Frequency or total number? A comparison of different presentationformats on risk perception during COVID-19

Abstract

Curbing the COVID-19 pandemic remains an ongoing global challenge. Institutionsoften release information about confirmed COVID-19 cases by citing the totalnumber of cases (e.g., 100,000), their (relative) frequency (e.g., 100 per1,000,000), or occasionally their proportion (e.g., 0.0001) in a region. Icompared the effect of these three presentation formats — total cases,frequency, and proportion — on people’s perceived risk. I foundpeople perceived a higher risk of COVID-19 from a total-cases format than fromfrequency formats when the denominators are relatively small, and the lowestrisk from a proportion format. Correspondingly, people underestimated totalinfections when given frequency and overestimated frequency when given totalnumber of cases. Additional comparisons were made among mathematicallyequivalent variations of frequency formats (e.g., 1 in 100, 10 in 1,000, 1,000in 10,000, etc.). The results provided qualified support for denominatorneglect, which seems to occur in bins into which denominators are grouped (e.g.,1–1000, 10000–100000), such that only across bins couldparticipants perceive differences. Finally, a mixed format of proportion andtotal cases reduced perceived risks from total cases alone, while a mixed formatof frequency and total cases failed to produce similar results. I conclude byproviding concrete suggestions regarding COVID-19 information releases.

1 Introduction

Since the beginning of the COVID-19 outbreak, the release of information regarding the spread of the virus has been a matter of critical importance. Many governments and media outlets have updated information about confirmed cases and SARS-CoV2 infection rates daily. For example, the U.S. Center for Disease Control and Prevention (CDC) releases information by 8 p.m. EST every day at both the national and state levels.¹ In the CDC’s information releases, total cases are expressed as a single number, while the infection rate is expressed as cases per 100,000. Similarly, websites such as https://ourworldindata.org/coronavirus,² among others, also have presented the total number of confirmed cases and infection rates, the latter expressed as confirmed cases per million. Are there significant differences among mathematically equivalent formats in conveying the risk of infection? The answer to this question has implications for the ongoing fight against the global pandemic (Bavel et al., 2020; Holmes et al., 2020).

In the remainder of this paper, I refer to the total confirmed cases as the total-cases format and the form of A/B (e.g., 1 in 1,000) as the frequency format. A third format of a decimal number between 0 and 1 is referred to as the proportion format (e.g., 0.001). Clearly, proportion and frequency can be converted to each other with no loss of information, and all three formats can be converted to one another if population information is available. My research question is: How do different presentation formats influence people’s perceived risk of contracting the coronavirus infection in a designated location? An understanding of differences in people’s perception of the two most frequently adopted formats in COVID-19 news releases, the frequency format and the total-cases format, has obvious practical significance in risk communication during the global pandemic. I also tested the phenomenon of denominator neglect — people’s tendency to attend to numerators and neglect denominators when processing probability ratios (Denes-Raj & Epstein 1994) — theoretically in the risk-perception context and delineate its boundaries.

Past research on risk perceptions has primarily focused on the frequency-proportion discrepancy, or the fact that people perceive higher risks from a frequency format than from a proportion format at a given rate. For example, a psychiatric patient was judged as posing a higher risk if their likelihood of committing harmful acts was provided as “10 out 100 of similar patients will cause harm” vs. “10% of similar patients will cause harm” (Slovic et al., 2000). The current research explores an additional format, total number of cases, which is arguably the most widely used metric to convey COVID-19 information.

Furthermore, although both the proportion format and the total-cases format can take only one form, there can be more than one frequency expression of the same rate when numerators and denominators are varied. For instance, 0.01 can be expressed as 1 in 100, 10 in 1,000, 100 in 10,000, etc. How do people respond to different frequency expressions with varying numerators and denominators? Though researchers have studied the frequency-proportion difference across medical, legal, business, meteorological, and psychological domains (Koehler, 2001; Peters et al., 2011; Reyna & Brainerd, 2008; Schapira et al., 2004; Slovic et al., 2000; Srivastava & Koukova, 2018), less is known about differences in perceptions of mathematically equivalent frequency formats.

In summary, this study simultaneously compares how three mathematically equivalent types of information-presentation formats — frequency, total cases, and proportion — affect people’s perception of the risk of COVID-19.

1.1 Theories and Hypotheses

Research on denominator neglect (also called the ratio bias) shows that “people who understand that proportion is a function of frequencies in both the numerator and the denominator still tend to pay less attention to the denominator as a default.” (Reyna, Nelson, Han & Dieckmann, 2009). In a classic demonstration of the denominator neglect, Denes-Raj and Epstein (1994), following Piaget and Inhelder (1951/1975), offered participants two transparent bowls of white and red jelly beans, and told them they would win money if they picked a red bean. One bowl contained 10 jelly beans, of which one was red; the other contained 100 beans, of which nine were red. The researchers found strong evidence for denominator neglect. That is, although participants knew that the former offered a higher probability of winning, because they focused on the numerator (9 vs. 1) rather than the denominator, they simultaneously felt that they were more likely to win in the latter case. In other words, participants were “following a mental strategy of ‘imagine the numerator’ and neglecting the denominator” (Slovic, Monahan & MacGregor, 2000, p. 285). If people partially or completely neglect denominators, we can conclude that the larger the numerator and denominator, the greater the risk people will perceive for a given rate. Arguably, the proportion format and the total-cases format could essentially be interpreted as a frequency format with denominator 1 and N (the entire population).³ In such cases, denominator neglect leads to two predictions regarding how different formats affect perceived risk.

H1:

For a given rate, perceived risk from the highest to the lowest are Risk _totalcases > Risk _frequency > Risk _proportion.

H2:

Among mathematically equivalent frequency formats, the larger the numerator (denominator), the greater the perceived risk.

When people are asked to estimate the total number of infected cases from proportion or frequency, they can rely on a cognition-based strategy or an affect-based strategy. The cognitive strategy involves simple arithmetic, the result of which can serve as a benchmark. For instance, a 0.0001 infection rate in a country of 1 billion people equals 100,000 total cases. However, extensive research finds that perceived risk is instead an embodiment of feelings and affect (e.g., Loewenstein, Weber, Hsee & Welch, 2001). At any fixed rate, due to denominator neglect, the feelings/affect resulting from a proportion (e.g., 0.0001) or frequency (e.g., 1 in 10,000)⁴ will be less strong than feelings resulting from total cases (e.g., 100,000). When proportion or frequency provided by outside sources produces momentary feeling/affect, and such feelings/affect are in turn adopted as a basis for the estimate of the total number of cases, people would underestimate the total number of cases.

H3a:

When presented with information in the form of proportion or frequency, people tend to underestimate the total number of cases of infection.⁵

In predicting people’s perceptions of a given infection rate presented in different formats, Hypotheses 1 and 2 assume that the participants have not been to the places mentioned in the scenario; their only source of information is the presenter (i.e., cognition→perception). How would the cognitions of those who live in places where the pandemic is ongoing be affected? That is, conditional on subjective perceptions, how would these individuals estimate the consequences of the pandemic when elicitation formats are varied (i.e., perception→cognition)?

Individuals often rely on feelings when making judgments, or an affect heuristic (Slovic et al., 2007; Kahneman, 2011). Thus, when they are expressing their subjective feelings/experience verbally in total number vs. frequency, a smaller total number of reported infections may be sufficient to match their emotional intensity and perceived seriousness of the situation, while a higher estimate in frequency is needed to match the same level of emotion/perception due to denominator neglect. Thus, conditional on individuals’ actual experience, one could find an opposite pattern of predictions in H1 in estimating the pandemic situation.

Formally, I predict:

H3b:

Individuals who are living through a pandemic in a certain place tend to estimate the seriousness of the pandemic of that place differently based on different elicitation formats, such that Estimate _frequency > Estimate _totalcases.⁶

H3a and H3b are indeed different manifestations of the same prediction. H3a keeps the objective information constant, while H3b keeps the subjective experience constant. What would happen when one uses the two presentation formats simultaneously in a presentation of A (B) or B (A) (e.g., the infection rate is 0.0001 (100,000 people in total))? If format A produces higher subjective risks than format B, two hypotheses could be proposed: an accumulation hypothesis and an average hypothesis. An accumulation hypothesis would specify that a mixed format of A and B would have a cumulative effect, while the average hypothesis would predict some sort of weighted average of A and B, such that people will perceive less risk from the “riskier” format but more risk from the “less risky” format. Hence:

H4a:

The mix of proportion or frequency format and the total-cases format will produce a cumulative level of perceived risks: Risk _mix > Risk _totalcases > Risk _{proportion/frequency}

H4b:

The mix of proportion or frequency format and the total-cases format will produce a medium level of perceived risks between the two: Risk _totalcases > Risk _mix > Risk _{proportion/frequency}

In addition to denominator neglect, a second theory, exemplar cuing theory, predicts a different pattern for risk perception (Koehler & Macchi, 2004). Koehler and Macchi (2004, p. 540) propose that “people attach more weight to unlikely events when they can easily generate or imagine examples.” They studied frequency formats in the context of incriminating DNA match, in which blood drops matched the DNA of a murder suspect, and found that a presentation with no exemplars greater than 1 (e.g., 0.001% of the people in a town of 500 who are not the blood source would match the blood drops of the suspect; 0.001%×500 < 1, no exemplar available) is more convincing evidence of a defendant’s guilt than a presentation with exemplars greater than 1 (e.g., 0.001% of the people in a town of 5,000,000 who are not the blood source would match; 0.001%×5,000,000 = 50 exemplars available). Crucially, their Experiment 2 showed that the availability of exemplars is important, but the number of exemplars is not (e.g., people are indifferent between 1 in 1,000 and 2 in 1,000). As the authors argued in a footnote on page 545, people are indifferent when a given ratio is provided as different frequency formats in the DNA match setting (e.g., 1 in 1,000, 10 in 10,000, and 10,000 in 1,000,000,000). If the finding extends to risk perceptions in the health domain, it would predict no differences among the different presentation formats, since all are mathematically equivalent in producing exemplars. For instance, if the infection rate in a country of 1,000,000,000 was 0.0001, then the exemplars in all three presentations are 10,000, equal to the total cases. Obviously, in order to imply at least one exemplar for a low-probability event, the reference class (i.e., how many people could be potentially infected) should be large enough that there exists at least one exemplar. Thus, a competing hypothesis of H1 is:

H5:

Given that there is at least one exemplar, all mathematically equivalent presentation formats of infection information will produce the same level of perceived risk because they imply the same number of exemplars.

1.2 Empirical Overview

In five studies, I explored the effects of different presentation formats on individuals’ risk perceptions of COVID-19. Study 1 kept respondents’ subjective experience unchanged and explored how their estimates of infection rates varied according to different elicitation formats within-person in a survey carried out in China (H3b). Studies 2–4, using between-subject designs and experimental manipulations, addressed my main research question of whether and how people’s perceived risk varies by presentation format (H1, H2, H3a, and H5); in these studies, I kept information mathematically equivalent across groups. In Study 2, I compared different frequency formats by varying the numerators/denominators and compared total-cases formats by adding and removing population information. In Study 3, I adopted a nine-group design by systematically varying the denominators and numerators of the frequency formats. Finally, Studies 4a and 4b further explored mixed formats on risk perception using 100-point scales (H4a and H4b).

2 Study 1

Study 1 used a non-experimental paradigm to explore how subjective experience mapped onto self-reported ratings. It tested H3b, which predicts that people would estimate the seriousness of the pandemic happening around them differently when two different elicitation formats of frequency and total cases were used. Study 1 adopted a within-subjects design in a natural setting during the peak of the COVID-19 pandemic in China. The within-subjects design provides a stringent test of the hypothesis that Estimate _frequency > Estimate _totalcases.

2.1 Method

Study 1 used a convenient online sample from China⁷ (N = 119; 54 females, M _age = 25.06, range = 18 to 54; located over 16 provinces) during the peak of the COVID-19 pandemic in 2020 (all data collected on January 29 and 30, 2020). The unique time window enables us to test H3b, for the pandemic was unexpected and sudden to citizens living in China, and no institutions were reporting infected cases daily yet. Therefore, participants answered their questions with little influence from official statistics.

Participants answered a series of questions related to the COVID-19 pandemic, two of which were of concern to this study. The first question concerned the total number of infections: “Among the 1.4 billion people in China, how many do you predict will get infected eventually?” (options: “fewer than 1,000,” “1,000–10,000,” ... “100 million–1 billion”). The second question concerned the frequency of infection against a smaller denominator: “On average, how many people will get infected out of 100,000 people in the entire nation?” (options: “fewer than 1,” “1–10,” . . . “10,000–100,000”). The order of the two questions was counterbalanced.

I examined whether participants provided consistent answers to these two questions. That is, given an answer to one question, there is a normatively correct answer to the other question. For instance, an answer of “1,000–10,000” to the first question should correspond to “fewer than 1” to the second question; an answer of “10,000–100,000” to the first question should correspond to “1–10” to the second question, and so on.

2.2 Results

Most of the answers to the first question were “1,000–10,000” (coded as 2, 29.41%) and “10,000–100,000” (coded as 3, 60.50%), with a mean of 2.89 and standard deviation of 0.94. Most of the answers for the second question were “1–10” (coded as 2, 41.18%) and “10–100” (coded as 3, 21.85%), with a mean of 2.62 and standard deviation of 1.24. In other words, most (89.91%) respondents to the first question thought that the total number of infected patients would eventually fall between 1,000 and 100,000; in contrast, most (63.03%) respondents thought that the infection rate was “1–100” people per 100,000, equal to 14,000 to 1.4 million of China’s population.

Next, I inspected the individual-level data. Among the 119 participants, 64.71% answered the questions inconsistently. If I used the total number as my benchmark, more than half (52.95%) of the respondents overestimated the frequency of infection, while the remaining 11.76% underestimated it. An exact binomial test of the equal likelihood of consistent and inconsistent predictions yielded a p = 0.0017, indicating that the majority were inconsistent; an exact binomial test of the equal likelihood of overestimation and underestimation yielded a p < 0.001, showing that participants were more likely to overestimate than underestimate infections in a frequency format.

2.3 Discussion

Study 1 showed that people were inconsistent within-subjects in estimating the pandemic consequences when the same question was asked as a total-cases estimate vs. a frequency estimate. Given the predicted total number of infected patients as the benchmark, people overestimated frequency. Likewise, if the predicted frequency serves as the benchmark, people underestimated the total number of infections. Having established that total cases and frequency can lead to different estimates even when the experienced event remains unchanged within-person, Studies 2–4 focus on my main research question: Do different presentation formats lead to different levels of perceived risk? Note that unlike in Study 1, in the subsequent studies, participants’ information about the focal place’s pandemic situation came only from the researcher.

3 Study 2

3.1 Method

I randomly assigned 355 CloudResearch-approved (https://www.cloudresearch.com/) MTurk participants into five conditions (frequency formats with numerators 1/10/100, total cases with/without population information; preregistered at https://aspredicted.org/blind.php?x=6ap9v7). Except for the total-cases-without-population-information condition, the other four conditions were mathematically equivalent. Overall, 350 participants (173 female; M _age = 39.61, SD = 11.25) passed the attention check; thus, I analyzed these participants in my subsequent analyses.

Participants read the following scenario: “Suppose you are doing business in a foreign country of 1 billion [bolded in the original material] population. You need to make an urgent business trip to this country, and travel to multiple cities. However, you are hesitating because of COVID-19 there. According to a credible information source, you know that 1/10/100 in every 10,000/100,000/1,000,000 people were currently infected.” In the total-cases groups, the bolded part read “100,000 people in total were currently infected in this country.” The country’s population information was not revealed in the total-cases-without-population-information condition.

After reading this scenario, participants were asked to answer three questions on five-point scales: (1) How risky do you perceive it is to travel in this country, from not risky at all to extremely risky? (measuring perceived risk) (2) How likely do you think you are to get infected, from not likely at all to extremely likely? (measuring perceived likelihood of infection), and (3) How likely are you to make the trip, from not likely at all to extremely likely? (measuring intention to make the trip). My focus is on perceived risks; I included the other two items to cross-validate that subjective risks were related to these two constructs. Additionally, I required participants in the three frequency formats to estimate “the total number of people who are currently infected out of 1 billion people.” All participants were explicitly required to “choose an answer as quickly as possible based on feelings without doing much deliberation,” from 1,000, 10,000, … , to 10,000,000.

3.2 Results

I first analyzed perceived risk group by group. For readers’ convenience, I abbreviate the five groups as num1 (numerator equals 1), num10 (numerator equals 10), num100 (numerator equals 100), totwithp (total cases with population information), and totwithoutp (total cases without population information) in the analyses below. The average perceived risk of five groups is shown in Figure 1. Contrary to the prediction of denominator neglect, I found no perceived risk differences among the three frequency formats (F(2, 204) = 0.53, p = 0.59), although there were significant differences among all five groups (F(4, 345) = 6.21, p < 0.001). To my surprise, I found no difference between groups in the total-cases format with and without population information (t(141) = –1.31, p = 0.19), providing evidence of denominator neglect. Post-hoc pairwise comparison using Tukey’s Honest Significant Test revealed that most of the difference among the five groups was driven by the two total-cases groups vs. the three frequency groups (totwithp vs. num100, p adjusted = 0.036; totwithoutp vs. num1, p adjusted = 0.013; totwithoutp vs. num10, p adjusted = 0.002; totwithoutp vs. num100, p adjusted = 0.0003).

Figure 1: Perceived risks of different groups in Study 2. The error bars represent ± one standard error. The first three groups refer to frequency formats with numerator 1, 10, and 100 respectively. The fourth and fifth group refer to total-case formats with/without population information.

Next, I calculated the effect size Cohen’s d of each group against the first group of num1: D _num10 = –0.10; D _num100 = –0.17; D _totwithp = 0.31; D _totwithoutp = 0.56 (see Figure 3 in Study 3 below). I further analyzed participants’ estimate of the total number of infections in the three frequency formats and found no significant differences (M _num1 = 3.00, M _num10 = 2.74, M _num100 = 2.83, F(2, 204) = 1.09, p = 0.34). Out of 207 participants, 45.9% estimated correctly, 30.9% underestimated total infections, and 23.2% overestimated. A test of equal distribution obtained results in the predicted direction: more participants underestimated vs. overestimated the total number of infections (60.4% vs. 39.6%, Exact Binomial Test p = 0.078).

Finally, perceived risk and perceived likelihood of infection were positively correlated (r = 0.77, t(348) = 22.84, p < 0.001), and perceived risk and trip intention were negatively correlated (r = –0.63, t(348) = –14.94, p < 0.001). The results of the perceived likelihood of infection and trip intention reflected a similar pattern: no significant differences emerged among the three frequency formats; there was a boost in perceived likelihood of infection and a reduction in trip intention for the two total-cases formats.

3.3 Discussion

Overall, Study 2 showed partial support for denominator neglect. I found no differences in perceived risk among the frequency formats when I varied the numerators from 1 to 1,000. When I adopted the total cases, participants perceived significantly higher risk, regardless of whether population was explicitly stated or not. This finding provides evidence for denominator neglect: the non-significant result implies that people largely ignored the total population — the denominator, in the total-cases format — when normatively they should take this piece of information into account. Past research showed that explicitly stating the frequencies by placing both numerators and denominators side by side could eliminate denominator neglect (Price & Matthews, 2009). This study has replicated this finding.

My next study has four purposes. First, I systematically increased the numerator in the frequency formats from 1 to 100,000 to replicate Study 2’s null results among frequency formats. Second, I have added a proportion format in an attempt to replicate the literature that proportion format conveys lower risk. Third, I tested Hypotheses 4a and 4b regarding the effect of mixed formats. Fourth, given that total cases can be expressed in two ways, as a mere total number or by the frequency format of “n out of N,” I examined whether this subtle difference affects perceived risk. The major difference is that in the total-cases format, the population information is a few sentences apart from the numerator, whereas in the frequency format, both pieces of information were placed side by side, possibly increasing the salience of the denominator and therefore decreasing denominator neglect (Price & Matthews, 2009). Note that population information is always provided in the remaining studies in the first sentence: “Suppose you are doing business in a foreign country of 1 billion population.”

4 Study 3

4.1 Method

Six hundred and thirty-seven CloudResearch-approved MTurk participants were randomly assigned to nine mathematically equivalent conditions (frequency formats with numerators 1/10/100/1,000/10,000/100,000, total cases with population information, proportion, mix of proportion and total cases; preregistered at https://aspredicted.org/blind.php?x=4dv2fn). Overall, 622 participants (322 female; M _age = 42.14, SD = 12.96) passed the attention check; thus, I analyzed these participants in my following analyses.

The instructions were similar to those in Study 2, with a few differences in the experimental manipulations. First, for the frequency format with numerator 100,000, I explicitly used an A out of B format — “100,000 out of the 1,000,000,000 people are right now infected.”⁸ Second, in all groups, I emphasized that infection rate referred to the “situation in the moment.” Third, in the mixed-format group, a proportion was followed by the total cases in parentheses — “the infection rate right now in this country is 0.0001 (100,000 people in total are infected).” Fourth, the differences between the total-cases and frequency formats with population as the denominator are (1) the numerator and denominator were presented apart in the total-cases format (see instructions in Study 2) and (2) the denominator was mentioned one more time in the frequency format by being placed side by side with the numerator.

4.2 Results

Study 3 replicated Study 2’s results: The likelihood of infection and trip intention were highly correlated with risk perception (r _{risk-infection} = 0.77, p < 0.001; r _{risk-intention} = –0.64, p < 0.001) and displayed similar patterns as perceived risk (more below). As a result, I focus on the analyses of perceived risk from here on.

The perceived risk of the nine groups is displayed in Figure 2. Overall, there were significant differences among the nine conditions (F(8, 613) = 9.29, p < 0.001). The first seven frequency conditions can be divided into two “Groups” (rendered as such to differentiate them from experimental groups) with the first four smaller numerators (num1-num1,000) constituting Group One and latter three (num10,000, num100,000, and total cases) constituting Group Two.

Figure 2: Perceived risks in different groups. Error bars represent ± one standard error. All groups convey mathematically equivalent information. The first six groups refer to frequency formats with a numerator of 1–100,000. The final groups refer to a mixed format of proportion and total cases.

Figure 2 shows the following results. First, there were significant differences among the first seven frequency groups (F(6, 474) = 2.46, p = 0.024). Second, there were no significant differences among the first four experimental conditions in Group One (F(3, 276) = 0.619, p = 0.603), replicating Study 2’s results. Third, there was a discernable increase in perceived risk from Group One to Group Two. That is, there was an increase in risk perception when the numerator increased from 1,000 to 10,000 (t(134) = 1.79, p = 0.038, one tail) or 100,000 (t(131) = 1.81, p = 0.036, one tail). Fourth, there were no significant differences among the three groups in Group Two (num10,000, num100,000 and total cases) (F(2, 198) = 0.043, p = 0.958). Fifth, the proportion format indeed lowered risk perception by a large margin as compared to all the other formats. Sixth, mixing the proportion and the total cases resulted in a medium level of perceived risk. I ran a planned linear contrast with the last three groups, with contrast code –1, 0, and 1 for the three groups of proportion, mixed format, and total number. There was a significant upward linear trend among the three groups (t(210)) = 7.30, p < 0.001). Figure 3 displays effect sizes across Studies 2 and 3. All effect sizes were computed as Cohen’s d against the frequency format of 1/10,000.

Figure 3: Effect size Cohen’s d in Studies 2 and 3. Cohen’s d was computed against the group of frequency format of 1/10,000. The first six groups refer to frequency formats with numerator 1–100,000. Totwithp and totwithoutp refer to total case formats with and without population information, respectively. The final group refers to a mixed format of proportion and total cases.

Finally, I analyzed participants’ estimates of the total number of infections in the first five groups (num1 to 10,000).⁹ Among the 346 participants, 45.7% correctly estimated the total number of infections. Among the rest of the participants, who estimated incorrectly, 67% underestimated the total number of infections, evidence that frequency display is more likely to result in underestimation (vs. overestimation) of the total number (67% vs. 33%, Exact Binomial Test p < 0.001).

4.3 Discussion

The results of Studies 2 and 3 provide qualified support for denominator neglect. I found that denominator neglect occurs in a stepwise rather than continuous manner. When the numerators were from 1–1,000, participants generally perceived similar levels of risk; when the numerators increased to 10,000 and above, their perceived risks increased and then remained stable. Both studies showed that total cases enhanced perceived risk over the frequencies with numerators up to 1,000, and the proportion format reduced perceived risk.

The literature also documented a 1-in-X effect, such that among frequency formats, individuals perceived higher risks when the numerator equals 1 (e.g., 1/200) as compared to other numbers (e.g., 5/1,000) (Pighin et al., 2011). A careful inspection of negative effect sizes of the first three groups in Figure 3 revealed that the 1-in-X effect may exist at a small magnitude, though no significant results were obtained. However, when the denominator is large (over 1,000), the 1-in-X effect is likely to be overshadowed by denominator neglect. This finding is consistent with evidence of a relatively small 1-in-X effect (Sirota et al., 2014). However, my effect size is still smaller than that of their meta-analysis (Hedges’ g = 0.42, 95% CI 0.29–0.54), possibly due to the fact that denominators in my groups grew exponentially (in the orders of 10); therefore, the 1-in-X effect was quickly confounded by denominator neglect.

One limitation shared by Studies 2 and 3 is that they used a five-point scale, where the majority of participants picked 2s (slightly risky) and 3s (moderately risky). To better calibrate the effect size, I adopted a 100-point scale in Study 4. More importantly, because the total-cases format is widely practiced, the finding that the mixed format of total cases and proportion can reduce perceived risk is worthy of replication.

5 Study 4a

Similar to Study 3, the mixed format in Study 4a is presented by placing two formats side by side — “Format A (Format B).” I counterbalanced the order of the total cases and proportion to control for any order effect. I have included the proportion format alone and the total-cases format alone as control groups.

5.1 Method and Results

Four hundred and seven CloudResearch-approved MTurk participants were randomly assigned to four mathematically equivalent conditions (proportion, proportion [total cases], total cases [proportion], total cases; preregistered at https://aspredicted.org/blind.php?x=f3g69g). For example, a “proportion [total cases]” item might read “The infection rate right now in this country is 0.0001 (100,000 people in total are right now infected in this country).” Three hundred and ninety-eight participants (217 females; M _age = 40.65, SD = 12.34) passed the attention check, so I analyzed this sample in my subsequent analyses. I adopted a similar scenario as in Studies 2 and 3, except that perceived risk was measured on a 100-point scale ranging from 0 (“not risky at all”) to 100 (“extremely risky”), with a starting position at the midpoint (“moderately risky”). I also included a question to check whether participants had been vaccinated against COVID-19.

Participants perceived similar levels of risk among the first three groups (Risk _proportion = 26.87, Risk _{proportion(totalcases)} = 32.17, Risk _{totalcases(proportion)} = 30.39, comparison among the three groups F(2, 293) = 0.89, p = 0.41), but the perceived risk rose significantly in the total-cases condition (Risk _totalcases = 48.76, comparison among all four groups F(3, 394) = 11.65, p < 0.001). Figure 4 shows the distributions of perceived risks across groups. The proportion format produced the lowest perceived risk, though statistically it did not differ from the other two mixed-format conditions; the total-cases presentation again significantly elevated risk perception.

Figure 4: Perceived risks related to different mathematically equivalent presentation formats of infection information. Horizontal bars show means, bands (around the means) show 95% confidence intervals, dots show raw individual data, and beans show smoothed density curves.

Next, I tested the hypothesis that proportion format would lead to underestimation (vs. overestimation) of the total number of infections. Among 94 participants in the proportion-format group, 68 (72.3%) wrongly estimated the total number of infections. Among these 68 participants, 88.2% underestimated; only 11.8% overestimated (Exact Binomial Test p < 0.001). Recall that in Studies 2 and 3, 60.4% and 67% of participants in the frequency groups, respectively, underestimated. The likelihood of underestimation was much higher in the proportion group than in the frequency group (an equal-proportion test against the result in Study 2 (χ²(1) = 14.37, p < 0.001); against Study 3 (χ²(1) = 10.27, p < 0.001), evidence consistent with a lower perceived risk from proportion than frequency.

Finally, perceived likelihood of infection was positively (r = 0.82, t(396) = 27.99, p < 0.001) and trip intention was negatively (r = –0.64, t(396) = –16.41, p < 0.001) correlated with perceived risk. Both measures reflect similar patterns as perceived risk. That is, participants’ trip intention was significantly lower in the total-format group than in the three other groups. I then ran the same tests with a sub-sample (N = 279) that excluded participants who had received vaccination. The pattern remained essentially unchanged (for perceived risk, Risk _proportion = 29.52, Risk _{proportion(totalnumber)} = 28.72, Risk _{totalnumber(proportion)} = 32.08, Risk _totalnumber = 48.76; comparison among the first three groups F(2, 206) = 0.26, p = 0.77; comparison among all four groups F(3, 275) = 9.13, p < 0.001). Surprisingly, vaccination did not lower participants’ perceived risk (Risk _vaccinated = 33.02 vs. Risk _{notvaccinated} = 35.43, t(396) = 0.74, ns)).

5.2 Discussion

Study 4a showed again that placing a proportion format side by side with total number of cases significantly reduced perceived risks from total cases alone, almost to a similar level of presenting proportion alone. I found no differences between the two mixed-format groups, so presentation order did not affect perceived risks. As shown in Figure 4, participants perceived the risk from the total cases as moderately risky (around 50), while participants in the first three groups perceived about 3/5 (around 30) of the risk level from the total-cases format. Interestingly, an almost identical ratio is obtained in Study 3: Risk _proportion = 1.70 vs. Risk _totwithp = 2.86, the former being three-fifths of the latter. In sum, the evidence indicated that people in a proportion format perceived only 60% of the risk level from the total-cases format, although the information contained in both formats was mathematically equivalent.

6 Study 4b

Study 4a found that the mixed format of proportion and total cases significantly reduced the perceived risk from the total cases. Study 4b explores the combination of frequency and total cases by adopting the same design as in Study 4a, only changing the proportion expression of “the infection rate right now in this country is 0.0001” to a frequency expression of “the infection rate right now in this country is 1 in 10,000.”

6.1 Method and Results

Four hundred and four CloudResearch-approved MTurk participants were randomly assigned into four mathematically equivalent conditions (frequency, frequency (total cases), total cases (frequency), total cases; preregistered at https://aspredicted.org/blind.php?x=f3g69g. Three hundred and ninety-one participants (198 females; M _age = 40.65, SD = 11.81) passed the attention check; I analyze these participants below.

I replicated the finding that total cases produce higher perceived risks than frequency (Risk _totalcases = 45.49 vs. Risk _frequency = 36.92, t(192) = 2.03, p < 0.05). However, Figure 5 shows that the two mixed formats (Risk _{totalcases(frequency)} = 45.61; Risk _{frequency (totalcases)} = 44.83) do not differ significantly from the other two groups. An omnibus test of differences among groups did not reach conventional significance (F(3, 387) = 2.0, p = 0.114 ).

Figure 5: Perceived risks related to different mathematically equivalent presentation formats of infection information. Horizontal bars show means, bands (around the means) show 95% confidence intervals, dots show raw individual data, and beans show smoothed density curves.

6.2 Discussion

Because risk perceived from mixed formats generally falls between the two non-mixed formats, the smaller reduction of frequency against total cases may contribute to the null effect of the mixed format: the magnitude of reduction in perceived risk by frequency alone is 8.57, compared with 21.89 by proportion in Study 4a. In conclusion, this study failed to detect any significant differences for the mixed formats (vs. frequency or total cases). In particular, when one adds frequency to total number of cases, it could not reduce perceived risk from the total-cases format.

7 General Discussion

How do different presentation formats of the same COVID-19 information affect people’s perceived risks of infection? Most of my results suggest that people are affected by denominator neglect, but important deviations emerged.

7.1 Frequency vs. Total Cases

Denominator neglect predicted a relationship of Risk _totalcases > Risk _frequency > Risk _proportion (H1), which is supported in my studies only when the numerators of frequencies are relatively small. The finding of Risk _totalcases > Risk _frequency is particularly important because most current information releases adopt these two formats, with some media using them simultaneously. Therefore, communicators should not treat them as equivalent; they need to carefully choose between them. Consistent with this finding, my results provided support for H3a, that people tend to underestimate the total number of cases when provided with the frequency/proportion data.

Below, I offer some advice for communicators choosing among formats when releasing information. A general rule of thumb is to report total cases to increase perceived risk and to report frequency/proportion to reduce perceived risk.

1. 1. If the goal is to make sure people know the total number of cases, directly providing the total number of cases is advised.

2. 2. If the goal is to reduce perceived risk to the minimum, a proportion format should be used.

3. 3. If the goal is to reduce perceived risk and the proportion format is for some reason not applicable, then adopting a frequency format with a small numerator is advised (below 1,000). In addition, the 1-in-X effect suggests not to use 1 as the numerator.

4. 4. If the goals are to make sure people know the total number of cases and to reduce perceived risk, then using a mix of proportion and total-cases formats is advised.

Denominator neglect also predicts higher perceived risks for larger numerators among mathematically equivalent frequency formats (H2). This prediction was partly supported. The results of Study 3 showed that perceived risks were similar among formats with numerators 1, 10, 100, and 1,000 (Group One), as well as between numerators 10,000 and 100,000 (Group Two). In addition, there was a discernable increase from the former “Group” to the latter. The fact that no differences were detected within “Groups” offers strong evidence that participants paid attention to both numerators and denominators when processing those numbers.

This null finding replicates Price and Matthews (2009), who found no detectable differences when participants saw both numerators and denominators in frequency formats when the authors varied denominators from 10 to 1000 with fixed ratio of 10% or 30% (Studies 3 and 4). Their interpretation is that when both numerators and denominators are made saliently by being displayed explicitly, denominator neglect disappears. In the current studies, I have varied the denominators from 10,000 to 1,000,000,000, using the same frequency format with a fixed ratio of 0.0001. I found null effect within “Groups” and denominator neglect across “Groups,” thereby extending Price and Matthews’ (2009) findings. In other words, denominator neglect won’t necessarily disappear when both numerators and denominators are made explicit; in some cases, when the numerators or denominators are large enough, the effect can still occur when comparing to frequencies of smaller numerators/denominators.

The null finding is unlikely to be explained by small differences among numerators, for the differences among the numerators were, in fact, large. It is also unlikely to be attributed to lack of statistical power, because my effect-size estimates from both Studies 2 and 3 showed nearly zero effect within “Groups.” Since exemplar-cuing theory predicted no difference among all formats (H5), my empirical results can also be interpreted as partially supporting both denominator neglect and exemplar cuing. Alternatively, because the fractions presented in Price and Matthews’ (2009) studies, as well as in the current study, afforded easy simplifications by eliminating zeros from both numerators and denominators, an explanation through an elimination heuristic might explain the null denominator neglect for numerators smaller than 10,000.¹⁰ If so, the present results suggest that the heuristic of eliminating zeros would be more difficult to apply for larger numbers.

Conceptually, frequency and total cases proposed in this project resemble the “day format” and “year format” in Bonner and Newell (2008), who found that presenting the number of deaths from different diseases per year instead of per day increased subjective risk. An important difference exists: the formats studied in the current paper offer a description of the same COVID-19 state of a designated location at a certain moment, and conversions among the three (proportion, frequency, total cases) are purely mathematical. In contrast, conversion between the day and the year format needed an additional time factor. Because the time factor is arbitrary (one could choose a week, month, year, decade, etc.), what is counted as a “total number” was not clearly defined in Bonner and Newell (2008). In our context, the total number is clearly defined and agreed upon by readers. One might even think of the day vs. year manipulation as a specific case of frequency formats with different denominators of 1 day and 1 year/365 days. Notably, the findings in this article also diverged from those in Bonner and Newell (2008): They found that subjective risk varies when denominators vary, while I identified exceptions when people are not affected by denominator neglect.

Finally, although I have provided format recommendations based upon empirical findings in this research and upon different goals one has adopted, there is a caveat: The recommendations proposed in this paper may bias or distort understanding of the true state of affairs (for instance, using a denominator larger than the actual population would be quite misleading). Future studies are needed to identify which representations are likely to provide the most accurate sense of the true state of affairs. Nevertheless, what constitutes a “true state of affairs” and whether we can really know it is an epistemic (or perhaps ontological?) issue beyond the scope of the current research.

7.2 Mixed Format of Proportion/Frequency and Total Cases

H4b predicted that a mix of proportion format and total-number format would lead to an intermediate level of perceived risks: Risk _totalcases > Risk _mix > Risk _{proportion/frequency}. This prediction for mixing proportion and total cases was generally supported in Studies 3 and 4a, whereas the intermediate hypothesis was not supported for frequency and total cases. One difference between Studies 3 and 4a is that Study 3 showed an intermediate level of risk, while Study 4a showed effects close to those in the proportion group. Future studies are needed to explore what factors determine the weight of each component format on the mixed format. From these results, one can conclude that people do not perceive accumulated risks from a combination of presentation formats (H4a rejected); rather, they perceive the information as if they are doing a weighted average (H4b supported).

This finding has high practical significance. Given that most institutions are currently using the total-cases format, which leads to the highest perceived risk among mathematically equivalent alternatives, a mixed format can be used if communicators aim to reduce perceived risk and prevent public panic. Again, if the goal is to reduce perceived risk while still objectively releasing the total number of cases of COVID-19, a mixed format of total cases with proportion is recommended; if the goal is to solely reduce perceived risks and prevent panic, a proportion or frequency format can be used.

Interestingly, although the proportion format leads to the lowest perceived risk, it appears to be much less frequently used in the real world, except in the case of fatality rates. One reason may be that infection rates are often small, and a rate expressed as a proportion is too difficult for people to process (Gigerenzer, 1994; Gigerenzer & Hoffrage, 1995). For instance, people may have a hard time understanding the difference between a rate of 0.0001 and 0.00001, while converting it back to frequency facilitates understanding (1 in 10,000 or 1 in 100,000). Regardless, I recommend the proportion format if the sole goal is to reduce subjective risk.

7.3 What Are Participants’ True Predictions?

In Study 1, I found that people overestimated frequency if the predicted total number served as a benchmark and underestimated the total number if the predicted frequency served as a benchmark (H3b). It seemed to the participants that the idea of 100,000 people becoming infected (out of 1.4 billion) was more difficult to accept than 10 people out of 100,000 getting infected, even though the former ratio is objectively smaller. It is, however, less clear whether this bias results from an underestimation of the total number of infections, an overestimation of the infection rate, or both. In addition, it is unclear which format reflects an individual’s true prediction. For participants who were inconsistent, one cannot argue that both predictions reflected their true cognition. Is it a sort of constructed estimate, similar to constructed preferences (Bettman et al., 1998) or constructed attitudes (Schwartz, 2007)? This is a question that future research should address.

A possible explanation is that people do not have a stable estimate of how many people are getting infected; instead, they perceive risks as feelings (Loewenstein et al., 2001) and translate them into different quantitative estimates for different scales. This interpretation is consistent with the finding in past research that response scales affect judgment (Slovic & Monahan, 1995; Slovic et al., 2000). There are two major differences, however, between the methods of Study 1 in this article and those of Slovic et al. (2000). First, I compared the two most frequently adopted formats in COVID-19 information releases, total cases and frequency, while they compared proportion and frequency. Second, my Study 1 adopted a within-subjects design, while Slovic and coauthors used between-subjects designs. The within/between difference is worthy of attention, for a within-subjects inconsistency strongly implies irrationality. Indeed, Kahneman (2011) predicted that denominator neglect “would surely be reduced or eliminated” (p. 329) when the two formats were directly compared. What is surprising here is that the effect persisted when the two questions were placed together.

7.4 Conclusion

When institutions release information about COVID-19, the two most frequently adopted metrics are the total number of cases and the frequency of cases. Research comparing these two formats has been lacking. My findings show that these two formats are not equivalent: the total-cases format generally led to higher perceived risk than the frequency format. Theoretically, I have provided support for the theory of denominator neglect in demonstrating Risk _totalcases > Risk _frequency > Risk _proportion, given that numerators/denominators in the frequency formats are relatively small. These findings can provide guidance for format choices in the release of information about COVID-19.

Denominator neglect alone, however, cannot explain my observations among frequency formats. In Studies 2 and 3, denominator-neglect-induced differences were only observed after crossing several orders of magnitude. For example, 1 in 10,000, 10 in 100,000, and 100 in 1,000,000 were perceived similarly in Study 3, while significant differences began to emerge when the numerator crossed a threshold of 10,000. In fact, if one sees proportion 0.0001 as a frequency format with denominator 1, a threshold theory could reconcile previous findings of both the existence of a frequency-proportion gap and null effects among mathematically equivalent frequency formats (Koehler & Macchi, 2004): when the numerators were within a certain range, participants perceived similar levels of risks; they started neglecting denominators when they crossed that range, usually by several orders of magnitude. Such an interpretation explains, to take a concrete example from Study 3, the difference between 0.0001 and 10 in 100,000, and no difference between 10 in 100,000 and 100 in 1,000,000. The numerator of the former crossed five orders of magnitude, while the latter crossed only one order.

Essentially, my interpretation is a categorization one: People may perceive a certain range of infections as small or acceptable and other ranges as severe or serious. However, it remains unclear exactly how such ranges function. Although in Study 3, risk perception made a discernable jump from numerator 1,000 to numerator 10,000, the evidence so far only supports the assertion that there may be a threshold between 1,000 and 10,000 where a qualitative change in risk perception takes place. We do not yet know whether this is a case specific to COVID-19 or a general rule for risk perception as numerators increase. Further studies are needed to determine the mechanism of change in risk perception when numerators change in finer scales. Although varying levels of difficulty in eliminating zeros may be one mechanism for the stepwise pattern, more process evidence is needed to validate whether this actually occurs and how exactly such elimination works. There may also be important individual differences. All these issues await further research. By contrast, I hold more confidence in the differences between total cases and frequency identified in this research. In addition, adding a proportion to total cases can reduce perceived risk from total cases alone. Both findings are likely to persist in settings beyond COVID-19.