Throughout this article we will use the following example:

Suppose we conducted a study and found out that moderate consumers of red wine have a 10-year risk of heart disease of 0.9%, and non-consumers have a risk of 1.2%.

Our objective is to find out whether red wine is good for the heart or not.

So our job is to compare these 2 proportions (call them p_{wine} = 0.009 and p_{no_wine} = 0.012). We can do this by calculating the following quantities:

- Risk difference
- Relative risk
- Odds ratio

## 1. Risk difference

The risk difference is simply the difference in proportions.

### How to calculate risk difference?

Above we saw that :

p_{wine} = 0.009

p_{no_wine} = 0.012

The risk difference is:

### How to interpret the risk difference?

So we found that the risk difference is -0.3%, which means that there is a 0.3% less risk of heart disease in the wine drinking group compared to the non-drinking group. Alternatively, we can say that there is a risk difference of -0.3% between the wine drinking and the non-drinking group, which is a more direct but less clear interpretation.

**Note**: For a positive risk difference, for example 5.7%, we say that there is a 5.7% greater risk of having heart disease in the wine consuming group compared to the non-drinking group.

## 2. Relative risk

Another way of comparing these risks is by dividing them.

### How to calculate relative risk?

The relative risk is the probability of the disease in the exposed group (wine consumers) divided by the probability of the disease in the unexposed group (non-drinkers):

### How to interpret the relative risk?

Since the relative risk (RR) is the ratio of 2 numbers, we can expect 1 of 3 options:

- RR = 1: The risk in the first group is the same as the risk in the second. So no evidence that drinking wine can either protect against or increase the risk of heart disease
- RR > 1: The risk of having the disease in the exposed group is higher than the unexposed group. So a person drinking wine has a greater risk of having a heart disease relative to someone who does not drink red wine
- RR < 1: The risk in the exposed group is less than the risk in the unexposed group. So a person drinking wine has a smaller risk of having a heart disease relative to someone who does not drink red wine

In our case, an RR of 0.75 implies that there is a 25% (1 – 0.75 = 0.25) reduction in the relative risk of heart disease in the wine consuming group compared to the non-consuming group. Alternatively, we can say that the risk of heart disease in wine consumers is 0.75 times that of non-consumers.

**Note**: If RR > 1, for example RR = 1.32, we say that there is a 32% greater relative risk of having heart disease in the wine consuming group relative to non-consumers. Alternatively, we can say that wine consumers have 1.32 times the risk of having heart disease compared to non-consumers.

### Risk difference vs relative risk

#### Similarities

- Both are means to compare risks
- They have the same direction of association: In our example above, both agreed that wine consumers have a lower risk of heart disease than non-consumers

#### Differences

They APPEAR to have different magnitudes.

In our example above, there is a 0.3% reduction in the absolute risk of heart disease for wine consumers and a 25% reduction in the relative risk, which can be misleading because one seems MUCH larger than the other. This difference in magnitude is more pronounced when we have a rare disease, as in our case (one group has a risk of 0.9% and the other 1.2%).

**To avoid any confusion/deception, medical studies should report both risk difference and relative risk.**

## 3. Odds ratio

This is the ratio of 2 odds.

### How to calculate odds ratio?

For a wine consumer, the odds of having heart disease are p_{wine} (the risk of having the disease) divided by 1 – p_{wine}.

For a non-consumer, the odds of having heart disease are p_{no_wine} divided by 1 – p_{no_wine}.

The odds ratio is the ratio of these 2 odds.

### How to interpret the odds ratio?

The odds ratio (OR) is a ratio of 2 numbers, like the relative risk we have 3 options:

- OR = 1: The odds in the first group are the same as those in the second. So no evidence that drinking wine can either protect against or increase the odds of heart disease
- OR > 1: The odds of having the disease in the exposed group are higher than the unexposed group. So a person drinking wine has greater odds of having a heart disease relative to someone who does not drink red wine.
- OR < 1: The odds in the exposed group are less than the unexposed group. So a person drinking wine has less odds of having a heart disease relative to someone who does not drink red wine

In our case, the wine consuming group has 0.752 times the odds of the non-consuming group of getting heart disease in the next 10 years.

Alternatively, we can say that the wine consuming group has a 24.8% (1 – 0.752 = 0.248) less odds of getting heart disease than the non-consuming group.

### When does odds ratio approximate relative risk?

As you can see, the interpretation of odds ratio is not as intuitive as that of the relative risk.

This is because most people tend to think in terms of risk (probability) and not odds.

#### So if the odds ratio is at best a mediocre version of the relative risk, then why use it at all?

It turns out that odds ratio can be an approximation of the relative risk when the latter cannot be calculated. For example:

- In case-control studies when the risk of a disease cannot be known
- When we use certain statistical methods (like logistic regression) that output results directly in the form of odds ratios

#### How well can odds ratio approximate the relative risk?

When the disease is rare, the odds ratio will be a very good approximation of the relative risk.

The more common the disease, the larger is the gap between odds ratio and relative risk.

In our example above, p_{wine} and p_{no_wine} were 0.009 and 0.012 respectively, so the odds ratio was a good approximation of the relative risk:

OR = 0.752 and RR = 0.75

If the risks were 0.8 and 0.9, the odds ratio and relative risk will be 2 very different numbers: OR = 0.44 and RR = 0.89

### Relative risk vs Odds ratio

#### Similarities

- They will always agree on the direction of comparison. In our example above, both will agree that wine consumers have less heart disease than non-consumers
- When RR = 1, OR = 1
- When RR > 1, OR > 1
- When RR < 1, OR < 1
- The smaller the risk of the disease is in both groups, the closer the OR is to the RR

#### Differences

Relative risk and odds ratio can be very different in magnitude, especially when the disease is somewhat common in either one of the comparison groups.

In cases where we cannot calculate the relative risk, sometimes we get stuck with an odds ratio that is a bad approximation the relative risk.