What’s the Risk? It’s all Relative
Introduction
For a number of years now, the public has been bombarded with the results of studies on the health risks associated with a range of substances and foods, from hair dyes to “transfatty acids.” The media often report something like, “Those eating food X were 50% more likely to be diagnosed with coronary artery disease.”
If you have been puzzled about just what is meant by terms like risk, absolute risk, relative risk, and odds, this article may help you develop a clearer understanding.
Important Note: The numbers used in this article, although realistic, were selected for demonstration purposes only, and are not based on actual studies. If you would like factual information about your own risk from smoking, please visit Learn About Your Risk from Smoking: How it Works & About the Data. 
Risk
Let’s start with the term risk. When we do something that could result in injury or illness, we have engaged in risky behavior. In this sense, the word risk has a negative overtone, which is not necessarily the case. Risk is simply the chance, or probability, of an event occurring.
A probability can be described by numbers, ranging from 0 to 1. An impossible event has a probability of 0, while an inevitable event has a probability of 1, with every other possibility in between. As an example, let’s say you have a pair of dice. There are 6 numbers on each die. The probability of rolling a single die and getting a 2 is one chance out of 6, which is 1 divided by 6, or 0.1667.
Risks and probabilities are usually expressed as a decimal, such as 0.1667, which is the same as 16.67 percent. In other words, if you were to roll a die 100 times, you would expect to get a 2 (or any other of the numbers) between 16 and 17 times.
Risks of illness are determined by observing a number of people over time and keeping track of how many come down with the illness of interest. In the same way that the people you associate with are your cohorts, when we assemble a group of people with common characteristics, we call it a cohort. The common characteristics might be certain health behaviors, for example smoking status. This is the basis of what is termed a cohort study, in which we observe the health outcomes of the group over time.
Absolute Risk
In order for a health risk to have meaning, we need to think about the time frame involved; a risk of 0.10 (10 %) of dying in the next decade looks much different than a risk of 0.10 of dying in the next minute! Often we’ll read or hear of lifetime risk, such as “. . .the lifetime risk of ovarian cancer is 1.8%.”
Absolute risk is the chance you will develop a certain disease during a specific time period. For example, let’s say that in a group of 20,000 people, 1,600 develop lung cancer. The risk of any one individual getting this disease would be 1,600 divided by 20,000 or 0.08. That means that your absolute risk would be 8 percent – or 8 chances out of 100.
Relative Risk
We can use relative risk to determine if certain behaviors cause more disease. Relative risk is when we compare one group of people to another. For example, we can measure the risk of getting lung cancer from smoking by comparing a group of 10,000 people who smoke with a group of 10,000 people who don’t smoke.
What is the chance that people who smoke will get lung cancer in 20 years of smoking? At the end of 20 years, let’s say that 1500 of the smokers get lung cancer. We divide 1500 lung cancer cases by the 10,000 total number of smokers, and get 0.15, or 15 percent. The 20year risk of lung cancer among smokers would be 15 percent.
Similarly, we might ask what the probability is of people who don’t smoke getting cancer in a 20year period. Let’s say only 100 out of 10,000 nonsmokers got cancer over 20 years. We divide 100 by 10,000 and get 0.01 or 1 percent. This would yield a “20year risk” of lung cancer among nonsmokers of 1 percent.
We can relate these two calculations to one another in a way that would give us an indication of the effect of smoking on the risk of lung cancer.
The risk among smokers was 15 percent and among nonsmokers was 1 percent. We divide 15 by 1 and get 15. This means the smokers were 15 times more likely to develop lung cancer than the nonsmokers. That is relative risk.
Keep in mind that while absolute risks of some diseases may be quite small, risks relative to one another may be very large. In our smoking example above, the absolute risk of getting lung cancer after smoking for 20 years was only 15 percent. But when you compare that to people who don’t smoke for 20 years, you see that you would be 15 times more likely to develop lung cancer if you were in the smoking group.
Excess Risk
Another term that appears in the literature is excess risk or relative excess risk, which is most commonly expressed by a statement like, “Those eating food X were 50% more likely to be diagnosed with coronary artery disease.” In the case of our smoking and lung cancer example, the Relative Risk was 15 (15 divided by 1), so the relative excess (or extra) risk due to smoking is 15 – 1 = 14 or 1,400% — quite a large effect!
Odds and Odds Ratios
The use of the word odds in the media is often synonymous with risk and probability. Though the two concepts are related, they have different meanings. The “odds” are defined as the probability that an event will occur divided by the probability that the event will not occur. If we let p equal the probability of an event occurring, then the probability of it not occurring is 1p (remember that the highest probability is 1). The odds would be p divided by 1 p.
Let’s think about the odds of the Buckeyes winning a particular football game. (Usually we talk about point spreads when it comes to football, but for the sake of illustration, we’ll talk about odds.) If it has been estimated that the Buckeyes have an 80% chance of winning a game, then p = 0.80. That means they would have a 20% chance of losing the game, which is 0.20 (or 1 p = 1 – 0.80). So the odds in favor of OSU winning the game are 0.80 divided by 0.20 = 4, or 4 to 1. For every 4 wins, we would expect 1 loss.
In our smoking example above, determination of the relative risk of developing lung cancer involved 20 years of observation of study subjects. As you can imagine, if a disease is very rare, it would take quite a long time for enough subjects to become ill with the disease of interest to calculate risks, and subsequently the relative risk. Casecontrol studies are used to assemble subjects and get results much more quickly.
In casecontrol studies, we begin with a group of people with a specific disease (cases) and a group of people without the disease (controls) and make measurements of the various risk factors we suspect may be involved in causing the illness.
Continuing with our smoking example, we can calculate the odds of smoking among the cases (who have lung cancer) and the odds of smoking among the controls (who do not have lung cancer). Comparing these two odds calculations by dividing them is analogous to calculating a relative risk; that is, we’re calculating a relative odds, or an odds ratio. Although odds ratios and relative risks are not identical, an odds ratio may be a good estimate of a relative risk.
Let’s take a look at some numbers again. The table below contains the same numbers we used before.
Smoking and Lung Cancer
CaseControl Design
Lung Cancer Cases 
Healthy Controls 
Total 

Smokers 
1,500 
8,500 
10,000 
NonSmokers 
100 
9,900 
10,000 
Total 
1,600 
18,400 
20,000 
Let’s assume now that we have recruited 1,600 lung cancer cases and 18,400 healthy controls. (It’s not very realistic to recruit so many controls, but let’s ignore that for now.) We ask the subjects about their smoking status and we find out that 1,500 or 93.7% (1,500/1,600 = 0.937) of the cases were smokers. That means that the probability of a case being a smoker is 0.937. Likewise, the probability of a case being a nonsmoker is 100/1,600 = 0.063 or 6.3%.
Of the 18,400 controls, 8,500 were smokers or 46.2% (8,500 / 18,400 = 0.462). So the probability of a control being a smoker is 0.462. The probability of a control being a nonsmoker is 9,900 / 18,400 or 0.538.
Based on the discussion of odds above, can you set up the calculation of the odds of smoking among cases? Among controls?
Recall that odds are the probability of an event happening divided by the probability of it NOT happening. For the cases, the odds of smoking are:
Probability of being a smoker = 0.937 = 14.9 Probability of not being a smoker = 0.063
For the controls, the odds of smoking are:
Probability of being a smoker = 0.462 = 0.859 Probability of not being a smoker = 0.538
Now we can calculate the odds ratio by dividing the odds of smoking among cases by the odds of smoking among controls, or 14.9 / 0.859 = 17.34. The interpretation of this is that the odds of smoking among the lung cancer patients are more than 17 times higher than the odds of smoking among the controls.
How does this compare to the relative risk from the cohort example? They are close. The odds ratio is the only measure of effect we can calculate from a casecontrol study, but if the study is done well and the disease is rare, the odds ratio is a good estimate of a relative risk from a cohort study.
A simpler way to calculate the odds ratio is to use what is called the crossproduct ratio. In our example above that would be (1,500 x 9,900) / (8,500 x 100) = 17.47 (the difference is rounding error). It is important when calculating the crossproduct ratio to have the table set up just the way we have it in the example.
Understanding terms like probability, risk, odds and relative risk, is essential to being able to navigate your way through health literature. We hope this introduction smoothes the way for you. Happy reading!
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