Linear Regression Lab

Linear Regression Clinical Lab

We predict a patient's Systolic Blood Pressure (SBP) from their Age and BMI using a linear equation.

Enter patient data → adjust coefficients → observe the live prediction and regression line update.

How Are β Coefficients Calculated?

In a real clinical model, β₀, β₁, and β₂ are never manually set. They are calculated from patient training data using Ordinary Least Squares (OLS).

OLS finds values of β that minimise the total squared gap between each patient's actual SBP and the model's prediction. This has an exact mathematical solution:

β = (XᵀX)⁻¹ Xᵀy

Training data → OLS → optimal β₀, β₁, β₂

The sliders let you explore the impact
of different values. Click "Auto-Fit" to watch
gradient descent learn the correct β values
from the data.

How MSE Is Calculated

Every patient has an actual SBP and a predicted SBP from the model. The gap is the residual. We square each residual (to punish large errors more) and average them:

                    MSE = Σ (yᵢ − ŷᵢ)² / n

                    Pink boxes = squared residuals · smaller = better fit

A perfect fit has MSE = 0. Squaring ensures large errors are penalised far more than small ones.

⚖️ Ethical Reality

Applying population-average coefficients to diverse patients can systematically under-predict SBP risk in specific metabolic subgroups.

① Patient Input

Patient Age

years

Patient BMI

kg / m²

② Model Coefficients

β₀ · Intercept 70.0

baseline mmHg

β₁ · Age Weight 0.30

mmHg per year

β₂ · BMI Weight 1.50

mmHg per kg/m²

③ Prediction Output

Predicted Systolic BP

—

mmHg

SBP = 70.0 + 0.30×Age
+ 1.50×BMI

—

MSE

—

BP Class

④ Regression Visualization — BMI vs Systolic BP (line shown at subject's age)