Multiple Regression

Model a continuous outcome while controlling for confounders

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Model

Y = β₀ + β₁X₁ + β₂X₂ + ... + ε

Each β coefficient is the expected change in Y per unit change in that predictor, holding the others constant.

Interactive: confounding intuition

Simulate an outcome (e.g., bone loss) driven by two predictors (age + smoking). Toggle smoking association and see how the simple (unadjusted) relationship between age and outcome can be distorted.

00.601
0.10.82.0
Takeaway: adjusted models help isolate effects when predictors correlate.

Simulated data (color = smoking)

Regression is usually done in software; this chart is for intuition.

Real Dental Scenario: Predicting Marginal Bone Loss

A periodontist wants to predict marginal bone loss (mm) around dental implants using patient factors. The fitted regression model is:

BoneLoss = 0.5 + 0.03 × Age + 1.2 × Smoking + 0.8 × Diabetes

Patient Inputs

3080
Smoking
No
Diabetes
No

Predicted Marginal Bone Loss

0.0 mm 0 5

Factor Contributions

Baseline (β₀)0.50 mm
Age effect0.00 mm
Smoking effect0.00 mm
Diabetes effect0.00 mm

Dental example

Predict marginal bone loss (mm) from age, smoking, diabetes, and oral hygiene score. Multiple regression helps estimate the smoking effect while accounting for age and comorbidities.