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
Marketers want to know if one advertisement causes customers to buy a certain item more often than another advertisement so they show each advertisement to 100 individuals. The following table shows the number of people who bought the item, based on which advertisement they saw:
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
Thus, the odds ratio for a customer buying the item after seeing the first advertisement compared to buying after seeing the second advertisement can be calculated as:
Odds Ratio = 2.704 / 1.857 = 1.456.
We would interpret this to mean that the odds that an individual buys the item after seeing the first advertisement are 1.456 times the odds that an individual buys the item after seeing the second advertisement.
In other words, the odds of buying the item are increased by 45.6% using the first advertisement.
Additional Resources
How to Calculate Odds Ratio and Relative Risk in Excel
How to Interpret an Odds Ratio Less Than 1
How to Interpret Relative Risk