Risk ratio: How much more likely is it?

In statistics, calculating the risk ratio is one of the four main ways of quantifying association. The other three are odds ratio, hazard ratio, and absolute risk reduction. Conceptually this is pretty simple: you determine the probability (this is different from the odds) of an outcome with and without a variable and then divide one by the other. If the answer is 1, your variable has nothing to do with your outcome. If it isn't 1, there is an association. Note that an association is NOT a causative relationship. This is a more robust statistic than odds ratio, but requires good random sampling. When the event being studied is rare, risk ratio and odds ratio are nearly interchangeable.

Let's do an example.
You might remember these from the odds ratio page. Watch the math, it's different this time!

A small number of children grow up to be vets. We suspect that living in a home with a cat makes it more likely that the child will be a vet when she grows up. To determine this we do a survey and we find the following:

	Cat	No Cat
Vet	24	34
Not a Vet	346	543

To calculate the risk ratio we need the two probabilities: the probability of being a vet with a cat and the probability of being a vet without a cat. That will be 24/(346+24) for cat and 34/(543+34) for no cat. Those reduce to 0.064 and 0.059 respectively. For the risk ratio we will take probability of vet with cat and divide that by probability of vet without cat: 0.064/0.059 = 1.08. That's greater than 1, so there is a (small) association with living with a cat as a kid and becoming a vet. Remember, the odds ratio was 1.11, in this example the two measures of association are very similar.

Let's do another example

We're doing two here because I want you to see how OR and RR are similar when the event is rare but very different when the event is common. So this time let's look at an association with a common event.

Many adults own cats. We suspect that growing up in a home with a cat as a child makes it more likely that the child will be a cat owner when she grows up. To determine this we do a survey and we find the following:

	Childhood Cat	No Childhood Cat
Current Cat Owner	453	164
Not a Current Cat Owner	47	346

To calculate the risk ratio we need the two probabilities: the probability of being an adult cat owner with a childhood cat and the probability of being an adult cat owner without a childhood cat. That will be 453/(47+453) for childhood cat and 164/(346+164) for no childhood cat. Those reduce to 0.906 and 0.33 respectively. For the risk ratio we will take probability of cat owner with a childhood cat and divide that by probability of cat owner without a childhood cat: 0.906/0.33 = 2.75. That's greater than 1, so there is an association with living with a cat as a kid and owning a cat as an adult. Remember, the odds ratio was 20.5, nearly an order of magnitude larger! When the event (cat ownership in this case) is common the odds ratio and risk ratio are very different, even if they express the same idea.