Visualizing Probability, Odds, and Log-Odds in Logistic Regression

This is the associative transformation visual you want for understanding probability, odds, and log-odds:

Left: Probability vs. Odds

• As probability increases (X-axis), odds increase slow at first, then explode as probability nears 1.

• The red line at odds = 1 (where probability = 0.5, i.e., 50:50 chance).
  

Middle: Odds vs. Log Odds

• Log-odds is the natural log of odds.

• At odds = 1 (red vertical), log-odds = 0.

• Log-odds go negative for odds < 1 (rare event), positive for odds > 1 (common event).
  

Right: Probability vs. Log Odds

• Log-odds increases linearly with probability in the middle range, but curves sharply at the ends.

• The red line is log-odds = 0 (which matches probability = 0.5).

• Logistic regression models this log-odds scale!
  

Clinical interpretation:

• When probability is low, odds/log-odds are tiny/negative.

• At 50% probability, odds = 1, log-odds = 0.

• Above 50%, odds and log-odds rise quickly.