Since 2001, the BBC’s Radio Four has broadcast More or Less. This weekly program breaks down statistics that have made their way into media headlines and is now also a podcast. A recent episode featured an alarming statistic concerning deaths after surgery: Women undergoing surgical procedures are 32 percent more likely to die if their surgeon is a man. Given that men outnumber women in most surgical specialties and that there were 6.6 million surgeries performed in 2019, a 32 percent increase in the risk of surgery-related death for women under the care of male surgeons grows all the more shocking.

But a quick fact check, in the spirit of More or Less, is in order. Of those 6.6 million surgeries cited in the linked NIH report, 52.9 percent were performed on women. From the linked Statista chart, the proportion of men in specialties with “surgery” in the title is 86 percent. A quick back-of-the-envelope calculation (6.6 million surgeries times 52.9 percent women patients times 86 percent men surgeons times a 32 percent mortality rate) yields 960,833 women who theoretically would have died from surgery in 2019. According to the CDC’s tally, though, the leading cause of deaths in the U.S. in 2019 was heart disease, which killed 696,692 Americans of all genders. What’s going on here?

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## Different Risks

Upon closer inspection, the headline figure reads “32 percent more likely to die.” The words “more likely” are critical here. They imply that there is a quantifiable difference in the respective likelihoods of either of two situations occurring. In this case, those situations are that a woman dies at the hands of a male surgeon or that she dies if the doctor is a woman. The 32 percent increase between the probability that the first situation occurs and the probability that the second situation occurs is what the headline figure represents.

This example best illustrates the difference between two types of likelihoods or, by another name, risks: absolute and relative risk. That 32 percent is a relative risk, while the underlying probabilities of male and female surgeons killing female patients are the respective absolute risks. Absolute risks are the probabilities that a situation occurs or doesn’t occur. In the example, the two absolute risks are the probability that male surgeon kills a female patient and the probability that a female surgeon kills a female patient. The proportionate increase from the second probability to the first represents the relative risk between these two scenarios.

The difference between absolute and relative risk might seem trivial or even pedantic. But understanding when and where to employ each type of risk is crucial to risk-based thinking in applied contexts. Alternatively, getting relative and absolute risks confused can lead to serious errors in judgment with ramifications for individuals, businesses and policymakers.

Getting back to the surgery headline example, the missing pieces of information are the absolute risks of patient deaths under the care of male and female surgeons. On More or Less, the medical statistician pointed out that these absolute risks in the cited study are 0.66 percent and 0.5 percent, respectively. In other words, the absolute increase in women’s deaths due to having a male surgeon is 1.6 deaths per every 1,000 surgeries. The relative increase, however, is 6.6 deaths per 1,000 surgeries minus five deaths per 1,000 surgeries divided by five deaths per 1,000 surgeries, or 32 percent. This translates into one more female patient dying for 625 more male-surgeon operations, as the host, Tim Harford, noted.

## Absolute Versus Relative Risk

Relative risk is the differential probability of an event occurring to a specific group in the data, while absolute risk is the probability that a situation occurs or doesn’t occur at all. The difference between relative and absolute risk might seem trivial or even pedantic. But understanding when and where to employ each type of risk is crucial to risk-based thinking in applied contexts.

## Relatively Speaking

In this case, confusing a relative risk for an absolute risk made the study’s results eye-catching when splashed across headlines in the popular media. But relative risks can change based on how a researcher constructs the relation of one event to another. For instance, what if the question was not how many women die due to having a male surgeon, but how many women survive thanks to a woman as their surgeon?

Nominally, this is the same question. The absolute risk that a female patient survives a female-surgeon operation is one minus five deaths per 1,000 surgeries, or 995 survivals per 1,000 surgeries. For male surgeons, the female-patient survival risk is one minus 6.6 deaths per 1,000 surgeries, or 993.4 survivors per 1,000 surgeries (that last one was an amputation). The difference in these absolute risks of female-patient survival is 995 minus 993.4 per 1,000, or 1.6 fewer women dying for every 1,000 operations by women. This is equivalent to the previous difference in absolute risk of death from having a male surgeon. The relative risk of surviving surgery from having a female surgeon as opposed to a male surgeon, however, is 1.6 per 1,000 divided by 993.4 deaths per 1,000 surgeries, or 0.16 percent.

Framing the research results this way is unlikely to generate a lot of general interest. In this case, presenting the most alarming relative risk is the headline finding of the study. It’s easy, then, to conclude that journalists writing up the results must have been engaging in a bit of selection bias about which of the study’s findings to highlight. Indeed, the study’s lead researcher, Dr. Christopher Wallace, alludes to the “degree of sensationalism in how things are presented” in the More or Less episode.

But are popular-media journalists really sifting through reams of recently published scientific articles and preprint archives, carefully scouring each work for headline-grabbing statistics? Most journalists are not highly trained scientists and certainly don’t have anywhere near the time to vet all new studies that come out for the most newsworthy findings.

The way members of the press typically discover striking numbers like the 32 percent figure is via press release. As psychologist and science communicator Stuart Ritchie points out, the very scientists performing the research often collaborate heavily in crafting press releases. These researchers have direct incentives, in terms of academic notoriety and future funding potential, to feature research findings most likely to grab journalists’ attention. This can include making the decision to present large relative risks without mentioning the underlying absolute risks upon which they are computed.

## A Theory of Relativity

All of this raises the following question: If relative risks can be spun so easily, why use them at all? Why not just report absolute risks and call it a day? As it turns out, there are situations in which relative risks are important. Absolute risks are tied to their base groups and don’t generalize well.

For instance, the absolute risk of death from heart disease is higher for elderly smokers than for elderly non-smokers. Likewise, the absolute risk of death from heart disease is higher for smokers 40 and younger than for non-smokers 40 and younger. The absolute risk of heart disease among all members of the elderly group is higher than for all members of the 40-and-younger group by a large margin, however. Does that mean that smokers aged 40 and younger are in the clear to light up until their elderly years?

More broadly, how can we compare the population-wide effects of smoking versus not smoking on heart disease risk across many disparately aged subpopulations? This is where relative risk shines. Relative risk accounts for the absolute risks of heart disease for smokers and nonsmokers across age cohorts. In doing so, it gives a more general risk profile of death from heart disease due to smoking for all ages society-wide.

Another recent example of the benefits of relative risk is the impact of vaccination on Covid-19-related fatalities. Weekly data from the U.K. for the period from August 8, 2021 to September 4, 2021 indicate that the absolute risk of dying from Covid-19 for those aged 50 to 79 was 55.3 deaths per 100,000 people for vaccinated individuals and 213.4 deaths per 100,000 people for unvaccinated individuals. For the purposes of the study, deaths are classified as Covid-19-related if they occurred within 60 days of a first positive test result.

These same absolute risks for British people aged 40 or younger were 0.3 deaths per 100,000 people and 1.7 deaths per 100,000 people, respectively. Here, the absolute risk of Covid-19 death is far lower for both vaccinated and unvaccinated individuals in the below-40 age group relative to just the vaccinated individuals of U.K. citizens aged 50 to 79. Computing relative risks of Covid death for vaccinated versus unvaccinated individuals, however, results in a 470 percent increase in death risk for people 40 and under and a 510 percent increase in death risk for people aged 50 to 79.

Furthermore, the relative risks across age cohorts above 18 years old in the surveillance report range from 217 percent to 808 percent Absolute risk ranges stretch from 0.4 per 100,000 to 129 per 1,000 and from 0.1 per 100,000 to 40.8 per 100,000 for unvaccinated and vaccinated individuals, respectively. Mass vaccination from a pandemic disease is a society-wide decision that policymakers and health authorities must coordinate, however.

The scale of the impact needs to be considered, which means potential risks must be generalizable. Taking an average across age cohorts of absolute risks for unvaccinated and vaccinated individuals over 18 years old in the surveillance report produces overall absolute risks of Covid-related death of 32.9 per 100,000 for unvaccinated individuals and 8.2 per 100,000 for vaccinated individuals. These absolute risks illustrate the fatality rates of two scenarios: one in which vaccines are either not available or not introduced that would result in four times as many deaths in the population as a second scenario in which vaccines become widely available.

The standard deviations for these absolute risks, however, are 47 deaths per 100,000 and 14.8 deaths per 100,000, respectively. Standard deviations of data being larger than their averages mean that data are either volatile or are being driven by some unconsidered factor. Here, the overall average elides the fact that different age cohorts have highly dissimilar absolute risks of Covid-related fatality. The average relative risk increase between vaccinated and unvaccinated individuals is 491 percent (another colloquial way to refer to relative risk is “4.9 times” higher).

In contrast to the huge standard deviations in the average absolute risks, the standard deviation of the 4.9-fold increase is 1.9-fold. In translation, for every vaccinated British person who suffers a Covid-related death, between 3 and 7 unvaccinated British individuals will die of Covid regardless of age group. This is a much more stable estimate that authorities can use in a general context and it is particularly useful when data on Covid deaths by age cohorts are thin or missing.

The main difference in relative and absolute risks comes down to outcomes’ scales and contexts. When the scale or context of the risks undertaken is at the individual- or group-level, it’s best to use absolute risk differences to guide decision-making. When decisions’ outcomes affect many groups, businesses, or industries differently; if absolute risks are highly disparate between specific groups; or if information on absolute risks for certain groups is thin or unknown, relative risk provides a better framework to evaluate the riskiness of potential decisions.

To draw a concrete example, the decision of whether to insure against some risk implies knowledge of the absolute risk of the event occurring. So, if a business purchases product liability insurance it would imply that the manufacturer understands the underlying absolute risk of the insured product’s failing. An apparel firm’s decision to purchase liability insurance on a new design of shirt, say, would be much different than a hardware company’s decision to purchase liability insurance on a new design of electric saw.

In contrast, if the absolute risk of a new saw design’s failure is unknown, the relative risk of newly designed products’ failure rates versus older products’ failure rates can be employed to understand the likely increase in the absolute failure rate of a newly designed electric saw versus these rates in older models. This would subsequently help to guide company stakeholders’ decisions about whether the insurance liability incurred will be worth mass manufacturing a new version of a relatively riskier product.

Finally, because it’s important for risk communication to build trust, where possible, both absolute and relative risk estimates should be reported, preferably in a way that contextualizes risks in terms of relevant outcomes at the decision level. Units matter, and individuals even more so.

## Resolution Economics

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