Patient Risk Prediction — Logistic Regression

Patient Cohort

Cohort size 180

Class separation 1.4

Decision Threshold

P(disease) cutoff 0.50

Live Performance

Sensitivity (TPR)

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Specificity (TNR)

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Accuracy

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AUC (est.)

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TP0

FN0

FP0

TN0

How it works

Each simulated patient has two biomarkers x₁ (e.g. a metabolic score) and x₂ (an inflammation marker). A logistic-regression model turns them into a disease probability:

z = w0 + w1*x1 + w2*x2
P(disease) = σ(z) = 1 / (1 + e^-z)

Weights (w0, w1, w2) are learned by gradient descent on the binary cross-entropy loss, the same objective used by real clinical risk scores fit on patient registries:

L = -Σ [ y·log(p) + (1-y)·log(1-p) ]
wj ← wj - η · ∂L/∂wj   (repeated ~300 steps)

The 3D surface is the fitted probability z = P(disease) over the (x1, x2) plane. The gold contour ring where P = 0.5·threshold-adjusted marks the decision boundary — a plane tilted by the weights, sliced at your chosen cutoff.

Class separation — how cleanly the two biomarker clusters (healthy vs. at-risk) split; low separation forces the model to trade sensitivity for specificity, exactly like a noisy real biomarker.
Threshold slider — lowering the cutoff catches more true cases (higher sensitivity, more false positives) — the same clinical trade-off as choosing an aggressive vs. conservative screening cutoff.
ROC curve — sweeps the threshold from 0→1 and plots TPR vs. FPR; the area under it (AUC) summarizes discrimination independent of any one cutoff.

Real-world relevance: this is the exact mechanism behind clinical risk calculators (e.g. sepsis-risk or readmission-risk scores) — a linear combination of features passed through a sigmoid, thresholded to a binary alert.

Healthy (y=0) At-risk (y=1) Decision boundary

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