# Risk ratios, odds ratios, risk differences: How do researchers calculate the risk from a risk factor?

## The effects of risk factors can be calculated in different ways. How are they calculated and interpreted?

Over their lifetimes, people are often exposed to various factors that make disease or death more likely to occur. These are known as health “risk factors”.

To guide decisions in public health, such as how to prevent premature deaths or how to allocate public health resources, it’s crucial to know about the impacts of different risk factors.

Risk factors that have a large effect, or are common in the population, can help us understand which interventions would make a big difference.

But how can we measure the risk that a given factor has on the outcomes we care about?

In this article, I’ll explain this with a simple example. I’ll focus on a hypothetical risk factor – a toxic pollutant – and show you how its impact can be calculated in different ways, and how this affects the interpretation of the underlying data.

## Ratios

One way to calculate the effect is by using ratios of the outcome between two groups.

For example, the **risk ratio** is aimed at answering the question: how *many times* higher is the risk of the outcome among people who are exposed to the risk factor?

In our example, the outcome is death, and the data show that people exposed to high levels of this hypothetical type of toxic pollutant were twice as likely to die within the next 20 years.

The opposite is the **survival ratio**. It answers the question: how many times higher is the chance of *avoiding* the outcome, among people not exposed to the risk factor?

In our example, people only exposed to low levels of this toxic pollutant were 1.5 times as likely to survive the next 20 years.

Researchers sometimes calculate the **odds ratio** instead of the risk ratio. The odds ratio answers the question: how many times higher were the odds of the outcome, in people exposed to the risk factor?

As you can see, the denominator is different from the risk ratio. Rather than calculating the proportion of people who died, it compares the *number* of people died to those who didn’t.

In the example, people who died had 3 times the odds of having been exposed to high levels of the toxic pollutant during the past 20 years.

The odds ratio is mathematically similar to the risk ratio when the outcome is rare, because A+B will be similar to B, and C+D will be similar to D. But when the outcome is common, the odds ratio and risk ratio can be very different.^{1}

## Differences

Another way to express the increase in risk is to measure how much of a difference the risk factor makes to the outcome.

The **risk difference** answers the question: how much higher is the risk of the outcome among people who are exposed to the risk factor?

In the example, people exposed to high levels of the toxic pollutant had a 25 percentage point higher chance of dying within the next 20 years.

Another metric is the **number needed to harm**. It aims to answer the question: how many people would need to be exposed to the risk factor, to see the outcome in one of them?

It is the inverse of the risk difference.

In our example, four people would need to be exposed to high levels of the toxic pollutant for one to die within the next 20 years, on average.

By clicking ‘show more’, you can see how they compare in a full visualization.

## Case study: How much does smoking increase the risk of death?

Let’s take smoking as a real-life example.

Researchers know from many studies that smoking increases the risk of death from cancers, heart disease, diabetes, tuberculosis, and other causes of death. But how much does it increase these risks?

In the chart, you can see an example using data from the United States.^{2}

It shows how smoking increases the risk of death from various causes, using the **risk ratio**. It shows, for example, that male smokers have 21 times the risk of dying from lung cancer as those who have never smoked.

## What to consider when calculating risks

There are several important things to remember when calculating risks from risk factors.

First, it’s important to have good data on the causes of death from death certificates, and epidemiological research to estimate the impact of risk factors on death.

In the example of smoking, data on smoking prevalence comes from national surveys – but people tend to underreport their smoking behaviors. This means that many smokers would be labeled as non-smokers, leading to underestimating the risk of death from smoking.

A related point to consider is that the risks can vary between people. For example, the category ‘current smoker’ is very broad, as people have smoked for different lengths of time and amounts.

A third point is that other confounding factors can be present. For example, smokers may also have other risk factors or behaviors that increase their risk of death. In the example above, the researchers had adjusted for other known confounding factors, but this is not always simple.

## Conclusion

Researchers can calculate the impact of a risk factor using different metrics – but they are interpreted differently.

In this article, we calculated these metrics using a simple hypothetical example – where there were only two possible levels of exposure and two possible outcomes. Researchers can use extended statistical methods for more complex questions.^{3}

These estimates also have important limitations. They depend on good underlying data and study designs to assess the causal effects of risk factors.

By estimating the effects of different risk factors, and how many deaths they cause, we can identify better ways to save lives.

#### Acknowledgements

Edouard Mathieu, Hannah Ritchie, and Max Roser provided valuable feedback on this article.

### Endnotes

Holmberg, M. J., & Andersen, L. W. (2020). Estimating Risk Ratios and Risk Differences: Alternatives to Odds Ratios. JAMA, 324(11), 1098. https://doi.org/10.1001/jama.2020.12698

Oza, S., Thun, M. J., Henley, S. J., Lopez, A. D., & Ezzati, M. (2011). How many deaths are attributable to smoking in the United States? Comparison of methods for estimating smoking-attributable mortality when smoking prevalence changes. Preventive Medicine, 52(6), 428–433. https://doi.org/10.1016/j.ypmed.2011.04.007

See this paper also for a discussion on the appropriate choice of metric depending on the situation.

Colnet, B., Josse, J., Varoquaux, G., & Scornet, E. (2023). Risk ratio, odds ratio, risk difference... Which causal measure is easier to generalize? https://doi.org/10.48550/ARXIV.2303.16008

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`Saloni Dattani (2023) - “Risk ratios, odds ratios, risk differences: How do researchers calculate the risk from a risk factor?” Published online at OurWorldInData.org. Retrieved from: 'https://ourworldindata.org/risk-ratios-odds-ratios-risk-differences-how-do-researchers-calculate-the-risk-from-a-risk-factor' [Online Resource]`

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```
@article{owid-risk-ratios-odds-ratios-risk-differences-how-do-researchers-calculate-the-risk-from-a-risk-factor,
author = {Saloni Dattani},
title = {Risk ratios, odds ratios, risk differences: How do researchers calculate the risk from a risk factor?},
journal = {Our World in Data},
year = {2023},
note = {https://ourworldindata.org/risk-ratios-odds-ratios-risk-differences-how-do-researchers-calculate-the-risk-from-a-risk-factor}
}
```

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