Model
logit(p) = ln(p/(1-p)) = β₀ + β₁x
p = 1 / (1 + e^{-(β₀ + β₁x)})
Exponentiating β gives an odds ratio. If β₁=0.7, OR=exp(0.7)=2.01 (odds roughly doubled per unit x).
Interactive: shape of the logistic curve
-6-3.02
00.802.5
OR per +1 unit x: -
OR = exp(β₁)
Probability vs predictor
Example x could be plaque score, age, or risk index.
Real Dental Scenario: Implant Failure Risk from Smoking
A study models implant failure probability based on the number of cigarettes smoked per day. The fitted logistic model is:
logit(p) = -3.5 + 0.15 × cigarettes/day OR per cigarette = exp(0.15) = 1.16
Check Patient Risk
02040
Adjust slider and see risk update in real time
Odds Ratio: Each additional cigarette/day multiplies the odds of failure by 1.16.
A 20-cigarette/day smoker has exp(0.15×20)=20× the baseline odds.
S-Curve: Failure Probability vs Cigarettes/Day
The red dot shows the current patient's position on the curve.
Dental example
Outcome: implant failure (yes/no). Predictors: smoking, diabetes, bone quality, clinician experience. Logistic regression estimates adjusted odds ratios for each predictor.