Metabolic syndrome is diagnosed clinically by the NCEP ATP III rule: a patient meets the diagnosis when 3 or more of these 5 criteria are true at once:
1. Waist circumference ≥102 cm (M) / ≥88 cm (F)
2. Triglycerides ≥150 mg/dL
3. HDL cholesterol <40 mg/dL (M) / <50 mg/dL (F)
4. Systolic BP ≥130 mmHg
5. Fasting glucose ≥100 mg/dL
Diagnosis: count(criteria true) ≥ 3
The 500-patient cohort is not five independent random draws. Each patient's 5 markers are sampled from a correlated multivariate normal: a fixed 5×5 correlation matrix (waist↔triglycerides +0.45, waist↔HDL −0.35, triglycerides↔HDL −0.40, etc. — insulin resistance drives abdominal fat, dyslipidaemia and glucose intolerance together) is factored with a real Cholesky decomposition R = L·Lᵀ, and correlated z-scores are built as z = L·u from 5 independent standard-normal draws u. Each z is then rescaled to that marker's clinical mean/SD (sex-specific for waist and HDL) to get real mg/dL, cm and mmHg values.
- Scatter (top) — every patient plotted at (waist, fasting glucose); color ramps cool→red with criteria count. The correlation shows up as a diagonal stretch, not a round blob.
- Histogram (bottom) — the distribution of criteria-met counts (0–5) across the cohort; the red-outlined bars (3, 4, 5) are the metabolic-syndrome-positive population.
- Individual risk tool — pick a "drive factor" and drag its value. The other four don't move independently: each one's conditional mean is recomputed from the exact multivariate-normal formula μj + ρ(drive,j)·(σj/σdrive)·(x−μdrive), i.e. the real statistical shift implied by the correlation structure, and your patient marker (white, ringed) moves accordingly on the scatter.
Real-world relevance: this is the same rule cardiologists and primary-care physicians use at a checkup — it flags roughly a quarter of adults in industrialised countries and roughly doubles cardiovascular-disease risk and increases type-2-diabetes risk five-fold when 3+ criteria are present. Modelling the markers as correlated rather than independent is what makes the synthetic cohort's clustering behaviour realistic.