Odds vs Risk: Mastering the Odds Ratio for Clinical and Epidemiological Insights

🎯 What Are Odds?

Odds are the ratio of the chance something happens to the chance it doesn’t happen.It’s not the same as probability or risk.

จริง ๆ คนไทยใช้คำนี้บ่อย คือ "หมอคิดว่า 50 50"

aka. "หมอคิดว่า odd = 1" ทั้ง 2 แบบความหมายเดียวกัน

Think of a coin toss:

Probability of heads = 0.5

\frac{0.5}{1 - 0.5} = 1

In epidemiology:

\frac{P}{1 - P}

Where P = the probability of the event happening.

This means if a disease occurs in 20 out of 100 people:

\frac{20}{100} = 0.20

\frac{0.20}{1 - 0.20} = \frac{0.20}{0.80} = 0.25

📌 Odds are less intuitive than risk, but they’re crucial in case-control studies and logistic regression.

Concept	Formula	Interpretation
Risk	$\frac{Events}{Total}$	Chance something will happen
Odds	$\frac{P}{1 - P}$ or $\frac{Events}{Non-events}$	Ratio of happening vs not happening

Note:
Odds > Risk if P < 0.5; Odds < Risk if P > 0.5.
Always check which measure your model uses.

If you want to compare odds between two groups (e.g., exposed vs. unexposed), you use:

OR =

\frac{Odds in exposed}{Odds in unexposed}

Odds in exposed:

\frac{A}{B}

Odds in unexposed:

\frac{C}{D}

So:

OR =

\frac{\frac{A}{B}}{\frac{C}{D}} = \frac{A D}{B C}

🔍 This is often how OR is calculated in case-control studies — because risk is not computable (no denominator for total exposed/unexposed).

If you model an outcome using logistic regression, the exponentiated coefficient is the Odds Ratio:

logit(P) =

log (\frac{P}{1 - P}) = β_{0} + β_{1} X \Rightarrow OR = e^{β_{1}}

This is why logistic regression is used in cross-sectional and case-control studies: it estimates relative odds, not absolute risks.

⚠️ When outcome is rare (e.g., <10%), OR ≈ Risk Ratio (RR). But when common, OR overstates the association.

Odds = P / (1 – P)

Odds Ratio = relative odds of event between groups

Odds ≠ Risk, especially when the outcome is common.
Use Odds Ratio when working with case-control data or logistic models.
For rare outcomes, OR ≈ RR — but don’t confuse them.
OR answers: “How much more likely (in odds) is the outcome in the exposed vs unexposed?”