Probability and Statistics > Probability > Odds Ratio

## What is the Odds Ratio?

An odds ratio (OR) is a measure of association between a certain property A and a second property B in a population. Specifically, it tells you how the presence or absence of property A has an effect on the presence or absence of property B. The OR is also used to figure out if a particular exposure (like eating processed meat) is a risk factor for a particular outcome (such as colon cancer), and to compare the various risk factors for that outcome. You could use the OR to find out how much alcohol use leads to liver disease. Or you might want to find out if cell phone use has some link to brain cancer. As long as you have two properties you think are linked, you can calculate the odds.

## How to Calculate the Odds Ratio

You have two choices for the formula:

**(a/c) / (b/d)**

or, equivalently:

**(a*d) / (b*c)**

**General Steps:**

Step 1: Calculate the odds that a member of the population has property “A”. Assume the person **already has** “B.”

Step 2: Calculate the odds that a member of the population has property “A”. Assume the person **does not have** “B.”

Step 3: Divide step 1 by step 2 to get the odds ratio (OR).

### Odds Ratio Example

The above image shows two levels of exposure to ice cream: those who ate it, and those who didn’t. The 2×2 table also shows two outcome levels: people who were ill (“cases”) and people who were not (“controls”). The odds ratio is calculated as follows:

**Ill people**: people who ate ice cream / people who did not = 13/17**People who are not ill**: people who ate ice cream / people who did not = 32/23- Dividing the two results, we get (13/17) / (32/23) = 0.55

The resulting odds ratio of .55 means that ill people were about half as likely to eat ice cream as well people.

## Odds Ratio Interpretation; What do the Results mean?

- An odds ratio of exactly 1 means that exposure to property A does not affect the odds of property B.
- An odds ratio of more than 1 means that there is a higher odds of property B happening with exposure to property A.
- An odds ratio is less than 1 is associated with lower odds.

However, it’s not *quite *as simple as that. You could think of the odds ratio as being a bit overly simplistic at describing real world situations. If, for example, you have a positive OR, it doesn’t mean that you have a statistically significant result. In order to figure that out, you need to consider the confidence interval and p-values (if you know it). The other issue is that even if you determine your results are statistically significant, that significance might not apply to all members of a population — there are nearly always a multitude of factors associated with risk. For example, this article points out that while overall, depression is strongly linked to suicide, “…in a particular sample, with a particular size and composition, and in the presence of other variables, the association may not be significant.”

## Population Averaged vs. Subject Specific Odds Ratio

**Population averaged models** compare marginal distributions and give an overview of the effect on a whole population. The margins of a contingency table contain the totals, so it makes sense for them to be used to calculate the **marginal odds ratio** for a whole population. On the other hand, **subject-specific models **look at joint distributions: specific conditions or experiences *within *the model. The joint distributions are used to calculate **conditional odds ratios.**

**Marginal Odds Ratio Example (for Population Averaged Models)**

Michael Radelet studied death sentence data from Florida from 1976-77.* Calculate the marginal odds ratio for the race of defendant and whether or not that made a different about if they got the death penalty:

**Solution**:

- Sum (marginalize) the values in the table. We’re interested in only the race of the defendant and whether or not they got the death penalty. Therefore, we can marginalize (sum up) values for the race of the victim. This creates a new 2×2 table:

- Use the information in the marginal table to find the OR (using the OR formula from above):

OR = (a/c) / (b/d) = (19/17)/(141/149) = 1.12/0.95 = 1.18.

The odds are 1.18 times higher that a white defendant will get the death penalty compared to a black defendant.

*If you’re interested in his findings, he concluded that there isn’t any clear evidence to support the hypothesis that the defendant’s race is strongly associated with imposition of the death penalty.

**Subject specific models** calculate the odds ratio using the same formula as all of the examples above. The only difference is that instead of summing all the variables together, you’ll hold one variable constant (i.e. you’ll use joint distributions).

**Reference**:

Radelet, M. L. Racial Characteristics and the Imposition of the Death Penalty. American Sociological Review, v46 n6 p918-27 Dec 1981

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