Gauge R&R (ANOVA)

In a crossed study every operator measures every part the same number of times. With p parts, o operators and n trials there are p·o·n readings. A two-way analysis of variance with interaction splits their total variation into four sources: parts, operators, the part × operator interaction and repeatability.

1. Work out the grand mean x̄, the mean of each part x̄_i, the mean of each operator x̄_j and the mean of each part and operator cell x̄_ij.
2. Work out the four sums of squares:

3. Divide each sum of squares by its degrees of freedom, p − 1, o − 1, (p − 1)(o − 1) and p·o·(n − 1), to get the mean squares MS_P, MS_O, MS_PO and MS_E. The F ratios are MS_P / MS_PO for parts, MS_O / MS_PO for operators and MS_PO / MS_E for the interaction.
4. Estimate the variance components from the mean squares. An estimate below zero is set to zero.

5. Add them up: reproducibility AV = Operator + Part × operator, GRR = EV + AV and total variation TV = GRR + PV. Each σ is the square root of its variance, in the unit of the readings.
6. Read the three indices:

The guide example, 10 sites × 3 operators × 3 trials with USL 52 and LSL 48, gives %GRR 31.6 %, %Tolerance 19.1 % and ndc 4. The details under the result list the ANOVA table and, for every component, its variance, σ, %SV and %Tolerance.

• Balanced, crossed studies only. Every operator measures every part the same number of times. Nested studies, such as destructive tests in which no part can be measured twice, and studies with missing readings need a different analysis.
• The interaction is not pooled. Some software drops the part × operator term when it is small and adds it to repeatability. This tool always keeps it, so its results can differ slightly from software that pools.
• Negative components are set to zero. When a mean square is smaller than the one it is compared with, for example MS_PO below MS_E, the estimate would be negative. It is reported as 0, which slightly raises the share of the other components.
• %GRR and ndc depend on the parts you chose. σ_PV comes from the parts in the study, not from the process. Parts close together make the measurement system look large. In the guide example the 10 sites span only about 1.2 µm against a 4 µm tolerance, which is why %GRR is 31.6 % while %Tolerance is 19.1 %.
• F ratios come without p-values. To judge whether a source is significant, compare its F ratio with an F table at the degrees of freedom shown in the details, for example F(18, 60) for the interaction in the guide example.
• %GRR answers: can the measurement system see the differences between the parts in this study? It matters when the gauge is used for process control or improvement work. Common guideline: below 10 % generally acceptable, 10 % to 30 % may be acceptable depending on the application, above 30 % generally not acceptable.
• %Tolerance answers: can the measurement system sort parts against the specification? It matters when the gauge makes accept and reject decisions. The same bands are commonly used.
• ndc answers: how many groups of parts can the system tell apart within the range of the study? 5 or more is the usual guideline.
• Decide which index governs before the study. The indices can disagree, as in the guide example. Write in the procedure which one decides for this gauge and why; choosing after seeing the result is not a test.
• Look at where GRR comes from. A large EV points to the gauge, the fixture or the handling between trials. A large AV points to how operators set up and read the measurement. A large part × operator component means some operators read certain parts differently.
• For the study design and a full worked example, read the Gage R&R after calibration guide.

• Deciding on one index alone. A poor %GRR with a good %Tolerance, or the reverse, says different things about the gauge. Report all three indices and the governing one named in the procedure.
• A sample that is too narrow. Parts or sites that all sit near nominal leave little part variation, so %GRR rises and ndc falls even on a good gauge. Choose parts that cover the expected spread of the process.
• Treating calibration as GRR. Calibration checks bias against a reference; a GRR study checks the spread of repeated readings. A calibrated gauge can still have poor repeatability or reproducibility.
• Reading a session difference as operator skill. When each operator measures in a separate session, operator and time are mixed: a drift between sessions shows up as reproducibility. Note the confounding in the report, or let operators measure in the same sessions.
• Too few parts or trials. With only a handful of parts or two trials the variance components rest on few degrees of freedom and change a lot from one study to the next. Ten parts, three operators and two or three trials is a common design.