📊 Understanding Odds Ratio and Risk Ratio

🎯 What is Risk Ratio?

Risk Ratio is a measure that represents the ratio of risks (incidence rates) between two groups. Let’s explain using hypothetical data:

💡 Calculating Risk Ratio

Risk Ratio is calculated using the following formula:

$Risk Ratio = \frac{Pa}{Pb} = \frac{0.9}{0.2} = 4.5$

In this case, the risk is 4.5 times higher for those not wearing masks. This indicates that the risk of contracting COVID-19 is 4.5 times higher without wearing a mask.

🎲 What is Odds Ratio?

Definition of Odds

Odds is defined by the following formula:

$Odds = \frac{P}{1-P}$

Calculating Odds Ratio

Let’s calculate the Odds Ratio using the previous data:

Odds Ratio is calculated using the following formula:

$Odds Ratio = \frac{\frac{Pa}{1-Pa}}{\frac{Pb}{1-Pb}} = \frac{9}{0.25} = 38$

🔍 When to Use Odds Ratio

1. Applicable in Both Cohort and Case-Control Studies

Cohort Study (Prospective)

• Collect records of individuals not wearing masks and examine COVID-19 incidence after a certain period
• Analyzes future events

Case-Control Study (Retrospective)

• Interview COVID-19 patients and non-patients about mask-wearing habits
• Analyzes past events

2. Approximation for Small Incidence Rates

When the incidence rate (p) is small, we can approximate $1-p \simeq 1$, allowing us to estimate Risk Ratio from Odds Ratio:

$Odds Ratio \simeq Risk Ratio = \frac{Pa}{Pb}$

📈 Relationship with Logistic Regression

Odds can be expressed using the sigmoid function as follows:

$ Odds = \frac{P}{1-P} = \frac{\frac{1}{1+exp(-\bf{w}^T\bf{x}+b)}}{1-\frac{1}{1+exp(-\bf{w}^T\bf{x}+b)}} = e^{-\bf{w}^T\bf{x}+b} $

Where $\bf{x}$ represents the vector of explanatory variables.

💡 Summary

• Risk Ratio: An intuitive measure that’s easy to understand
• Odds Ratio: Can be used in various study designs
• When incidence rate is small, Odds Ratio can be used as an approximation of Risk Ratio
• Odds Ratio plays a crucial role in Logistic Regression