings across all studies suggest that the effect of numeric format is manifested when the ratio is described by relatively small numerators (i.e. 1 or 2 out of 20) but not when the numerator increases (i.e. 4 out of 20). This is consistent with our conceptualization that relatively small numerators are more concrete, facilitating individuals to imagine or identify themselves with the situation.

Importantly, Study 2 sheds insight into the underlying reasoning by showing that the extent to which individuals imagine or identify themselves with the situation tracks risk perceptions. The finding that a ratio format facilitates more imagination than a percentage, particularly when the valence is negative, is consistent with the idea that ratio information is processed more concretely. Study 3 provides a more direct test for the underlying reasoning by examining whether construal level moderates the effect of numeric format on risk perceptions.

Study 2a shows that individuals can imagine or identify themselves more with the situation when it is described as a ratio than as a percentage when the valence is negative. Although the finding is consistent with the idea that ratio information is more concrete, is clearer and easier to understand (Brase, 2002), and more vivid (e.g. Newell et al., 2008), there are no explicit tests of whether ratio information is more concrete than percentage information. If ratio information is more concrete relative to percentage information, individuals’ construal level in terms of concrete versus abstract thinking is likely to moderate the effects of frequency format on the extent of imagining, negative affect and risk perceptions.

Two hundred and 21 participants from Amazon’s Mechanical Turk (Male = 51%; $MAge$ = 35.22, $SDAge$ = 10.95,$RangeAge$⁠: 20–85) were randomly assigned to one of four conditions of a 2 (numeric format: ratio and percentage) × 2 (construal level: abstract and concrete) between-subject design. Participants considered the purchase of a new digital camera as in Study 2. While the valence was always negative, frequency format was manipulated by citing Consumer Report’s independent testing of the performance of the new digital camera. The review stated that “5% or 1 out of 20 of the cameras manufactured by this manufacturer are/is expected to require repairs.”

Participants’ construal level was manipulated by adapting a common priming task used in the construal literature (e.g. Carrera et al., 2020; Freitas et al., 2004). Prior to being exposed to the excerpt about the camera and the frequency format manipulation, participants in all conditions were told the following: “Before evaluating the digital camera, we would like to gather more information about the purchase situation you are facing. We are particularly interested in your thoughts about buying a new digital camera.” In the high-level, abstract condition, participants were asked “In particular, we are interested in your thoughts about WHY you would buy a new camera. For example, what are the underlying reasons for buying a new camera, what benefits would you look for in a new camera, etc.? Please take some time below, and list at least five specific reasons, to explain WHY you would buy a new camera.” In the low-level, concrete condition, participants were asked “In particular, we are interested in your thoughts about HOW you would buy a new camera. For example, what actions would you take to buy a new camera, what specific steps would you take, etc.? Please take some time below, and list at least five specific actions, to explain HOW you would buy a new camera.”

Participants’ likelihood of buying the camera was captured with a seven-point item: “How likely are you to buy the new Profoto camera?” (1 = Not likely at all; 7 = Very likely). Perceptions of risk (α = 0.94), negative affect (α = 0.96), positive affect (α = 0.96) and extent of imagining (r = 0.86) were measured as in Study 2a. The results of a confirmatory factor analysis supporting the discriminant validity of the scale measures are reported in Appendix F (supplementary material).

Two items were included as manipulation checks for the concrete versus abstract manipulation: “To what extent did you focus on the reasons (vs focus on the steps) when thinking about buying a camera?” (1 = Very focused on the details; 7 = Very focused on the big picture), and “To what extent did you think about buying a camera in a high-level (vs low-level) way? (1 = Very low level; 7 = Very high level). Participants were also asked to recall how many cameras were expected to require major repairs (5%, 1 / 20, other) ^[6]. Finally, to explore the role of numeracy, following Lipkus et al. (2001), we measured numeracy by asking participants to solve three problems (see Appendix A in supplementary material for details).

The objective numeracy score was not significant for any of the dependent variables reported next (all p’s > 0.153) and was not included in the analyses reported next. A new factor, match, was created to analyze the data. Match assumed a value of 1 (0) when numeric format was percentage and construal level was abstract (concrete) or when numeric format was ratio and construal level was concrete (abstract). We also report the results of Study 3 as 2 (Numeric Format: Ratio and Percentage) × 2 (Construal Level: Abstract and Concrete) ANOVA in Appendix E (supplementary material).

The two items were not significantly correlated (r = 0.12, p = 0.073). We used the first measure, focus on the big picture, in a 2 (frequency format) × 2 (construal level) ANOVA in our analysis. The results revealed a marginal effect of construal level (⁠$MAbstract$ = 4.06 vs $MConcrete$ = 3.59; F(1, 217) = 3.37, p = 0.068, $ηp2$ = 0.015). No other effects were significant (p’s > 0.794).

Given the non-significant correlation between the two manipulation check items, we decided to run a post-test with a sample of students to provide additional support for the concrete versus abstract manipulation. Two hundred and forty-nine student participants (Male = 49%; $MAge$ = 19.81, $SDAge$ = 1.58,$RangeAge$⁠: 18–33) were randomly assigned to one of four conditions of a 2 (numeric format: ratio and percentage) × 2 (construal level: abstract and concrete) between-subject design. We used the exact manipulation from the main study (Study 3). Three items were included as manipulation checks for the concrete versus abstract manipulation (α = 0.61), with higher numbers indicating more abstract processing: “To what extent did you focus on the reasons for buying a camera versus on the steps you would take to buy a camera in the above purchase situation?” (1 = Very focused on the steps; 7 = Very focused on the reasons), “I thought about why I wanted to buy a camera. (1 = Not at all; 7 = Very much), and “I thought about the specific actions I would take to buy a camera.” (1 = Not at all; 7 = Very much); reversed coded). A factor analysis confirmed that the three items loaded on a single factor (explaining 56.31% of the variance). A 2 (frequency format) × 2 (construal level) ANOVA revealed a significant effect of construal level on the manipulation check scale (⁠$MConcrete$ = 3.85 vs $MAbstract$ = 5.01; F(1, 245) = 57.27, p < 0.001, $ηp2$ = 0.189) only; no other effects were significant (p’s > 0.117).

An ANOVA on risk perceptions revealed a significant effect of numeric format (⁠$MRatio$ = 3.81 vs $MPercent$ = 2.86; F(1, 217) = 21.86, p < 0.001, $ηp2$ = 0.092) and a significant interaction between numeric format and match (F(1, 217) = 10.64, p < 0.001, $ηp2$ = 0.047). As expected, planned contrasts revealed that when there was a match between frequency format and construal level, risk perceptions were significantly higher in the ratio versus the percentage format (⁠$M1 out of 20−Concrete$ = 4.22 vs $M5%−Abstract$ = 2.60; F(1, 217) = 31.64, p < 0.001, $ηp2$ = 0.127), but were not significantly different when there was a mismatch (⁠$M1 out of 20−Abstract$ = 3.40 vs $M5%−Concrete$ = 3.11; F(1, 217) = 1.00, p = 0.320, $ηp2$ = 0.005). These findings provide support for H4a and H4b.

A 2×2 ANOVA on purchase likelihood revealed a significant effect of frequency format (⁠$MRatio$ = 3.95 vs $MPercentage$ = 4.88; F(1, 217) = 16.22, p < 0.001, $ηp2$ = 0.070) and a significant interaction between frequency format and match (F(1, 217) = 6.36, p = 0.012, $ηp2$ = 0.028). When there was a match between frequency format and construal level, purchase likelihood was lower in the ratio than in the percentage format (⁠$M1 out of 20−Concrete$ = 3.60 vs $M5%−Abstract$ = 5.11; F(1, 217) = 21.55, p < 0.001, $ηp2$ = 0.090). The difference was however, not significant when there was a mismatch between frequency format and construal level (⁠$M1 out of 20−Abstract$ = 4.30 vs $M5%−Concrete$ = 4.64; F(1, 217) = 1.13, p = 0.289, $ηp2$ = 0.005). The effects on behavioral intentions thus mirror the effects on risk perceptions.

An ANOVA on the extent of imagining revealed a significant effect of numeric format (⁠$MRatio$ = 5.10 vs $MPercent$ = 3.64; F(1, 217) = 18.94, p < 0.001, $ηp2$ = 0.080) and a significant interaction between numeric format and match (F(1, 217) = 5.63, p = 0.019, $ηp2$ = 0.025). As predicted, planned contrasts revealed that when there was a match between numeric format and construal level, participants’ extent of imagining themselves in the situation was significantly higher in the ratio versus the percentage format (⁠$M1 out of 20−Concrete$ = 5.69 vs $M5%−Abstract$ = 3.45; F(1, 217) = 22.72, p < 0.001, $ηp2$ = 0.095), but the difference was not significant when there was a mismatch (⁠$M1 out of 20−Abstract$ = 4.50 vs $M5%−Concrete$ = 3.84; F(1, 217) = 1.95, p = 0.164, $ηp2$ = 0.009). These findings provide support for H4a and H4b.

An ANOVA on negative affect revealed a significant effect of numeric format (⁠$MRatio$ = 3.00 vs $MPercent$ = 2.28; F(1, 217) = 9.33, p < 0.003, $ηp2$ = 0.041) and a significant interaction between numeric format and match (F(1, 217) = 5.98, p = 0.015, $ηp2$ = 0.027). Planned contrasts revealed that when there was a match between numeric format and construal level, negative affect was significantly higher in the ratio than in the percentage format (⁠$M1 out of 20−Concrete$ = 3.39 vs $M5%−Abstract$ = 2.10; F(1, 217) = 15.20, p < 0.001, $ηp2$ = 0.065); there was no difference when there was a mismatch (⁠$M1 out of 20−Abstract$ = 2.60 vs $M5%−Concrete$ = 2.46; F(1, 217) = 0.18, p = 0.669, $ηp2$ = 0.001). These findings provide support for H4a and H4b.

Two separate moderated mediation models investigated whether extent of imagining mediated the effect of numeric format and match/mismatch on negative affect and risk perceptions (Hayes, 2022; Model 8; 95% confidence interval; 10,000 bootstrap resamples). First, the results suggest that extent of imagining mediates the effect on negative affect (index of moderated mediation = 0.70, se = 0.30, CI95 [0.14, 1.31]). Specifically, the conditional indirect effect of numeric format on negative affect through extent of imagining was significant only when there was a match between numeric format and construal level (b = 1.00, se = 0.21, CI95 [0.61, 1.44]) but not significant when there was a mismatch (b = 0.30, se = 0.23, CI95 [−0.14, 0.77]). Similarly, extent of imagining mediates the effect on risk perceptions (index of moderated mediation = 0.72, se = 0.30, CI95 [0.14, 1.31]). Specifically, the conditional indirect effect of numeric format on risk perceptions through extent of imagining was significant only when there was a match between numeric format and construal level (b = 1.02, se = 0.22, CI95 [0.61, 1.45]) but not significant when there was a mismatch (b = 0.30, se = 0.24, CI95 [−0.15, 0.79]).

We also tested a moderated sequential-mediation model using Hayes (2022) bootstrapping method (Model 85; 95% confidence interval; 10,000 bootstrap resamples). The results revealed a significant indirect path (index of moderated mediation = −0.19, se = 0.09, CI95 [−0.40, −0.03]) such that the conditional indirect effect of numeric format on risk perceptions through extent of imagining and then negative affect was significant when there was a match between numeric format and construal level (b = −0.27, se = 0.08, CI95 [−0.45, −0.14]) but not significant when there was a mismatch (b = −0.08, se = 0.07, CI95 [−0.23, 0.04).

Study 3 provides direct evidence that a relatively low frequency is more concrete when it is represented as a ratio than an equivalent percentage. When ratio information was paired with a concrete, low-level construal and percentage information was paired with an abstract, high-level construal, the extent of imagining, negative affect and risk perceptions were higher across the ratio and percentage formats. The effects attenuated when the ratio information was paired with an abstract, high-level construal and the percentage information was paired with a concrete, low-level construal. These findings suggest that concrete thinking or low-level construal leads to more congruent and fluent processing of the ratio information than abstract thinking or high-level construal.

Our conceptualization suggests that frequency information in the form of a ratio is more concrete, more easily imagined, thus leading to higher negative affect and risk perceptions than an equivalent percentage when the valence is negative. The evidence thus far supports our reasoning that ratio information is processed more concretely (Study 3), and that extent of imagination mediates the effect of numeric format on risk perceptions (Studies 2a and 3). Since extent of imagination is the critical aspect in our reasoning, Study 4 sheds more insight into the underlying reasoning by manipulating the manner in which individuals imagine or visualize the numeric information. If individuals are asked to process the numeric information in a manner that impairs their ability to imagine themselves in such situations, the effects of numeric format should attenuate. Said differently, any factor that impairs individuals’ ability to spontaneously imagine or identify with the situation is likely to moderate the effects of numeric format. Study 4 thus examines whether influencing the extent to which individuals can spontaneously imagine themselves in the situation moderates the effects of numeric format.

Two hundred and 27 participants from Amazon’s Mechanical Turk (Female = 49%; $MAge$ = 37.22, $SDAge$ = 11.20,$RangeAge$⁠: 20–76) were randomly assigned to one of four conditions of a 2 (numeric format: ratio and percentage) × 2 (spontaneous processing: no and yes) between-subjects design. The procedure and stimuli were similar to those used in Study 3 with no construal level manipulation.

The “no spontaneous processing” condition was intended to promote a more analytical way of thinking about the frequency information. The manipulation was created based on the cognitive-experiential self-theory (CEST; Epstein, 1990) that has been used to study ratio bias (Denes-Raj and Epstein, 1994). CEST postulates that people use two systems to process information: analytical-rational (deliberate, slow and logical system) vs intuitive-experiential (fast, automatic and emotionally driven system). Thus, in the “no spontaneous processing” condition we encouraged participants to switch to a more analytical-rational processing (vs intuitive-experiential).

Individuals’ attention was drawn to process the frequency as ratios between objects (i.e. randomly drawing from a box with numbered balls), and thus impair the extent to which individuals spontaneously imagine themselves or identify with the situation. Specifically, participants read the following instructions prior to responding to the dependent measures in the 5% (1 out of 20) conditions: “Before proceeding, please visualize the moment at which you are considering the purchase of the camera. In considering the risk involved in purchasing the camera, it is helpful to imagine a box containing 100 (20) balls, numbered from 1 to 100 (1–20). If a ball randomly drawn from the box has a number from 1 to 5 (has the number 1), then the camera will require repairs; if the ball has a number from 6 to 100 (2–20), then the camera will not require repairs. Imagine this process and write what comes to mind since the likelihood of randomly drawing a ball which has a number from 1 to 5 (the number 1) is identical to the risk that the camera would require repairs. In writing what comes to mind, you can include any other detail you think is relevant.” The anticipation was that the instructions would not only focus individuals’ attention on processing the frequency information analytically but also make them cognitively busy, thus impairing the extent to which they spontaneously imagine themselves in the situation. Participants in the “spontaneous processing” condition did not read these instructions.

Risk perceptions (α = 0.96), negative affect (α = 0.97), purchase likelihood, (α = 0.98), positive affect (α = 0.96) and the extent of imagining (r = 0.88) were measured as in Studies 2 and 3. A factor analysis confirmed that the six risk items loaded on a single factor (explaining 83.95% of the variance. The results of a confirmatory factor analysis supporting the discriminant validity of the scale measures are reported in Appendix F (supplementary material). To check how concretely participants visualized the information across the spontaneous and no spontaneous processing conditions, they were asked: “Please indicate how concretely you visualized the risk that the camera will require repairs” (1 = Not at all concretely; 7 = Very concretely). As in Study 3, we asked participants to recall how many cameras were expected to require major repairs according to the Consumer Reports, and all respondents recalled the information correctly.

A 2×2 ANOVA on concrete thinking revealed only a significant effect of spontaneous processing (⁠$MYes$ = 5.94 vs $MNo$ = 5.46; F(1, 223) = 5.78, p = 0.017, $ηp2$ = 0.025). No other effects were significant (all p’s > 0.169). This finding suggests that the risk was more concretely visualized in the spontaneous than in the no spontaneous condition where cognitively busy participants were asked to follow instructions to inhibit spontaneous processing.

An ANOVA on risk perceptions revealed significant effects of numeric format (⁠$MRatio$ = 3.75 vs $MPercent$ = 2.88; F(1, 223) = 16.66, p < 0.001, $ηp2$ = 0.070) and spontaneous processing (⁠$MYes$ = 3.00 vs $MNo$ = 3.63; F(1, 223) = 8.56, p = 0.004, $ηp2$ = 0.037), which were qualified by a significant two-way interaction (F(1, 223) = 6.91, p = 0.009, $ηp2$ = 0.030). According to our expectations, planned contrasts revealed that in the spontaneous processing condition, risk perceptions were significantly higher in the ratio relative to the percentage format (⁠$M1 out of 20$ = 4.35 vs M_5% = 2.91; F(1, 223) = 24.16, p < 0.001, $ηp2$ = 0.098), but there was no difference in the no spontaneous processing condition (⁠$M1 out of 20$ = 3.16 vs M_5% = 2.84; F(1, 223) = 0.99, p = 0.321, $ηp2$ = 0.004). Risk perceptions were significantly higher in the spontaneous versus the no spontaneous processing conditions when the probability was represented as a ratio (F(1, 223) = 15.76, p < 0.001, $ηp2$ = 0.066) but not when it was a percentage (F(1, 223) = 0.04, p = 0.835, $ηp2$ = 0.000). Thus, as intended, the no spontaneous processing manipulation lowered risk perceptions in the ratio condition (vs the ratio condition when there was spontaneous processing) but resulted in similar risk perceptions across the two percentage conditions.

A 2×2 ANOVA on purchase likelihood revealed significant effects of frequency format (⁠$MRatio$ = 3.89 vs $MPercent$ = 4.47; F(1, 223) = 6.55, p = 0.011, $ηp2$ = 0.029) and spontaneous processing (⁠$MYes$ = 4.43 vs $MNo$ = 3.93; F(1, 223) = 4.85, p = 0.029, $ηp2$ = 0.021) and a significant interaction (F(1, 223) = 4.10, p = 0.044, $ηp2$ = 0.018). In the spontaneous processing condition, purchase likelihood was significantly lower in the ratio versus the percentage format (⁠$M1 out of 20$ = 3.41 vs M_5% = 4.45; F(1, 223) = 11.27, p = 0.001, $ηp2$ = 0.048), but were no different in the no spontaneous processing condition (⁠$M1 out of 20$ = 4.37 vs M_5% = 4.49; F(1, 223) = 0.13, p = 0.715, $ηp2$ = 0.001). The results for purchase likelihood thus mirror those for risk perceptions.

An ANOVA on the extent of imagining revealed a significant effect of numeric format (⁠$MRatio$ = 5.10 vs $MPercent$ = 3.81; F(1, 223) = 14.75, p < 0.001, $ηp2$ = 0.062), a marginal effect of spontaneous processing (⁠$MYes$ = 4.14 vs $MNo$ = 4.76; F(1, 223) = 3.37, p = 0.068, $ηp2$ = 0.015) and a significant two-way interaction (F(1, 223) = 4.05, p = 0.045, $ηp2$ = 0.018). As expected, planned contrasts revealed that in the spontaneous processing condition, the extent of imagining was significantly higher in the ratio relative to the percentage format (⁠$M1 out of 20$ = 5.73 vs M_5% = 3.78; F(1, 223) = 18.38, p < 0.001, $ηp2$ = 0.076), but the difference was not significant in the no spontaneous processing condition (⁠$M1 out of 20$ = 4.45 vs M_5% = 3.84; F(1, 223) = 1.56, p = 0.212, $ηp2$ = 0.007). As with the other measures, the extent of imagining was significantly higher in the spontaneous versus the no spontaneous processing conditions when the frequency was represented with a ratio (F(1, 223) = 7.56, p = 0.006, $ηp2$ = 0.033), but not with a percentage (F(1, 223) = 0.02, p = 0.902, $ηp2$ = 0.000). In other words, the no spontaneous processing instructions inhibited individuals’ ability to imagine themselves in the situation in the ratio condition but not in the percentage condition.

Another ANOVA on negative affect revealed a significant effect of numeric format (⁠$MRatio$ = 3.05 vs $MPercent$ = 2.08; F(1, 223) = 18.65, p < 0.001, $ηp2$ = 0.077) and a significant interaction between numeric format and spontaneous processing (F(1, 223) = 4.00, p = 0.047, $ηp2$ = 0.018). As expected, in the spontaneous processing condition negative affect was significantly higher in the ratio relative to the percentage format (⁠$M1 out of 20$ = 3.45 vs M_5% = 2.03; F(1, 223) = 21.43, p < 0.001, $ηp2$ = 0.088), but there was no difference in the no spontaneous processing condition (⁠$M1 out of 20$ = 2.64 vs M_5% = 2.12; F(1, 223) = 2.52, p = 0.114, $ηp2$ = 0.011). Further, negative affect was significantly lower in the no spontaneous versus the spontaneous processing conditions when the frequency was represented as a ratio (F(1, 223) = 6.61, p = 0.011, $ηp2$ = 0.029), but not when it was a percentage (F(1, 223) = 0.08, p = 0.777, $ηp2$ = 0.000).

As in Study 3 we estimated two separate moderated mediation models to test whether the extent of imagining mediated the effect of numeric format and spontaneous processing on negative affect and risk perceptions (Hayes, 2022; Model 8; 95% confidence interval; 10,000 bootstrap resamples). The results suggest that extent of imagining mediates the effect on negative affect (index of moderated mediation = −0.59, se = 0.29, CI95 [-1.17, −0.02]). Specifically, the conditional indirect effect of numeric format on negative affect through extent of imagining was significant when there was spontaneous processing (b = 0.85, se = 0.21, CI95 [0.47, 1.29]) but not when there was no spontaneous processing (b = 0.27, se = 0.22, CI95 [−0.15, 0.70]). Similarly, extent of imagining mediates the effect on risk perceptions (index of moderated mediation = −0.70, se = 0.35, CI95 [−1.40, −0.02]). Specifically, the conditional indirect effect of numeric format on risk perceptions through extent of imagining was significant when there was spontaneous processing (b = 1.02, se = 0.24, CI95 [0.54, 1.49]) but not when there was no spontaneous processing (b = 0.32, se = 0.26, CI95 [-0.18, 0.84]).

We also tested for a serial mediation such that the effect of numeric format influences the extent of imagining, which affects negative affect, and then risk perceptions, when there is spontaneous processing (Hayes, 2022; Model 85; 95% confidence interval; 10,000 bootstrap resamples). Consistent with our reasoning, the results showed a significant indirect path (index of moderated mediation = 0.17, se = 0.10, CI95 [0.01, 0.40]) such that the conditional effect of numeric format on risk perceptions, through extent of imagining and then negative affect was significant when there was spontaneous processing (b = −0.25, se = 0.09, CI95 [−0.45, −0.11]), but not when there was no spontaneous processing (b = −0.08, se = 0.09, CI95 [−0.23, 0.05]).

The introductory illustration where two consumers, contemplating the purchase of a new laptop, evaluate objectively equivalent product performance information that differs in whether the likelihood of failure is described as a ratio or a percentage, exemplifies the core issue examined in this research. Despite the large literature on how the numeric format of risk information affects judgments and decisions, as well as the literature on the effects of framing on judgments, the answers to the questions are not clear. Further, since different theoretical perspectives suggest competing predictions, the current research seeks to examine how the numeric representation of relatively low risks in terms of a ratio versus percentage affects risk perceptions. Another, perhaps more important, objective is to provide insights into the underlying reasoning for the effects.

This research contributes to the literature in at least four ways. First, consistent with the numerous demonstrations of the violation of the principle of descriptive invariance (Tversky et al., 1988), a robust finding was that risk perceptions were higher when a frequency in the negative valence is represented as a ratio (1 out of 20) than an equivalent percentage (5%). Second, our findings suggest that the effect of frequency format on risk perceptions is generalizable to relatively low risk levels other than ones represented by “1 out of some number,” when the valence is negative. Perceptions of risk were higher when the ratio was represented as 2 out of 20 than as an equivalent percentage of 10%. Our findings also suggest that there is a threshold of low risk at which the effects of numeric format attenuate. Study 2b indicates that the threshold appears to be 20%.

Third, a consistent finding in the context of low-risk likelihoods was that valence moderated the effects of numeric format on risk perceptions such that while risk perceptions were higher in the ratio (vs percentage) format when the valence was negative (1 out of 20 vs 5%), there was no difference when the valence was positive (19 out of 20 vs 95%). Study 2a attested to the robustness and generalizability of the findings by showing a similar pattern for another relatively low risk level (2 out of 20 vs 10% and 18 out of 20 vs 90%) and for a ratio that is not represented as “1 out of” some number. Together, the results of Studies 1 and 2a also suggest that valence had a stronger effect on risk perceptions when the numeric information was represented as a ratio than as a percentage. Consistent with prior research, although valence did not affect risk perceptions in the percentage format when the risk was extreme (5% vs 95%; Levin et al., 1988), the effect of valence emerged when the risk level was not as extreme (10% or 20% vs 90% or 80%; Levin and Gaeth, 1988). Future research should examine the underlying reason for why valence does not affect evaluations when the numeric level represented as a percentage is at the extreme but has an effect at less extreme levels. In contrast, valence affects risk perceptions when the risk is represented as a ratio even at extreme levels (1 out of 20 vs 19 out of 20). While our results have practical implications in the context of product performance evaluations, future research should disentangle valence and magnitude in examining the effects on risk perceptions.

Fourth, this research provides insights into the underlying mechanism for the observed findings. The insights into the underlying reasoning are particularly important given the competing a priori predictions regarding the effect of numeric format on risk perceptions. Our finding that risk perceptions are higher when the frequency information with a negative valence is represented as a ratio (1 out of 20) than as a percentage (5%) is in contrast with the idea that a specific likelihood of occurrence will be perceived as less likely and thus less risky when it is denoted by a ratio of 1 out of some number. Instead, we find that frequency information described as a ratio evokes greater negative affect than an equivalent percentage. The differential effect of numeric format on negative affect is consistent with the affect heuristic (Slovic et al., 2005) and the risk as feelings model (Loewenstein et al., 2001). The risk as feelings model is however silent with regards to the underlying reasons for the differential affect. We trace the reasons for the differential negative affect across the ratio and percentage formats. Because concrete representations are more vivid, we argue that the extent to which individuals can imagine themselves or identify with the situation is higher with a ratio than a percentage format. A ratio format is thus likely to evoke higher negative affect and risk perceptions relative to an equivalent percentage, particularly when the valence is negative. Study 2 shows that a ratio (vs percentage) format engenders the extent to which individuals imagine themselves in the situation when the valence is negative.

Further, Study 3 provides the most direct evidence that ratio information is more concrete than percentage information. The findings showed that pairing ratio information with concrete thinking and percentage with abstract thinking had a significant effect on extent of imagining, negative affect and risk perceptions, but these effects attenuated when ratio information was paired with abstract thinking and percentage information was paired with concrete thinking. Finally, Study 4 demonstrates that it is the ease of imagining or identifying with the situation, engendered by concreteness, that underlies the effects of frequency format. Specifically, if individuals are asked to process the frequency information in a manner that inhibits the extent to which they imagine themselves in or identify with the situation, the effect of frequency format on the extent of imagining, negative affect and risk perceptions are attenuated. The no spontaneous processing instructions decreased the ability to spontaneously imagine or identify with the situation and lowered risk perceptions and negative affect in the ratio condition but did not affect extent of imagining, negative affect and risk perceptions in the percentage condition.

Together, the studies reported in this research not only address the questions raised in the introductory illustration but also contribute to the literature on judgments of risk based on numeric information, affective decision-making and risk perceptions. We propose a conceptual framework that explains the differential effects of numeric format, based on how ratio and percentage formats are processed. Our main contribution is in outlining the process behind the differential effects of how risk is represented numerically and demonstrating boundary conditions. We experimentally show that a ratio representation is more concrete than a percentage, engendering more pronounced mental imagery, higher negative affect and risk perceptions, thus extending the research in the domains of risk and risk perceptions, numeric framing of risk information, affective valuations and the role of affect-laden imagery.

Our findings have important implications for the theory of planned behavior (Ajzen, 1991). Specifically, our findings highlight that perceptions of risk could mediate the effect of behavioral beliefs on intentions and behavior. For example, to the extent that communicating the risk of texting or speeding in terms of a ratio versus a percentage increases risk belief, it could discourage texting or speeding.

Finally, our findings also add to the substantive body of research on numerosity effects that have been documented in a variety of contexts. For example, changing the scale in which attribute information is provided (expanded vs contracted scales, different units, unit granularity) affects consumer preferences and decisions (e.g. Burson et al., 2009; Monga and Bagchi, 2012; Pandelaere et al., 2011; Zhang and Schwarz, 2012). Further, framing of price promotions (e.g. dollar vs percentage terms) and order of presentation of multiple percentage discounts influence consumers’ perceptions and purchase intentions (e.g. Chen et al., 1998; Davis and Bagchi, 2018; Darke and Chung, 2005; González et al., 2016; Krishna et al., 2002). Plus, perceptions of unit price will be anchored on the first piece of information presented (price or item) when it is difficult to compute unit pricing in larger packages (Bagchi and Davis, 2012). Finally, people judge the magnitude of a quantitative expression by focusing on foreground information (i.e. the numerator) at the expense of background information (i.e. the denominator; Raghubir, 2008). We contribute to this stream of research by showing that some numerosity effects could be affect driven based on how numeric expressions are perceived and encoded. Our findings are different from Raghubir (2008) who argues that the more salient the denominator, the more attention is paid to the numerator. We do not look at denominators in our studies. We demonstrate that consumers differentially perceive the numeric formats because they focus on the numerator of the ratio, which then affects the extent to which consumers imagine themselves in or identify with the situation (vs percentage format). We are the first to provide an explicit test of the underlying process.

Our findings could inform future research in the area of services. Perceived risk has been found to moderate attitudes and behaviors such as trust, satisfaction, loyalty and willingness to pay (Casidy and Wymer, 2016). Specifically, financial, social, performance and psychological risks have significant negative effects on the relationship between satisfaction and willingness to pay, whereas only financial risks have significant negative effects on the relationship between loyalty and willingness to pay. Future studies could focus on how the numeric framing of attributes affect risk perceptions in the service domain, and distinguish between the relative importance of different types of risk (e.g. financial, performance, social and psychological risk; Casidy and Wymer, 2016).

Future research could also explore the effects of numeric format in other decision-making domains (e.g. health, betting). As discussed above, research could investigate risk perceptions in the service domain in a more nuanced manner. Second, more research is needed on how to mitigate the effects of numeric format when the valence is negative. We show that one way to attenuate the effect is to impair individuals’ spontaneous imagining by asking them to cognitively process the frequency information in a more rational manner, but there could be other ways to do so. For example, spontaneous processing could be inhibited by imposing cognitive load, or reducing the time given to make the decision; and conscious and controlled processing could be encouraged by increasing justifiability ^[7]. Third, future research could also consider a larger range of likelihoods (e.g. from 5% to 95%) using more neutral events that have the same valence (e.g. drawing a blue chip from a bag).

One of the limitations of our research is that the stimuli in our studies all refer to products and do not include services. The intangible nature of services could lead to higher risk perceptions in general compared to products. Nevertheless, we expect the same pattern of results and underlying process based on our theoretical arguments. Further, we do not distinguish between different types of risk in our studies (e.g. financial, social, performance, psychological, physical risk). Our risk measure captures financial and performance risk, but more nuanced measures could be investigated in future studies.

From a managerial and public policy perspective, communicating risk in a manner so as to encourage people to take preventive measures may be better achieved with numeric information presented as a ratio as compared to a percentage. For example, policymakers, government agencies and not-for-profit organizations focused on decreasing credit card debt, unhealthy food consumption or texting while driving should focus on ratios when communicating risk to encourage risk reducing behaviors. Similar to our recommendation, to discourage drinking and driving, the NHTS states, “Every day, about 32 people in the USA die in drunk-driving crashes – that’s one person every 45 min” (Link to Drunk Driving | Statistics and ResourcesLink to the Website). Behavior modification campaigns could focus on providing statistical information in a ratio format to discourage people from other negative behaviors such as speeding, texting while driving, smoking or vaping.

Further, University of Utah Health Care implemented a campaign promoting breast cancer screenings using statistical information about breast cancer occurrence and lives saved with screenings in ratio format, emphasizing that “1 in 8 women will develop breast cancer” and that “screening can save up to 37 lives every day in the U.S.” (Link to Humour Appeal (Advertising)Link to the The Humour Appeal (Advertising). Finally, the National Education Association launched a campaign to recruit educators to help fight cyberbullying stating that “more than 1 in 3 young people have experienced cyberbullying online” and “more than 25% of adolescents and teens have been bullied repeatedly” (Link to Bullying resources: learn, prevent, stopLink to the Homepage). These practical examples are in line with our empirical findings, and highlight the importance of numeric framing in various domains. Finally, our findings could be applicable in the context of more effective marketing communications. Overall, the understanding of behavioral biases in the context of risk communications can be instrumental in helping consumers make better judgments and decisions.

Study procedures were in accordance with the ethical standards of the institutional research committee and with the 1964 Helsinki declaration and its later amendments.

The supplementary material for this article can be found online.