U01.06.006 Odds ratio

This guide explains essential risk measures used in medical research, along with their formulas and how they relate to contingency tables (2×2 tables).

Contingency Tables: The Foundation

Contingency tables, also known as 2×2 tables, provide a structured way to analyze the relationship between two categorical variables. In the context of quantifying risk, these variables are often:

Disease or Outcome: Presence or absence of a specific disease.
Exposure or Intervention: Whether an individual has been exposed to a potential risk factor or received an intervention.

Here’s a sample contingency table layout:

Disease or Outcome	Exposed (a)	Unexposed (b)	Total
Present (c)			c
Absent (d)			d
Total	a	b	a + b

Key Risk Measures:

Several risk measures can be calculated from a contingency table, each offering a unique perspective on the association between exposure and disease:

Odds Ratio (OR): Used in case-control studies. Compare the odds of having a risk factor in cases (individuals with the disease) to the odds in controls (healthy individuals).
- Formula: OR = (a * d) / (b * c)
Relative Risk (RR): Often used in cohort studies. Compare the risk of developing a disease in the exposed group to the risk in the unexposed group.
- Formula: RR = a / (a + b) / c / (c + d)
Attributable Risk (AR): The difference in disease risk between exposed and unexposed groups.
- Formula: AR = (a / (a + b)) – (c / (c + d)) or AR% = (RR – 1) x 100% (easier for percentages)

Example:

Let’s consider a study on smoking and lung cancer. We create a contingency table to analyze the data:

Disease (Lung Cancer)	Exposed (Smokers)	Unexposed (Non-Smokers)	Total
Present (Diagnosed)	20	1	21
Absent (Not Diagnosed)	10	19	29
Total	30	20	50

Calculating Risk Measures:

Odds Ratio (OR):
- OR = (20 * 19) / (1 * 10) = 38
Relative Risk (RR):
- RR = (20 / 30) / (1 / 20) = 13.33 (approximately)
Attributable Risk (AR):
- AR = (20 / 30) – (1 / 20) = 0.67 – 0.05 = 0.62 or AR% = (13.33 – 1) x 100% = 1233% (high percentage due to small sample size for non-smokers with lung cancer)

Interpretation:

In this example, smokers are 38 times more likely to have lung cancer compared to non-smokers (based on the OR).
Smokers have a roughly 13 times greater risk of developing lung cancer compared to non-smokers (based on the RR).
Smoking is associated with an increased risk of 62% for lung cancer (based on the AR).

Important Notes:

Contingency tables only show associations, not causation.
The choice of risk measure depends on the study design (case-control vs. cohort) and research question.
This is a simplified example, and real-world studies might involve larger sample sizes.

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U01.06.006 Odds ratio

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