Gage R&R

Gage repeatability and reproducibility (gage R&R) asks one question: how much of the variation you see is the part, and how much is the measurement system? mfgQC's default method is the crossed ANOVA decomposition from the AIAG MSA manual, which — unlike the older Average-and-Range method — estimates the part×operator interaction explicitly and pools it into error only when it is not significant.

This page documents what gage_rr(method="anova") actually computes, formula by formula, pinned to the implementation in mfgqc/gage_rr.py. For the end-to-end study workflow (planning the design, attaching roles, reading the verdict) see the Gage R&R workflow.

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1. The model and its decomposition

The crossed study has \(p\) parts (random), \(o\) operators (random), and \(r\) replicate trials in every part×operator cell. The two-way random-effects model is

\[ y_{ijk} = \mu + P_i + O_j + (PO)_{ij} + \varepsilon_{ijk}, \qquad i=1\dots p,\; j=1\dots o,\; k=1\dots r . \]

Sums of squares and the ANOVA table

mfgQC builds a textbook balanced two-way ANOVA (_anova). With \(\bar y\) the grand mean, \(\bar y_{i\cdot\cdot}\) the part means, \(\bar y_{\cdot j\cdot}\) the operator means, and \(\bar y_{ij\cdot}\) the cell means:

Source	Sum of squares (code)	df
Parts	\(SS_P = or\sum_i(\bar y_{i\cdot\cdot}-\bar y)^2\)	\(p-1\)
Operators	\(SS_O = pr\sum_j(\bar y_{\cdot j\cdot}-\bar y)^2\)	\(o-1\)
Part×Operator	\(SS_{PO} = SS_\text{cells}-SS_P-SS_O\)	\((p-1)(o-1)\)
Equipment (error)	\(SS_E = SS_\text{total}-SS_\text{cells}\)	\(po(r-1)\)
Total	\(SS_\text{total}=\sum(y_{ijk}-\bar y)^2\)	\(por-1\)

where \(SS_\text{cells}=r\sum_{ij}(\bar y_{ij\cdot}-\bar y)^2\). Each mean square is \(MS = SS/df\). The F-ratios are taken against the equipment (error) mean square — \(F=MS_\text{source}/MS_E\) — and the interaction p-value is \(p_\text{int}=\Pr\!\big(F_{(p-1)(o-1),\,po(r-1)} > F_{PO}\big)\) (scipy f.sf).

Variance components from the expected mean squares

The components are solved from the expected mean squares of the random model. Which solution is used depends on the pooling decision (see §2).

=== "Interaction pooled (default when not significant)"

A pooled error mean square is formed and the no-interaction expected mean squares apply:

$$
MS_\text{err} = \frac{SS_{PO}+SS_E}{(p-1)(o-1)+po(r-1)}
$$

$$
\sigma^2_\text{repeat}=MS_\text{err},\quad
\sigma^2_\text{oper}=\frac{MS_O-MS_\text{err}}{pr},\quad
\sigma^2_\text{part}=\frac{MS_P-MS_\text{err}}{or},\quad
\sigma^2_\text{int}=0 .
$$

=== "Interaction retained (significant)"

The full model uses $MS_E$ for repeatability and $MS_{PO}$ for the interaction:

$$
\sigma^2_\text{repeat}=MS_E,\quad
\sigma^2_\text{int}=\frac{MS_{PO}-MS_E}{r},\quad
\sigma^2_\text{oper}=\frac{MS_O-MS_{PO}}{pr},\quad
\sigma^2_\text{part}=\frac{MS_P-MS_{PO}}{or}.
$$

Every component is floored at zero (max(..., 0.0)) — negative variance estimates from the method-of-moments solution are reported as exactly \(0\), never as a negative number.

Rolling the components up

The reported quantities are standard deviations (the square roots), assembled exactly as the code does in compute:

\[ \begin{aligned} \text{EV (repeatability)} &= \sigma_\text{repeat} \\ \text{AV (reproducibility)} &= \sqrt{\sigma^2_\text{oper} + \sigma^2_\text{int}} \;=\; \sigma_\text{oper}\ \text{(pooled case, } \sigma^2_\text{int}=0) \\ \text{GRR} &= \sqrt{\sigma^2_\text{repeat}+\sigma^2_\text{oper}+\sigma^2_\text{int}} \;=\;\sqrt{\text{EV}^2+\text{AV}^2} \\ \text{PV (part variation)} &= \sigma_\text{part} \\ \text{TV (total variation)} &= \sqrt{\text{GRR}^2 + \text{PV}^2} \end{aligned} \]

Reproducibility folds the interaction in

mfgQC defines \(\text{AV}^2 = \sigma^2_\text{oper}+\sigma^2_\text{int}\), so the operator and the part×operator interaction both count as reproducibility — the appraiser's contribution to spread. In the pooled (default) case \(\sigma^2_\text{int}=0\) and AV is just the operator term. This matches the AIAG convention that GRR \(=\sqrt{\text{EV}^2+\text{AV}^2}\).

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2. The interaction-pooling rule

mfgQC follows the AIAG MSA 4th ed. convention: pool the part×operator interaction into error when its F-test is not significant at \(\alpha = 0.25\) — that is, when \(p > 0.25\) — and retain it otherwise.

_INTERACTION_POOL_ALPHA = 0.25
...
pooled = p_int > _INTERACTION_POOL_ALPHA   # pool when interaction is NOT significant (p > 0.25)

Why 0.25 and not 0.05

AIAG uses a deliberately permissive \(\alpha = 0.25\) rather than the conventional 5%. The interaction term, when real, inflates reproducibility (AV); pooling a genuine but weakly-powered interaction into error would understate the appraiser's contribution. The high \(\alpha\) retains the interaction unless the data give fairly strong evidence it is absent (\(p > 0.25\)), guarding against that understatement.

The alpha parameter of gage_rr() is not the pooling threshold — it controls the confidence level of the component intervals only (see §4). The pooling cutoff is the fixed constant _INTERACTION_POOL_ALPHA = 0.25.

Whether pooling happened is reported in the text (interaction pooled into error (not significant) vs retained (significant)) and exposed as result.pooled.

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3. The reported metrics

Let \(\sigma_c\) be the standard deviation of component \(c\) (EV, AV, GRR, PV) and \(\sigma_\text{TV}\) the total. mfgQC reports three families of percentages.

%study variation — a ratio of standard deviations

\[ \%\text{study}_c = 100\cdot\frac{\sigma_c}{\sigma_\text{TV}} \]

This is a ratio of standard deviations, not variances. The AIAG study-variation multiplier (commonly \(6\sigma\) or \(5.15\sigma\)) is applied to every component including the denominator TV, so it cancels — the code computes the ratio directly (100.0 * sd / tv). The percentages therefore do not add to 100% (standard deviations don't add in quadrature).

%contribution — a ratio of variances

\[ \%\text{contribution}_c = 100\cdot\frac{\sigma_c^2}{\sigma^2_\text{TV}} \]

Because variances do add, the %contribution column does sum to 100% (EV + AV(+INT) + PV). This is the column to use when you want shares that partition the total.

%tolerance — GRR against the spec window

When a two-sided spec is attached (tol = upper − lower), each component is also expressed against the engineering tolerance, using a fixed \(6\sigma\) multiplier:

\[ \%\text{tolerance}_c = 100\cdot\frac{6\,\sigma_c}{\text{USL}-\text{LSL}} \]

pct_tol is None when no two-sided spec is present. The headline number practitioners quote, %tolerance (GRR), is the GRR entry of this dict.

Number of distinct categories (ndc)

\[ \text{ndc} = \Big\lfloor 1.41\cdot\frac{\text{PV}}{\text{GRR}}\Big\rfloor \]

The factor \(1.41\approx\sqrt 2\), the ratio is truncated to an integer (int(...), toward zero), and the result is floored at 1 (max(1, ...)); if GRR \(\le 0\) it returns 0. ndc estimates how many non-overlapping part groups the gage can reliably tell apart.

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4. Confidence intervals

For the ANOVA method, mfgQC attaches component confidence limits (EV, AV, GRR, PV) on the standard-deviation scale, computed by the Modified Large Sample (MLS / Burdick–Larsen) method that reproduces AIAG MSA Table III-B.9. The confidence level is \(100(1-\alpha)\%\), and alpha defaults to 0.10 — i.e. 90% limits, the AIAG convention. (Note this differs from capability's 95% default.) The xbar_r method reports point estimates only.

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5. The verdict

mfgQC renders a single AIAG-style acceptability verdict from %study(GRR) and ndc (_verdict):

def _verdict(pct_grr, ndc):
    if pct_grr < 10 and ndc >= 5:
        return "acceptable"
    if pct_grr > 30 or ndc < 2:
        return "unacceptable"
    return "marginal (conditionally acceptable)"

%study(GRR)	ndc	Verdict string
\(< 10\%\)	\(\ge 5\)	acceptable
\(> 30\%\) or ndc \(< 2\)	—	unacceptable
anything else	—	marginal (conditionally acceptable)

How this maps to AIAG canon

AIAG's headline %GRR bands are: < 10% acceptable, 10–30% conditionally acceptable, > 30% unacceptable, with ndc ≥ 5 desirable. mfgQC encodes those bands but couples ndc into the verdict: a system with %GRR < 10% but ndc < 5 is reported as marginal, not acceptable; and ndc < 2 alone forces unacceptable. The standalone ndc adequacy check (ndc_adequacy, AIAG ndc ≥ 5) is also reported as a separate assumption so you can see the ndc shortfall even when %GRR is fine.

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6. Assumptions checked

The ANOVA method populates these structured assumption checks and reports them — it never silently changes the analysis:

• Normality of residuals (Anderson–Darling) on the cell-centered residuals \(y_{ijk}-\bar y_{ij\cdot}\).
• Homogeneity of variance across operators (Levene) — equal measurement spread per appraiser.
• ndc adequacy — passes when ndc \(\ge 5\) (NDC_MIN); the failing recommendation tells you the gage cannot reliably distinguish parts.

Two further assumptions are structural to the design and enforced as hard errors, not soft checks:

Balanced crossed design required

The study must be balanced — every part×operator cell must have the same number of trials — and have \(r\ge 2\) replicates. An unbalanced design raises a ValueError that names the trials-per-part for each operator; \(r<2\) also raises. The part, operator, and replicate roles must all be attached.

Beyond what the code can test, the study is only meaningful if the parts span the actual process range (parts are a random sample of production, not hand-picked) and operators are representative appraisers — assumptions of the random-effects model that no software can verify for you.

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7. Worked example

The canonical AIAG MSA 4th-ed. study: 10 parts × 3 operators × 3 trials. Here we attach a spec of \([-3, 3]\) so the %tolerance line appears.

import pandas as pd
import mfgqc

# AIAG MSA 4th ed. data: (operator, trial) -> measurement for parts 1..10
_AIAG = {
    ("A", 1): [0.29, -0.56, 1.34, 0.47, -0.80, 0.02, 0.59, -0.31, 2.26, -1.36],
    ("A", 2): [0.41, -0.68, 1.17, 0.50, -0.92, -0.11, 0.75, -0.20, 1.99, -1.25],
    ("A", 3): [0.64, -0.58, 1.27, 0.64, -0.84, -0.21, 0.66, -0.17, 2.01, -1.31],
    ("B", 1): [0.08, -0.47, 1.19, 0.01, -0.56, -0.20, 0.47, -0.63, 1.80, -1.68],
    ("B", 2): [0.25, -1.22, 0.94, 1.03, -1.20, 0.22, 0.55, 0.08, 2.12, -1.62],
    ("B", 3): [0.07, -0.68, 1.34, 0.20, -1.28, 0.06, 0.83, -0.34, 2.19, -1.50],
    ("C", 1): [0.04, -1.38, 0.88, 0.14, -1.46, -0.29, 0.02, -0.46, 1.77, -1.49],
    ("C", 2): [-0.11, -1.13, 1.09, 0.20, -1.07, -0.67, 0.01, -0.56, 1.45, -1.77],
    ("C", 3): [-0.15, -0.96, 0.67, 0.11, -1.45, -0.49, 0.21, -0.49, 1.87, -2.16],
}
rows = [
    {"part": p, "operator": op, "trial": t, "y": v}
    for (op, t), parts in _AIAG.items()
    for p, v in enumerate(parts, start=1)
]
df = pd.DataFrame(rows)

qc = (mfgqc.load(df, measure="y",
                 roles={"part": "part", "operator": "operator", "replicate": "trial"})
          .spec(lower=-3.0, upper=3.0))

print(qc.gage_rr().report())

Gage R&R (method=anova)
=======================
Design: 10 parts x 3 operators x 3 trials
Verdict: marginal (conditionally acceptable)   ndc = 4

component          std dev   lower 90%   upper 90%  %study var   %contrib
Repeatability(EV)   0.19993       0.177       0.231      18.42%      3.39%
Reproducib.(AV)    0.22684       0.128       1.014      20.90%      4.37%
GRR                0.30237       0.235       1.033      27.86%      7.76%
Part(PV)            1.0423       0.759       1.717      96.04%     92.24%
Total(TV)           1.0853                             100.00%    100.00%

interaction pooled into error (not significant)

%tolerance (GRR) = 30.24%

Assumption checks:
  [PASS] normality (Anderson-Darling): AD=0.64, p=0.0924; skew 0.386; n=90
  [PASS] homogeneity_of_variance (Levene): variance ratio 1.16, p=0.994; n=90
  [FAIL] ndc_adequacy (ndc (AIAG ndc>=5)): ndc 4; n=90

Recommendations:
  - ndc = 4 (< 5 AIAG): the measurement system cannot reliably distinguish parts. Improve gage resolution/repeatability.

Reading the verdict:

• Interaction pooled. The interaction p-value here is \(\approx 0.974\), far above the AIAG \(0.25\) cutoff, so the part×operator term is folded into error and AV is the operator term alone.
• %GRR = 27.86% of study variation — inside the 10–30% band, so conditionally acceptable, not good enough to wave through.
• ndc = 4 (\(\lfloor 1.41\cdot 1.0423/0.30237\rfloor = \lfloor 4.86\rfloor = 4\)), below the AIAG target of 5 — the gage struggles to resolve the parts, and the ndc_adequacy check fails.
• PV dominates (96% of study variation; 92% of variance contribution): most of what the study sees is genuine part-to-part difference, which is the healthy part of this result.

The headline numbers are available as a flat dict via .summary() — consume that or .to_dict() from code; never parse .report() text.

g = qc.gage_rr()
g.summary()["pct_study_GRR"]   # 27.860650881142313
g.summary()["ndc"]             # 4
g.summary()["verdict"]         # 'marginal (conditionally acceptable)'
g.pooled                       # True

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Source standard

The ANOVA decomposition, the variance-component expected-mean-square solution, the %study / %contribution / %tolerance metrics, the ndc formula, the component confidence limits (Table III-B.9), and the acceptability bands all follow:

AIAG. Measurement Systems Analysis (MSA) Reference Manual, 4th ed. Automotive Industry Action Group, 2010.

See the bibliography. The interaction-pooling threshold follows AIAG's \(p>0.25\) convention (see §2). One documented refinement remains: the verdict couples ndc into the %GRR acceptability bands (a system with %GRR < 10% but ndc < 5 is reported marginal, not acceptable), pinned to the implementation in §5.

See also

• Gage R&R workflow — running a study end to end.
• Reading the assumption report — what the guardrail checks mean.
• Bibliography — the AIAG MSA reference.