# How to Interpret Odds Ratios

In statistics, probability refers to the chances of some event happening. It is calculated as:

PROBABILITY:

P(event) = (# desirable outcomes) / (# possible outcomes)

For example, suppose we have four red balls and one green ball in a bag. If you close your eyes and randomly select a ball, the probability that you choose a green ball is calculated as:

P(green) = 1 / 5 = 0.2. The odds of some event happening can be calculated as:

ODDS:

Odds(event) = P(event happens) / 1-P(event happens)

For example, the odds of picking a green ball are (0.2) / 1-(0.2) = 0.2 / 0.8 = 0.25.

The odds ratio is the ratio of two odds.

ODDS RATIO:

Odds Ratio = Odds of Event A / Odds of Event B

For example, we could calculate the odds ratio between picking a red ball and a green ball.

The probability of picking a red ball is 4/5 = 0.8.

The odds of picking a red ball are (0.8) / 1-(0.8) = 0.8 / 0.2 = 4.

The odds ratio for picking a red ball compared to a green ball is calculated as:

Odds(red) / Odds(green) = 4 / 0.25 = 16.

Thus, the odds of picking a red ball are 16 times larger than the odds of picking a green ball.

## When Are Odds Ratios Used in the Real World?

In the real world, odds ratios are used in a variety of settings in which researchers want to compare the odds of two events occurring. Here are a couple examples.

### Example #1: Interpreting Odds Ratios

Researchers want to know if a new treatment improves the odds of a patient experiencing a positive health outcome compared to an existing treatment. The following table shows the number of patients who experienced a positive or negative health outcome, based on treatment. The odds of a patient experiencing a positive outcome under the new treatment can be calculated as:

Odds = P(positive) / 1 – P(positive) = (50/90) / 1-(50/90)  = (50/90) / (40/90) = 1.25

The odds of a patient experiencing a positive outcome under the existing treatment can be calculated as:

Odds = P(positive) / 1 – P(positive) = (42/90) / 1-(42/90)  = (42/90) / (48/90) = 0.875

Thus, the odds ratio for experiencing a positive outcome under the new treatment compared to the existing treatment can be calculated as:

Odds Ratio  = 1.25 / 0.875 = 1.428.

We would interpret this to mean that the odds that a patient experiences a positive outcome using the new treatment are 1.428 times the odds that a patient experiences a positive outcome using the existing treatment.

In other words, the odds of experiencing a positive outcome are increased by 42.8% under the new treatment.

### Example #2: Interpreting Odds Ratios The odds of an individual buying the item after seeing the first advertisement can be calculated as:

Odds = P(bought) / 1 – P(bought) = (73/100) / 1-(73/100)  = (73/100) / (27/100) = 2.704

The odds of an individual buying the item after seeing the second advertisement can be calculated as:

Odds = P(bought) / 1 – P(bought) = (65/100) / 1-(65/10)  = (65/100) / (35/100) = 1.857