Multivariate Control Chart (Hotelling T²)

Hotelling T² — Multivariate Control Chart

When a process has several correlated characteristics (e.g. pressure and temperature, hardness and density), running a separate Shewhart chart on each one is not enough. It misses shifts that only show up in the correlation structure: pressure up, temperature down — each unremarkable individually, together a clear alarm.

T² statistic: T²ᵢ = (xᵢ − x̄)′ S⁻¹ (xᵢ − x̄). Squared Mahalanobis distance of each observation from the sample mean vector, weighted by the inverse covariance matrix.

Control limit (Phase II, individuals): UCL = (p(m+1)(m−1)) / (m²−mp) · F_{α, p, m−p}, where m = baseline sample size, p = number of variables, α = false-alarm probability (typically 0.0027 as the 3σ equivalent).

Benefit: catches correlation breaks that single charts miss. Drawback: an alarm does not say which variable is out of control — follow-up analyses (T² decomposition or parallel single charts) are needed for that.

Pitfalls

Highly correlated variables: If two columns measure essentially the same thing (correlation > 0.99) the covariance matrix becomes singular or near-singular and the inversion fails. Verify each variable is independently informative.

Too small m: When m ≤ p the covariance matrix is not invertible. Rule of thumb: m should be at least 5×p, better 10×p.

Hard to diagnose: A T² alarm says "something in the variable vector is out". It does not say which variable — or combination — is responsible; that needs contribution decomposition or parallel per-variable Shewhart charts.

α choice: α = 0.0027 matches the classic 3σ convention. Larger α increases sensitivity but multiplies false alarms — multiplicatively in p.

Examples

This module ships with the following example datasets — load any of them in the app with a single click.

Available in the following cycles

• DMAIC: Control
• DMADV: Verify
• 8D: D6 — Implementation