Measurement System Analysis

MSA Type 6 — Stability / Long-term Behavior

Stability study of a measurement system: reference part over time on an I-MR or x̄-R control chart, plus a linear drift-test.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Quality Tools > Gage Study > Gage Run Chart / Stability Study (I-MR chart of a reference/master part over time)

Description

MSA Type 6 repeatedly measures a reference part (master/standard) with the same measurement system over weeks or months and checks whether it remains stable over time. The engine is a thin wrapper around the SPC control-chart engine: control limits (from baseline subgroups, or supplied via µ₀/σ₀) and the Nelson rule evaluation (incl. ±1σ/±2σ zones) are reused 1:1 from the `control-chart` Algorithm Lab entry — no duplication. The only MSA-specific piece is the linear drift test (simple regression of the primary metric over the time axis, followed by a two-sided t-test on β₁ = 0) and the combined verdict from Nelson violations and the drift p-value.

Formulas

β₁ (Steigung)

β_{1} = \frac{\sum _{i} ( t _{i} - t ˉ ) ( y _{i} - y ˉ )}{\sum _{i} ( t _{i} - t ˉ ) ^{2}}

Slope coefficient of the linear regression of the primary metric y_i over the time axis t_i (timestamps or 1..N index)

SE(β₁)

SE (β_{1}) = \frac{s ^{2}}{\sum _{i} ( t _{i} - t ˉ ) ^{2}}, s^{2} = \frac{\sum _{i} ( y _{i} - β _{0} - β _{1} t _{i} ) ^{2}}{N - 2}

Standard error of the slope, from the residual variance s²

t, p

t = \frac{β _{1}}{SE ( β _{1} )}, p = 2 (1 - F_{t} (∣ t ∣, N - 2))

Two-sided t-test on β₁ = 0 (no drift) with df = N − 2

KI(β₁)

β_{1} \pm t_{1 - α /2, N - 2} \cdot SE (β_{1})

Two-sided confidence interval for the slope at the chosen α

Verdikt-Ampel

⎩ ⎨ ⎧ stable unstable marginal 0 Nelson-Verletzungen \land p \geq α \geq 3 Nelson-Verletzungen \lor p < α /10 sonst

Combined verdict from Nelson rule violations and the drift-test p-value. `unstable` at ≥ 3 violations OR a strong trend (p < α/10, one tenth of the significance level); `stable` only at 0 violations AND p ≥ α; anything in between is `marginal`.

Assumptions

A single reference part is measured repeatedly with the same measurement system over the whole study duration (multiple reference parts → multiple Type 6 instances side by side)
The measurement order is time-ordered; real timestamps should be monotonically increasing (otherwise a warning fires; the engine sorts automatically)
The drift test assumes a linear trend — cyclical or step-shaped drifts are not reliably detected by the t-test, though the Nelson rules do catch many of them

Limitations

Baseline subgroups < 20 yield unstable control limits (warning `W_BASELINE_LT_20`); Nelson violations within the baseline may contaminate the limits (`W_LIMITS_FROM_UNSTABLE_BASELINE`)
Cp/Cpk from a stable window, EWMA/CUSUM, and multiple reference parts per study are deliberately out of scope (see module spec § 12)
Nelson rules 7 and 8 are disabled by default (see the `control-chart` entry) — enabling them raises the false-alarm rate

References

Nelson, L. S. (1984) — The Shewhart Control Chart — Tests for Special Causes, Journal of Quality Technology, 16(4), 237–239
Robert Bosch GmbH — Qualitätsmanagement in der Bosch-Gruppe, Heft 10: Fähigkeitsuntersuchung von Messsystemen
AIAG — Measurement Systems Analysis Reference Manual, 4th Edition (MSA-4)