**Disclaimer: This post is intended for undergraduate students. Therefore, I will be discussing Odds Ratios from that perspective.**

Students often encounter the term Odds Ratio when learning about Case Control studies.

[At the undergraduate level, students are taught that an Odds Ratio is obtained from a Case Control study, while Relative Risk is obtained from cohort studies.]

Before we understand Odds Ratio, we need to understand ‘Odds’.

**What are ‘Odds’?**

**Example**: Assume I have a six sided dice.

**a.** What is the probability of obtaining a ‘four’ on a single throw of the dice?

Answer: The dice has six sides. The probability of obtaining any side after a single throw is 1/6. Therefore, the probability of obtaining a ‘four’ on a single throw of the dice is 1/6.

**b.** What are the odds of obtaining a ‘four’ on a single throw of the dice?

Answer: The dice has six sides. If obtaining the number of choice is a success, then the total number of successes after a single throw is one. The total number of failures (obtaining other numbers) after a single throw is five. The odds of obtaining a ‘four’ after a single throw of the dice is given by:

**Odds= Total number of successes/ Total number of failures**

Odds = 1/5

It is easy to see that the Odds of an event are different from the probability of that event.

**What is an Odds Ratio?**

**In epidemiology, Odds Ratio= Odds of Disease among exposed/ Odds of Disease among unexposed**

Therefore, the first step in the calculation of an Odds Ratio is the estimation of the corresponding Disease Odds.

In a 2×2 contingency table,

**Odds of Disease among exposed = a/b**

**Odds of Disease among unexposed= c/d**

Therefore,

Odds Ratio= Odds of Disease among exposed/ Odds of Disease among unexposed = (a/b)/ (c/d)

When solved, we obtain

**Odds Ratio= ad/bc**

**Interpretation of Odds Ratios**

Let us consider a few examples to understand how to interpret Odds Ratios:

**Example 1**: A case control study has found that the Odds of developing lung cancer are 10 and 1 among smokers and non-smokers, respectively.

What is the Odds Ratio? Interpret the value.

Answer: The Odds Ratio is given by

Odds Ratio= Odds of disease among exposed/ Odds of disease among unexposed = 10/1 =10

Interpretation: Those with lung cancer are 10 times more likely to give a history of smoking as compared to those without lung cancer. Since the Odds Ratio is greater than 1, exposure to smoking is harmful with respect to developing lung cancer.

**Example 2**: A Case Control study found that the Odds of having acute cardiac events in those engaging in regular physical activity is 1. The corresponding odds for those not regularly engaging in physical activity is 3. What is the Odds Ratio? Interpret the value.

Answer: The Odds Ratio is given by

Odds Ratio= Odds of disease among exposed/ Odds of disease among unexposed = 1/3 =0.33

Interpretation: Those having acute cardiac events are 0.33 times likely to give a history of regular physical activity, as compared to those without acute cardiac events. Since the Odds Ratio is less than 1, exposure to regular physical activity protects from having acute cardiac events. (OR Regular physical activity is beneficial with respect to having acute cardiac events.)

**Example 3**: The odds of developing cardiovascular disease among those who smoke is 5. The odds of developing cardiovascular disease among those who use smokeless tobacco is 5. What is the Odds Ratio? Interpret the value.

Answer: The Odds Ratio is given by

Odds Ratio= Odds of disease among exposed/ Odds of disease among unexposed = 5/5 =1

Interpretation: There is no difference in risk of developing cardiovascular disease between those who smoke and those who use smokeless tobacco.

**Summary**

**1. The Odds Ratio is the ratio of Odds of disease among exposed to the Odds of disease among unexposed.**

**2. It is given by the notation ad/bc**

**3. An Odds Ratio of less than 1 indicates that the exposure protects from occurrence of disease/ event.**

**4. An Odds Ratio of more than 1 indicates that the exposure increases the risk of disease/ event occurrence.**

**5. An Odds Ratio of 1 indicates that there is no difference in risk (of developing disease/ event) between exposure and non-exposure**

**Exercise**: The findings of a case control study conducted in Florida, USA are given in the 2×2 table below. Calculate Odds Ratio and interpret the value.

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