Gage Rr Anova

Two-Way ANOVA Gage R and R (manufacturing-quality/as9100/gage-rr-anova)

Use when a replicated gage repeatability and reproducibility (Gage R and R) study must be scored with the two-way analysis of variance estimator: balanced readings of several parts by several operators are decomposed into part, operator, part-by-operator interaction and equipment variance components, and the combined measurement error is scored against the percent GRR acceptance bands. This leaf implements the balanced two-way random-effects ANOVA in pure Python, stdlib only, deterministic and offline. It pairs with manufacturing-quality/as9100/measurement-systems-analysis, which runs the range-based Gage R and R estimator for studies without replicated operator-part interaction; the ANOVA estimator is the choice when every operator-part cell carries two or more trials and the interaction must be tested. The gage-linearity-bias-study, attribute-agreement-analysis and statistical-process-control leaves cover the adjacent study types and the process monitoring that follows gage approval.

Domain quick reference

• Balanced layout: O operators, P parts, n trials per operator-part cell, with readings passed as a dict {operator: {part: [trials]}}. Total degrees of freedom OPn - 1.
• Sums of squares around the grand mean, with part_mean, op_mean and cell_mean the level means: ss_part = nOsum over parts (part_mean - grand_mean)^2; ss_operator = nPsum over operators (op_mean - grand_mean)^2; ss_interaction = n*sum over cells (cell_mean - part_mean - op_mean + grand_mean)^2; ss_equipment = sum over readings (reading - cell_mean)^2. The four effect sums add to the total sum of squares.
• Degrees of freedom: df_part = P-1, df_operator = O-1, df_interaction = (P-1)(O-1), df_equipment = OP(n-1); each mean square is ms = ss/df.
• Variance components by the expected-mean-square equations with the non-negative floor on the interaction and operator components: var_equipment = ms_equipment; var_interaction = max(0, (ms_interaction - ms_equipment)/n); var_operator = max(0, (ms_operator - ms_interaction)/(Pn)); var_part = max(0, (ms_part - ms_interaction)/(On)).
• Variation chain: ev = sqrt(var_equipment), av = sqrt(var_operator), iv = sqrt(var_interaction), combined grr = sqrt(ev^2 + av^2 + iv^2), pv = sqrt(var_part), total tv = sqrt(grr^2 + pv^2).
• Percent GRR = 100*grr/tv with the 10/30 bands: under 10 percent is acceptable, 10 through 30 percent is conditional (usable only for specific applications), over 30 percent is unacceptable.
• Number of distinct categories: ndc = floor(1.41*pv/grr); five or more categories is the common adequacy threshold.
• F statistics: f_part = ms_part/ms_interaction tests the part effect against the interaction; f_interaction = ms_interaction/ms_equipment tests the interaction against repeatability. A large part F means the parts differ relative to the operator-part interaction; a small interaction F means operator offsets are additive across parts.
• Interaction-free data: when operator offsets are purely additive the interaction mean square falls below the equipment mean square and the floor clamps iv to zero, exactly the worked example below.
• AS9100 frames monitoring and measuring resources as controlled and fit for purpose (paraphrase of clause 7.1.5 practice); the gage study is the common aerospace evidence that a measurement system is fit for its task, summarized here without clause text.

Workflow

1. Tabulate the study as a balanced dict {operator: {part: [trials]}}: every operator must measure every part with the same number of trials, two or more.
2. Validate the layout first with validate_study; it raises ValueError for fewer than 2 operators, fewer than 2 parts, fewer than 2 trials per cell, ragged cells and non-numeric readings.
3. Run the decomposition with anova_grr_study(data) to get the full result dict: sums of squares, degrees of freedom, mean squares, variance components, ev/av/iv/grr/pv/tv, percent_grr, ndc and the F statistics.
4. Read the verdict: percent_grr under 10 is acceptable, 10 through 30 is conditional, over 30 is unacceptable. Score any percent value directly with verdict_for_percent_grr(pct) when you already have it.
5. Pull the five-source table (part, operator, interaction, equipment, total) with anova_table(data) for the report.
6. Interpret the F statistics: a large f_part confirms the parts span real variation, and a small f_interaction confirms the operator offsets are additive, so grr is driven by ev and av only.
7. Judge resolving power with ndc (five or more categories is the usual adequacy threshold) before approving the measurement system.
8. Confirm the deterministic checks with the contract test scripts/test_gage_rr_anova.py.

Worked example

Three operators A, B, C, three parts P1..P3, two trials per cell, operator B offset 0.01 low and operator C offset 0.04 high relative to A:

A: P1 [0.30, 0.32], P2 [0.50, 0.52], P3 [0.70, 0.72] B: P1 [0.29, 0.31], P2 [0.49, 0.51], P3 [0.69, 0.71] C: P1 [0.34, 0.35], P2 [0.54, 0.55], P3 [0.74, 0.75]

Running anova_grr_study on this fixture gives the module outputs below (deterministic, reproducible):

• Grand mean 0.5183; sums of squares ss_part 0.4800, ss_operator 0.00670, ss_interaction about 4e-31 (the offsets are additive, so the interaction is numerically zero), ss_equipment 0.00135; the four sum to the total 0.48805.
• Degrees of freedom 2, 2, 4, 9 (total 17); mean squares ms_part 0.24, ms_operator 0.00335, ms_equipment 0.00015.
• Variance components: var_equipment 0.00015, var_operator 0.000558, var_interaction 0.0 by the floor, var_part 0.04.
• Variation chain: ev 0.01225, av 0.02363, iv 0.0, grr 0.02661, pv 0.2000, tv 0.20176. The identity tv^2 = grr^2 + pv^2 holds.
• Percent GRR 13.19, inside the conditional band; verdict "conditional".
• Number of distinct categories ndc = floor(1.41*0.2/0.02661) = 10, above the adequacy threshold of five.
• F statistics: f_part about 2.3e30 (huge, the parts dominate) and f_interaction about 7.0e-28 (essentially zero, no operator-part interaction), so the study is driven by ev and av alone.
• ANOVA table rows: part (0.48, 2, 0.24, f_part), operator (0.0067, 2, 0.00335, F not applicable), interaction (~4e-31, 4, ~1e-31, f_interaction), equipment (0.00135, 9, 0.00015, F not applicable), total (0.48805, 17).

Verification

• Run the module on the worked fixture and confirm the anchors fall in the spec bands: percent_grr in [12.5, 14.0], ev in [0.011, 0.014], av in [0.021, 0.027], ndc == 10 and verdict "conditional".
• Confirm the closed-form identities: ss_part + ss_operator + ss_interaction + ss_equipment equals the total sum of squares around the grand mean, and tv^2 equals grr^2 + pv^2.
• Confirm the floor: an interaction-free fixture (additive operator offsets) yields var_interaction 0.0 and iv 0.0, and a second run returns a dict identical to the first (determinism, no RNG).
• Confirm the result dict keys match the documented set exactly (grand_mean, the four ss and df and ms keys, the four variance components, ev, av, iv, grr, pv, tv, percent_grr, ndc, f_part, f_interaction, verdict, distinct_categories).
• Confirm ValueError rejection: one operator, one part, one trial per cell, ragged cells (a missing part or unequal trial counts), and non-numeric or boolean readings all raise ValueError.
• Confirm verdict band edges: 9.99 acceptable, 10.0 and 30.0 conditional, 30.01 unacceptable.
• Run the contract test offline: python3 scripts/test_gage_rr_anova.py (34 tests, deterministic, under a second).

Related leaves

• manufacturing-quality/as9100/measurement-systems-analysis: the range-based Gage R and R estimator (EV from the average cell range, AV from the appraiser-average spread); choose it when each cell has few trials and no interaction term is needed.
• manufacturing-quality/as9100/gage-linearity-bias-study: the operator offset and linearity regression study for a single gage over its working range.
• manufacturing-quality/as9100/attribute-agreement-analysis: agreement scoring for go/no-go and categorical judgments, not continuous readings.
• manufacturing-quality/as9100/statistical-process-control: the process monitoring that runs once the measurement system is approved.

Pitfalls

• Routing a single-trial or short study here: without replicated trials per cell there is no within-cell error term and no interaction test, so the range-based estimator in measurement-systems-analysis is the right tool; the ANOVA path needs n >= 2 trials per cell.
• Reporting the equipment component without the interaction: the combined grr is sqrt(ev^2 + av^2 + iv^2), so an unmodeled interaction understates measurement error when operator offsets differ across parts.
• Forgetting the non-negative floor: a negative raw interaction estimate (ms_interaction below ms_equipment) means no interaction is detectable, and the component is clamped to zero, not reported negative.
• Reading ndc as zero for a perfect gage: when grr is zero the ratio 1.41*pv/grr is undefined and the module returns None for ndc and distinct_categories, with percent_grr 0.0 and an acceptable verdict.
• Treating a tiny interaction F as proof the operators agree: the interaction test only shows the offsets are additive; operator spread still enters av and can push percent_grr into the conditional or unacceptable bands by itself.
• Interpreting F statistics on degenerate data: with no equipment or interaction variation the denominator mean square is zero and the F value is None (both terms zero) or infinite (effect present), never a meaningful finite number.
• Routing operator offset or linearity studies here: those belong to gage-linearity-bias-study, and go/no-go agreement studies belong to attribute-agreement-analysis; this leaf scores only the ANOVA variance-component decomposition of continuous replicated readings.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_gage_rr_anova.py

The test covers the worked-example anchors against the spec magnitude bands (percent_grr in [12.5, 14.0], ev in [0.011, 0.014], av in [0.021, 0.027], ndc 10, verdict "conditional"), the balanced-design identities (sum-of-squares decomposition, tv^2 = grr^2 + pv^2, percent GRR ratio), the exact result-dict keys, the five-source ANOVA table with matching F values, the 10/30 verdict band edges, the non-negative interaction floor on a 2x2 fixture, degenerate zero variation data, deterministic reruns, and ValueError rejection of one operator, one part, one trial, ragged cells and non-numeric readings.

Compliance

• Standards referenced, not reproduced: AS9100 clause 7.1.5 frames monitoring and measuring resources; the balanced two-way ANOVA expected-mean-square equations and the 10/30 percent GRR bands are common MSA methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.