Odds ratio: How much more likely is it?

In statistics, calculating the odds ratio is one of the four main ways of quantifying association. The other three are risk ratio, hazard ratio, and absolute risk reduction; I'll add pages on those as they come up. Conceptually this is pretty simple: you determine 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. Generally the risk ratio is the statistic most people are interested in; however, if a researcher doesn't have good random sampling (s)he can't calculate risk ratio. As long as the event being studied is rare, the two are nearly interchangeable.

Let's do an example

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 odds ratio we need the two odds: the odds of being a vet with a cat and the odds of being a vet without a cat. That will be 24/346 for cat and 34/543 for no cat. Those reduce to 0.069 and 0.062 respectivly. For the odds ratio we will take odds of vet with cat and divide that by odds of vet without cat: 0.069/0.062 = 1.11. That's greater than 1, so there is a (small) association with living with a cat as a kid and becoming a vet.

Let's do another example

We're doing two here because I'm going to repeat them when I get around to risk ratio and 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 odds ratio we need the two odds: the odds of being an adult cat owner with a childhood cat and the odds of being an adult cat owner without a childhood cat. That will be 453/47 for childhood cat and 164/346 for no childhood cat. Those reduce to 9.63 and 0.47 respectively. For the odds ratio we will take odds of cat owner with a childhood cat and divide that by odds of cat owner without a childhood cat: 9.63/0.47 = 20.5. 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.

It would be inappropriate, however, to say that a childhood cat makes you 20-times more likely to be a cat owner. (It's 6 times; we need to use the relative risk for that statement, not the odds ratio). This is often confused in literature. The issue is the difference between odds and probabilities. You probably don't think much about the difference, but statisticians do. If you survey 10 people about kittens and find out that 8 like them you have discovered that there is an 80% (8/10 --> 0.8) probability that a person is a fan of kitten or that the odds are 4:1 (8/2 --> 4) that a person is a fan of kittens.